Youtube comments of MC116 (@angelmendez-rivera351).
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PopeLando Well, not necessarily. Mathematically, it is necessary to treat 9*13 and 13*9 as separate calculations, and in order for the method to be valid, it is necessary that both calculations have the same output, since we know multiplication is commutative. Hence, we can consider this as a teat of sorts. If the result for calculating 13*9 fails to be equal to the result for calculating 9*13, then the method is invalid - the converse is not true, though, so if this test is passed, more tests are needed to determine sufficiency. However, this is the first step.
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Hunter Walker You can still express that using cosines and sines. In fact, using tangents and secants is not necessary for this, because you can use the hyperbolic sine and cosine instead, which are objectively superior functions to work with. Remember that cosh(u)^2 – sinh(u)^2 = 1, so sinh(u)^2 = cosh(u)^2 – 1, or simply sgn(u)sinh(u) = sqrt(cosh(u)^2 – 1). Therefore, to integrate 1/[x·sqrt(x^2 – 1)], let x = cosh(u), so sqrt(x^2 – 1) = sgn(u)sinh(u), and dx = sinh(u)du, so the integrand simplifies to 1/cosh(u) with sgn(u) multiplying from the outside, and this is completely trivial because cosh(u) = [exp(u) + exp(-u)]/2.
Grant is correct. The cotangent, secant, and cosecant are not particularly important functions, nor are they the most useful.
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Actually, yet again we stand corrected. Not even division by zero is a problem difficult enough to remain unsolved. As of 1998, mathematicians who study group and field theory have finally managed to build a form of algebra which modifies the distributive rules of commutative and associative rings such that division by zero is defined. This is what they have called wheel algebra theory, and it is merely an extension of the current number system that exists, not a brand new one from scratch. There are at least two possible ways in which this functions properly: we either assign a value to all the indeterminate forms, and then let ξ = 1/0, where M < ξ < L, M representing a number of infinite magnitude within the positives and L representing a number of infinite magnitude within the negatives. We get rid of multiplicative associativity in the general case and allow an additive nullifier ξ to exist uniting the positives with the integers (turning the real number line into a circle and the complex plane into a torus).
The other form we can do this is to define a divisor operator / as unary rather than binary, we modify the basic algebraic distribution laws so that they are consistent both in the general case and in the special cases (which would simplify to the ones we currently have in non-zero division fields) and it would leave us with a number system known as the extended/completed Riemann sphere. /0 = Infinity ("Infinity" as in the symbol used to represent Infinity, not as in the actual concept of Infinity. The two are very distinct. Our concept of infinity still isn't a number, but /0 produces a number of infinite magnitude which is unsigned, meaning it is neither positive nor negative. Thus the real number line is still folded and united in a circle. The complex plane is united in a sphere since /0 is both real and imaginary. Then, in the very center of the sphere, lying outside the plane itself, exists the element W = 0/0. So division by zero is a matter of modifying distributive laws involving the division operator, leaving division as unary, and introducing two new numbers to the system, that way it preserves every other law of algebra.
Both systems are consistent in themselves and the axioms are regular and stable.
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@Vespyr_ No, that is horribly incorrect. Firstly, that is not how division actually works: division is not repated subtraction, and multiplication is not repeated addition. Secondly, even if division did work that way, your answer is still wrong, becaue 20/0 would be equal, by your definition, to the number of times you have to subtract 0 from 20 to achieve 0. The problem is that, even if you subtract 0 an infinite amount of times from 20, you still do not achieve 0. The answer is not 0, nor is it an infinite number. It is just impossible to achieve 0 via such repeated subtractions, hence 20/0 is undefined.
Nevermind this, because as I explained firstly, division is not repeated subtraction. The reason division by 0 is problematic is because, in order for division by a quantity A to be possible, you need to have the following property: if A·x = A·y, then x = y. This does not occur with 0. 0·1 = 0·(1 + 1), but 1 = 1 + 1 is false, in general. So division by 0 is hopeless.
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@avr2fp4 I regret to inform you that it is mostly a matter of those works being fraudulently attributed to authors, containing contrary theology, and/or containing historical inaccuracies.
I regret to inform you, as a non-Christian, that this makes them not particularly different from the canonical books, many of which suffer of the same problem. Besides, us seculars are interested in the books for what they are, not for what religious people want them to be or do not want them to be. I do not come to videos in this channel with an agenda of supporting arguments in favor or against a particular canonization, unlike you guys. I come to these videos because, regardless of what value (or lack thereof) they have to the religious minds of society, they are intrinsically interesting by their own merit, simply by virtue of the fact that the make otherwise difficult-to-access information more accessible to the general public. It is unfortunate that some religious people, like yourself, are unable to understand this, and so are willing to discourage people from looking into these books which you strongly see as heretical, but I have no emotional investment in whether books are heretical or not. People like me seek knowledge for the sake of knowledge, not for the sake of the ego. And by the way, I tell you this not out of hatred for Christianity, but out of appreciation for skepticism, and the act of seeking knowledge. If you were Muslim, or Hindu, or Buddhist, I would still tell you the same thing. Even if you were just another secularist, I would still tell you the same thing.
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David Larson Actually, not at all. The fundamental theorem of arithmetic (not algebra) can easily be expressed in the following form: for every integer x > 0, there exists some a(n) such that x = 2^a(0)*3^a(1)*5^a(2)*•••, and that there is a bijection between N and the set of all such a(n). Then, it makes perfect sense that 1 is not a prime. Why? Because x = 1 if and only if a(n) = 0. Meanwhile, for every prime number, there exists some n such that a(n) = 1. And from this, we get a corollary that the set {ln(2), ln(3), ln(5), ...} is a vector basis for the vector space of all ln(n), and using the fundamental theorem of arithmetic, it becomes trivial that ln(1) = 0 is the zero vector, which is consistent both with the fact that 0 is the additive identity and 1 is the multiplicative identity (in fact, one implies the other), and therefore, that 1 is the empty product. But this is only possible if and only if 1 is not a prime number. So, if anything, it's not so much a matter of definition as it is a matter of keeping mathematics working consistently. It's basically proof that 1 should not be a prime.
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blownspeakersss I'm a mathematics student too. I also do know that mathematicians all over the world with different specializations and with post-doctorate degrees agree that this sum does equal -1.
Your argument that the sum diverge proves nothing, since you are assuming that convergence is a necessity for any sum to have a value. While it is true that in calculus we define S = lim n -> (infinity) : S(n), this definition is akin to defining exponentiation as iterated multiplication, which not only is an incomplete definition, but useless for most mathematical applications since we must deal with fractional exponents, and in occasion complex-imaginary exponents. However, the availability of the Taylor series allows us to expand the domain of the function, and we can redefine exponentiation in terms of its Taylor series. Or equally, we can use the limit definition of e to achieve this, but the result is the same, because the binomial expansion of the limit simplifies to the Taylor series.
A similar concept is being applied here. Defining an infinite series in terms of its limit is actually somewhat of a flaw, especially because it assumes continuity. In other words, it assumes lim (x -> c) f(x) = f(c) for all infinite quantities c, even though most functions fail to satisfy this equation at least at one point. Mathematicians have developed the notion of discontinuities at infinity and it is well-defined and rigorously so too. In fact, one of the different ways to equivalently define a conics section is via a discontinuity at infinity in the complex hyper-plane.
Ramanujan summation, Cèsaro summation, and different other methods of summability extend the summability of functions to divergence, allowing for analytic continuations of functions such as the Riemann-Zeta function, and it extends the domain. Furthermore, these sums have been rigorously defined and we even understand their properties and their limiting behaviors very well via several proven theorems of the 20th century. The details are too many to discuss here and my response is already long enough, but the fact is that this all proven and consistent with the standard mathematical axioms we assume when we define finite summation.
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32:15 - 32:28 This is just false, and again, this suggests that you may just have a fundamental misunderstanding here. Computers use approximations, but humans do not, so when we do pure mathematics, we use exact numerical values. Can we express those numerical values with decimal digital strings? No, but this is irrelevant: pure mathematiciams are rarely concerned with those things. Those are just representations, not the number itself. We are still working with exact values. The fact we do so symbolically is entirely inconsequential: decimal digital strings are still symbolic. Why would you be concerned with what computers can and cannot do if you are doing pure mathematics? This is the issue. You almost seem to really not understand what pure mathematics is about, or the fact that humans need not be like computers, or that numbers as objects are different from the objects that represent those other more abstract object, or that decimal string representations are still a symbolic tool.
32:30 - 32:34 Because this is pure mathematics, not applied mathematics. Why should we care about what a computer does? That a computer cannot do something does not imply that something is "fake." C'mon Mr. Wildberger. You are smarter than this,... right? You are, right? Arithmetic operations are not defined as "something you do algorithmically," so there is no reason why computability has to be relevant to pure mathematics. A computer cannot do sandpile arithmetic, but this does not invalidate the mathematical research on the set of objects known as "sandpiles."
32:36 - 32:41 They absolutely can. The fact that you are too dishonest to accept the answers or incapable of understanding why they are valid answers is your problem, not ours. Honestly, this is just starting to sounds like one super-dragged out and elongated argumentum ad ignorandum. In effect, you are showing that you fail to understand something, bland because you fail to understand it and it fails to obey your poor misconceptions, it must be false. This is just a logical fallacy.
32:56 - 33:09 Yes, although infinite objects are interesting to study for their own sake too. You need to stop it with this misconception that everything that is studied in pure mathematics has to serve a practical purpose as opposed to being studied for the sake of studying it. However, you are correct that the study of infinity happens to be incredibly useful to us. The entirety of physics is dependent on the idea, for example. Did you know that it is only because of this "fake" arithmetic that we are able to study quantum theory and create the Internet so that you can have a place to upload a video to present your lackluster lecture? Now you know if you did not know already.
33:16 - 33:23 Yes, hence why it is called pure mathematics or theoretical mathematics and not applied mathematics, Mr. Wildberger. I would have imagined that this alone would be sufficient to dismantle your entire worldview, but I guess life sometimes just surprises you.
35:00 - 35:14 False. In the definition of a derivative, there is absolutely nothing that requires calculating a limit to infinity. This limit can be calculated without having at all to appeal to sequences.
36:05 - 36:28 This is an incredibly misleading description of how limits work. Despite what the name suggests, the definition of what a limit is has nothing to do with the repetition of a process over and over again. In fact, a limit has nothing to do with any type of process. It can, in certain situations, be informally and unrigorously described as such to create a visual intuition of what results look like, but this is merely done as a pedagogical tool, not a mathematical tool. Limits are not processes. They do not involve processes, and nothing is being repeated. Supertasks, while somehow related, are an entirely different subject of study in mathematics.
36:32 - 36:50 I will reserve my opinion on this since I have to look more into this, but independently of whether your project is successful or not, your misleading and dishonest thesis that the standard real analysis and arithmetic is "fake" and that infinity is problematic has been thoroughly debunked, and I would hope you acknowledge this mistake humbly and move on.
37:03 - 37:10 Standard calculus has never required us to assume anything about tasks.
37:12 - 37:24 You still have not proven this arithmetic is fake. Also, just because it is not needed does not mean it is not better. Technically, you do not need to work with any sets that are not the set of natural numbers, but this does not mean the set of natural numbers is the best set to work with.
37:45 - 37:53 I will gladly do so.
38:37 - 38:56 No, there is nothing inherently sociological about this. This paradigm has been maintained for the simple reason that it works, it is the most useful, and is perfectly valid, and you have utterly failed to present a convincing case otherwise.
38:57 - 39:00 And I am Ángel Méndez Rivera, and thank you for reading this.
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Trying to solve the Collatz conjecture simplifies to answering the question, "Can you get to 1 if we start from any odd number?", simply because that if we start with an even number, it will go to 1 if it is a power of 2, and it will go to an odd number otherwise. Furthermore, we know odd numbers are of the form m = 2n - 1 for all natural n. Hence 3(2n - 1) + 1 = 6n - 2 is always an even number, so we can divide by 2, obtaining 3n - 1 for all n. Then, every odd number of the form 2m - 1 = (2^p + 1)/3 is guaranteed to be go to 1 eventually since 3m - 1 = 2^p, and powers of 2 are guaranteed to go to 1. Thus, now we are interested in the question, "Starting from any odd number, can we always get to a number of the form 2(2^n + 1)/3 - 1 = 2^(n + 1)/3 + 2/3 - 1 = (2^(n + 1) - 1)/3 for some n?" An affirmative answer would prove the Collatz conjecture. Now, is this not beautiful?
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Silly Sausage In the purest sense? There is no "purest sense." There are definitions, and by definition, 1 is not a prime number. There is no such a thing as a "pure prime number" or a "fake prime number."
Excluding 1 because it does not fit is not horribly weak. Excluding 1 because it does not fit is quite literally how definitions work. If it fits or satisfies the definition of a prime number, then it is a prime number, and if it does not fit it or satisfy it, then it is not a prime number. The same holds true for all definitions. 4 is an even number because it fits the definition of an even number. 3 does not satisfy the definition of an even number, so it is not an even number. By your extremely horribly flawed logic, excluding any number from the set of primes hust because it does not fit "is "horribly weak." You may as well declare that all numbers are prime numbers while you are at it. To finish it off, you should also declare that all numbers are square numbers, cube numbers, perfect numbers, Fibonacci numbers, Lychrel numbers, whatever.
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Trevor B “Until the p-atic number system is shown to be completely related to the number system we use, and follow all the same rules, we cannot assume it does.”
No, we CAN assume it does, and we DO assume it does, because that is precisely how mathematics work. Just leave it to the experts before you say nonsense. The p-adic numbers are analogous to the real-numbers, except with a different metric space. In other words, both systems are special cases of a metric choice. In any case, in mathematics, as long as a claim is consistent with the axioms and can be derived, it is valid and allowed.
“We do not make the rules in math, we find them for various types of number systems.”
You have no clue of what you are talking about.
1. Number systems is a minor component of what mathematics is about.
2. We CREATE the number systems, and we CREATE the rules for them. We decide the axioms we want to work with, and only from these axioms can we prove these theorems. There cannot be a theorem with an empty set of axioms. 2 + 2 = 4 only because we decided to define addition axiomatically in such a way that resulted in this equality. We can redefine and obtain different results. Math is a formal logic tool, abstract and symbolic, which we created for our survival.
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Jeffrey Black “The only rules “i” doesn’t follow is the rule of not having square root for negative numbers...”
Actually, this was never a rule, this was a theorem especifícale about the field of real numbers.
“...and the rule that square roots should always be positive.”
The set of square roots never has the cardinality of 1 except for input 0, so this was never a rule nor a theorem.
Also, your statement is false. i breaks many rules. Namely, i^2 = (-1)^(1/2)*(-1)^(1/2) = [(-1)*(-1)]^(1/2) = |1|^(1/2) = 1, which is a contradiction. In other words, (ab)^c = a^c*b^c is not true in general. Also, i breaks the rule of uniqueness of logarithms.
“What he is doing is redefining addition and distance.”
No, what he is doing is working with the group of p-adic numbers. There is no redefinition, as your previous sentence had stated.
“It relies upon infinity - infinity which is undefined.”
No, it does not. Infinity is not a number, so there is no such a thing as infinity - infinity, because arithmetic with “infinity” does not exist. Watch the video again.
“Something being done before doesn’t it makes sense.”
Mathematics doesn’t care about what makes sense or not. In mathematics, you start with a set of axioms a set of laws of inference. With those, you then prove theorems, and sentences are either true or not.
“Once upon a time bloodletting was deemed a medical treatment even though it was pure nonsense.”
Once upon a time, an uneducated troll on YouTube called a divergent series nonsense, even though there was rigorous and expertly careful treatment which endorsed its valuation. Guess who am
I talking about.
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@JOELRODRIGUEZ-lk9gu 1 is prime, its literally the first prime number,...
No, it is not. 2 is the smallest positive prime number (with –2 being the greatest negative prime number).
...and it does fit the definition of what a prime number is.
No, it does not, and I would bet real money that you do not know what the definition of a prime number is. You probably think the definition of a prime number is "a positive integer whose only positive divisors are 1 and itself." If you do believe this, then you are in the wrong, because this is not the definition of a prime number. It has not been the definition of a prime number for at least a century now. A prime number is an integer whose only positive proper divisor is 1. The number 1 has no proper divisors at all, so it is not a prime number, and for the same reason, neither is –1.
Prima materia = first matter = prima = primero = first
I am glad you know the etymology of the name "prime number," but ultimately, the etymology is irrelevant to the mathematical definitions. My name is Ángel, and the etymology of my name is the Greek word άγγελος, which means "messenger." I am, however, not a mailman, nor a courier. The etymology of my name has nothing to do with the characteristics that define me as a person. Similarly, the etymology of a word used as an English name for a mathematical concept has nothing to do with the definition of the mathematical concept.
2 is not first lol, its secunda materia, second matter
Numbers are not made of matter. Numbers are abstract concepts which exist only in the mind of sentient beings which are capable of doing mathematics. That being said, the reason the prime numbers were named after 'prima materia' as a metaphor is that they took the atomic theory of matter as an inspiration: atoms, also called first matter, were considered to be fundamental, elementary particles comprising all matter. Any particle of matter could eventually be reduced to atoms, and atoms built all other particles. The prime numbers play a similar role with the integers: the prime numbers are analogous to atoms, in that they cannot be multiplicatively composed of other numbers, and in turn, the integers are multiplicatively generated by them. –1 and 1 are not prime numbers, because they do not fit that role: they actually do not generate any integers at all, since 1•(–1) = (–1)•1 = –1, 1•1 =
(–1)•(–1) = 1. –1 and 1 are trapped in their own bubble, incapable of generating the other integers. The same happens with 0. The prime numbers 2, –2, 3, –3, 5, –5, etc., are the ones we need to generate integers. 1 is the empty product of primes, and then we have 2, 3, 2•2, 5, 2•3, 7, 2•2•2, 3•3, 2•5, 11, 2•2•3, 13, 2•7, etc. In the other direction, we have –1, with (–1)•(–1), –2, –3, –2•2, –5, etc.
Prime numbers are extremely important.
Yes.
I know there is a why it was changed, the real reason behind it, but will we ever find out the why?
I strongly recommend you watch Another Roof's video on the topic, because the truth is, for most of history, 1 was not considered to be a prime number. Prior to 1600s, it was not considered to be a prime number. It began to be considered a prime number due to a misunderstanding of the concept, but in the 1800s, they stopped considering it a prime number again. Furthermore, we know exactly why both things happened, and we have known ever since they happened. There is absolutely no mystery at all. I am sure this has actually been explained to you before, but you just were not paying attention.
I can prove 1 is a prime already visually, without any fancy formulas.
False. I know you cannot do that, because the definition of a prime number is given by a formula, not by any geometric concepts. You think you can prove it, but this is how like children think Santa Claus is real.
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@shivanshpandey3905 No, the term "product"
does not refer to that. While many group operators denoted "product" are strictly binary, and they may or may not be associative or commutative, depending on the algebra, in mathematics, mathematicians operate with a more general definition of a product that is not restricted to group theoretic axioms. Products can have 0 or 1 terms. If it has 0 terms, then it is called "the empty product," and it is equal to the multiplicative identity. If it has 1 term, then it simply behaves like the identity operator. These are trivial products, so in applications, they are not referenced often, but in mathematics, empty products and unary products are built in and appear everywhere. n^0 = 1 is a product with 0 terms (n appears 0 times). So is 0! = 1. The Pi notation for products and Sigma notation for summations is even designed to reflect this. Ultimately, products are merely operators which are distributive over another set operator, which you may call addition for semantic purposes.
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@OptimusPhillip Yes, in terms of numerical value, indeterminate forms are considered undefined. But they are very useful in calculus because of how they affect limits.
No, they are not. Indeterminate forms arenot valid mathematical concept, and they have no effect on the concept of limits whatsoever. Real analysis works just fine without indeterminate forms.
(f(x+h) - f(x))/h = 0/0 when h=0, so it's undefined. But the limit as h approaches 0 is very much defined (when f(x) is continuous),...
You mean "when f is continuous." It makes no sense to talk about f(x), an individual real number, being continuous. Also, you are wrong: the continuity of f is not sufficient to guarantee that the above limit exists. A counterexample to your claim is the Weierstrass function, which is known to be continuous everywhere and differentiable nowhere. Anyway, even if you delivered your point incorrectly, I understood what you meant: for some values of x, the above limit as h —> 0 may exist. Yes, this is true, but indeterminate forms are irrelevant here.
...and is in fact the definition of the derivative.
Yes, I am aware. I already took calculus in college. I am far beyond that stage in my education, right now.
If 0/0 is just undefined, derivatives don't exist, and calculus doesn't work.
False. 0/0 is indeed just undefined. This has absolutely no effect on calculus, because 0/0 never occurs in calculus. lim f(x)/g(x) (x —> p), where lim f(x) (x —> p) = lim g(x) (x —> p) = 0, does occur, but this has nothing to do with 0/0. These are completely different things. It seems to me as though you fundametally misunderstand how limits work, hence why you mistakenly think 0/0 and lim f(x)/g(x) (x —> p) are the same thing. They are not.
That's why we have indeterminate forms, at least when working with limits.
No, that is not why. The reason is because of historical tradition, and because the education system for mathematics is notorious for being garbage at a worldwide level.
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Herbert Vilhjálmsson Every number cubed is equal to {0, 1, 8} mod 9. To prove this, simply evaluate (9n + m)^3. By the binomial theorem, this is 9^3·n^3 + 3·9^2·n^2·m + 3·9n·m^2 + m^3 = 9k + m^3, since the first three summands are all divisible by 9. If m = 0, then m^3 = 0. If m = 1, then m^3 = 1. If m = 2, then m^3 = 8. If m = 3, then m^3 = 27 = 0 mod 9. If m = 4, then m^3 = 64 = 1 mod 9. If m = 5, then m^3 = 125 = 8 mod 9. If m = 6, then m^3 = 216 = 0 mod 9. If m = 7, then m^3 = 343 = 1 mod 9, and if m = 8, then m^3 = 512 = 8 mod 9. As you can see, any natural number cubed is equal to 0 mod 9, 1 mod 9, or 8 mod 9.
Why is this last fact important? Because this implies the sum of three cubes can only ever equal the sum of these three numbers mod 9. And you can never get 4 mod 9 or 5 mod 9 from those sums. So the sum of three cubes will never 4 or 5 mod 9, meaning that if a number is 4 or 5 mod 9, then it has no representation as a sum of three cubes.
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19:25 - 19:30 No, this is not true. There are plenty of real transcendental numbers which are computable numbers. For example, this is true of π and e. π + e is a computable number, as are π & e themselves. More impressively, πe & π/e are also computable numbers. Not only are they computable, they are efficiently computable, because there exist fast-converging sequences of partial sums of a function-generated sequence with which you can calculate these. Computing them is easier than proving their rationality or lack thereof.
19:31 - 19:39 Yes, you can. Your statement has been thoroughly debunked. The mathematical literature on the computational analysis and computer science, as well as the existence of irrational numbers that are also computable numbers is almost a century old and is quite enormous. I have nothing else to add here because, as I said, the literature is overwhelmingly large and comprehensive, and it does far more than just addressing the false claims you are presenting here.
19:44 - 19:51 What? Nonsense. Because uncomputable numbers exist and you fail to understand how they work, this means real number arithmetic is fake? That is just now how it works. An operation need not be defined algorithmically. In fact, they are not defined algorithmically. Some functions are computable, and some are not, but what makes a function satisfy the definition of a function has nothing to do with computability. All a funcion is, it is a subset of the Cartesian product of two sets, such that for every element of the first factor, there exists at most one element of the second factor which the former forms a pair that is an element of the subset.
19:53 - 19:56 They are properly defined. They are the supremum of a set of numbers and the infimum of its complement, they are the limit of a set of Cauchy sequences. The limit operator, the supremum operator, and infimum operator, as well as the sets they act on, are all properly defined. Cauchy sequences are well-defined. Therefore, the numbers are properly defined. There is nothing problematic here. I fail to understand what the problem is. Uncomputable numbers exist? What exactly is the problem with the existence of uncomputable numbers? Mathematics is a collection of formal theories, not a prescription or description of how the physical world operates.
19:57 - 20:03 Your presentation of real analysis as it is understood by mathematicians today is completely inaccurate, as is your presentation of the understanding that we have of set theory and group theory. I cannot tell if this is deliberate dishonesty, or deliberate ignorance and misunderstanding of the literature, but regardless, the fact that you are presenting these obviously incorrect claims to an audience you yourself acknowledged may not have the sufficient education to be able to discern as true or false for themselves in order for to present constructive criticism and logical refutations, is completely abhorrent.
There is no denying that some of your other work is respectable, and being allowed to present skeptical viewpoints is necessary for the community to work, but when you actively try to mislead an unknowing audience that is seeking to be educated on a topic for the sake of getting a validation card, that is when the conversation stops being about skepticism and it starts being about your ethics. This is not a critical argument you are presenting about mathematics, you are merely trying to fill an agenda disguising it as a presentation on mathematics you happened to construct very poorly.
20:20 - 20:28 Yes. Sometimes, some things are too complicated to explain properly. You want to tell you how the Big Bang works? Well, I cannot explain that to you if you lack certain prerequisite knowledge to understand the explanation you are asking for. That is just how life works. You cannot run without first learning how to walk. This is supposed to be a flow of the educational system? The education system is completely broken, I agree with this, but not for this reason. This is also by no means a flaw with real analysis, this is ultimately just a mere inconvenience inherent in the way learning fundamentally works. I cannot teach you about computable numbers if you are still trying to learn how to add numbers, so of course I would have to tell you "that is for a more advanced course."
20:28 - 20:38 Such as?
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TurboCMinusMinus “”i” is at least a “newly invented number”. 0 and -1 are not.”
Did you fail you history courses in high school? -1 and 0 were discovered not many centuries ago, and i is at least two or three centuries old. Get over yourself.
“Saying the sum of increasing positive numbers approaches -1 is objectively wrong.”
1. No one said that the sum of increasing positive numbers approaches -1. Just to be clear, the phrase you used is nonsensical, incoherent. A sum is an operation, and you evaluate it and complete the evaluation. Sums do not “approach anything”, because the act of approaching is reserved to limits. If you claim that the sum of all elements of an ordered set is something, then you either evaluate it to that, or you do not. And whatever you evaluate it to be has, without loss of generality, no relation to the limit of a sequence, be it the sequence of partial sums or not. In vector calculus, we state that IF the sequence of partial sums of a summation over an ordered set converges, THEN this summation equals the limit. This is a statement about continuity, not about identity. If the sequence does not converge, that tells us nothing about whether the sum has a value or not, or whether it can have one or not, or what it should be.
2. Objectively wrong? By what set of axioms? If you are going to talk about objectivity, then you better start by specifying an effectively axiomatized formal theory. Otherwise, you are just being an idiot by throwing out useless terminology.
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@davidgould9431 Ancient philosophers were justified in making the mistake, though: axiomatic set theory did not exist in their day, so it was quite literally impossible to have known that their understanding of infinity, which is based on intuition, was actually inaccurate. WLC, on the other hand, is certainly not justified in making this mistake, as he lives in the 21st century, during an era where we have a better understanding of infinity, and of set theory, than we ever did before.
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@jakethemistakeRulez That's exactly what it is.
No, that is exactly what it is not, and if that was what it was, then multiplication axioms to define a ring would not be necessary, as they would follow from the addition axioms.
And ok, multiply (add the sum of one of them, the sum of the other times) these two infinite series:...
Oh, so you have not read the memo, huh? Series are not addition. They are not sums. Series are sequences. Specifically, we say that the sequence of partial sums of a sequence f is a series, and we denote this series by s[f], and we say that the series of f converges if and only if lim s[f] exists. The two expressions you have provided me with are the limits of a sequence, even though the traditional notation for these sequences is to denote them as indetermined addition of elements of f. If I multiply both series to evaluate the product π·e, I am not doing repeated addition, I am simply evaluating lim (1 + 1/n)^n·p(n).
it makes total sense because say you have x^0.25, it's the fourth root because 0.25 is actually 1/4 so what times itself 4 times is x?
You either do not understand what the objection is, or you do not understand what repeated multiplication means. There is no amount of x's I can multiply together to obtain the 4th root of x. 0 x's multiplied together gives 1, 1 x multiplied together gives x, 2 x's multiplied together gives x·x, etc. So x^(1/4), categorically, and strictly, is not repeated multiplication. The fact that x^(1/4) is one of 4 solutions to the equation y·y·y·y = x is irrelevant, because it still does not comprise repeated multiplication, since x is the product here, not the factor.
It's maybe not obvious but it makes total sense and is still addition.
No, it is not, and no mathematician agrees with you on this.
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@AShaif ...but suddenly we go all skeptic when it comes to the cause of the universe, because there is 1% chance there is no cause, despite arguments from fine-tuning, irreducible complexity, contingency, truth, language origin, the evolutionary argument against naturalism, and other cosmological, teleological arguments for the first cause.
You have an idiosyncratic usage of the word "despite." Why would you use the word "despite" only to proceed to mention a crap ton of extremely bad arguments that are straight up just much worse than the Kalam cosmological argument in terms of validity and soundness? You can present 100 bad arguments for the existence of God if you want. Present 1000. A million. Hell, you can present an infinite amount of bad arguments, if you want to. It will not change a thing. 0·ω is still equal to 0. What we need is a good argument, not a trillion extremely poor ones.
So if induction is a problem, why don't we act skeptical with things that are far more taken-for-granted.
Whatever do you mean? I cannot think of anything more taken for granted than the existence of God. There is more evidence for the existence of UFOs than there is for the existence of God. And this is really saying a lot. The 100 bad arguments you listed above do not make it less taken-for-granted. All they do is demonstrate that not only is the belief taken-for-granted, but it is also indefensible.
But my favorite is the contingency argument and the argument from language origin.
Seriously? Your favorite arguments are the worst of the bunch? Well,... you are a creationist, so I guess this makes sense.
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Christopher Sewell “Your continued uses of phrases like ... constitute weasel words...”
I’ll admit this. Not only is this true, but it also happens that these weasel words constitute appeal to authorities, which still imply some sort of fallacy at hand. Truth be told, I could provide specific references, but I’m a lazy person when it comes to most online debates, and this point isn’t fundamental to the core of my argument, so I’m willing to simply move on.
“I specifically said in my example that it would be wrong to say that if f(1 + 2 + 4 + 8 +...) = -1, then 1 + 2 + 4 + 8 + ... = -1”
I agree. Whenever the condition given applies, the proposition is false. However, do note that it is a condition. Note the IF at the beginning of the word. If the premise is false, the conclusion is indeterminate. And the premise is indeed false. Ramanujan summation is not a function that assigns a value to some summation input. Rather, the summation input IS the Ramanujan sum. There are several ways to illustrate this point:
1. If we naively assume convergence must be true in order for any series to have a value, then 1 + 2 + 4 + 8 + ... = Infinity. Therefore, if Ramanujan summation were some function (which it isn’t), then f(infinity) = -1. However, this reasoning leads to a contradiction, because functions can only have one out put. Yet, for f(1 + 2 + 3 + 4 + 5 + ... = Infinity) = -/12, which means there are two different outputs for the same input. Hence Ramanujan summation isn’t a function as you claim it is.
2. We know that 1/2 + 1/4 + 1/8 + 1/16 + ... converges to 1. This is uncontroversial. In a common calculus setting, it is assumed that convergence implies equality. Which means the correct equation is 1/2 + 1/4 + ... = 1, not f(1/2 + 1/4 + ...) = 1. This is true for all convergent sums. Now, Ramanujan summation actually evaluated to the same value convergent sums evaluate to, yet it also sums divergent series as well. It extends the capabilities of convergent evaluations, so we may as well replace it with (in some countries they do). Hence, Ramanujan summation, serving the same role as convergent summations, is not a function. It is wrong to treat it as such.
3. Also, fundamentally, just looking at the definition of what a Ramanujan sum is should tell you that all Ramanujan does is not assign a finite value to every infinite value produced by an infinite series, but rather it simply tells you what a series is equal to. Ramanujan sums are a method of evaluation, not a function.
“The problem with a statement like 1+2+4+8+...=-1 without ANY context, besides being absurd on its own, is that the series on the left is clearly divergent.”
The series on the left is divergent: true. This truth is not particularly relevant. There are no mathematical theorems or calculus theorems that declare that divergent sums can’t’ve finite values. Our definition is a conditional statement itself: IF the sequence of partial sums for a series converges to a limit, THEN the series equals this limit. This Conditional by itself however holds no bearing on what the consequences are if the premise is false: namely, it doesn’t say anything about series that don’t converge. It doesn’t say anything about these series being summable or what the summation should be. Hence in theory, as long as any methods can be developed while remaining consistent with mathematical axioms, you can use whatever method you want to assign divergent series whatever values as long as this method can assign to every convergent series a value which agrees with the limit of the sequence of partial sums.
“Then 1 + 2 + 4 + 8 + ... = 1 + 3 - 1 + 7 - 3 + 15 - 7 + ... = 1 - 1 + 3 - 3 + 7 - 7 + 15 - 15 + ... = 0 + 0 + 0 + 0 + ... = 0 = -1”
These operations are not exactly valid. I’ll explain why just next.
“...or you disagree that addition is commutative...”
Addition is NOT commutative. It may come as a surprise to you, but addition is only commutative in general if you’re adding a finite amount of numbers or elements in the real numbers. In the more general case of infinite addition though, commutativity does not hold in general, not even for many convergent sums. The Riemann Theorem of Rearrangement states that for any conditionally convergent series (a.k.a, a series with alternating signs which happens to converge, but which would not converge without sign alternation), the rearranging of elements being added results in altering the value of the series itself. This is true for the alternating harmonic series, for instance, which converges to Ln(2). By switching terms around, such as -1/2 + 1 - 1/4 + 1/3 - ... you actually change the sum even though all the elements remain exactly the same. Yet you have no problem with these non-commutative convergent series because they still converge. Which shows that non-commutativity isn’t what causes problems in evaluating divergent series, but rather the non-convergence. Yet I already explained why non convergence isn’t sufficient of a condition for declaring a series as unsummable. So it is perfectly consistent with mathematics and acceptable to say that addition isn’t commutative in general for any arbitrary number of addends, and that divergent sums may be summable.
“This is why such identities are an abuse of notation”
Claiming something is an abuse of notation required authoritative justification.
“If you’re using an analytic continuation, you can say statements like, “let f be the analytic continuation of 1/(1-z)””
1. You don’t need the analytic continuation of 1/(1-z) to evaluate 1 + 2 + 4 + ... = -1.
2. No, it never needs to be specified. I’ve never seen a mathematical paper specifying such a thing. Analytical continuations typically are taken for granted from my understanding, and often times functions end up being redefined as their analytic continuations. For example, exponentiation isn’t really defined as iterated multiplication. No one uses this definition to evaluate the exponential function. We either evaluate the limit of the definition of e or use the Taylor series, so much that it is an alternative correct definition.
3. Actually, now that I rethink my response, f(z) = 1/(1-z) isn’t a function which requires continuation, since it is defined everywhere except for the pole z = 1 (which can be independently renormalized anyway as f(1)renom = -1/2.
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The hotel IS full, though, because every room is paired exactly with one guest, or one set of guests, anyway. What this paradox proves is that the map n |-> n + 1 is bijective for the set of natural numbers. It proves the set {0, 1, 2, ...} and the set {1, 2, 3, ...} have the same number of elements despite the fact that the latter is a proper subset of the other.
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9:58 - 10:09 I am glad you acknowledge this, because a very popular misconception among non-mathematicians is that this is not the case.
10:45 - 10:54 Presenting this as the key point is somewhat misleading, though I understand what you were trying to get at. The set of real numbers are defined via limits, be it indirectly with Dedekind cuts, or directly as equivalence relations of Cauchy sequences. In either case, these numbers that are expressible as an infinite string in decimal digital representation are just the limit of a sequence, and it so happens that this limit has at least one decimal representation.
10:55 - 11:09 I seriously hope you are not going to present this video with the premise that infinity is a problematic concept. Such a premise is fundamentally flawed.
11:45 - 12:00 This is completely and beyond inaccurate. π & e as numbers are much older than what you are claiming them to be. The irrationality of numbers such as e & π was proven centuries ago, but in addition to that, you are neglecting to mention that algebraic irrational numbers, such as sqrt(2) and φ, were known to exist and to be rational by the Greek millennia ago. Constructible numbers in general include those irrational numbers, called quadratic irrational numbers, and they were of great importance to the Greek. I am no sociologist, but I do know also that in other ancient cultures, some of the metallic ratios were considered important as well for at least some applications in architecture and the visual arts. As for proving that their decimal digit representation required an infinite string, yes, this did occur later - though not that much later, but the existence itself of these numbers was known for a very long time. So saying that it was only now that we had to acknowledge that other types of numbers besides the irrational numbers exist is just false.
17:56 - 18:00 Challenging? Yes. Problematic? No.
18:05 - 18:14 This is false. Others in the comments section have already given examples of this, so I myself will not bother having to repeat what they said, but you should know you can also do a quick search on Google and see for yourself that such algorithms do exist, and applied mathematicians will tell you this. Many of these algorithms have already been implemented in computing, though to a limited degree, obviously. Stating there is no algorithm is inaccurate, and this is stemming from the fundamental misconception that there is only one valid sequence of steps to add any two numbers expressed in decimal notation with a finite string. Actually, you need not appeal to decimal strings to add real numbers at all.
19:19 - 19:24 Sure, for computable numbers, this is true. I fail to understand how this is problematic. At best, all this implies is that string representation is limited, and so are computers, which is a moot point, because computers are already limited anyway by virtue of physics. You can never accomplish with a computer all that a human can accomplish because that is just the nature of computers: they are different from humans. However, you need not know what the decimal string representation of a number is to be able to work with the number in the relevant context and understand it. Also, you already discussed earlier how, in applications, it is strictly necessary to have a flexibility for truncations and for approximations, so this is not a problem. These numbers are still definable, and from the definition alone, you can always make some amount of progress. Also, it should be noted that even the formal definitions of computable functions from any given computing model appeal to a concept of theoretical lack of limits about a given thing.
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20:38 - 20:41 You cannot seriously say that given the hundreds of thousands of number theoretic proofs that exist, not only on the theorems of arithmetic themselves, but even on the effective finite axiomatization of arithmetic or impossibility thereof. This is just more dishonesty. If you could at least bother to give examples, then that would somewhat help your case.
21:33 - 21:38 What does that actually mean? If by "evaluate," you mean "to compute the complete decimal digit string representation of the number," then sure, it is true that we cannot write such a number in a complete decimal digit string representation. However, this is an entirely arbitrary, unnecessarily restrictive, unhelpful, and misguided definition of the verb "to evaluate." Such a definition would also imply that numbers such as 10^(10^100) and TREE(3) are not actually numbers that exist, because it is impossible to write the complete decimal digit string representation on a paper. This is particularly true for uncomputably large numbers, such as the busy beaver numbers and the Rayo numbers. You may as well deny the existence of any natural number larger than 10^(10^80).
Also, this claim seems to point to a fundamental conflation between the symbolic representation of a number and the number itself. The symbolic machinery you so despise is employed to represent every number, because numbers are fundamentally abstract conceptualizations not present in the physical world, so a physical actualization of the pure concept of the number is impossible. When I say every number, I really mean every number. The symbol "2" is just that: a symbol for the number. Other languages have other symbols for it as well. The symbol should not be confused for the number it represents. In this regard, if your logical deductions are consistent, then using the symbol "2" in Kindergarden arithmetic is no less problematic than using the symbol "log(3)" to represent the irrational numerical value it represents. In fact, why should you care about decimal digital string representations when computers operate in binary digital string representations? We should be representing the number as 10, not 2, at least if we want to maintain logical consistency with what your comments seem to imply.
By any sane and reasonable definition of the verb "to evaluate," these symbolical representations are already trivially evaluated, because we already know what abstract object to identify the symbolic representation with. The equality relation, when written as part of a symbolically-written equation of two expressions, merely identifies a representation of a number with another representation. The number itself is already known from recognizing the value represented by either string of symbols, or from using definitions and axioms to simplify the expressions via identification with other expressions: other string-symbolic representations of the same number, or object, more generally, since this also applies for matrices and vectors, for examples. This process of identifying an expression written with a string of symbols representing an object with another expression whose corresponding number value it represents is already known and recognized, is what mathematicians call an evaluation. It has nothing to do with computing a string of digits in any partocular base, be it decimal or binary.
Mathematicians do not think of digital strings as being numbers. These digital strings are merely how we choose to represent them because they are a very historically convenient way of actually keeping track of tallies and counts. You need to stop thinking as if numbers had to be fundamentally representable this way to be valid. Plenty of non-mathematicians, and especially younger students, think of the decimal representation as the number itself. So if you tell them 0.(9) = 1, they lose their mind, because they are incapable of conceptualizing the idea that a number can be represented by a decimal digital string in two different ways or more, because they think of each representation as a number, not as a representation. This would be akin to thinking that 2/4 and 2/1 are different rational numbers because the integers being divided are different. Strangely enough, they also may find themselves thinking that 1 and 1.0 are different numbers. This is a mistake that we need to address.
22:15 - 22:18 It is interesting that you mention γ in your list of examples, because it actually demonstrates why your previous claims are silly. Did you know that it is not known whether γ is rational or not? It has been proven that if γ were to be rational, then it would have an astronomically large denominator. Writing the number in quotient form would be completely unfeasible. It also would have a decimal digital string representation of infinite length. This would be an example of a rational number that, according to your argumentation, could not exist, it would be a "fake" number from a "fake" arithmetic. Also, independently of whether γ itself is rational or not, such rational numbers do exist, you even mentioned them earlier.
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@theodoregossett9230 It does have to do with surjective functions, though, since the "size" of a set (technically, its cardinality or numerousity) is defined in terms of bijective functions. Two sets X, Y are equinumerous if and only if a bijection f : X —> Y exists. This is how equinumerousity is defined. Equinumerousity is an equivalence relation, and it partitions the universe of sets into equivalence classes, called cardinality classes. The cardinality class of a set X can be denoted |X|, and with the axiom of choice, these cardinality classes can be totally ordered. Now, we can say that for two sets X, Y, |X| =< |Y| if and only if there exists an injective function g : X —> Y. If X and Y are finite sets, then in order for them actually be finite, it must be the case that there exist functions p0 : X —> N, p1 : Y —> N which are injective, where N is the set of natural numbers, since this is how the finitude of a set is defined. To be able to say that |X| < |Y| is to be able to say that that some injective function g : X —> Y exists, but that no surjective function h : X —> Y exists. Your "formulation" of the pigeonhole principle, that if |X| < |Y| < |N|, then there is no injective function g : X —> Y, is a tautology, since the consequence is precisely the definition of the condition, and thus, this is not actually a formulation of the pigeonhole principle.
Technically, the pigeonhole principle actually is the following statement: if |X| < |N|, and |Y| < |N|, and there exists a surjection h : X —> Y which is not injective, then |X| < |Y|. In essence, this is just saying that surjectivity and injectivity are dual, in the sense of category theory, at least when it comes to finite sets.
Another formulation is to say that if |X| < |Y|, and Y is a subset of the power set of X, then there exists some S in Y such that |S| > 1. This one is the more straightforward formulation from the English wording.
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Jeffrey Black No one claimed that -1 was invented to address the problem presented in the video. You are using a straw man argument fallacy here, and it is fairly easy to detect. What the video is talking about has little to do with what -1 is or does or why it was invented. Hence your objection to the OP does not actually say anything relevant.
No, the axioms of mathematics do not say anything about the content of this video. In particular, the axioms of mathematics do not say anything about infinite series. You also say this as if there existed one unique set of axioms of mathematics, but this is false. There are different mathematical formal logic theories, each with its own set of axioms and inference-deduction rules. In most of those, the axioms say nothing about infinite series, because, in particular, the axioms of arithmetic are all limited to talking about addition of elements of finite sets. Statements about infinite summation in those theories are undecidable, as proven by Gödel’s First Incompleteness Theorem, and such, they do not disprove nor prove any claims about infinite summation, not even those where the elements in the set are of order-type ω.
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This equation is actually a better example of a homogeneous equation. Given y' – y/Id = –(y/Id)^2, the better substitution here is z = y/Id, equivalent to Id·z, hence Id·z' + z = y', hence y' – z = Id·z'. Thus, Id·z' = –z^2, and this is separable. Hence z = 0, or –z'/z^2 = 1/Id, hence 1/z = ln(–Id) + A if Id < 0, 1/z = ln(Id) + B if Id > 0, so z = 1/[ln(–Id) + A] if Id < 0, z = 1/[ln(Id) + B] if Id > 0, hence y = Id/[ln(–Id) + A] if Id < 0, y = Id/[ln(Id) + B] if Id > 0. This definitely works better using initial conditions, though. If –z'/z^2 = 1/Id. Integrate over [x, –1] and over [1, x], respectively. We have that 1/z(–1) – 1/z(x) = ln(–1/x) for x < 0 and 1/z(x) – 1/z(1) = ln(x) for x > 0, which translates to 1/z(x) = 1/z(–1) + ln(–x) for x < 0 and 1/z(x) = 1/z(1) + ln(x) for x > 0. Hence, for every x, y(x) = 0, or, for every x < 0, y(x) = x/[ln(–x) – 1/y(–1)], and for every x > 0, y(x) = x/[ln(x) + 1/y(1)]. More concretely, what we have is, for every x < 0, y(x) = y(–1)·x/[y(–1)·ln(–x) – 1], and for every x > 0, y(x) = y(1)·x/[y(1)·ln(x) + 1], or for every x, y(x) = 0. Even more succintly, for every x, y(x) = 0, or y(x) = y[sgn(x)]·x/{y[sgn(x)]·ln(|x|) + sgn(x)}. As for the latter, notice that lim y (x —> 0) = 0. Since y' – y/Id = –(y/Id)^2, we have that lim y' – y/Id (x —> 0) = lim –(y/Id)^2 (x —> 0), and since lim y/Id (x —> 0) = 0, we have that lim y' (x —> 0) = 0. So, if we define y(0) = 0, then for every nonzero real x, y(0) = 0, y(x) = y[sgn(x)]·x/{y[sgn(x)]·ln(|x|) + sgn(x)}, and with y defined in this fashion for every real x, y is continuously differentiable everywhere, and includes y(x) = 0 in the case that y[sgn(x)] = 0. This is what thr video misses.
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This is a good question. There is no authoritative source that uses the symbol N or the name "natural numbers" and excludes 0, and the ISO does not even recognize the "whole numbers" nomenclature, since in most of the rest of the world, "whole numbers" refers to the set of integers Z.
Mathematically, the only sets you make axiomatically are N and Z, and N includes 0, with the definition that 0 := {}, and if n is a natural number, then Union(n, {n}) is a natural number. There is no "whole numbers" nonsense, which is largely an invention of North American schools of the 20th century.
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Are some of these questions unanswerable?
A better answer is "are these questions meaningful and coherent?" Typically, if a question has no answer, then it is because the question itself is nonsensical. The fact that you can throw words together to form a grammatical correct interrogative sentence does not imply that the sentence itself is meaningful. "Why is the Sun equal to the number 5?" This sentece is grammatically correct and mechanically correct too, but the question is literally nonsensical. It is meaningless. Is it unanswerable? Yes, it is unanswerable, but not because there is some fundamental flaw in human reasoning; rather, it is because the question is, in a sense, not even really a question at all, just a nonsensical string of words that happens to be grammatically correct. It is because the person who asked the question is not very bright and does not have the best understanding of how questions are supposed to work. Questions like "Why is there something rather than nothing?" seem, superficially, like they deserve an answer, because they appeal to the very flawed and unreliable thing that we call "common sense", but upon further and careful inspection, such questions are equally as nonsensical as the example I gave. There is nothing to answer, because the question is meaningless: ultimately, it is not asking anything at all.
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@lexinwonderland5741 I agree with you that more should have been mentioned on it, since the distinction between irreducible elements and units in a ring is ultimately at the root of why 1 cannot be considered a prime number.
Look, do not get me wrong. I think that understanding how mathematical concepts from antiquity involved into mathematical concepts today as our understanding of mathematics improved and became more refined is very fascinating, and certainly an important kind of knowledge to have in general. However, as far as answering the question "is 1 a prime number?," the history is not enlightening at all: it ultimately does not answer the question. Yes, I know that mathematicians in the 1700s thought of 1 as a prime number, this is all well and fine, but that tells us nothing as to whether 1 actually is or should be considered a prime number or not. These questions are questions regarding the relationships between various mathematical concepts at a foundational level, not questions about names and conventions that mathematicians vote on. If you want to get at the question of whether 1 is a prime number or not, then you ought to compare the prime numbers with 1, analyze their properties and their roles within the integers, then compare how these things extend or fail to extend when you move on to other mathematical structures, like polynomials and Gaussian integers. This is how you answer the question.
Appealing to the history of mathematics actually reinforces most people's misconception that 1 should be considered a prime number, and reading the comments to this video has resoundingly confirmed this suspicion. I think that discussing the history is perfectly fine when addressing the question "why did we ever consider 1 a prime number?" or "how has our understanding of prime numbers changed?" But neither of those questions is the question the video claims to address.
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@SplendidKunoichi The pigeonhole principle is neither obscure nor confusing, and it does serve a purpose, as it is used in multiple applications, several of which were mentioned in this very video, which I realize you may have not watched at all.
Your claim about the reason these are called principles is completely unsubstantiated. In fact, the reality is, the way these theorems are named is completely arbitrary. There is no rhyme or reason behind things being named laws, principles, lemmas, corollaries, etc. These names are merely a matter of historical relic, not of actual epistemic substance.
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The reason we use degrees, rather than hours, is because 360 hours are meant to denote not a full rotation of the Earth, but rather, a half-period of the lunar cycle. 360 hours are 15 intervals of 24 hours, which is half a lunar month. As such, one hour is 15°, which is trigonometrically a very important and fundamental constant.
With all of that being said, I have learned that thinking of degrees as a unit of measurement of angles is not the conceptually appropriate way to think of them, if only because, a ratio of two quantities with the same dimensionality is dimensionless, and thus, is numerically independent of the units it is measured in: there should not exist such a thing as different units for a dimensionless quantity, mathematically speaking. So what is actually going on instead? Well, what is going on is that degrees are a scale factor for the trigonometric functions. Writing sin(1°) is the same thing as writing sin(π/180). In other words, sin(x°) = sin(π/180·x). So when you change from degrees to radians, you are not changing units of measurements. What you are doing instead is rescaling the trigonometric functions.
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This is exactly right, and it explains why Christianity is currently (though not for long) most popular religion in the world. Christianity is such an ill-defined, ambiguous-or-incoherent mess of a belief system, that it is very malleable, and thus, very easy to reinterpret in a way that agrees with your already pre-existing intuitions and conceptions of the world. It can easily adapt to any culture, and most mindsets. Other religions are a lot more rigid with what exactly defines their belief system, so they are not as malleable, and so, not as easy for people to adopt as a way of justifying their pre-existing intuitions and conceptions. Virtually no Christian is actually willing, or capable, of aligning their beliefs in accordance to the actual history of how Christianity evolved from Judaism. Christianity functions as nothing more than a tool for confirmation bias.
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nightsinger81 A product can have any number of inputs you want it to. It can have 0 inputs, in which case, the product is 1. You have already seen this before, with the equations n^0 = 1 and 0! = 1. n^0 is literally, by definition, a product with 0 factors, because n^m is literally defined as the value of the product in which n is a factor m times. You can also represent this using Π product notation, although doing this on YouTube is extremely inconvenient. 1 is the product of 0 primes because it is the empty product.
On a similar vein, every prime number is a product of prime numbers. Each of those products is a product with 1 input, much like how 1 is a product with 0 inputs. Again, you already have seen this reflected with the equation n^1 = n, because n^1 is literally the value of the product in which n appears as a factor 1 time, and there are no other factors.
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1. Suppose God is omniscient.
2. Because God is omniscient, God knows all false statements, all true statements, and all absurd statements.
3. Because God knows which statements are true and are not, this implies God knows why they are true. Better said, God understands the criteria that a statement must meet in order to be assigned the propositional value "True".
4. Suppose the set of such criteria can be signified by K, and it has multiple elements K(1), K(2), ..., K(N).
5A. If there cannot be no such criteria, then the assignation of the values "T, F, A," are arbitrary, ergo truth and/or knowledge are subjective.
5B. If there are such criteria, then God must know them, because God is omniscient by premise #1.
6. This implies God knows that "Any proposition P(0) that meets the set of criteria K is true" is in itself a true proposition. We call this proposition P(1).
7. If P(1) is true, then it meets K.
8. Therefore, P(2)="P(1) is true because it meets K" must itself meet K.
9. Therefore, P(3)="P(2) meets K" is also true, which implies P(3) meets K.
10. By induction, P(X)="P(X-1) meets K" also meets K for any X.
11. This induction implies that for any statement to be true, the statement must be true because it meets a criteria, and this conclusion must be valid because it is known to be true, and as such it meets the criteria which make it true, and the above must also be true...
12. Premise 11 is an infinite regression.
13. Infinite regression is a logical fallacy, Argument Ad Infinitum.
14. Therefore, conclusion of premise 11 is invalid.
15. Premise 11 is invalid because premise 5B must necessarily yield an argument ad Infinitum.
16. Therefore, 5B is invalid.
17. Therefore, 5A is true because there either is a set of criteria or there is not, it is a Boolean conditional.
18. Therefore, truth is arbitrary and subjective.
19. If truth is subjective, then any statement cannot be known to be true, false or absurd.
20. Because no statement can be known, God cannot be omniscient.
C: Therefore, God is not omniscient.
Q.E.D.
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0:32 - 0:37 Contention in the community? I have no idea of what you are talking about. Arithmetic is not at all controversial right now in the mathematical community.
0:46 - 0:48 I assume you will define what "fake arithmetic" is later, correct?
2:41 - 2:51 This is not really correct. To begin with, the phrase "number system" is completely meaningless and ambiguous as you have presented it. Also, arithmetic operators are defined axiomatically, thus it is inaccurate to present operators and their axioms separately. To further worsen your argument, in mathematics, we often perform arithmetic with objects that, generally, are not considered numbers, such as sets, vectors, matrices, higher-order arrays, and other types of formal abstract structures. Many of them have no absolutely no relationship to the real numbers. A more accurate presentation of the topic that is fit even for non-mathematicians, such as sociologists, for example, would be to say that an arithmetic underlies (1) a collection of objects (2) a collection of rules that prescribe how these objects behave.
3:37 - 3:52 No, this is completely incorrect. Firstly, there is no such a thing as "the set of decimal numbers," and pretending that such a set exists, when you are presenting this to an audience which you yourself acknowledged is not necessarily mathematically inclined, is very dishonest. All this does is create misconceptions and confusion for the sociologists and anger the mathematicians and logicians. The set should merely be presented as the set of real numbers, and then just provide some examples, such as the numbers π or the golden ratio φ, numbers a non-mathematician may be familiar with, and this will give them an idea of what you mean without being misleading.
Secondly, those numbers are not contained in the set of rational numbers. To the contrary: the set of rational numbers is a subset of the set of real numbers.
5:26 - 5:34 Once again, this is misleading, because as I stated earlier, mathematical operations are defined based on certain properties, not the other way around.
6:20 - 6:41 You need to be careful with how you are wording what you say, because first you said these properties need to be proven, but then you stated that you need very precise definitions, not acknowledging that operations are themselves defined by properties. The problem is that you are starting this discussion without even having introduced the terminology properly to sociologists, and you have not explained to them what is the distinction between an axiom, a definition, and a theorem, and how those are all related.
9:01 - 9:23 I am not sure if this claim is supposed to be general about arithmetic, or this is only specific to floating-point systems and other similar things in applications. However, just in case, I need to clarify that this claim is only true precisely in an applied mathematics context. In general, though, this is not the case. Mathematicians generally work with exact, precise arithmetic, not approximations, unless they are studying approximation theory.
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26:12 - 26:19 π·e·sqrt(2) = 12.07700795676661899945694128307507080267577798989681051119841053. Just somewhat bigger than the previous answer.
26:21 - 26:26 π/[e/sqrt(2)] = π·sqrt(2)/e = 1.634445292479835509920482389420615372333245322395692016471053262. This is just the previous answer divided by e^2.
26:29 - 26:36 Yes, you are allowed to. It is not merely a belief system, as these are rigorously well-defined. π^[e^sqrt(2)] = π^exp[sqrt(2)] = exp{exp[sqrt(2)]·log(π)} = 110.8924304703034108980408007027890870545498126130158109541567996.
26:45 - 26:49 This would be a meaningful argument if not for the fact that your definition of "explicit answer" is founded on a flawed premise and misunderstanding the topics you are presenting. The definition is ultimate both misleading and misguided.
26:54 - 26:59 Except your silly definition of "answer" only allows for finite decimal digital string representations, which in this case is impossible, because 1/3 = 0.(3), which is an infinite string. This is precisely the problem with asking loaded, dishonest questions to your opponents in the face of the lack of logical arguments, without allowing for any discussion about the question itself. All you accomplished was shoot yourself in the foot, and expose your lecture as not only dishonest, but incoherent with itself. Still, you asked for the answer and nothing else, so when we give you the answer, you have to accept it. You are not allowed to have it any other way, because you asked a question and allowed for no discussion about the question, you wanted an answer and nothing else. This what you get. To bad and stupid questions, you get bad and stupid answers. That is how life works. Maybe present an actual argument instead of pretending these questions will earn you any sort of "gotcha!" moments against us. Then we can talk.
Anyhow, I easily give you a symbolic answer with no infinite decimal digital string representations: 1/2 + 1/3 + 1/5 = (3·5 + 2·5 + 2·3)/(2·3·5) = (15 + 10 + 6)/30 = 31/30. This is still symbolic. This still is not the number itself, merely a representation of it. You should still be mad about this if you are to stay consistent with the founding premise of your arguments. Here is the answer with the infinite string you so despise: 1/2 + 1/3 + 1/5 = 0.5 + 0.2 + 0.(3) = 1.0(3).
27:09 - 27:015 This is a classic example of the false dilemma fallacy. You conveniently ignore the fact that it is possible for both the understanding to be correct and for the framework to be genuine, and that the reason the answer fails is because the question is postulated in such a dishonest, loaded manner, that it cannot be answered without the other party butting in. This is why you are not allowed to ask loaded questions without allowing for discussion about the question only to be later picky about the answers that have already been established repeatedly to be completely valid in spite of your irrational denial. Remember that the burden of proof is on you anyway, so if you have to resort to asking such questions and rejecting the answers arbitrarily and at will, then this proves your viewpoint has absolutely no merit.
27:20 - 27:23 cos(7) + tan(2) = -1.43113760891821435350210858059450053121370435476753632501934199. If you want a symbolic answer, then here is a nice one: cosh(7i) – i + 2i/[1 + exp(4i)]. How do you like that?
27:31 - 27:35 log(3)·sin(4) = -0.83143252143997940236766565165617378511021231163412925895934121
27:36 - 26:42 I am not going to bother typing this into my cell phone. However, I can tell you this is equal to an alternating sum of hypergeometric functions, which is exciting.
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27:49 - 27:50 cos(5)^2 + sin(5)^2 = 1. This is basic trigonometry, though it seems the choice for this exercise was intentional. Oh well.
27:55 - 28:11 Ah, so this was intentional. Thank you for confirming.
29:32 - 29:38 Yes? Then let me ask you a question, Mr. Wildberger. What is 24? Can you reduce it any further for me?
29:43 - 29:51 But what IS 24? That is actually a very interesting question.
30:25 - 30:33 Yes, it is! There is no moral or axiomatic obligation to simplify the string of symbols 1/2 + 1/3 + 1/5 to another string of symbols representing the same value. We only do so because we can and we find it convenient. However, you do not have to do it, there is no rulebook created by God that states this. 1/2 + 1/3 + 1/5 is just as valid an answer as 62/60, just as valid an answer as me denoting with Σ, just as valid as 1.0(3). These are valid representations of the same number. Proving it is not difficult. However, you would, for whatever reason, claim they are not a valid representations, a claim that have purportedly already demonstrated, but never actually tried to justify. All you have done is try to be sneaky and get us "analysts" as you pejoratively call us with "gotcha!" challenges that actually fail to achieve their purpose, because to us, these are super easy to answer, regardless of whether you want to accept those answers or not.
30:35 - 30:45 Why are you saying "evaluates to"? The "=" symbol is read "equal to," not "evaluates to." Evaluation is a procedure, equality is a relation. You are severely conflating these concepts. Also, YOU stated that 1/2 + 1/3 + 1/5 cannot be evaluated to 1/2 + 1/3 + 1/5. Us analysts never said it. You did. If anything, all you are doing is deny the symmetric and reflexive properties of the equality relation, not actually prove anything.
31:06 - 31:11 You can do this with rational numbers too. Have you never heard of the concept of Egyptian fractions? Number theorists study how rational numbers can be expressed as a sum of other types of specific fractions. In those contexts, using the reflexive and symmetric properties of equality, as well not simplifying the sum, it not only permitted, but completely required.
31:30 - 31:43 That is not things work. You do not get to make a logically fallacious demand, and then claim that if an argument cannot satisfy that logically fallacious demand, then your stance has merit. This is just another fallacy in itself. Your definition of "calculation" is not only founded on an invalid premise, but the accompanying claim that these numbers are indeed not computable in the Turing sense as you claimed earlier is also demonstrably false. Once again, this is more dishonesty for the sake of an agenda. There is absolutely no merit to your stance, at least not in the way you have presented it here, although even then, finitism as a whole does not have a lot of merit, and many refutations of classical finitist arguments have been provided by mathematicians and philosophers in the last few decades. I am not going to recite the entire literature here in my comment, and I assume you know how to browse the Internet, so we leave it at that and move on.
31:44 - 31:50 A computer is not even able to do complete natural number arithmetic, as I mentioned earlier. Uncomputably fast-growing functions exist, so uncomputably large natural numbers exist. Go ahead and tell a computer to calculate Graham's number for me. Come back to me once you have obtained a string of digits as an answer.
Also, you have yet to explain why computers not being able to do this is problematic for pure mathematics. A computer is not able to have genuine emotions, does that mean emotions are "fake" and do not exist? Is the study of psychology problematic because computers cannot conceptually handle emotions, which are very much not in line with Boolean logic? This is a seriously a ridiculous argument, you are presenting here.
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All hyperoperations can be defined in this fashion. Given the μth hyperoperation, denoted %, the S(m)th hyperoperation, denoted #, can be defined by having m#0 = 1, m#S(n) = m%(m#n). Even more direcrly, we can define a function H : N^3 —> N such that H(m, n, 0) := S(m) & H(m, S(n), S(μ)) := H(m, H(m, n, S(μ)), μ). This uniquely defines every hyperoperation all at once. This is related the Ackermann function.
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@dlevi67 The difference between string theory and philosophy is that scientists who dedicate themselves to study and push forward string theory do not necessarily adhere by it. *For the most part*, scientists treat string theory for what it is: a set of hypotheses and conjectures, not as fact, which is contrary to what news media tend to do, because news media do not understand science. There is nothing inherently problematic about theoretical physics any more than there is anything inherently problematic with abiogenesis research in biology. The only thing that is problematic is the fact that the Internet places the experimental burden on people whose only job is to build models instead of placing it on people whose job is to carry out experiments. Why? Because 90% of the Internet is just idiots who think they understand what they are talking about. News media are particularly more problematic due to the importance that they place on the 5% of string theory researchers who do speak about string theory very strongly as if it was fact. They ignore the other 95% of researchers who are perfectly sane and impartial about their topic of research. Why? Because news media do not care about genuinity, they want to create spicy headlines and get the attention of people, which they will not be able to do if they focus on the 95% of researchers. Theoretical physics is doing just fine. Building models is still part of the scientific method, and although the headlines of the articles were presented with the purpose of being clickbaity and not with the purpose of being honest and informative, it does not change the fact that a theoretical calculation, even if only theoretical, is still meaningful and progressive. It is still part of science. An experiment cannot be carried out without first having a hypothesis to test. Of course, we can have a discussion about whether the hypothesis is actually testable or not, but this is clearly outside the scope of what we are currently talking about. What we are talking about is the legitimacy of theoretical physics and how it differentiates itself from philosophy. And that is precisely what I am doing: explaining the difference. And I repeat: there is literally nothing wrong with theoretical physics. The only thing wrong is how the Internet as a whole willfully misrepresents both the contents of theoretical physics and the purpose and role of theoretical physics, and the fact that everyone likes to pretend like the 10 leading researchers who fanatically treat string theory like factual represent the 10 000+ researchers in the community, when they obviously do not.
And by the way, I am not saying we should not be combatting that string theory fanatism. But also pretending that theoretical physics is no different from philosophy is equally dishonest and equally extremist and misguided.
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@piderman871 No, that is completely inaccurate. The reality is, the fundamental theorem of arithmetic is true, regardless of whether 1 is classified as a prime number or not. The definition of a prime number affects the precise phrasing of the theorem in the English language, but the actual formulae in logic are completely unchanged by any labels we assign to the quantities. Also, the theorem holds for a rather large class of structures known as unique factorization domains. The integers are neither unique nor special in this regard. The integers form a unique factorization, but so do the Gaussian integers, the rings of polynomials on the integers, etc. It holds for all fields, trivially, and such. Also, I am of judgment that the explanation given in the video is simply incorrect. 1 not being a prime number has nothing to do with how easy it is to state the fundamental theorem of arithmetic in English. It just has to do with the fact that, well, 1 really is not a prime number. This is not a matter of semantic convention, but a mathematical fact. We can very easily demonstrate that a conceptual, fundamental distinction exists between integers such as –1 and 1, and the integers we call prime numbers. For starters, the prime numbers multiplicatively generate the rest of the integers. This concept can be generalized to all commutative rings. How these irreducible elements generate the rest of the ring may not be unique, unlike in the integers where it is unique, but this is irrelevant: they still generate the ring multiplicatively. This is to say, the irreducible elements of a ring are not multiplicatively closed. 2 and 5 are prime numbers, but their product is necessarily not, by definition. Can we generate any integers with –1 and 1? No. (–1)•1 = 1•(–1) = –1, and (–1)•(–1) = 1. This is to say, the set {–1, 1} is multiplicatively closed. Further insight can be extracted from this: –1 and 1 are the only integers which have multiplicative inverses. As a result, –1 and 1 divides all integers trivially. This is not true of the prime numbers. The prime numbers are not invertible, and if they were, then they would not be able to satisfy their fundamental property: that of generating the integers (or more generally, the ring they are a part of). Them not being invertible is also at the core of them not being multiplicatively closed. As a result, the prime numbers do not divide all integers. They do not divide –1 or 1. –1 and 1 are only divisible by themselves, and so they have no proper divisors. The prime numbers do have proper divisors: –1 and 1, and those are the only proper divisors of a prime number. As you can see, –1 and 1 are fundamentally different from the prime numbers, and this distinction is not unique to the integers: again, it applies to all commutative rings. Therefore, insisting that 1 is a prime number is like insisting that Ferrari is a fruit, or that Toyota is an an animal. It makes absolutely no sense. However, the nail in the coffin is revealed when you compare the integers to the rational numbers. In the rational numbers, the rational numbers 2, 3, 5, etc., are all invertible, and they actually lose all the defining properties they had as prime numbers. Nonetheless, they actually gain all the properties that –1 and 1 had in the integers, and now, they all have the same algebraic properties. –1 and 1, meanwhile, have their properties unchanged. This means that –1 and 1 were never prime numbers to begin with, they were never irreducible elements of the ring of integers. Quantities such as 2 and 5 were, but in the ring of rational numbers, there are no irreducible elements: there are no prime numbers in the rational numbers. Instead, all the rational numbers, except 0, are in the same category as –1 and 1. They are called units, or unitary numbers, or invertible numbers. The set of units of a ring forms a group. In the integers, the group of units is {–1, 1}. For Q, the group of units is Q\{0}.
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@ruaraidh74 The result of throwing a die is a digit from the set {1, 2, 3, 4, 5, 6}. Therefore, the result of throwing a die is a 10^120-tuple from the set {1, 2, 3, 4, 5, 6}^(10^120). A 10^120-tuple of digits is not an epistemically meaningful message, so what you describe is impossible. Also, what is this supposed to be analogous to? Although we tend to model dice throws with probability theory, the reality is, the physics of a die throw are deterministic. With sufficiently precise control of the initial conditions of the throw, one can control the outcome of the throw, and accurately predict it, with the accuracy improving as the amount of throws increases. As such, in principle, one could throw a die 10^(10^(100)) times, and have it always come out a 6. Dice throws are not random. The reason we often have to model them as if they are random is because, under most circumstances, it is not possible to determine the initial conditions with sufficient precision, and as dice are examples of chaotic systems, any predictions you end up making on the basis of the initial conditions is redundant. Instead, we assume a uniform distribution for the initial conditions, and provided that there are no irregular forces acting on the die during the throw, as is usually the case, the uniformity is preserved by the physics, so we can model the outcome of a die as a random variable distributed uniformly. Therefore, in practice, we have sufficient justification to state that the outcome of a die is approximately probabilistic, and that the outcome of each digit during a single throw is approximately 1/6. This is analogous to what can be said for all other physical interactions in the universe.
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I should note that, technically, the definition given in the video is not the definition of the integral. If f is a function on [a, b], then you can partition [a, b] into intervals [x(i), x(i + 1)] with x(0) = a and x(n + 1) = b, and you can tag this partition letting t(i) be an element of [x(i), x(i + 1)]. Then the Riemann sum over the tagged partition is the sum of f[t(i)]·[x(i + 1) – x(i)]. The mesh of a partition is given by max[x(i + 1) – x(i)]. The integral is equal to the limit as max[x(i + 1) – x(i)] —> 0 of the Riemann sums.
Here, we have that f(x) = 4·x^2 is a function on [1, 4]. So the Riemann sums are the sums of 4·t(i)^2·[x(i + 1) – x(i)]. Now, we have that max[x(i + 1) – x(i)] —> 0, so it is quite natural to have x(i + 1) – x(i) = max[x(i + 1) – x(i)] – d(i), with d(i) —> 0, so x(j) – x(0) = j·max[x(i + 1) – x(i)] – Sum{0 =< i =< j – 1, d(i)}. Let t(i) = x(i) + s(i) = x(0) + max[x(i + 1) – x(i)]·i + s(i) – Sum{0 =< j =< i – 1, d(j)}, with s(i) —> 0. Hence 4·t(i)^2·[x(i + 1) – x(i)] = 4·[1 + s(i) – Sum{0 =< j =< i – 1, d(j)} + max[x(i + 1) – x(i)]·i]^2·{max[x(i + 1) – x(i)] – d(i)} = 4·{[1 + s(i) – Sum{0 =< j =< i – 1, d(j)}]^2 + 2·[1 + s(i) – Sum{0 =< j =< i – 1, d(j)}]·max[x(i + 1) – x(i)]·i + max[x(i + 1) – x(i)]^2·i^2}·{max[x(i + 1) – x(i)] – d(i)} = 4·max[x(i + 1) – x(i)]·[1 + s(i) – Sum{0 =< j =< i – 1, d(j)}]^2 + 8·max[x(i + 1) – x(i)]^2·[1 + s(i) – Sum{0 =< j =< i – 1, d(j)}]·i + 4·max[x(i + 1) – x(i)]^3·i^2 – 4·[1 + s(i) – Sum{0 =< j =< i – 1, d(j)}]^2·d(i) – 8·max[x(i + 1) – x(i)]·[1 + s(i) – Sum{0 =< j =< i – 1, d(j)}]·d(i)·i – 4·max[x(i + 1) – x(i)]^2·d(i)·i^2. Since max[(i + 1) – x(i)] is asymptotically equivalent to K/n for some real K > 0, one can use repeated applications of Tannery's theorem to conclude the limit as max[x(i + 1) – x(i)] —> 0 of the sums of the above is equal to the same limit of the sums of 4·max[x(i + 1) – x(i)] + 8·max[x(i + 1) – x(i)]^2·i + 4·max[x(i + 1) – x(i)]^3·i^2. The sums are given by 4·(n + 1)·max[x(i + 1) – x(i)] + 8·max[x(i + 1) – x(i)]^2·(n^2 + n)/2 + 4·max[x(i + 1) – x(i)]^3·(n^3/3 + n^2/2 + n/6), and as we have max[x(i + 1) – x(i)] —> 0, we get 4·K + 4·K^2 + 4/3·K^3.
For reference, we have, from the fundamental theorem of calculus, that the integral is equal to 4/3·(4^3 – 1^3) = 4·(4^2 + 4·1 + 1^2) = 4·(16 + 4 + 1) = 4·21 = 84. So we know 4/3·K^3 + 4·K^2 + 4·K – 84 = 0. This has only one real solution, K = 3. In fact, the way one would go about computing K is by having K = x(n + 1) – x(0) = b – a = 4 – 1 = 3, but proving that this is the case is tedious and very laborious in itself. This is why we do not use the definition of the integral to compute them, and instead, we prove the minimal integral theorems, which are easier to do than computing any given integral, and then use the theorems instead.
Also, if someone is wondering how exactly is this more correct, this is because t(i) is not being taken to simply be equal to x(i) or x(i + 1), but is an arbitrary number in the enclosed interval. Otherwise, s(i) = 0 or s(i) = x(i + 1) – x(i) respectively; and x(i + 1) – x(i) is not constant, hence t(i) is not merely a first-degree polynomial function of i. Otherwise, d(i) = 0.
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@karlwinkler4223 Of course, this does seem to be in conflict with "Love your neighbour,"...
How so? The Greek word translated as "neighbour" has essentially a meaning equivalent to today's "fellow Jew" phrase. The person who suggested that this was not meant to be about Jewish people, but all humans, was the apostle Paul, but Paul never met Jesus, and all of the other leading proto-Christian preachers of the time clearly disagreed with Paul, since Paul said so himself in the letters that he wrote to those leaders. The apostle Peter, in particular, who Paul claimed to have met, and who was presumably one of the 12 Disciples, was described as having believed that Christianity was only for the Jews, and that you still needed to observe the Torah (i.e., be a religious Jew) to be Christian. This was a different religious sect from the one Paul started, and the only reason Paul's sect won was because Roman theologians and political leaders liked that Paul was antagonizing Jewish people, so they began to subscribe to his belief system, and they weaponized it to justify persecuting the Jews. It eventually became the Church of Roman Empire, and the preachers were all Roman priests who persecuted the Jewish Christian sects that still persisted. It was a very openly anti-Semitic view, and they treated Jewish people as "Christ-killers," all in view of the Gospels, which portray the Roman government as essentially being innocent and having "washed" their "hands," since the Jews were the ones who asked for Jesus' crucifixion.
Does this mean Jesus himself was racist? No. No one knows what Jesus actually preached, but it seems that Jesus was still some type of Messianic, apocalyptic Second Temple preacher, so solely from a socio-cultural viewpoint, it is unlikely that he was spreading any message of universal salvation like Christians today would have you believe.
...but I would think it would be foolish to say that those ideas align with the spirit of Christianity.
Do they? Slavery was openly endorsed in the New Testament by Paul. Even Jesus said nothing against slavery, not according to the Bibles, anyway. Besides, what do you mean by "the spirit of Christianity"? What exactly makes you think we get to define what that is? As soon as 15 years after the death and crucifixion of Jesus, there were dozens of Christian sects competing with each other, some more universalist and less ethnocentric than others. All sects of modern Christianity are descended from only one of those sects: Paul's sect. Does Paul get to define "the spirit of Christianity," solely because his sect is the only one that did not get persecuted into evolution or extinction? That is a pretty ridiculous standard, I would say. Moreover, at the end of the day: no one actually knows what is it that Jesus taught or believed. We do not know even the general gist, much less the details. Pretending that we can know or define "the spirit of Christianity," especially in light of the history of Christianity as a whole, is very arrogant, and I would say, naïve.
Either way, if anything is undeniable, it is that slavery has always been a part of the history of Christianity. Jesus himself was never said to oppose slavery, and evidently, no Christian leader had the motivation to use their "Christian values" to oppose slavery, so, trying to pretend that Christianity can be meaingfully construed as somehow being "anti-slavery" is dishonest at best. Let's not give credit to Christianity that it does not deserve. I am tired of irreligious people trying to defend indefensible things that religion has been directly involved with.
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0:20 - 0:30 What type of object does x represent? Simply calling it a variable does absolutely nothing.
0:30 - 0:38 This implies x is the element of a near-semiring, or some similar structure.
0:39 - 1:26 For an arbitrary element x of a near-semiring, x can only be guaranteed to be always expressible as a sum of 1s for all x in the near-semiring if the near-semiring is the free near-semiring generated by {0, 1}, which is actually the semiring of natural numbers {0, 1, 1 + 1, ...}. In other semirings, the statement just made by the video is false. For example, in the ring of Gaussian integers, 2 + 3·i cannot be written as a sum of 1s. In the ring of algebraic numbers, sqrt(2) cannot be written as a sum of 1s.
2:27 - 3:00 Once again, this assumes x is an element of the natural numbers. If x is an element of some other semiring that is not a subsemiring of the natural numbers, then this statement is false.
3:00 - 5:05 Many comments to the video have indicated that you cannot differentiate using this rule, because the definition of the derivative is only applicable to the real numbers, and here, x is restricted to the natural numbers. Technically, this is true, but it does not genuinely address the issue at hand. Why do I say that? Because given a function f[m] : N —> N such that f[m](x) = x^m, there is nothing stopping me from defining an operator D such that D{f[m]} = m·f[m – 1] for m > 0, and D{f[0]} = 0. Whether this operator deserves to be called "the derivative" or not is a discussion that is not relevant to the argument in the video. Choosing to define such an operator is perfectly valid, and so this is not where the proof goes wrong. The proof goes wrong later.
5:05 - 6:17 This is where the proof goes wrong. Let g : N —> N be such that g(x) = x + ••• + x (x times). D{g}(x) is not equal to 1 + ••• + 1 (x times). Why not? Because the (x times) part was ignored. Instead, D{g}(x) = 1 + ••• + 1 (x times) + x + ••• + x (1 times) = x + x = 2·x. Another way to see the mistake is that x^2 = x·x, and so, if D{x^2} = 2·x, then D{x·x} = 2·x as well, but instead, in the proof, it is assumed that D{x·x} = x·D{x} = D{x}·x = x, which is incorrect. D{x·x} = x·D{x} + D{x}·x = 2·x. This also highlights why saying x·x = x + ••• + x (x times) is a mistake. Yes, for natural x, this is correct, if sloppy and imprecise, but the moment you bring the operator D into this, it changes things, because we are no longer dealing with x as a natural number, we are dealing with functions. x is not a function, and neither is x^2: these are natural numbers in their own right. However, the functions f[1] and f[2] defined earlier, such that f[1](x) = x and f[2](x) = x^2, do indeed represent functions, and so, it is meaningful to talk about D{f[1]} and D{f[2]}. However, it is not meaningful to say f[2] = f[1] + ••• + f[1], or anything silly like that. f[2] cannot be written as a sum of f[1]s only. f[2] = f[1]·f[1], but this cannot be written as a sum, since f[1] is not a natural number, it is a function. This incorrect proof shows that the way schools teach symbolic phrases in mathematics is very misleading. Schools teach students to treat expressions such as sin(x) or x^2 as functions, but this is just incorrect: sin(x) and x^2 are just numbers, in most instances. You can define functions g and f[2] such that g(x) = sin(x) and f[2](x) = x^2, but one most not conflate f[2] with x^2 or g with sin(x). g is a function, sin(x) is just a number. f[2] is a function, x^2 is just a number. Since they are different types of mathematical objects, how you do mathematics with them changes quite significantly. You cannot take derivatives of numbers. You can take derivatives of functions. You can multiply numbers, and often, though not always, those products can be written as sums. You can multiply functions, but unlike with numbers, these products can almost never be written as sums, unless one of the factors is a constant function whose output is a natural number. The distinction seems pedantic after you have already been seeing this misuse of notation for years, but despite how it may seem, I think this paradox demonstrates how important it actually is.
6:17 - 7:05 While you can definitely conclude 2 = 1 at this stage, division is actually not mathematically valid, because we are dealing with natural numbers here, and division is not an operation you can perform with natural numbers. If we were talking about rational numbers, that would be a different story. Instead, what you need to realize is that you have a functional equation here, 2·f[1] = 1·f[1]. Since f[1] is not the 0 function, it is scalar-cancellable, and so 2 = 1. Again, realizing that we have functions, rather than just numbers, is important.
7:14 - 9:57 This entire explanation is misleading. x is a number, not a function, so even calling it a constant is not quite accurate. It is, however, variable. Variables and functions refer to different things. A variable just refers to some quantity that could be different: it could be any of many of the same type. The crucial issue is understanding that writing x·x as a sum of x's is acceptable (though imprecise, and should not really be done either way), but differentiating numbers is nonsense. Instead, you want to realize that f[1](x) = x for all natural x, and so x·x = f[1](x)·f[1](x) = (f[1]·f[1])(x) = f[2](x), thus f[1]·f[1] = f[2], which can be differentiated (in the strange sense in which I have clarified earlier), but unlike with x·x, f[1]·f[1] cannot be written as a sum of f[1]'s, because f[1] is a function, not a natural number. So, technically, there are multiple mistakes in the proof.
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With all due respect, it is problematic to state that atheists should be regarded as having equal burden of proof as theists do, and this showcases how your religious bias is still somewhat getting in the way of some of your conclusions, even if you have been otherwise neutral and objective for the most part. While most atheists in the West do share a common worldview, it is not at all the case that all of them do, and not all atheists you interact with on the Internet are going to be from the Western world. Also, and more importantly: atheists are not going around openly promoting their worldview to other people. We do not proselytize our humanism, and we do not go around trying to force to make laws that conform to hyperspecific moral codes based on entirely unfalsifiable and undecidable claims. Do you know which group of people do this, though? Religious people. You are presenting discourse between theists and atheists as being a symmetric social dynamic. It is not. Most of us our living our lives, minding our own business, until a religious fanatic randomly shows up, and starts being irritating with their proselytism. I kid you not, I have spent time watching videos about mathematics from college professors, and you find Christians and Muslims trying to spread their faith forcefully. Nowhere is safe from this type of invasive discourse. See, if I want to have a conversation where I discuss naturalism, then yes, the burden of proof is on me to demonstrate that naturalism is true. But that is not what is happening here. If you guys want everyone else to be religious, then the burden of proof is on you to demonstrate that your religion is the correct worldview. If you want to be held on an equal standard as atheists and not treated like you have an obligation to prove your unprovable beliefs to us, then stop being so public about everything you believe, stop trying to make laws based on your beliefs, and stop treating atheists like we are freaks of some sort (which most religious people do). This is a simple concept that even intelligent religious people seem to grasp with, and I find it strange, because I know for a fact you guys do not apply this kind of double standard when it comes to things not having to do with theism vs atheism.
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Also, we DO have enough information. You are just personally ignorant about the information. For example, the I.O.C. is, as far as is currently known, far more reliable a source than the I.B.A. Additionally, we can corroborate by realizing that Imane Khelif is an Algerian athlete. In Algeria, being transgender is illegal, and if she had been born intersex with male genitalia and a propensity for high testosterone production, she simply would have been raised as a man and her birth certificate and passport would reflect this, because the government would have dictated for this to be the case. The only reason she has female in her birth certificate, female in her passport, and female in any other legal document used for her to be allowed to compete in Algeria's women's sports is because she is biologically female. The government knows she is biologically female, and this is about as strong confirmation as you can get from a country like Algeria. So there is no significant risk of the I.O.C. having made a mistake when most of the screening and testing simply would have occurred long before she was even allowed to be a boxer in her country of residence.
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Keldor314 You are incorrect. The example you gave with rolling the dice and usig each indexed result as a digit in a sequence constitutes an example of a definable number. How so? Because in providing the example, you actually constructed a definition of the number! The initial role of any given dice can be specified a by a set of spacetime coordinates, which are unique. Then, for each set of spacetime coordinates, there is a unique dice constant that you can form according to your example, and we can define the constant as the constant whose canonical real number representation is the sequence of strings formed by the indexed result of the rolling of a K-dice, a K-dice defined as the dice whose sequence of infinite rolls began at spacetime coordinates K. Then, the number is necessarily definable.
Also, the other mistake you make is that you assume that definitions are restricted to finite sequences of strings. They are not, and as such, the set of all sequences is uncountable. Nonetheless, it is true that the number of definitions possible is countable, but proving this is extremely non-trivial and has little to do with any bounds on possible sequences.
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@youwillwin7107 For many, the regularity of the universe and the precision with which the universe exploded ( expands ) into being provides even more evidences for the existence of God.
I have no clue why you present "exploded" and "expanded" as being synonymous. They are not synonymous. Also, the universe did not "expand into being." This is nonsense.
The Teleological argument goes like this: 1. Every design has a designer 2. The universe has high- complex design 3. Therefore, the universe has a designer.
Premise 2 is false. The universe is not a design, and does not exhibit signs of being a design.
Scientists are finding the universe is like that watch ( anology of William Paley ), except even more precisely designed.
Citation needed.
These highly-precise and interdependent environmental conditions (called "anthropic constants") make up what is known as the "Anthropic Principle"-- a title for the mounting evidence that has many scientists believing the universe is extremely fine tuned (designed) to support human CONSCIOUSNESS on earth.
No, this is a misrepresentation of what the anthropic principle is.
Thats why some notorious atheists including Antony Flew later believed in God.
Antony Flew converted right as he developed a severe case of dementia.
Some Anthropic constants example include: birth date of the star-planetary system if too early: quantity of heavy elements would be too low for large rocky planets to form if too late: star would not yet have reached stable burning phase; ratios of potassium-40, uranium-235, -238, and thorium-232 to iron would be too low for long-lived plate tectonics to be sustained on a rocky planet flux of cosmic-ray protons (one way cloud droplets are seeded) if too small: inadequate cloud formation in planet’s troposphere if too large: too much cloud formation in planet’s troposphere rotation period if longer: diurnal temperature differences would be too great if shorter: atmospheric jet streams would become too laminar and average wind speeds would increase too much fine structure constant (a number, 0.0073, used to describe the fine structure splitting of spectral lines) if larger: DNA would be unable to function; no stars more than 0.7 solar masses _if larger than 0.06: matter would be unstable in large magnetic fields _ if smaller: DNA would be unable to function; no stars less than 1.8 solar masses oxygen to nitrogen ratio in atmosphere if larger: advanced life functions would proceed too quickly if smaller: advanced life functions would proceed too slowly Jupiter’s mass if greater: Earth’s orbit would become unstable; Jupiter’s presence would too radically disturb or prevent the formation of Earth if less: too many asteroid and comet collisions would occur on Earth.
This is all mere speculation. We have no way of knowing what the universe would be like if the universe were different. These claims are unscientific, as they are unfalsifiable.
For more evidence: https://reasons.org/explore/blogs/tag/fine-tuning/page/2 https://reasons.org/explore/publications/articles/rtb-design-compendium-2009
This is not a scientific source, so this is dismissed at hand.
What are the chances? It's not there just a few broadly defined constants that may have resulted by chance. There are more than 100 very narrowly defined constants that strongly point to an Intelligent Designer.
No, there are not. Currently, per quantum field theory, there are only 20 degrees of freedom in the constants of the universe. Also, it is impossible to know how narrowly defined these constants are, since it is impossible to measure them with infinite precision. Also, determining the width of definition is not sufficient for determining the probability of these constants occurring.
Astrophysicist, Hugh Ross, calculated the probability these and other constants would exist for any planet in the universe by chance (i.e, without divine design). To meet all conditions, there is 1 chance in 10^1038 (one chance in one with 1038 zeroes after it)-- essentially 0% chance. According to probability theory, odds of less than 1 in 10^50 equals " zero probability".
Citation, or it did not happen.
Check:https://reasons.org/explore/publications/articles/probability-for-life-on-earth
Not a citation, so dismissed at hand. I doubt Hugh Ross actually ever did any of this: at best, he is being taken out if context and misunderstood, but if he had ever made such a sloppy, mistake-filled calculations, then his credibility as an astrophysicist would be lowered tremendously, as it would display ignorance of probability theory at a basic level.
It only proves that atheism is just a dogmatic belief.
What it proves is that you do not understand what citing your sources means.
Important: The term “entropy” describes degree of thermodynamic “disorder” in a closed system like the universe.
It does not. What it does actually describe is the information that we have about the microstates of a system in extrapolation from the known macrostate.
Amazingly, our universe was at its “minimum entropy” at the very beginning,...
No, it was not. The classical laws of thermodynamics are known to not be applicable at the Planck scales at which rapid cosmic inflation began.
...which begs the question “how did it get so orderly?”
This is very simple: entropy is not a description of orderliness. "Order" and "disorder" are not scientifically valid concepts, not within physics.
Looking just at the initial entropy conditions,...
The initial conditions of cosmic inflation are literally unknown, what the hell are you talking about?
...what is the likelihood of a universe supportive of life coming into existence by coincidence? One in billions of billions? Or trillions of trillions of trillions? Or more?
This is a meaningless question. To ask what the probability of an event is, you must assume that other events are possible (not known to be true), and what the probability of the other events is (not knowable).
Sir Roger Penrose, 2020 Nobel prize winner and a close friend of Stephen Hawking, wondered about this question and tried to calculate the probability of the initial entropy conditions of the Big Bang...
No, he did not. Once again, you are misrepresenting well-known scientific ideas. In this case, you are strongly misrepresenting Penrose's work.
According to Penrose, the odds against such an occurrence were on the order of 10 to the power of 10^123 to 1.
Citation needed.
But Penrose's answer is vastly more than this: It requires 1 followed by 10^10^123 zeros It’s important to recognize that we're not talking about a single unlikely event here. We’re talking about hitting the jackpot over and over again, nailing extremely unlikely, mutually complementary parameters of constants and quantities, far past the point where chance could account for it.
Alright, cool story. Now, show me some nonfiction. Show me some science.
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@AShaif The composition fallacy is not a decisive defeater,...
It is, though. Formally speaking, the composition fallacy is the claim that if S is the mereological sum of the parts in collection P, and each element of P has a property Q, then S also has the property Q. Even just from naive set theory, this does not follow, obviously. For example, every natural number is finite, but the set of natural numbers is infinite.
Induction is taken for granted for everyday life, but suddenly we go all skeptic when it comes to the cause of the universe...
Everyday life, and the origins of the universe, are fundamentally different circumstances, and the distinction is relevant. So it is reasonable to not overextend induction here.
...because there is 1% chance that there is no cause,...
The probability is much bigger than 1%.
...despite the arguments from fine tuning,...
I provided a systematic debunking of the strongest version of the fine-tuning argument below.
...irreducible complexity...
Irreducible complexity is mutually inconsistent with the fine-tuning argument, as the fine-tuning argument assumes evolution is true. Also, the concept of irreducible complexity is unscientific.
...contingency...
I already addressed the argument from contingency you presented above. However, there are better, strongers versions of the contingency argument. The strongest version I know of is the modal ontological argument presented by Alvin Platinga. However, the argument is flawed, in that the starting premise is speculative assumption, and also, the jump from "Possibly, necessarily, a maximally excellent being exists" and "necessarily, a maximally excellent being exists" requires accepting modal axiom 5, which is very much questionable.
...truth...
The existence of truth is entirely explained by the theory of evolution.
...consciousness...
Idbit.
...language origin...
Idbit.
...evolutionary argument against naturalism...
The evolutionary argument against naturalism betrays a misrepresentation of the theory of evolution, so it is unscientific. Also, this is mutually inconsistent with the argument from irreducible complexity.
...and other cosmological, teleological arguments for the first cause.
These all have the same unsoundness problems that the Kalam has: they all fundamentally misunderstand causation, and all assume that the universe "began," which is false.
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@MZONE991 If causal finitism is true, then we end up with a simple non-composite thing which is being itself.
This is nonsensical, as "being" is not a thing in itself, nor can it be.
You just showcasing your ignorance of classical theism.
You can make up your own worldview and slap a catchy name onto it. It does not make any less made up. You are just showcasing your ignorance of all concepts philosophical.
because the first cause of the universe cannot be a composite thing, because this eternal composition of parts is also infinite causation.
No, it is not the case that an eternal composition of parts is infinite causation.
because that's what we mean when say "God"
No, it is not. I assure you most Christians do not define "God" in such a manner, and neither does WLC, or most apologists. All you are demonstrating is that you are sufficiently intellectually dishonest, that you are willing to let the word "God" mean whatever you want it to, even change its meaning, as long as your ego thinks it is helping you prove a point. To define "God" in such a manner is pointless, as someone who accepts the existence of "God" defined in this manner is not even a theist, only a deist. You have yet to demonstrate that "God" is worthy of worship, is accurately described by the Bible, etc. You cannot call it "God" until you demonstrate such things, lest you admit you are intellectually dishonest.
this is like asking "why call the molecule with 2 hydrogen and oxygen atoms water?"
No, it is not like that at all, because "water" is what we call just a particular liquid, though to what extent the name is applicable is nebulous, vague, ambiguous, fuzzy, and ill-defined, since it is just a colloquial categorization based solely on intuition; but it just so happens that this liquid is composed of molecules of 2 hydrogen atoms ionically bonded with 1 oxygen atom, and so by extension, we also call it "water," whenever applicable. This is not at all analogous to taking multiple completely different definitions, and slapping the label "God" onto them without (0) proving any of the definitions is satisfied or even satisfiable, (1) the definitions are equivalent.
Because if they don't, then by definition, they are not one composite whole, and do not form one being.
That is not what the definition of composite being is.
Analogy: if each mechanism in a car does not interact with the other, then is this really a car?
This is not a valid analogy. Built into the definition of "car" specifically is the fact that certain parts interact in a specific way, but there is no reason to suggest this ought to be true of all composite things.
Language is a human construct.
No. Language is a construct of a social-emotional species. Language exists in other animal species, and it even exists in plants, strangely enough. Anyway, this is just a minor nitpick on my part. But what is actually important is that you are failing to undertand how this social-emotional construct actually works. Words are signifiers, but signifiers serve no purpose without making reference to a signified thing. A correct use of language only declares a same usage of meaning for a signifier when the signified is the same in the different usages. If I take a word two refer to two different things, while pretending that it actually only refers to one thing, then I am using language incorrectly, and it demonstrates I am ignorant or dishonest.
but the actual cause of lightning is not identical to what Greek mythology calls Zeus.
You are so close to understanding the point that Kanna-chan is making.
however, the fundamental origin of reality is exactly th God described by classical theism.
The problem is that this is just not true, whatever you think classical theism is. Also, it is nonsensical to talk about the fundamental origin of reality, since whatever that origin is must itself be a part of reality. That is, unless you think the origin is fictional.
Beyond this, there are multiple other problems with your argument.
0. You have yet to demonstrate that "atemporal" is a coherent property.
1. That "causation" is not inherently temporal.
2. That causal finitism is true.
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@timothyhicks3643 I do not know what your sources are, but Mormonism does not worship Jesus as a god at all. The mainstream Mormon theology worships Elohim, whom they also call God the Eternal Father. They view Jesus as human, and divine, but not as God, and instead, they call him the son of Elohim, also known as Jehovah in their view. They also do not worship the Holy Spirit as God. Furthermore, and more importantly, they have a very unusual theology reminiscent of Scientology, in which they think that all humans actually can become essentially God, or be "like God." This makes their eschatology and soteriology fundamentally different in many ways from that of any sect of Christianity. Jesus is not special in this regard. What makes him special is in the fact that he was chosen to be the ultimate sacrifice, making him "the redeemer of the world," as they call him. They view Jesus and the Holy Spirit as part of a "holy council" they call the Godhead, but in this case, Jesus operates more like an angel, than like a god. This is also not particularly special, as they also view John the Baptist, Peter, and James as angels. In fact, they view many of the disciples as angels, as well as many of the other important figures in Christianity. Finally, the fact that they put the Book of Mormon on equal footing with the Bible means they essentially rely on an entirely different holy book, which makes the worldview worthy of being considered a different religion. Many scholars of religion would agree this is the case. For more details, see Shipps, Jan (2000). Sojourner in the promised land: forty years among the Mormons. Chicago: University of Illinois Press. I think Mormonism has much more in common with Islam than with Christianity, even though Mormonism has stayed closer to Christianity than Islam has. However, this can easily be explained by the fact that Mormonism and Christianity have stayed in very close geographical and sociopolitical contact, which is not so much true with Islam and Christianity. Their views about Elohim as God the Eternal Father also differ from that of any Christian sect. They believe that Elohim actually has a physical body, for example. See Roberts, B. H., ed. (1909). History of the Church of Jesus Christ of Latter-day Saints. Vol. 5. Salt Lake City: Deseret News. Within circles of Mormon fundamentalism, they believe that God (Elohim) is a resurrected man who ascended to godhood. In this aspect, Mormons disagree with all the Abrahamic religions. See Ostling, Richard; Ostling, Joan K. (2007). Mormon America: The Power and the Promise. New York: HarperOne. ISBN 978-0-06-143295-8. and Alexander, Thomas G. (1980). "The Reconstruction of Mormon Doctrine: From Joseph Smith to Progressive Theology" (PDF). Sunstone. Vol. 5, no. 4. pp. 24–33.
My knowledge of Montanism is limited (to be fair, historians know little about Montanism as well since it went extinct), but in Montanism the belief was that the prophets of antiquity were not merely inspired by God, they were God himself speaking, and that Jesus was actually only one of the many prophets. They believed that any given prophet being possessed directly by God at the time has authority over even the words of Jesus.
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The idea that the existence of God is a proper axiom to sustain objective morality is flawed.
Suppose God really was omniscient, which it could not be, both because omniscience is logically impossible and because the Bible presents some instances where God is portrayed as omniscient.
If that were the case, that would mean God does know right from wrong.
However, why does that matter? God can say whatever it wants without regard for that knowledge. Yes, maybe God does know right from wrong, but when God tells us, “Thou shalt not kill,” how know we that God isn't lying to us about this being a moral obligation? How know we that God actually does know morality? The only possible way to connect the premise to its conclusion is be proposing that God is infinitely good, but this is impossible since omniscience is not compatible with ultra benevolence. Hence, morality, as it is dictated by God, is not objective. It fails to the Euthyphro dilemma.
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EANTY Crown They are not the same word. Rather, the meaning of the word "primo" as in "cousin" is semantically derived from the word "primo" as in prime. The etymology is that, in medieval Hispanic culture, the theory of families was that there exists a nuclear family, and the siblings from this nuclear family is what used to be more formal called the zeroth order siblings, because the degree of separation is 0, because they are the children of your own parents, so the number of generations you have to rewind back before you have a common ancestor is 0. Then cousins are "primo hermanos" since the degree of separation is 1: you have to go back 1 generation to achieve a common ancestor: you do not share parents, but do share a set of grandparents. This makes sense since the word prime, etymologically, is directly linked to the number 1. Then you have "segundo hermanos", which in English would be the equivalent of second cousins, which would be second-order siblings, because the degree of separation is 2: sharing great-grandparentsc but not grandparents or parents. However, this terminology became completely lost after some time simply because second cousins tended to become irrelevant in the family life for practical matters, so the phrase "primo hermanos" was reduced to simply "primos". Of course, you still do see the phrase primo hermanos occasionally in books and in some parts of South America, but for the most part its obsolete, or it otherwise became corrupted
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Ligeia D.Aurevilly psychologists are constantly trying to become scientists.
No, they aren't, because they already ARE scientists, and you aren't the one to decide otherwise. There is scientific consensus that psychology uses a scientific method, ironically.
They only can use statistics, since each case is different.
ALL science can only use statistics. There do not exist any non-statistical inferential-deductive methods in science. Stating otherwise is a demonstration of ignorance on how science works.
Science is now more based on correlation than causation.
No, it is not. I can say that... because I'm a scientist, and my entire job is to read and analyze statistics as well as analyze the methodologies utilized in the different projects and fields of scientific research. Correlation does not imply causation. This is precisely what makes science powerful and useful: it goes beyond correlation and into discussing causations rigorously and in a verifiable, demonstrable fashion.
Reviews, cohort studies, are all based in previous studies...
You think the scientific methods rely only reviews and cohorts? Geez, you definitely need to take several courses about science, because your knowledge on this is apparently null. Review, while a necessary part of the method, is only a very small part of it. Most studies are independent of one another, both in content-design and in research team.
The conclusions taking by indirect ways are not always true or valid.
This is just simply false. There is no explaining to be done here, this is just a fascetious lie. Next time you make outrageous claims, give examples, will you? But of course, you can't give examples, because your statement is false.
That's why statistics are not equivalent to science.
That's a strawman argument. No one claimed they're equivalent. However, there is no science without statistics.
They are only a way to justify concepts by manipulating numbers.
You obviously know nothing about statistics if you believe this. There is no manipulation of numbers in statistics, and whenever someone does manipulate them, we call that fraud, and we report the doer, which will get their title removed or suspended, or it will certainly ruin their credibility as a researcher. In statistics, you have some data, and all inferences are made from that data. No manipulation. Then, you conduct a statistical analysis on the conclusions that different teams of research make to assess the sets of conclusions. Afterwards, you have several meta-analysis to determine whether there is consensus on these assessments. The statistics are applied in multiple layers. Statistics involves the calculation of quantities from the data, not manipulating the data.
That we all know.
No, because it is false. Are you so ignorant that you are unaware of your ignorance?
Standard course material does not mean that students really understand statistics.
He never said that. Strawman #2.
I did study lots of them in college...
And you obviously failed the courses despite studying, because one does not study "lots of statistics". That's not how it works. Statistics is jargon that refers to the theory of probability and statistical inference in mathematics, which is different from the statistical inferences from surveys and studies that are done.
but I have a permanent feeling that there is always a way to justify anything...
The only reason you have that feeling is because you don't understand how statistics work, or how science works, or how formal deductive logic work. You can't justify anything arbitrarily, it genuinely is impossible. The claim "2 + 2 = 5 in base 10 in standard language" is objectively wrong and unjustifiable. It can be proven that the claim cannot be proven BECAUSE it is false. That alone is a counterexample to your claim that anything can justified. That's being a little pedantic, but the reality is that I don't need to prove the self-evident statement that you can't justify anything you want to. Believing that you can implies a lack of understanding on the subject of justification and truth.
depending on the method you choose to write your papers...
That simply is not true. The way you write your papers has nothing to do with justification and everything to do communication. And if you read a dictionary, you'll realize those aren't synonymous.
Statistics are no exact sciences.
Sciences are, by definition, inexact. There does not and cannot exist such a thing as an "exact science", because science is inherently statistical and approximational. Statistics are not a science. You said these yourself. I'll leave it to ignorant people to make arguments that contradict their own claims.
There can't be science unless you can compare groups that are identical to compare differences.
This is just false. And if you think this is true, then I assume you don't physics is scientific, or archaeology, or paleontology, or meteorology, or nuclear chemistry, or geology, or oceanography, or many subfields of biology. Basically, what you are declaring is that science does not exist. How? Because in every science, valid inferences can be made without having identical groups. You do know that is the literal point if statistics, right? Oh, wait, I forgot you don't really understand statistics. My bad.
Every evidence-based method is inexact and can be biased...
Keyword: can. They CAN be, but usually are actually not. Why? Because we have rigorous procedures in place to eliminate bias. The entire point of statistics is to eliminate bias. Just read any textbook on statistics and one of the first chapters you will find will be dedicated to discussing biases and their mitigation.
therefore whoever uses this popular method to talk about "science" is lying.
Beautifully by someone so ridiculously ignorant that they have no idea what science is. Let me give you a quick lesson: ALL SCIENCE is inheretly evidence-based. By definition, science is inductive and empirical. That is why it is different from mathematics. And when I say ALL SCIENCE, that is no hyperbole. Mention literally any field of science whatsoever and I'll give examples on how it is evidence-based.
Nobody says absolute truth.
This is 100% irrelevant and a strawman. The scientific method does not depend on every scientist being an objective person.
Also, aside from being a strawman, the claim is also false. No one is 100% objective, but everyone believes in at least one objevtively true statement. For example, the claim "1 + 1 = 2" is objective. It is an absolute truth.
It is up to you to go further, analyse and decide what you want to keep or not from his speech.
It is indeed, and the only thing I choose to keep from his speech is 1. The fact that he is full of BS. 2. The fact that he is wrong and perhaps delusional altogether.
How do you know who is credible source?
By studying the source. If the source is able to consistently make true statements, present information accurately without hiding facts or presenting them differently from how the researchers did, and if the claims are verifiable, then the source is reliable. That wasn't so hard, was it?
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Euquila No, that is not true nor relevant to the problem. 1. Infinite arithmetic and performing arithmetic with infinitely many numbers are both forms of math that are completely sensical, rigorous, intuitive, and widely applicable when considered in the framework of set theory. This is true for both convergent series and divergent series. 2. It is an empirically verifiable fact that most mathematicians who speak against these claims have little to no expertise in the mathematical theories that concern divergent theories. In fact, most of these mathematicians actually are not mathematicians to begin with as they do not conduct much research in mathematics, and instead dedicate themselves to teaching, which is an entirely different domain of work altogether separate from mathematics, and they may or may not be good at it, but this logically gives them no credence on the topic of how divergent series should be interpreted. Opinions should be left to the actual experts who have bothered to study the theory in detail as well as those who use these results in applications - because, yes, they appear in multiple applications, this is not just some mere abstract meaningless math. What I am saying here is pure common sense and it reflects on the same reason why no mathematicians ever talks about wheel theory: few mathematicians are actually qualified to do so. 3. The fact is that the mathematical community is hypocritical and it has always been. They want to pretend there are no rigorous foundations for these results even though there is an entire literature full of it. The reason is because historically, many mathematicians have cared about rigor for the sake of rigor and not for the sake of the conveniences that rigor actually does have. It does not help that often times they take for granted results which are less rigorous. They are also hypocritical for the very simple fact that the somehow want to consider arithmetic addition per set theory and convergent sequences of partial sums to be one and the same thing, but refuse to accept any attempt to legitimize any divergent summation as a summation to begin with, even though objectively these summations are very natural and intuitive extensions to summation that not only perfectly obey all of the field axioms as well as the algebraic manipulability of summation that series should obey in the first place despite the fact that convergent series in general fail to obey it, but which is also vastly more practical than any restriction to convergent series that there could be. It is aggravating and it can only be considered a very anti-logical attitude, much like those that existed prior to well-founded set theories.
The problem has nothing to do with the notation of Sigma and Pi. It has to do with the message that the mathematical community wants to send to the world with the inconsistent attitudes. I’m sorry, but the mathematical literature supersedes anything that any group of mathematicians will ever have to say, because it is the literature that has the formal logic verified for itself. That is that.
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themanwiththepan You can lose at the game because when the game ends, you only take what is in the pot, and if you paid more to play than what was in the pot, then you lose. Of course, this would be avoided if you paid $1 to play, since then you can never lose, but no casino would ever ask this. Furthermore, that misses the point, because the theorem states that even if the casino asks you $1000000000000 to play, you should still play, since even in this situation, the expected winning is infinite, even if you can lose a lot.
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23:26 - 23:48 Wait, so you ARE arguing that these rational numbers are "fake" as you would call them because they cannot be written? This seems to contradict statements you made earlier in your presentation. Anyhow, I am glad you are at least now choosing to stay logically consistent, although you had to go quite late into the video to make the decision.
24:05 - 24:09 This infinite sum is an example of a computable real number, so it is inaccurate to say you cannot calculate it. Anyhow, yes, that symbol is the number's "name," whatever that means. Numbers do not have names. We assign them names because we humans need a way to refer to them, but the names depend on the language we choose to communicate with. As far as the conceptual abstract object that we call number itself is concerned, though, it has no name, and it does not care if it as a name or not. It just is what it is: an abstract object that has the value of a quantity.
24:39 - 24:42 To this moment, you have failed to successfully explain why it does not work. All you did was claim there is no algorithm to compute these numbers, which is false, and then proceeded to say that the entire system is ill-defined without any proof of this, and then began to proliferate arguments that are based on flawed premises and misunderstandings of concepts and of the current paradigm. A lot of finitist ideas are based on misconceptions and a lack of understanding of concepts as well, but here, as I stated earlier, I get the impression that you maybe do understand the concepts, and just pretend not to for the sake of an agenda. I still have not been given sufficient reason to forgive the amount of dishonesty in the video, so pardon me for continuing to repeatedly call you out on your intentions and your dishonest presentation of the concepts to an unknowing audience. Anyhow, this was all a rant just for to explain that, no, you have not proven the system does not work. All you did was present a few claims that can be easily shown to be false, and decided to move on from there.
25:14 - 25:21 So you are telling sociologists to prevent us from being dishonest in the exact same way to decided to be dishonest throughout most of the video by completely dodging the question and failing to present a proof of concept? Yes, I approve of that, although it makes you seem like a hypocrite.
25:35 - 25:39 Hearing you say this is quite hilarious, when in reality, the entire premise of finitism is reliant on philosophical obfuscation and not any sort of operational formal theory of logic.
26:03 - 26:09 π + e + sqrt(2) = 7.274088044421933522624619578841863460524088368452013469118591958
What? You wanted more digits? You never said how many digits you wanted the answer to have. This is what you get for asking a loaded dishonest question and not giving any room to explain why it is a silly question. You wanted an answer? I gave an answer, and you have to accept it, because you said you would not allow for any jargon or philosophical arguments about the validity of the question or the answers
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Let us stay with the real numbers before going into the complex numbers. Above, I explained that you can use the ordering of the real numbers to extend to the real numbers into a system where every subset has a supremum, and this introduces the objects T and B, which Wikipedia (and most non-mathematicians) call ∞ and –∞.
However, there is a different extension you can do with the real numbers. Rather than using the ordering, you can use a notion in the theory of topological spaces called a one-point compactification. The idea is similarly to that of the affinely extended real line, except that you take the two endpoints that this line has, and you join them together into one point. It is almost as if you are saying "let –∞ = ∞," but you can do this very rigorously, and it is actually very useful as well. While you do give up the ability to do arithmetic, like with the affinely extended real numbers, and while you also give up the ordering, unlike the affinely extended real numbers, it gives you one huge benefit in exchange: the ability to do projective geometry. As such, this structure is called the projectively extended real line. You can study Möbius transformations (whenever people write 1/0 = ∞, they are talking about a special case of an elementary Möbius transformation, they are not talking about division, and this is clearly another example of abuse of notation, which again, is by no means universal).
If you think about it visually, it amounts to turning the real line into a real "circle" of sorts, but a circle with an infinite circumference. This is intentional, because one grand motif in projective geometry is that lines are treated like circles, and parabolas are just treated like ellipses, and since circles are ellipses, they are all just ellipses in the context of projective geometry. Lines are just ellipses with infinite eccentricity, and parabolas are just ellipses with eccentricity 1. You can even get funky and treat hyperbolas like ellipses as well if you allow complex numbers. More importantly, this type of structure allows you to systematically study the different kinds of asymptotes that exist. One intuition behind this is that it makes calculus more symmetric, so to speak. Rather than speaking about approach T or B (∞ and –∞ respectively), you instead speak about approaching ∞ (again, we really should use a different symbol) from the right (the positive real numbers) or the left (the negative real numbers), which feels rather natural.
The projective extended complex plane, also known as the Riemann sphere, is actually just a trivial extension of the projectively extended real line: it is just the union of the complex numbers and the projectively extended real line. The Riemann sphere is extremely useful in complex analysis as it simplifies many different concepts, and it serves as the main inspiration of wheel theory. In your comment, you talked about the complex numbers having only one "infinity," and it refers precisely to this, the Riemann sphere.
This brings me back to why I said it is confusing to use the word "infinity" here. You said "the real numbers have 2 infinities, while the complex numbers have 1 infinity." This is clearly not the case, though. As you can see, there are two different ways you can extend the real numbers, one of them introducing two new objects (which you can think of as infinite if you want, but please do not call them "infinity"), and the other one introducing only object (this object completely unrelated to the T and B of the other extension). In principle, there are other rigorous ways you can extend the real numbers too to introduce infinite objects, though they may not necessarily be useful. As for the complex numbers, I only talked about the projective extension, but you absolutely can use an affine extension of the complex numbers if you want to, it is perfectly valid, just nowhere as useful as the Riemann sphere. This introduces not two new objects, but infinitely many new objects, actually. The intuition is that, in analogy to how the real line was closed off by two endpoints, the complex plane is being closed off by a border which, intuitively, is the shape of a circle with infinite circumference. Each new infinite object corresponds to a direction in the complex plane. You can come up with a toroidal extension, where the real axis and the imaginary axis both get projectively extended, but separately, so that you get two new different infinite objects: one imaginary, one real. You can do all sorts of other extensions, as long as they are mathematically coherent.
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0:03 - 0:12 You would need to specify what ring you are working in. In the trivial ring, 0 = 1 is true. For the sake of this video, then, I suppose we should assume we are working in a non-trivial ring, so that the proof could be conceivably incorrect.
0:32 - 0:43 This is not actually necessary, nor does it add more correctness to the proof. The restraint that x is nonzero ends up being irrelevant.
2:50 - 3:12 It should be noted that a^2 – b^2 = (a – b)·(a + b) is only true if a and b commute, which is to say, if a·b = b·a. Since x = y, x·y = y·x is indeed true, but this should be stated.
4:10 - 4:11 At this stage in the proof, we most definitely have a 0 = 0 situation. Specifically, since x = y, it follows that x – y = 0, and so (x – y)·(x + y) = 0·(x + y), while y·(x – y) = y·0 = 0·y, and so we have 0·(x + y) = 0·y.
4:12 - 4:41 This is where the proof went wrong. As I noted in my previous paragraph, the equation (x – y)·(x + y) = (x – y)·y is equivalent to 0·(x + y) = 0·y, since x = y implies x – y = 0. What the video is thus effectively doing is declaring that 0·(x + y) = 0·y implies x + y = y, which is not true. This is because, even when a is not equal to b, 0·a = 0·b = 0, and this is true in all rings. This is equivalent to just saying that 0·2 = 0·1 implies 2 = 1, which is obviously not the case.
5:13 - 5:29 This means x is idempotent with respect to addition, and since this is a ring, it implies x = 0. This contradicts the fact that x is arbitrary, though.
5:39 - 5:45 This extra restriction is not necessary, all you need is for x to be arbitrary
6:18 - 6:21 Even if it were true, it would not show the universe will end.
9:04 - 9:08 It is not that you are "not allowed to divide by 0." Rather, it is that, since we are working in a ring, as is required for distributivity to apply, and addition, subtraction, and multiplication to be well-defined, it must be the case that 0·x = y·0 = 0 for all x, y, and so 0 is not cancellable, which means that even if a is not equal to b, 0·a = 0·b.
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@Thefamilychannel723 The question you ask does not make sense, if you analyze it carefully. You see, in physics, we analyze worldlines. A worldline is just some path in spacetime, and the laws of physics just tell us what the restrictions are for these paths. These paths are meant to describe physical systems, and we can interpret the endpoints of the path as the "beginning" and "end" of the systems' lifetime. At each point (t, x, y, z) in the path, you can ask question about the state of the system at that point. If you have two systems, you can compare their worldlines, and you can compare their beginning points. This means you can compare the time coordinates of their beginning points. So, it makes sense to ask "which came first?," in the sense that you can ask "if t0 is the beginning time of system A, and t1 is the beginning time of system B, then, is t0 less than t1, are they equal, or is t0 greater than t1?" This question is perfectly coherent.
So, why is your question not coherent? Is it not analogous to the scenario I just presented? No, it is not analogous. The problem is, spacetime, as a physical system, cannot be represented by a worldline... because it is the collection of all possible worldlines to begin with! As such, it really makes no sense to ask about the "beginning" or "end" of spacetime as a system. Your question is analogous to taking the set of real numbers, treating it like a number in itself, and then asking "Does the set of real numbers come before 0, after 0, or is equal to 0?" That question does not make any sense. You can ask if a real number comes before another real number, but you cannot ask if the set of real numbers itself comes before or after a particular real number included in the set. Your question is completely analogous to this: the example I presented is the 1-dimensional analogue of your question, since spacetime is basically just 4 copies of the set of real numbers multiplied together.
The lesson here is this: you can meaningfully ask about spatiotemporal properties of physical systems embedded within spacetime, but those questions stop making any sense when you ask them about spacetime itself. Spacetime is fundamentally different from all other physical systems which exist, so you need to think about it separately. By the way, I do not know if you are religious or not, but I will just say this: Christian apologists consistently fail to understand this fundamental distinction between spacetime, and other systems embedded in spacetime. This is why the Kalam cosmological argument fails miserably: the argument insists that the universe (and therefore, spacetime) had a beginning point. But if you study the mathematics and physics of spacetime, then one can see that this is simply not true.
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@Lincoln_Bio Kilograms measured on Earth already have gravity in them.
No, they don't. An object with a mass of 1 kg on Earth also has a mass of 1 kg on the Moon. The mass does not change because the internal conservative forces and its material content isn't affected by the gravity of the exterior environment. A kilogram-force does have gravity factored in, but a kilogram-force is a different unit than a kilogram.
If you'd have divided the weight in kg by the rate of acceleration in m/s
Weight is measured in Newtons, not kilograms. Mass is measured kilograms.
and came up with a unit of "mass" that you called Newtons, I'd have been on board.
Uh, what? Newtons are a unit of force, not of mass. Mass is measured in kilograms.
But here I am in physics class with 2 identical scales in my hand except one notch on one is 9.8 on the other. Genius mate.
No, I think you are a little confused. The scales may be externally identical in appearance, but internally, they're different, because one is designed to measure the force between you and the Earth, the other one is designed to measure your mass. They measure different things. Actually, strictly speaking, the one measuring force isn't a scale, but it's colloquially known as a scale even if it is incorrect. It's just called a Newtonmeter.
Hence, arbitrary scale.
It's not arbitrary. One measures force, the other measures mass. And the reason there is a factor of 9.8 between the two is because the formula for the force of gravity exerted on you by the Earth is F = mg, where g = 9.8 m/s^2. The confusion comes from the fact that North America uses the imperial and/or the US customary system of units instead of the SI system of units, so rather than having scales give you the weight in Newtons, they give you weight in units of pound-force, where a pound force is defined as a pound of mass with an acceleration of 9.8 m/s^2, or whatever the corresponding constant is in feet/s^2. People confuse the pound-force with the pound: the former is a unit of force, the latter a unit of mass. This confusion would be avoided if schools and the economy just used the SI units consistently in every country altogether instead of silly imperial units and offshoots thereof.
and I've come across the same mathematical problem of gravity that frustrated me as a child
Well, I don't understand where the frustration lies. To me, it seems as though you still have very bad misconceptions about physics ever after you started retraining yourself. You're confusing potential energy with kinetic energy, weight with mass, and Euclidean geometry with hyperbolic geometry. I know that's not your fault, but that's why I'm here to tell you these things. I don't want you to believe that your education in Newtonian physics is accurate when even you've realized that it isn't.
it turns out that until we have a solid theory of quantum gravity, current generally accepted physics still can't give me the answers nearly 30 years later
No, I don't think that's the case. Like I've already said previously, and like you said yourself, your education on Newtonian physics seems very mediocre, and your knowledge of general relativity is probably not much better, since it'd be impossible to accurately understand general relativity without understanding Newtonian physics. That's why I told you that you should first improve your understanding of Newtonian physics first and make it very solid before you move on to general relativity, and only after you've understood general relativity can you start trying to look for a theory of quantum gravity. You also need to learn quantum field theory, but that's a whole other separate discussion. And I'm not saying this to be condescending, though I admit my tone is probably more blunt than it should be because I have no social skills. I say it because I do want you to be able to get the answers that you want. I just don't want you to take the wrong path to do that and waste your own time. And I think that the only you can get to where you want to be is if you try to have a solid grasp of Newtonian physics and general relativity instead of jumping to quantum gravity right away.
So I want to try & get to the bottom of it, questioning assumptions as I go, because science.
I'm not telling you to not question those assumptions. I'm telling you to master and deepen your understanding of those assumptions before you start to question them. You can't write a book review about a book you've never read. Similarly, you can't critique an argument or theory without first having thorough understanding of what the argument or theory is saying and claiming and how it has been historically justified. If you're questioning a hypothesis without actually understanding what the hypothesis is saying, then like I said earlier, that's not science, that's just mindless skepticism. Mindless skepticism is a form of ideological zombieness. I doubt that this is what you want to accomplish. That's why I'm telling you this.
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Ashish Ahuja Every cube of an integer is equal to 0 mod 9, 1 mod 9, or 8 mod 9. You can also write 8 mod 9 as -1 mod 9. This can be proven very easily. Simply cube the expression 9n + m using the binomial theorem. The result will be of the form 9k + m^3, where k is irrelevant. Run m from 0 to 8, and the resulting set of cube numbers is 0, 1, 8, 27, 64, 125, 216, 343, 512, and these numbers are all 0 mod 9, 1 mod 9, and -1 mod 9.
This is important, because it means the sum of three cubes will be equal to some summation permutation of 0, 1, and 1, in mod 9. In total, 27 permutations exist, but the only possible results of these 27 permutations when you check them is 0 mod 9, +/ 1 mod 9, +/- 2 mod 9, and +/- 3 mod 9. You never get +/- 4 mod 9, and since -4 mod 9 = 5 mod 9, this means the sum of three integer cubes will never be 4 mod 9 or 5 mod 9, which means any number that is 4 mod 9 or 5 mod 9 cannot have a representation of 3 cubes of integers.
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@cidien *...but he didn't really "prove" or "disprove" much of anything.*
He did.
What he did was give scientific reasons why he believes he's right.
Yes... in the scientific community, this is exactly what we call "proving" a point. I suggest you learn how to use a dictionary.
He just reasonably disproved very specific theories...
Yes, but you just said he disproved nothing. You contradicted yourself within the same paragraph.
...where he's assuming he has all the data on how it works.
The data he is using is experimentally accurate. There are millions of experiments which confirm it. There is nothing be "assumed" here. There are only facts.
I'm not trying to argue in favor of hollow Earth here,...
This is pretty much exactly what you are doing. Insisting you are doing otherwise does not mean it is true. At most, all this amounts to is you deluding yourself.
...but people are becoming dangerously blind these days when the term "science" is being thrown around.
This is just a baseless assertion with no evidence to support it. You have met less than 0.01% of the entire human population within your lifetime. You have possibly no way of knowing what is true about "people these days." What the statistics do show, though, is people are better informed about scientific matters today than ever before.
If someone they perceive as an authority figure tells them something and that it's proven with science, people tend to stop any and all critical thinking and just accept it...
That is their fault for being stupid, not the authority figure's fault, and this has no bearing on whether what was said by the figure is scientifically factual or not.
I mean, heck, we're STILL teaching our kids the pyramids were the burial grounds for pharaohs in school for some reason.
Uh... yes, because this is factually true. You are just mad because they are not teaching the wild conspiracy theories you want them to teach, like "it was aliens!" or something which has absolutely no evidence in its favor.
It's just crazy what people tend to accept just because they're told something by authority figures.
You act as if you are any different, even though you probably just belief random stuff spouted on the Internet by random people who have absolutely no knowledge about the topics and have an explicit agenda to lie to others. Stop being a hypocrite.
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Objects in Motion Well, the constants are there so the dimensions match out. Remember that constants are simply there for the proportionality, their units usually mean nothing unless they are fundamental constants of nature, such as the speed of light. The constant in F = kx, for instance, is absolutely irrelevant and it doesn't matter what its units are. It doesn't matter what it's value is. It's simply there to highlight the fact that F is proportional to x when it comes to springs (provided certain conditions, of course). Now, of course, we calculate these values because they are necessary in engineering, but in terms of what the units mean, they're irrelevant.
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@Jeremy-wp4yh As expected, the comments are filled with people quoting scripture out of context or not understanding what they've read.
I seriously doubt you have the high literacy skills necessary for your opinion to hold any weight in that subject-matter. Besides, the criticism is completely worthless, as it fails to point out examples, and as it fails to actually provide corrections anyone could learn from. You are exhibiting the behavioral traits of an Internet troll, and not the traits of an intellectually honest individual who is willing to have a discussion in which you and other people help each other learn and improve yourselves. Why should anyone even take you seriously, when you are only here to antagonize, and not to be civil? Do you seriously believe anyone is going to convert to Christianity, when the only thing you are doing is be a bit of an annoying pest?
But nevertheless, why are atheists always debating Christians.
I know this sentence is meant to be a question, but you clearly forgot to replace the period with a question mark. When you make mistakes like this, do you expect me to believe that you have the sufficient literacy skills to actually be a reliable judge of other people's comprehension of the text that they are reading? Because if that is your expectation, then my advice to you is: stop it.
Anyway, to actually answer your question: we are not always debating Christians. Most of the time, we are just living our lives, and not debating anyone at all. The problem is, people like you always show up, and start cooking up some trouble. People like you are always trying to overthrow democracy by imposing religious laws onto the country, and are always promoting that we indoctrinate our children and brainwash them with mythology. You expect us to sit back and allow that? Of course not. People like the Jehovah's Witnesses are always knocking on people's doors and trying to annoy us with beliefs they are incapable of supporting with evidence, all while promoting anti-scientific drivel. Are Muslims doing that? At least in the West, no, they are not. Are Jewish people doing that? Again, at least in the West, no. Are Buddhists doing that? No. Are Satanists doing that? No.
Why not other religion?
Again with the linguistic mistakes. You are really not inspiring confidence, as far as literacy goes.
Anyway, we DO debate people from other religions. We even debate fellow atheists. We do not debate people from other religions as commonly as Christians, because they do not bother us anywhere nearly as much Christians with their nonsense, but we still do debate them whenever they try.
Why not Satanism?
Satanism is a lot like Daoism, in that while it can technically be argued to be a religion, many of the adherents are essentially irreligious, and only hold Satanism philosophically, and not ritually. Satanism and Daoism are also like Buddhism in that these religions are functionally atheistic. Sure, you can be a Buddhist and believe in gods, but this is not actually a requirement of Buddhism. The Great Dao in Daoism is essentially a metaphor for the divine, incomprehensible aspects for the universe, and so, Daoism is basically a form of pantheism, but at the end of the day, pantheism is essentially a form of superstitious or panpsychic atheism that reinterprets religious language and tries to make that language work within the confines of atheistic worldviews. Basically, pantheism and atheism are identical in what they assert, they only differ in the details of the semantics they use when making their assertions. Satanism is like Daoism in this regard. Satanists, generally speaking, treat Satan more like an abstract symbol than like an actually existing, concrete entity. It is basically a modernity-resurrected form of ancient "pagan" pantheism. So, when it comes down to it, people who self-identify as atheists do not have all that much to disagree with when it comes to Satanism.
Besides, every Satanist I have met has been far friendlier and far more ethical than almost all Christians I know. Also, these people hold no political power at all, given how much of a minority they are. They are not even trying to legislate our lives with religious laws infringing upon human rights, so there is not much of a point to debating them.
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Brett Carwile To be honest with you, you bring up many good reasons not to move to Japan, but the thing is that those exact same issues exist in any country in the planet. There doesn't exist a single place on Earth that doesn't discriminate brutally against foreigners and treats them like shit. That's just how it is. I agree with you that people shouldn't be delusional about Japan becoming their new ideal home forever. But at the same time, I think you're underestimating how poorly treated foreigners are treated everywhere else in the world. I'm Puerto Rican, but I lived in the U.S.A for quite some time, and the U.S.A isn't any better than Japan in that regard. Many groups of people - not just immigrants - are treated worse than most animals. And I also lived in France too. And again, France isn't any better. They treated me a lot better than I was treated in the U.S.A, but other immigrant communities were not as lucky, and France also has its own sets of issues to resolve aside from discrimination against immigrants.
The way I see it, it doesn't matter how poorly they treat foreigners in Japan. I'm already 100% used to being treated like an orangutan for a number of reasons, so they experience won't be anything new to me. I also have plenty of Plan Bs and Cs just in case Japan ends up not working out for me. I don't want to invalidate your traumatic experience in Japan. But like I said, most countries in the world aren't any better. One way to think about it is: every country in the world is complete shit. But some shit is slightly better suited for some people than for others.
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Let f : [0, ∞) —> R, where R is the ordered field of real numbers, and [0, ∞) := {t in R : t >= 0}, and f(x) = sqrt(x) everywhere. We say lim f(x) (x —> ∞) = L for some real number L, if and only if for all real numbers ε > 0, there exists some U > 0, such that if x > U, then |f(x) – L| < ε. |f(x) – L| < ε is equivalent to L – ε < sqrt(x) < L + ε. Since sqrt(x) >= 0, it follows that max(0, L – ε) < sqrt(x) < L + ε. This is equivalent to x >= 0 and max(0, L – ε)^2 < x < (L + ε)^2. Hence, if max(0, L – ε)^2 =< U, then if x > U, x > max(0, L – ε)^2. However, x < (L + ε)^2 for some ε, L, and x > U. Therefore, lim sqrt(x) (x —> ∞) = L is false for every real number L, which means lim sqrt(x) (x —> ∞) does not exist.
On the other hand, for all real numbers B > 0, there does exist a real number U > 0, such that if x > U, then sqrt(x) > B. This would be the case whenever U >= B^2. Therefore, one may say that as x —> ∞, sqrt(x) —> ∞. This is just special notation to say what I said above.
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Let sqrt : [0, ∞) —> [0, ∞) be a bijection, such that sqrt(x)^2 = x everywhere. Let g : [5, ∞) —> [0, ∞) be such that g(x) = x – 5 everywhere. g is a bijection. Let f := sqrt°g. Since sqrt and g are bijections, f is a bijection. Therefore, dom(f) = dom(g) = [5, ∞), and range(f) = codom(f) = codom(sqrt) = [0, ∞). Since g(x) = x – 5 everywhere, f(x) = (sqrt°g)(x) = sqrt(x – 5) everywhere. Therefore, graph(f) = {{{x}, {x, sqrt(x – 5)}} in [5, ∞) cross [0, ∞) : x in [5, ∞)}.
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Mathematics are not static either. How mathematics were understood in the 1800s was very different than it is today: so different, it would be irrecognizable.
By your extremely flawed logic, nothing should be taught at all. Science is not static, languages are not static, literature and art are not static, social skills are not static. Nothing is static, and again, I would like to emphasize, mathematics are no exception to this. So by insinuating that there is something wrong with teaching skills that are contemporary, you are insinuating that teaching any skills at all is wrong, since all skills are contemporary and change drastically over time.
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@MZONE991 I did not make up classical theism nor did I come up with the name. Further proof you are ignorant of it
Oh, c'mon. I was being fascetious. Of course I know you did not make-up classical theism. Poe's Law, I guess. My point is, though, that proclaiming to adhere to classical theism, as opposed to any other form of theism, adds absolutely nothing to the conversation. It does not make any claims or premises you present any more true or any better substantiated.
When I say "we define God this way" I am reffering to us classical theists, not people like WLC or most popular apologists.
I know that. But God is not well-defined in classical theism, any more than in other form of theism. This is what I was getting at. Immanence and transcendence are not well-formed properties, at least not in the presentation that most philosophers who adhere to classical theism present; let alone simplicity and timelessness.
But anyway, in contignent entities, it is always the case that they are a composition of essence and existence.
On basis of what? There is no universal agreement among philosophers that this view is true. Between essentialists and existentialists, there is a disagreement as to what roles essence and existence play, and how they are related. In Quinean ontology, though, existence is not even a property. Hence, it is incoherent to speak of essence and existence on the same grounds. Here, I can agree that for any given thing, (0) there is a set of properties, encoded in well-formed formulae in higher-order logic, that uniquely defines a thing, or classification of things, and (1) there is an existence clause that speaks of said set of properties being satisfied (instantiated) or not being satisfies, and this corresponds to the existence or non-existence of a thing in question. This much I can agree to.
Suppose we call the parts A, B and C. If A, B, and C don't interact with each other at all then it cannot be said that A and B and C compose one entity.
Says who? This is completely arbitrary, ontologically speaking. What if I consider every mereological sum of entities to be itself an entity? There is nothing incoherent about this. It may not be practical, but there is nothing fundamentally problematic with such an ontology.
If such parts interact from past eternity then this is literally a causal Loop, A interacts with B, B interacts with C, C with A and so on....
That is not what a causal loop is, though. This very obviously ignores the possibility of relative simultaneity of the parts interacting, in which case, there is no loop.
A causal loop is an infinite regress And since it has been shown to be impossible then the ground of all reality is an entity whose essence is identical to existence.
Okay, there are two issues here.
0. An infinite regress has not been shown to be impossible. To the contrary, I consider infinite regress to be the most plausible description of reality, mathematically and physically.
1. The fact that it is without parts does not imply that it is an entity whose essence is identical to existence. I know classical theism subscribes to the doctrine of divine simplicity, but that doctrine is precisely what we are scrutinizing here. It is precisely what you are being asked to demonstrate. Simply making the assertion will get us nowhere.
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@MZONE991 There are schools within classical theism, but we all agree that God is divinely simple.
Good job missing the point. The agreement is irrelevant. The notion of divine simplicity is ill-defined. This would not change if every single human on the planet Earth accepted it as infallible doctrine.
Every entity that exists contingently has a certain essence.
Yes, but this is because every entity, contingent or not, has a certain essence. By definition, for something to be an entity, it must have an essence that defines it, and the existence of that essence must be satisfied.
Nothing can exist without an essence, thus we have a composition of essence and existence,...
No. That is not how composition works. Composition is a function of tuples of essences, not of essences and non-essences. Existence is not an essence, because existence is itself part of the description that qualifies what an essence is. So it is incoherent to talk about an entity being composed "of existence" and something else. No, the being merely either exists or does not exist, and the Boolean description of existence here is not a defining component of the being itself.
...and this is a composition because it is possible for that entity to not have existed in some possible world.
That is a baseless assertion. Leaving aside the fact that you are completely wrong as to how composition works, there is also the issue that there is no well-defined ontological characterization of what comprises a possible world.
Causal loops don't need to be time dependent,...
Yes, they do, by definition.
...even if the causation is timeless, this is still a causal loop.
No, it is not. Causation is temporal, by definition. Whatever it is you are describing, it is not a causal phenomenon. You are equivocating terminology here.
And Alexander Pruss has shown in his work that they are impossible using many arguments and paradoxes.
No. Alexander Pruss has claimed to prove their impossibility, but I am confident in saying he is mistaken. His argumens are not sound.
So a car that exists is a composition of its essence and existence.
No. There is no distinction between a car that exists, and a car that does not, because existence is not part of a car's identity/definition. A distinction between any two entities can only exist in essence, because the essence is literally what makes the entity in question. To put it more plainly: an entity is its essence. To say that an entity exists makes a difference in the ontological description of the world, but not of the entity.
Following the causal finitism principle, the first cause of everything has to be simple.
You are begging the question. The causal finitism principle is the very thing we are challenging and asking you to prove.
God is something that we can never fully understand.
This is a self-contradicting claim. In order to be able to claim this with sufficient justification, you would have to have sufficient knowledge about God, since this is not a claim that can be derived from first principles.
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@yoichiswiftshot902 Fundamentally, people choose everything,...
Not according to the Bibles. According to the Bibles, Elohim, or alternatively, YHWH, is sovereign over all the world. Nothing happens unless YHWH permits it to happen. "When a prophet is deceived, I have deceived that prophet," says the beginning of Ezekiel 14:9.
...even in the Bible, Adam ignored God.
Adam did not choose to ignore Elohim. Elohim created Adam to be prone to deception. Adam being deceived was an inevitable consequence of Elohim's choices. Adam did not even have knowledge of "good and evil," so Adam had no awareness that he was doing such a thing as "ignoring" someone.
Do you not murder because your parents told you?
No. I do not murder, because, as I am not a psychopath, I do not enjoy murder. I would hate to do such a thing, and I find such a thing to be deeply unloving.
Were you born with perfect knowledge of good and evil?
No. Neither was Adam.
Do you not choose what you want?
We have the illusion of choice, but the scientific evidence does not support the conclusion that we have freedom of choice.
If you did it, then you chose it,...
This assumes we have freedom of choice, which I have already dismissed as a baseless assertion.
...there's rewards and social pressure for every action you take,...
No, not necessarily. The only actions that are rewarded are those that satisfy two conditions: (a) someone is willing to provide the reward, (b) that same someone is aware that the action has taken place, and knows how to find the person who committed it.
If God tells you to love your neighbor as you love yourself, it's like a parent teaching their children how to see the world.
No, it is not. YHWH has never told me anything. Delusional people insist that YHWH has said something, all without being able to prove it. Also, my parents are flawed people. My parents could be trying to teach me incorrect things - as many parents in the real world actually do. Therefore, accepting their teachings is not necessarily ethical. In fact, many of the things I learned from my parents were false. On the other hand, you are not willing to accept that anything YHWH says could be false, and you also have no way of proving that YHWH has ever said anything.
We can let the void of meaningless atheism guide us to nothing, since meaning is a made up social construct for human survival,...
No. "Meaning" is a concept we apply to words. We say that a word has meaning if there is a definition for the word that is known by someone.
...or we can let a God guide us into goodness, compassion, charity and all these other values that are getting hallowed from society, thanks to your secular worldview being so prominent.
0. No "god" is guiding you to do anything. Even if you have been brainwashed to delude yourself into thinking that someone else is responsible for your intuitions about what is ethical, you are still ultimately relying on those socio-biological intuitions, regardless of whatever fictional name you are choosing to attribute to those intuitions.
1. Compassion and charity are not being hallowed from society. To the contrary: compassion and charity are at an all-time high.
If your idea is so great, why are the results so bad?
The results are not even slightly "bad." You are thoroughly misinformed.
Suicide, divorce, mental health, baby killing, isolation, has all gotten worse in modern times.
No, it has not. This is factually incorrect, and you have no sources to support your claim.
Church means community,...
You can be without religion, not believe in the existence of mythological beings, and still belong to a community.
...it's not good for people to be alone and not apart of something,...
That is not for you to decide. Some people prefer being alone. Some people need to be alone.
If atheism has anything to say about it, then it's just for random chemicals to shoot in our brain for survival sake, and that's depressing just to hear.
0. "Atheism" has nothing to say about it, because "atheism" is not a worldview. "Atheism" is merely the stance of not being convinced that a mythological being exists in the actual world.
1. Chemicals in the brain are not random. I suggest you take an introductory course to chemistry. Maybe I can buy you a textbook. Chemistry is deterministic, just as all physical processes are.
2. There is nothing "depressing" about hearing that the entirety of our bodies can be accurately described as a physical system. In fact, most people who have this understanding are mentally healthy.
The smartest people ever were agnostic or religious, along with the greatest people.
This is false. Some of the "smartest" or "greatest" people in history (whatever this means) were indeed religious, but many were deists, and some were atheists. Also, the "worst" people in history were religious.
Now we have transgenderism becoming prevalent,...
No, transgender people always existed, and they probably comprised the same proportion of people they do today. Their existence not being recorded in writing as commonly is not indicative of them not having existed. The fact that society has evolved towards becoming more accepting of people's gender identity is one thing, but this has no bearing on how prevalent transgender people have been historically.
...we're so stupid without god...
There is no evidence to support this assertion. The evidence indicates that there is a strong positive correlation between irreligiousity, and higher education. This does not demonstrate a causal link, but it does categorically disprove your assertion. Your assertion is false.
...we somehow argue there's more than men and women by saying it's a social construct.
No, more like, we argue that there is no such a thing as "men" or "women." Biologically speaking, people have different sexual characteristics, but these characteristics are impossible to identify for a layperson beyond, superficially, their genitalia, and for 99% of the people that you will ever meet, you will never know what their genitalia is, or accurately be able to guess what it is, anyway. Moreover, as your biological characteristics are literally irrelevant outside of the bed, and irrelevant to anyone besides your mating partner and the medics responsible for your healthcare, they have absolutely no bearing on how you should be named, how should you speak, what jobs you should have, what sports you should be allowed to participate in, what people you should be allowed to participate with, how you should dress, what social traits people should expect you to exhibit, or how you should be treated by others in general. As such, "masculinity" and "femininity" are entirely fictitious concepts, examples of a societal delusion. They are the remnants from a time when people lacked an understanding of human biology, and incorrectly believed that people with some sexual characteristics were literally a different biological species from people with certain other sexual characteristics, and were thus classified as "not human," and were thus treated like property. They are essentially mythological, archaic constructs, no different than the belief in YHWH or Zeus.
Every single word is a social construct.
Every word is a social construct, but the concepts being represented or named by those words are not necessarily social constructs. For example, the existence of electrons is not a social construct. Their existence is demonstrable, independent of the delusions that society holds to. Gender cannot be demonstrated to exist in such a manner. Gender is nothing more than what society has deluded itself into thinking it is.
Atheism only accomplished making everyone feel no accountability, because everything means nothing, and now we're getting dumber for it.
0. "Atheism" has accomplished no such thing, because "atheism" is not a worldview.
1. I can assure you that there are no atheists or irreligious people who assert that "everything means nothing." The sentence "everything means nothing" is not even a coherent utterance.
2. There is no evidence that we are getting "dumber," this is a completely baseless assertion.
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Here is the correct explanation to the Monty Hall problem: the Monty Hall problem consists of a sequence of 2 sample spaces. The first sample space is the sample space {Door A, Door B, Door C} = {Picks Car, Picks Goat 1, Picks Goat 2} = S1. The second sample space is the sample space {Monty reveals Goat 1, Monty reveals Goat 2} = S2. Monty will never reveal the car: this is one of the rules of the show. What we are interested in is the set of ordered pairs (X, Y) such that X is one of outcomes in S1, and Y is one of the outcomes in S2. This is because events in S1 come before S2 always, this is another rule of the show: you always choose a door first, he reveals only after this. This set of ordered pairs is {(Picks Car, Monty reveals Goat 1), (Picks Car, Monty reveals Goat 2), (Picks Goat 1, Monty reveals Goat 1), (Picks Goat 1, Monty reveals Goat 2), (Picks Goat 2, Monty reveals Goat 1), (Picks Goat 2, Monty reveals Goat 2)}. This set has 6 elements, and two of the elements have probability 0. Namely, (Picks Goat 1, Monty reveals Goat 1) & (Picks Goat 2, Monty reveals Goat 2) are event sequences with probability 0, because the third rule of the show is that Monty never opens a door you have picked. In other words, P = 1/6*0. Meanwhile, there are 2 goats out of 3 doors, so the probabilities of (Picks Goat 1, Monty Reveals Goat 2) & (Picks Goat 2, Monty Reveals Goat 1) are both 1/6*2 = 1/3. The other two elements have probability 1/6 each since the probability of any individual element is 1/6, and since there is only 1 way for each element to happen, their probabilities are 1/6*1. This gives a probability distribution of {0, 0, 1/6, 1/6, 1/3, 1/3}. These add to 1, as expected. Since (Picks Car, Monty Reveals Goat 1) and (Picks Car, Monty Reveals Goat 2) have combined probability of 1/6 + 1/6 = 1/3, then the other 2 possible events have combined probability 1/3 + 1/3 = 2/3. Switching wins with probability 2/3 because the event sequences in which one would switch if one knew what they are have probability 2/3.
Yes, there are only two doors with a goat discarded, but the goat discarded had a higher probability of being discarded than of not being discarded, so the weighting of the choice of two doors is not fair, but rather 2/3.
EDIT: This can be generalized to N zonks with N + 1 doors. You can pick N + 1 doors, and there are N revealing sequences Monty can make following that. This means that there are N(N + 1) ordered pair of events. The probability of an individual pair will occur is the probability of your pick multiplied by the probability that Monty will do a particular reveal conditionally provided your pick. If you pick a goat, which has probability N/(N + 1), then Monty is forced to show you the other N - 1 goats, and as such, the probability of this ordered reveal sequence is 1, hence the probability of the pair is N/(N - 1)*1. This accounts for N(N - 1) of the pairs. N of the pairs have probability 0 since they involve revealing a goat you already picked. The other N pairs are winning pairs, so their probability is N/[N(N + 1)] = 1/(N + 1). As such, switching has significantly a higher probability of wins.
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@DoofusChungus ...if God wanted to create a universe that works, it would be in a way that makes sense in this universe.
This statement is hardly coherent. If God wants to create a universe, then They are free to create it however They want to, since they are omnipotent. On top of that, even if the type of life God wants to create is physically incompatible with the just created universe, God can still choose to create that life in that universe, and have it be self-sustained within that universe. Again, God can do this, since They are omnipotent, which means that They are not limited by physical restrictions. Bold of you to assume that physical restrictions could ever put a hamper on God's all-creative powers. Also, if a particular universe created is physically incompatible with a type of life being created, then why are these two incompatible? Is God not the one choosing to make them physically incompatible?
If God created the universe with certain logic and physics purposefully, why would he choose to go back on it?
This is a strawman. No one in this thread (or anywhere) has suggested that God should go back on Their creation and start all over. The argument being presented is simply that, in the presence of an omnipotent creator, there cannot exist such a thing as fine-tuning. You see, by definition, fine-tuning a phenomenon to an external parameter refers to calibrating the parameters of the phenomenon to be compatible with the external parameter. This act of calibration for compatibility insinuates that there are restrictions that need ti be accounted for to make a given phenomenon-external paramater combination possible. However, God is omnipotent. Therefore, no such restrictions exist. Therefore, there is calibration that is possible, by definition, since no restrictions exist in the combinations existed. Therefore, there is no fine-tuning. If classical theism is true, then it is, by definition, impossible for fine-tuning to exist when it comes to creation. Since fine-tuning does not exist, it quite literally cannot serve as evidence for the existence of God.
For me to be typing this comment on this phone, trillions upon trillions of coincidences had to happen for trillions upon trillions of other coincidences to happen,...
Coincidences? According to who? Because, I know you do not believe these are coincidences, you believe these are consequences of God's creative will. And I also know that I, like most non-theists, do not believe these are coincidences either. I cannot speak for the other non-theists in the thread, but I can speak for myself, so I will just clarify this right now: I am a physicalist. As a physicalist, I do not believe there exists such a thing as a "coincidence," because the word "coincidences" implies that events in the universe occur randomly. They do not occur randomly, though. As far as all evidence available points to, the universe is deterministic, which means that every physical interaction that happens happens with a probability of 1. Well, that is an oversimplification, though. I am not a classical determinist, because as a physicalist, I account for the existence of quantum phenomena. As such, I instead hold that quantum determinism holds. Even then, the conclusion is essentially the same: in the sense that you apparently meant it, I do not believe there exists such a thing as a "coincidence." So, again, I ask you: trillions of coincidences had to happen, according to who?
Once is an accident, twice is a coincidence, three times is a pattern.
Yes. And?
I believe the point of that was to show that, despite all of the coincidences that needed to happen for any of us to to be here, we are still here.
Well, no. There were no coincidences that happened. Every event that happened along the way can be entirely explained via the scientific method.
By some miracle our parents, out of billions of people in the world somehow happened to be in the right place at the right time.
There is absolutely nothing miraculous about it. Do you not know how many sexual couples exist on planet Earth? There are literally billions of them. Also, the right place at the right time? Hm... I was born in a colony, into poverty, into a land with high crime rates, with several chronic health issues, to a dysfunctional family, and raised in an unsanitary environment. Sorry, but that is definitely not "the right place, at the right time."
Not to mention, for each generation, the correct sperm [cell] had to make it.
There is no "correct" sperm cell. The probability that in any given generation, some sperm cell was going to fertilize an ovum, would have been extremely high, and you would have been whatever that sperm cell was, because it necessarily would have been impossible for you to not be, since clearly, you were born.
It would be "finely tuned" so that everything that has happened or existed would happen or exist.
No, it would not be finely-tuned. God is omnipotent. You believe this, right? If God is omnipotent, then the exact sequence of events that led to our existence would have been possible in all universes, because God could have simply chosen to trigger the sequence of events at will, without having to worry about whether it is physically impossible or not: They transcend physical constraints. Am I wrong in claiming that you believe God is omnipotent? Am I wrong in claiming that you believe God transcends physical constraints?
Who knows if God has to abide by the laws of this universe or not? That part of the question really does not matter.
Who knows? Does this mean you do not believe God is omnipotent, or am I misunderstanding the question? As to whether it matters or not: it absolutely does matter. If God is omnipotent, then They are not constrained by the physical incompatibility between a given universe, and the life They create in it. As such, it is genuinely impossible for any God-created universe to be uninhabitable, since God can always create life in such a universe.
The real question is, why wouldn't he?
Why would They?
If he created the universe a certain way, with the whole of the future already planned out, then what would be the point of changing anything?
Once again, no one is arguing that God should change anything. We are discussing the impossibility of fine-tuning in the presence of an omnipotent creator. We are not discussing the problem of evil, nor making any judgments on whether God should have created the universe the way creationists claim They did or not.
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@x-popone6817 Whatever begins to exist has a cause.
This is completely meaningless unless you define what you mean by "begin to exist", and what you mean by "cause". That being said, by most definitions of the word "beginning", this premise is at least debatable.
The universe began to exist.
By most definitions of the word "beginning", this premise is completely unsubstantiated. It is an assumption, not a fact.
Therefore, the universe has a cause.
Yes, this does follow from the premises, but the premises are questionable, and even if I grant the premises too, this only establishes that the universe had a cause. The argument fails to establish the nature of the cause.
there are implications of what this cause is and what properties it has
No, there are not. Any claims concerning the properties of such a cause are necessarily speculative and unprovable.
It does get you to a mind, as I already explained.
No, it does not, and you have not explained anything. All you have done is claim that it does, and followed it up with "please, trust me", which is not an explanation.
The Big Bang was the beginning.
No, it was not, and I would know this, since I am a physicist, as I already said in a previous comment.
Even if it wasn't necessarily and it was just a singularity,...
Here you go, using the word "singularity" without having any idea of what it means. No, it could not have been a singularity, because singularities are not physical objects that exist, singularities are mathematical artifacts. You can literally find this in the Wikipedia article on singularity.
...you would still face the problem of how an impersonal force like that can expand at a point and not have the effect permanently.
Effect permantly? You are just linking nonsense together. As for expanding from a point, there is no evidence that the universe's expansion began from a single point.
The universe can't be because then we would be eternal.
No, that is a non sequitur. Our eternality does not follow from the universe's eternality. The age of the objects contained in the universe is not beared upon by the age of the universe. The universe is 13.8Ε9 years old, but that does not mean we humans are that old. The universe has average temperature 2.3 K, but that does not mean we have said temperature. A property that is had by the universe as a whole does not need to be had by its constituent parts.
Its finite.
There is actually no conclusive evidence demonstrating that the universe is finite in volume.
The universe is expanding, if you reverse that, you get a beginning.
No, you do not. If you reverse the expansion of the universe, you get to an immeasurably hot and dense state of the universe, such that any further contraction is not coherent with current physical theory, and this state of the universe is known to occur at the end of the Planck epoch.
If there is a mind with free will, he can create the universe, no problem.
No, not necessarily. If we have free will, that does not mean we can create a universe.
if the cause of the universe was impersonal, that can't happen since there's no person, no free will.
This is just an speculation, not a fact. You have not proven that this is true, you have merely assumed it is true, and then you expect us to agree. Non-intelligent processes can cause things. Meteorites cause craters. Stars cause hypernovas. Stars are not intelligent beings, and neither are meteors. Besides, you are begging the question. You are trying to prove the universe has a cause, but in order to do so, you are assuming that the universe had to be created, which is more specific than just being caused. "Caused" and "created" mean different things. If something was created, then it was caused, but being caused does not imply it was created. You do not get to just smuggle non-synonyms into your argument.
The effect should've been clearly eternal as well,...
No. Causes need not transfer their properties to their effects. This is just another assumption. Actually, not only is this not substantiated, but it is demonstrably false. Humans can cause objects to exist that are not themselves sentient. There. Cause-to-effect transfer of properties has been falsified.
How am I dishonest?
You make false claims, knowing that they are false, and you proclaim yourself to be more knowledgeable on topics of science despite not being a scientist. You also refuse to define your terms rigorously, but insist in still presenting an argument. These are all defining characteristics of dishonesty.
Atheists ALWAYS claim us Christians or theists are dishonest.
No, not always, but very often, yes. Is this surprising to you? It should not be. We make this claim so often because it tends to be true far more often than not.
Atheism is not some ultimate standard where everyone that disagrees is dishonest.
You are right, and not every theist is dishonest. Some theists are merely ignorant. You, in particular, just so happen to be both.
I wouldn't be surprised if you think almost all YouTube apologists are dishonest.
They ARE dishonest. Apologetics actively requires that scientific data and mathematical concepts be misrepresented to prove a point.
This is not to mention the philosophical grounds for the finitude of the past.
All the philosophical arguments for the finitude of the past are flawed. In fact, professional philosophers generally do not take them seriously. Only theologians and apologists specifically do.
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IngRand'lo Russell I never claimed P(1) is true independent of the other propositions. It does require proposition P(2) to be true, and P(2) requires P(3), ...P(N) requires P(N+1), ad infinitum. So it is an infinite regress.
"What is truth?"
That question can't be answered precisely because of the infinite regress. If the answer is objective, that means that there exists an objective determinant that operates by objective standards to define truth, and when the definition is met by a proposition, that proposition is necessarily true. However, accepting this necessarily yields premise 11 as a consequence. Thus truth cannot be objective. Instead, we must say "Truth is X, Y, Z." and we must declare this definition to be true a priori: it's an axiom. It cannot be argued a posteriori because of infinite regress. But because the question of what is truth can only be answered a priori, it is subjective. So truth can literally actually be anything. That's the problem: that's why omniscience can't exist, because axioms are a necessary logical consequence and axioms make omniscience impossible.
Yes, the statement "Truth is subjective" must necessarily be subjective, which means that in some inconmensurable domain of axioms, Truth is objective, and perhaps God is in fact omniscient and does exist, but this domain of axioms is not logical: that is to say, infinite regress and contradictions aren't assumed to be logical fallacies. However, that isn't how we operate and the reasons are very intuitive, tautological by definition.
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demoeder That isn't a problem, given that semantical and grammatical rules are subjective. Thus, whether a sentence is semantically correct and grammatically correct is subjective.
In any case, that is irrelevant, because the semantic structure and grammatical structure does not make the sentence true. One can construct the proposition "I am in Mars.", and this proposition is both grammatically correct and semantically valid (no oxymorons), but by no means is it true. The truth value of a sentence is clearly not determined by its grammatical structure, the content itself goes beyond grammar. The sentence “an electron has negative charge” cannot be determined to be true or false based on grammar, and the semantics of the term "electron" is extremely debatable and non objective. We humans assign names to objects, so it's subjective, they don't have inherent names or definitions. So there is no liars paradox here. And there is no problem with proving common assumptions wrong. Science does that on a daily basis (Humans don't have 5 senses, but more).
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IngRand'lo Russell You've not replied to my question, but I'll proceed with my explanation anyway, since a specific answer given by you is actually not necessary.
You made the proposition P(0)="Truth is Objective", and you claim P(0) is true. What makes P(0) true though? Why is your claim justified within the domain of logic? You can claim "P(0) is T BECAUSE it satisfies K". Okay. But then my following question is, what makes this last claim, P(1) true? And this question is in fact relevant. If satisfying K does NOT imply being true, then satisfying K is irrelevant, which means that P(0) remains an unjustified claim and thus dismissible. P(0) can only be true if P(1) is true. But that's a conditional. So the question is what makes P(1) true. Now you get a composition of propositions, because the answer would then again be "P(1) is true because it satisfies K". Now, you fail to understand why this is a problem. It begs the question, what makes this new proposition, P(2), true? Once again, P(1) is only justified if P(2) is. What you're failing to understand is that, I'm postulating the definition P(N) is T <=> P(N) satisfies K, but this definition is arbitrary and by no means is it necessarily true. So I must justify [P(N) is T <=> P(N) satisfies K]:=P(N+1) by saying P(N+1) is T <=> P(N+1) satisfies K. But then this itself needs to be justified. Postulating that two things are equal doesn't make it true. "P is true because it is true" can't be concluded from "P is true because it satisfies K" like you think: it fails because the conclusion requires the premise to be proven true. Remember: any claims about the nature of truth are themselves presumed to be a truth. If I say "Truth is subjective", then this very statement must necessarily be a subjective truth. Similarly, when you provide any standard for justifying a claim as true, the claim that the relationship of a claim to such a standard makes the claim true is itself another claim that must be true, which means the very relationship we evaluate to determine that P(0) is true must itself be true before we conclude P(0) actually is true.
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IngRand'lo Russell What you're failing to understand is that every claim needs justification. Arbitrary claims are dismissible and this is why arguments with no logic in them are regarded as invalid. But giving a justification, while it is enough in practice, it is for all logical purposes never enough. See, if you make the claim, "The sun is yellow" and you justify it by saying "science", most people will accept this justification because people a priori assume science is true. But that a priori assumption is still just an assumption, it isn't justified. Why should we actually assume science is indeed true and it works? You can try to justify that by saying "well our senses", or whatever justification you want to give it. But the justification is in itself a claim that requires justification... a.k.a, it is indeed an infinite regress. In practice, we never take it that extent because it is impractical to be unnervingly skeptic, but that by no means does it mean objectivity actually exists on a fundamental reality. This is exactly what premise 11 meant to say, but of course I wrote as a formal logic proof so it wasn't obvious. But now I explained in layperson terms and this what it comes down to: any justification to a claim is in itself a claim that requires justification. This is why objectivity can't exist, and this is why knowledge doesn't exist in the way we think it does. Thus that's why omniscience isn't possible. We're used to thinking of knowledge as just storing facts in our mind and relating them, and distinguishing them from fiction and falsehood. But that's not how it works: in order to determine whether something is true or not, you need a reference frame, a set of axioms to assume. Depending on the set of axioms, the statement will either be true or not. It works very much in the same way one can't simply say a distance is long without referring to a scale, an instrument of measurement, and the conditions at which the measurements are made. It's not so counterintuitive: it is already known absolute time doesn't exist either and that reference frames in physics are relative. But physics is merely mathematics mixed with observations to fix variables and elements, and mathematics are merely a branch of logic. So relativity in physics, which is proven, can only really work out if logical proofs are themselves relative to the assumed set of axioms, and this conjecture was what I proved more formally above.
And yes, such a proof is itself relative to the axioms of logic I assumed, but that isn't a problem and it does not make the proof dismissible, because the axioms I assumed are the so-called common axioms that are pretty much universally assumed (although also universally misunderstood).
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IngRand'lo Russell IngRand'lo Russell You have the guts to try to pull a Philosophy 101 argument when I'm a philosophy major, HA! Pathetic.
Listen. Let me make this as clear as I can.
The statement "Truth is ___." cannot be completed. Do you want to know why it can't be? Attempt it yourself. Complete the statement "Truth is ___." yourself, it seems to be so important to you. Once you complete it, I'll explain why it can't be completed.
Saying "Believing X is absurd" isn't an argument Ingrad. Now it's my turn to retort Philosophy 101. Why is believing that truth is subjective absurd? Why is the belief in God not absurd? What is your criteria for saying that a proposition is absurd?
No, you keep misunderstanding the truth argument. The proposition ("Truth is objective" satisfies K) is assumed a priori, but by no means is this what I was challenging. Yes, P(0) satisfies K. But why does that make P(0) true? Why is K the set operator that makes P(0) true and not a different criteria L or M or N? See, it doesn't matter whether ("Truth is objective" meets K) is true or not, and it doesn't matter whether it is true a priori or a posteriori. What does matter is that meeting K is what makes "Truth is objective" a true statement, according to you, but such a claim is not justified. The claim (Meeting K makes "Truth is objective" true) must and is itself being claimed to be objectively true. But this implies the claim itself must both meet K and be converted to a true statement by virtue of meeting K. Now, meeting K is arbitrary and can be assumed a priori, but you must still prove that [Meeting K makes (Meeting K makes "Truth is objective." true.) true.]. And now you get a circularity, because this new proposition can once again be assumed a priori to meet K, but it'll beg the question about whether meeting K makes it true because that is what you were trying to justify in the first place. We're not trying to justify it meets K, we're trying to justify that meeting K makes a claim true. And yes: it needs to be justified. It can't be true a priori because if L, M, and N all exist, I can just claim that P(0) meets M and make it a priori, but then I still must also justify that meeting M makes P(0) true. Otherwise, by allowing a priori assumptions, you get arbitrariness, contradictions and subjectivity. Because that means that whether meeting K makes true or meeting L makes true is arbitrary and can be chosen. No, in order for objectivity to hold, one must PROVE (a.k.a a posteriori) that meeting K is what makes something true as opposed to meeting L or M. And trying to prove that is what leads to infinite regress. I was never trying to dispute that P(0) meets K: in fact, assuming a priori that P(0) meets K is necessary in order for my argument to proceed, and I did assume it all along.
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IngRand'lo Russell And that's exactly why "Truth is __." CANNOT be completed. Because saying "Truth is X" is a truth claim itself that therefore must be justified by saying ("Truth is X" is X), and this therefore by [("Truth is X" is X) is X], but this again will need be justified, because if it isn't, then by no means can we ever conclude that ("Truth is X" is X) to be true, and since this must be true and justified, but it isn't, we can't conclude "Truth is X" is itself true.
Think of a real life example. Let's visualize rape. Why is rape wrong. Instead of dealing with truth, let's deal with wrong. "Why is rape wrong?" Well, rape is wrong because it is an action that meets criteria K. But why does meeting criteria K make rape wrong? I mean, yes, in the K-ist current of ethic thought, rape is wrong, but not in the M-ist current of ethic thought. So then you must justify choosing K over M. So you say "K is not wrong and justified whereas M is in that K meets K and M does not". Okay, K meets K, but why does that make it true? See, it's impossible to properly answer "Why does meeting K make P(N) true?" without engaging in infinite regress. See, you said "Truth is objective" is true because believing "Truth is subjective" absurd. But what makes the statement "Believing 'Truth is subjective' is absurd" itself true? Therein lies the problem. Every justification needs a justification, but the question isn't how do I justify it, the question is why does this justify it as opposed to that other thing. Of course, there is one way to prevent this regress from happening. And that is to make "Meeting K makes P(0) true" an axiom. In other words, you must say that P(1) is true because I declare it to be true a priori. But that is equivalent to subjectivity because K is arbitrary, so you can axiomatize absolutely any definition you want and choose for completing the statement "Truth is ___", and that is exactly what subjectivity is. A priori truths aren't much different from opinions.
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IngRand'lo Russell 1. No, being a philosophy major does not imply I have it all figured out, but your retort was condescending so it deserved the typical response of declaring credentials. You can pretend you understand philosophy better than I do, but that is irrelevant if you're incapable of posing simple argument.
2. What is truth subjective to? Well, I already answered this. It is subjective the set of axioms you assume and the circumstances under which you assume those axioms.
3. 2+2=4 is not an a priori truth, so your argument is not even sensical. I can prove 2+2=4 starting by assuming the Peano axioms, which are very intuitive to assume.
4. "Truth is knowledge of the things as they are, as they were, and as they are to come." Truth and knowledge are synonymous? That's just as bold of a claim, and most people would disagree, obviously. Anyhow, that statement already fails to the argument I've been posing. For the claim to be true, it must satisfy the criteria that assign it the truth value. You have given no such criteria, actually, because the concept of things "being as they are" is already subjective and it depends on whether you subscribe to solipsism, Platonism, empiricism, realism, etc. So that is already a claim saying that truth is subjective. Had you made a claim that appeared more objective, you'd've more of an argument. But here is the issue.
5. Let's assume you did give such criteria K for saying the claim you made was true. We could verify that your claim did satisfy K, but then we would need to show that meeting K made the claim true by also subjecting such a relationship to K and proving that would make the relationship true, but then this even larger relationship... and you get the point.
6. You misunderstood claim 6. I'm not questioning why the claim is true per se. I'm instead pointing out the fact that if God knows that P(0) meets K, and if God knows that K is supposed to exist as objective determinant, then the relationship "P(0) true because K" must itself be true. God knows it is true. But whether he knows it or not is not particularly relevant. P(1) being true is a logical consequence of the biconditional requiring that P(1) itself meet K. In other words, God's knowledge of P(0) being true is independent of God's knowledge of P(1) being true, but the act of P(0) being true is not independent of P(1) being true. And similarly, P(1) being true is directly dependent upon P(2) being true, etc. See, God knows that such a sequence of P(N) is true all the way to infinity, so that's not the issue. The issue is the existence of such a sequence in the first place. In order for it to be logically valid, it can't be infinite, but if it isn't infinite, then at some point one must merely declare any arbitrary P(N) to simply be true without justification, to simply say "Let P(N) be true" a priori and make it an axiom on its own account. And then K would have to also be determined arbitrarily, at the discretion of the subject making the claim. So it is either subjective, or an objective infinite regress, but an infinite regress isn't a valid option so the only option is subjective.
7. You declared that saying that Truth is subjective is equivalent to claiming Truth does not exist. That is only true if you assume a priori truth must necessarily be objective and should not be otherwise, an assertion which is itself arbitrary and not really justified. And here is the problem: "Truth exists not" can only prevent self-contradiction if the claim itself is not a truth. Yet if it is not a truth, then it is either absurd or false. So, not much of an argument, because the axioms we assume with logic do not really allow it to be an argument.
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IngRand'lo Russell "Let me ask you..." (I'm just stopping the elipsis here because I'm addressing the entire paragraph)
That's clearly not how justification works. Not because I'm incapable of providing a justification for a random claim does it mean EVERY justification does not require a justification of its own: that's VERY non sequitur on several levels. See, science is in itself an assumed set of axioms, and one can very much "choose" to not choose them. Of course, people do choose them because it's practical and instinctual and it appears counterintuitive to humans to assume other axioms. Scientists ADMIT this: they know science is based on a bunch of assumptions that can't ultimately be proven at all. They understand that for any logical framework to arise, axioms must be assumed. This is also true of mathematicians: most of them understand mathematics is a human construct that works by axiomatization. There are literally thousands of papers based on this. How do we know natural numbers can be added together to form other natural numbers? By the Peano AXIOMS. And can you choose to reject such axioms? Of course you can! Entire fields of mathematics are based on this, and even they have practical applications. Is "Division by 0 is undefined" an objectively true statement simply because it is mathematical? No. It's true subjective to the Peano axioms. There is a different set of axioms you can assume where division by 0 is allowed, and the resulting structure is called a Wheel. This is called wheel Algebra. And even amateurs of mathematics know of wheels existing.
So, your point about my inability to justify science is moot, because in practice we don't try to justify science, we assume it a priori. However, objectivity does require justification, because objectivity requires that there's a way by which all claims can be concluded a posteriori. So the infinite regress argument holds.
I already explained what it means for truth to be subjective in the paragraph above.
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Shea O'Donnell the Bible contradicts itself though. There are several chapters in which God lies.
That said, perfection and omniscience are both logically impossible, so such an argument is wrong anyway.
Additionally, even if God were truly perfect, that would not mean that whatever God says is not an opinion. If morality is determined God, then it is arbitrary, hence an opinion. If God determines morality based on knowledge about the universe, then objective morality can only come from the universe itself, not from God. Hence, either way, it still fails.
Just accept it, damn it. It isn't logically possible for God to be the basis of objective morality. There isn't anything wrong with that.
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Paul Elpers The problem with that argument is that the choice of God as an objective determinant is completely arbitrary, hence subjective itself, because there is no objective standard that allows us to choose between the morality of God as presented in Islam or of God as presented in Christianity, or of God as presented in Judaism. Additionally, God does change his morality quite a few times before reaching the current one, according to the Bible. Objective morality under Christian theism really just means the opinion of God dictates morality, which once again falls under the objective category. The problem is that objectivity cannot stem from any entity whose existence entails a form of consciousness, because by definition if it does, it becomes subjective. That is just what the definition of objectivity and subjectivity are. God would need to not be a consciousness, but rather some sort of force, except that contradicts the properties of God as presented in the Bible.
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Paul Elpers The reason why infinite knowledge won't imply objective morality nor authority is not related to knowledge, but rather to the self-contradicting nature of objectivity itself. The problem is that Dennis is posing the argument with a question of epistemology (a.k.a, How do you know murder is wrong), but in reality the correct question to ask is, "Why is it wrong?" What makes murder wrong, what characteristic of the act of murdering makes it a wrong moral action? We can see it as a spectrum: one side of the spectrum dictates that actions that meet the standard corresponding to such side are wrong, and the ones on the other side are correct. I can appeal to the axioms I assume and my instincts to justify my knowledge of morality, but I can't justify saying that something is wrong without first answering, "What makes it wrong?" What does it mean for something to be wrong? No matter what answer U give or God gives, not because of knowledge, but because of the fact that objectivity fails to be possible. So you see now why infinite knowledge is irrelevant.
See, the issue here is, no matter what justification you give to the answer of that question, the justification will itself always require justification. This problem arises from the nature of objectivity itself and how truth operates. The burden of proof and the nature objectivity require that justification exists and that there is no arbitrariness. But this then inevitably leads to an infinite regress, which s a logical fallacy. Which is why objectivity isn't a thing.
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coolguy 284 If you are talking about the modern definition, sure, but the historical origin is far more complicated than this. In reality, what happens is that you draw a circle and then a triangle using a central angle. There are many limes that have some special relationship to the circle, e.g a tangent line or secant line. The ratios are then defined in terms of how the special line is parametrized by the angle and the radius of the circle. Only a few of these ratios are taught today, though, which are the sine, cosine, tangent, cotangent, secant, and cosecant.
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Ayush Sharma “Gödel’s Incompleteness Theorem proves that every logical system that is complex enough to describe itself must suffer with undecidable statements and internal paradoxes.”
This is actually incorrect. For starters, there are two Gödel Incompleteness Theorems, not one. The First Gödel Incompleteness Theorem states that for any effectively axiomatized formal theory T of arithmetic with a language L sufficiently rich to capture every property semantically, if T is consistent, then T is negation-incomplete, or equivalently, there exists some proposition P in L such that it is undecidable in T. The theorem says nothing about paradoxes, actually. Also, the Second Incompleteness Theorem states that, for theories T which satisfy the conditions of the antecedent of the First Incompleteness Theorem, if T is consistent, then T cannot prove its own consistency.
“...but it can itself contradict itself if you apply it on undecidable statements.”
No, you are wrong again. By definition, if a statement is undecidable, and if the theory is incomplete, and if the theory is consistent, then adding this statement as a new axiom of the theory cannot lead to a contradiction. We are not dealing with maximally consistent sets of axioms, precisely because the theories are incomplete, and furthermore, incompletable.
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Stefan It couldn't possibly be. All the Queens must be in different rows in order to form a solution. The first Queen can be placed in exactly 8 positions of the first row, since the entire row is available. The second Queen can be placed in exactly 6 positions of the second row if the first Queen is in a corner, since it attacks the two positions at the end, or exactly 5 positions if the first Queen is not in a corner. Thus, you can legally place 2 Queens in 5*6 + 6*2 = 42, since there 2 corners. More explicitly, (8 - 3)*6 + (8 - 2)*2 = 42. To place a third Queen in the third row, you would be able to occupy 4, 3, or only 2 squares, depending exactly on how the first 2 were positioned and which squares they attack. You have to count cases separately. But for sure, the answer would have to be less than 2*8^2 because of this.
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@Dhen Phu Now, improper limits are defined a little differently. Improper limits are the type of limit that your teachers read as "as x approaches ♾, f(x) approaches L." These are the ones people have trouble with, because the notation that we use is lim f(x) (x —> ♾), which makes people think there is some sort of infinite process happening. But once again, this is not the case, despite what the notation may suggest. Here is the definition: lim f(x) (x —> ♾) = L means "for every ε > 0, there exists some real number δ, such that for every x in the domain of f, x > δ implies |f(x) – L| < ε." Notice how this definition is almost entirely analogous to the definition of a proper limit: there are two quantifiers, and a proposition that needs to be true in accordance to those qualifiers. The only distinction is in the proposition that needs to be true. Notice how, once again, no notion of process or algorithm is built into this definition. Notice how this definition never invokes any "infinite quantities" or ideas of a "never-ending task." This is, just once again, a definition about the existence of numbers satisfying some inequalities and bounds.
This type of limit is important, because as mentioned in the video, limits of Cauchy sequence is how we define real numbers. But as you can see, limits of a Cauchy sequence are not actually defined in a problematic way.
A Cauchy sequence is defined as a sequence that, for every ε > 0, there exists some natural number N, such that for all n & m, if n > N, then |a(n) – a(m)| < ε. Once again, no processes or algorithms here, just statements about whether numbers exist satisfying some inequality. So Cauchy sequences are perfectly well-defined as long as you accept that a sequence exists and is a well-defined idea.
A sequence is just a function that maps every natural number to some number of some set. The only possible way you would have a problem with how sequences are defined is if you wanted to assert that that the set of natural numbers does not exist, or that you cannot have such a map. But you can obviously have such a map, just take the identity map. So the only way you can assert that any of these things is problematic is if you assert that the set N does not exist. And this is why ultrafinitism is absurd.
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To start with, you should probably stop using the word "infinity" to talk about these objects, because it is mostly nonsensical, and it does not correspond to the way the mathematical theory actually works, and so, you will likely just end up confusing yourself by trying to think of these objects all under the name "infinity." "Infinity" is not a mathematical object, it is just some vague, pseudo-mathematical concept. Yes, in mathematics, we work with infinite objects all the time, but these objects all have different names, they are not "infinity," and they are well-defined within a given context, but they are not denoted by the symbol ∞ that everyone uses. These objects all do different things, and what it means for them to be "infinite" means different things, depending on what they are. This is why, if you want to understand what is really happening, you should really stop thinking of "infinity" as more than just a concept, and instead, just at mathematical concepts for what they are. I believe this is the most useful advice I can give you to move forward with understanding these topics.
The mathematical object that we call the "real number system" is defined in such a way that it has two fundamental components. One of the components makes the real numbers form a field. This means you can add these objects, you can multiply them, and you can (mostly) divide them, and there are certain strict rules for how to do that. The other component is that of an ordering: you can order the real numbers, you can compare them. It makes sense to say that 0 is less than 1. It makes sense to say that 1/3 < 1/2. This is important, because not every mathematical structure has this. Now, the way the real numbers are ordered is actually very, very special. In what way is it special? In three ways: (a) addition and multiplication are compatible with the ordering; (b) the ordering is total, meaning that for any two real numbers x, y, you can always, without exception, compare them; (c) the ordering satisfies the least upper bound property. This last one may confuse you. What is the least upper bound property? It is the property that says, that if I take any arbitrary (non-empty) subset S of the real numbers, if S is bounded from above (it has an upper bound), then it must have a least upper bound. For example, consider the interval (0, 1) (the endpoints are excluded). This interval has an upper bound. 2 is an upper bound. 10 is an upper bound. π is an upper bound. However, of all the infinitely many upper bounds, one of them is the smallest one possible, and this is the real number 1. Why is this the smallest upper bound? Because any real number less than 1 is either in the interval (0, 1), or smaller than 0, and so, not an upper bound of (0, 1). Even though (0, 1) has no greatest real number, it does have a least upper bound, which is 1. By the way, the least upper bound is also called the supremum, and this is the name I will be using from now on.
This should make you think a little more about (c). The ordering is such that every nonempty set of real numbers bounded from above has a supremum, but it feels as though we should be able to make this even stronger. What happens if we extend the real numbers, in such a way that all sets of real numbers have a supremum? For this to be possible, there needs to exist some object greater than all real numbers, and this object will be the largest object in the set. I will call this object T, which stands for "top." There also needs to exist some object smaller than all real numbers, and this object will be the smallest in the set. This object needs to exist so that the empty set can have a supremum in this ordering. This object, I will call B, which stands for "bottom." Hence, we have the set of all real numbers, and also, the objects T and B. Together, this new ordered system is called the affinely extended real line, and the geometric, visual idea is that the line of real numbers has been extended in such a way, that it now has endpoints, B and T. If you read the Wikipedia article, or some other popular but non-scholarly source, then you will find that the symbols for T and B that they use are ∞ and –∞, and they usually are read "infinity" and "negative infinity." However, as I already explained to you, this notation/language is very confusing, and it is misleading, if not outright incoherent, and this is not universal in the mathematical literature either. So, although I do want you to be aware where the symbols –∞ and ∞ come from when doing calculus, I will not be using them, unless it amounts to clarifying something, because if I do use them, it will confuse you. Every time you see a statement of the form lim f(x) (x —> ∞) = L, you should replace this with lim f(x) (x —> T) = L, and similarly with –∞ and B. Also, every time you see lim f(x) (x —> p) = ∞, you should replace this with lim f(x) (x —> p) = T, and again, this analogous with –∞ and B.
One thing that is very important to understand is that, while you can do arithmetic with real numbers, you cannot do arithmetic with T and B (which is why people usually say "infinity is not a number"). Yes, these are valid, well-defined mathematical objects as far the ordering system is concerned, but you cannot perform addition, multiplication, or division with these objects, without creating a bunch of contradictions. This can be proven carefully, but I will not do that here. And I know that often, you will see these strange "conventions" where you see things like 1/∞ = 0 and x + ∞ = ∞, but these are just abuse of notation, and are not universal - different contexts use different conventions for abusing notation. Properly speaking, there actually is no such a thing as arithmetic with these objects. You can still do certain operations with these objects, like the supremum operation, of course, and you can still define many classes of functions for these objects whenever you are not relying on analysis rather than arithmetic, but these funcions and operations are not the operations we usually call "arithmetic." This is why it is very tricky and complicated to evaluate limits when it comes to expressions where these objects become involved in some way or another, and actually, you are not required to have these objects to do any calculus at all.
Anyway, this is all to say that the conceptual idea I just explained, of making sure all sets of real numbers have a supremum (an idea that is indeed very useful, as long as you are not trying to turn it into an arithmetic number system), is where the symbols –∞ and ∞ come from. Meanwhile, the usage of the symbol ∞ in the context of complex analysis, is completely, completely unrelated to this, and again, it just boils down to abuse of notation. But what is the actual underlying concept behind it? I will explain that in the next comment.
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0:52 - 0:55 Saying "I can keep adding 0s forever will continue giving me 0" is a mathematically incoherent statement, because of the non-concept of "adding forever." It is extremely easy to make mistakes in mathematics when using colloquial notions and intuition, when you should be using precise language instead. I think this here is the actual mistake in the "proof," rather than anything else. There is nothing that can be discussed at all about the validity in the proof until the terms in the proof are actually precisely defined, to begin with, so nothing else could possibly be meaningfully identified as the mistake. This is not very different from how, if I say "Gooblydegock is heavy and of the color martin," it amounts to nothing but literal gibberish, since the words "gooblydegock" and "martin" are undefined terms. There is no meaning in say the sentence is true or false: conceptually, it is not even really a sentence at all, to begin with. It is merely a nonsensical string of symbols. On that note, the idea being conveyed has multiple inequivalent ways of being formalized, but the most basic of those formalizations, and the one most commonly used, as well as the one most relevant to this video, would be to consider a sequence f(n) = 0. You can find the sequence of partial sums of f, and that gives you still s[f](n) = 0. This also means lim s[f](n) = 0. Yes, I do understand that colloquial language and intuition are important in the context of teaching. However, as far as proofs are concerned, those are completely inappropriate.
2:20 - 3:37 What is truly happening here is that we are talking about the partial sums of two different sequences. Earlier, we had f(n) = 0. Here, we have g(0) = π, g(n) = 0 otherwise. What the video is claiming is that lim s[g](n) = lim s[f](n). This is clearly false. The reason this is confusing, though, is that the notation used in the video (which is inappropriate when doing mathematics) makes it seem as though you are still talking about the sequence f and its partial sums, but in reality, it has been replaced by the sequence g with its partial sums, without the viewer noticing, because the notation being used is simply misleading and ill-defined. This goes back to what I said in my first paragraph. Since it is not even clear what the notation being used is even supposed to mean, it can be easily used to deceive people. This is not a matter of unnecessary pedantry, it lies at the very core of the problem.
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@kurtu5 I know ring theorists have proof that you cannot. All commutative rings can be decomposed into 4 substructures: the zero divisors (always includes 0 itself), the invertible elements (also called the units; always includes –1 and 1), the irreducible elements (which we call the prime numbers when it comes specifically to the integers; for a given ring, this substructure could be empty), and the composite element (for a given ring, this substructure could be empty). There is no exception to this. In the case of the integers, the classification simplifies as follows: the set of zero divisors is {0}, the set of units is {–1, 1}, the set of prime numbers {..., –5, –3, –2, 2, 3, 5, ...}, and the composite numbers, which are the remaining integers. Because of the additive symmetry, though, when studying number theory, we very often ignore the negative integers, so we speak as if only the positive integers could be prime numbers. In this case, we would have four substructures, {0}, {1}, {2, 3, 5, ...}, and whatever remains. Theorems in number theory are just special cases of theorems in ring theory, which hold for very general classes of rings, not just specific chosen rings like the ring of integers. As such, we can literally just classify the theorems in accordance to what rings they apply to. We can even make statements about what theorems can be proven about rings at all.
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Jeffrey Black No, the rules of summation do not apply generally to finite summands as they do to infinite summands. Notice that for finite quantities N, it is true that N + 1 = N, despite the fact that 1 is not the identity element of summation. This is not true for finite N. Additionally, infinite summation is not considered to be associative or commutative, while finite summation is. Infinite summation is not closed under any subset of the real numbers, which is why you can obtain irrational numbers from adding infinitely many rational numbers (as in the definition of the number e). This is not the case for the sum of finitely many rational numbers.
Addition is defined by the axioms of Peano arithmetic. Peano arithmetic works with natural numbers. We can extend the natural numbers to the real numbers trivially, while still preserving the properties of addition as derivable by induction from the axioms. We can prove theorems about addition when dealing with a finite natural N amount of summands, using induction. Examples of these theorems are commutativity, associativity, closure, distributivity, etc. However, this does not work once you have infinitely many summands. Why? Because infinity does not exist in Peano arithmetic, and there are no quantities with infinite absolute value in the set of real numbers. You cannot use Peano induction to prove a claim about an infinite number, so commutativity and associativity no longer hold. You can add the convergence axiom of summation such that we can define infinite summations, but even with this axiom, we cannot prove any of the properties of finite summation applying to infinite summation. And in fact, by the Riemann theorem of rearrangement, this properties generally do not hold for infinite series even when they converge.
In summary, you are simply wrong in saying that the sum of infinitely many positive numbers must be positive, because such a claim cannot be proven from the axioms. Yes, if a and b are positive, then a + b is positive, and this provable from the axioms because there are only two summands, and two is finite, so I can use induction. The same cannot be said about an infinite number of summands. It objectively cannot.
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Jeffrey Black “Some of the general rules do still apply.”
Burden of proof. Prove that there is at least one rule of summation which applies for both a finite number of summands and an infinite number of summands.
Good luck doing it, you will never succeed, but I would like to see you try.
Otherwise, your claim is meaningless and dismissible.
“If you take any number and you add a positive number, the result will be a positive number.”
-200 + 5 = -105. I just disproved your claim.
If I add TWO positive numbers together, then the total is a positive number. If I add INFINITELY many positive numbers, the total is not generally positive. Do you understand the difference between TWO and INFINITELY MANY? EHHHH???? Tell me, do you know the difference between FINITE and INFINITE? Is that a question you know how to answer? I really hope so.
“There is no way for it to magically loop back to 0”.
Burden of proof.
“And yes, my argument wasn’t a straw man.”
No, I never said that. Marks my words. I said that the argument you claim I presented, which I never did, was not a straw man. I never said the video talked exclusively about series, but even if I had said that, it would not be a straw man argument. I do not appreciate you misrepresenting my words. You either don’t know how to read English, or are a dishonest prick. Or both.
So tell me, do you know the difference between FINITE and INFINITE? Because the axioms know the difference very well. They know the difference so well that one of the above doesn’t exist for the axioms. So do you know the difference? From now on, I am going to keep asking this question, and until you answer it, I (probably) will not respond to anything else you say, so you’d be wasting your time. So tell me, do you know the difference?
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Jeffrey Black “By logical induction, it doesn’t matter how many you add...”
There is no such a thing as logical induction. There is mathematical induction, which is an axiom of Peano arithmetic, but the number does matter for mathematical induction, because it is limited to natural finite numbers. You cannot prove claims about adding infinitely many positive numbers with induction. It is impossible, and this has been proven with set theory.
“This because instead of adding infinitely many numbers, it can be replaced with infinitely many summations.”
How dense are you? Those two things are the same thing.
“This gives you a partial sum. You then take an add it to the partial sum. This will continually produce positive numbers, if all your numbers are positive.”
Wow, what a strawman. There you go again. I never said anything about the sequence of partial sums. My comments were very specific in the distinction. I was talking about the ordered summation over an entire infinite set. This is different from having a sequence of partial sums, all of which are sums of only finitely many members. There exists a difference between infinite and finite.
“The only issue would become where infinite is real, where if it isn’t, then there is no infinite sum in the first place.”
Do you ever bother to make a statement that is mathematically coherent, or do you always invent your own bullshit terminology and butcher all of maths by writing semantically nonsensical sentences?
“Are you capable of honestly representing what I say?”
I am. You obviously aren’t. Enough said.
“Where did I claim the video talked exclusively about series?”
I never said that you said that. Nice try, but I think you might wanna brush up on your reading comprehension. Because you just misrepresented my claim yet again, as you have been doing throughout the entirety of this conversation.
“It was a joke, about you blatantly lying about what I said.”
You call quoting someone verbatim lying? Hah. How delusional.
“So good job projecting your own inadequacies.”
You’re the only one doing that here. Last I checked, I am not the one calling quotes verbatim “a lie”, nor am I the one who is incapable of using mathematical terminology correctly and unable to formulate sentences with some sort of sense in them. The only way I can help you is to tell you to pick up a dictionary.
“Good riddance to bad garbage.”
Indeed, not responding to you would constitute a good riddance to your bad garbage, especially the nonsensical garbage. But after hearing the usage of a phrase like “logical induction”, I was stricken by humor and could not help but say something about it. Some things are too hilarious to ignore. A mathematician needs their comedy once in a while too.
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@jeffreyblack666 "If my claims were objectively, provably false (and you could do so), you would have done that by now. Dismissing them is not proving them false."
I did disprove them, by showing that by your own admission, the proof requires mathematical induction, yet mathematical induction can only prove statements about natural numbers, while your claim is one of infinities, not of natural numbers.
"And really? So pointing out it is just asserted without any backing isn't enough to prove it is baseless?"
According to your own previous comments, it obviously is not, as you just used this objection yourself against me when I called you out first.
"Are you seriously that pathetic?"
Much ad hominem.
"If you really want to be that pathetic we can go quite deep down that rabbit hole, by asserting back and forth that the other person hasn't proven the claim to be baseless."
You have already been doing this for at least half of the conversation. This is why I told you that the conversation is futile and that you are wasting time. Any rational person would have listened and stopped doing this, but instead you chose to keep the conversation going on for weeks doing this. The fact that you only realized that this has been happening just now instead of much earlier further demonstrates my point.
"Also, that isn't a contradiction, so in what way do you think I am full of contradictions?"
I will allow you to figure it out yourself.
"If there was mathematical literature supporting your claim, why haven't you cited it? Like an actual citation/reference?"
Because this is a YouTube comment section in which such formalities are not necessary, but also because you are so irrational that you are unwilling to listen to me and to decide to verify a basic definition of induction you could have verified by briefly reading some dictionary, so it would be worthless for me to cite a mathematical literature you would have even less of a probability to listen to. This is especially so, since earlier on you dismissed the validity of my degree, saying that a degree should not prop up my claims, which entails that any work or words that any mathematicians like me or higher do or say will not matter to you anymore than my words matter to you.
"Just saying there is is just another baseless claim. Unless that literature actually has the proof, which would require it to be responding to my proof or otherwise prove that the sum of positive numbers is a negative number, then it would either be completely irrelevant, or just a baseless appeal to authority."
Such a proof does exist in the literature, which is easily accessible online, if you are interested. Yet I know you are not interested, since you are not even willing to check your definitions.
"I don't need literature to support my claims."
You are correct. You do not need anything to support your claims, which is why you have not bothered to give support for any of your claims, which is why they are all invalid.
"I'm not so pathetic and childish that I need someone high up to support me."
1. Ad hominem.
2. The fact that you call the usage of mathematical literature to support a mathematical argument childish itself suggests you are indeed childish. The fact that you spent so much time derailing from the topic using straw man and then lying about it to frame it on me suggests you are childish. The fact that you refuse to acce[t the burden of proof despite the fact that you initiated this conversation suggests you are childish. Calling a rigorous definition pathetic suggests you are childish. Not respecting me as a person suggests you are childish. The fact that you have refused to listen to my advice and you have decided to keep this conversation lasting for weeks, and even reviving it after you had not responded because you had no valid arguments, yet out of desperation returned and decided to recur to straw men and ad hominem to try to regain control in a conversation that is hopeless for you, it all suggests you are childish. Almost everything about this conversation supports the claim that you are childish. I would love to say that you must be no older than 15 years old, because I do feel like I am talking to a child throwing a temper tantrum. However, age is irrelevant for immaturity and lack of ability to listen and have civil conversations. Also, the only relevant aspect of the conversation about you is the argument you present, and as of now, your argument is invalid, and you have yet to prove anything about anything.
Now, if you excuse me, I am in fact very tired of your nonsense. This conversation stopped being fun a while ago. I have no intention of continuing a never ending conversation with a child-like person with zero skills for having civil discussion and who have no willingness to respect a higher-up or listen to what anyone has to say. You have drawn out this conversation to the point of pettiness, and it is a waste of time for the both of us. As such, I will not do you the favor of indulging you anymore in your desperation to win the debate you already lost. So I will gladly mute you and even maybe block you. I hope to never encounter someone like you again. Ciao.
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@AcademiadePlaton it seems to me, that when people speak like this, mathematics is in a very similar place of physics.
I don't think mathematics is quite as controversial as physics currently is, but since you do have a reason to claim this, I'll listen to what you have to say.
No one questions its usefulness or if it actually works.
This may be only your personal experience, because I know for a fact that in my personal experience, everyone is constantly asking, for literally any mathematics problem that we solve, why is it useful, and why should anyone care. It can become quite frustrating, especially when coming from a place of anti-intellectualism as opposed to genuine curiosity, which is more often thecase than not. As for physics, I do agree that this happens far less often, although the controversies surrounding physics aren't really related to the idea of usefulness, but other philosophical issues, instead.
But rather its philosophical congruence, for example the current crisis in quantum physics.
Philosophical congruence is an important subject. Humans like learning for the sake of learning, regardless of how useful this may be. The mere act of exercising our minds with fictitious constructs is good for our mental development and problem solving skills. If anything, that is a reason why you should care about the philosophical implications of anything regardless of inherent usefulness.
Few physicists talk about deterministic inconsistencies.
This is true, but most people with an education in quantum physics would agree that classical determinism is false anyway, in favor of quantum determinism.
Could be what he is referring to?
I honestly doubt it. As I said, the issues that're controversial in physics are different in essence to those in mathematics - although this isn't actually an issue in mathematics, nor is it controversial. He just happens to be on the side of a very tiny minority of people who have qualms with a subject that has been discussed to hell and back already in philosophical circles. What he is presenting here is what is known as "ultrafinitism," a worldview which almost no mathematician subscribes to because it currently has very little merit both in concept and in practice. Of course, there is active research in formal systems that reject a notion of infinity, which is legitimately interesting, but to actually take the motivation of the research as truth and claim that therefore all other mathematics are "fake" is absolutely unjustified, and frankly, maybe also a little insulting to other mathematicians. The problem is that the arguments in favor of ultrafinitism are typically based on a flawed premise and a basic misunderstanding of how infinity is used in non-finitist mathematics.
Is there a fundamental crisis in mathematics?
According to ultrafinitists, yes, but almost every other mathematician would say otherwise. This is different from physics, where physicists and philosophers at large agree that physics is currently having a crisis. Honestly, to me, this just seems to me like a tiny minority of mathematicians just have qualms with the concept of infinity without really having good reason for it, and maybe they're too highly influenced by emotional passion and intuition, to the extent that would rather put those things above rigor, which is what mathematics is based on. I don't mean this is as an insult either.
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@buycraft911miner2 It could, but we don't have a way to know that, as we don't know everything about primes, and probably never will.
I hate to burst your bubble down, but despite how much you want to insist that mathematicians are highly ignorant about prime numbers, they are not. We know more than enough about prime numbers to know that our definition is the correct one. In fact, our definition not only encapsulates the concept of prime numbers perfectly in the integers, it does so in all commutative rings. Tested and proven. We have thousands of theorems on the matter reinforcing this conclusion, together with over 200 years of studying ring theory formally to back it up. So, no, you are completely wrong about our inability to know even basic facts about prime numbers, and I wish you were not so arrogant as to pretend you can tell mathematicians what they can and cannot know.
Therefore, there is room for discussion, especially with a case like this one where, technically, 1 should be prime, but isn't because it's redundant.
No, this is factually incorrect. 1 not being a prime number is not a technicality. 1 literally does not satisfy the definition of a prime number. 1 is not a prime number, and should not be considered one. Redundancy has nothing to do with it, and in my comments above, I laid out a perfect line of reasoning behind the definition of prime numbers, and why –1 and 1 are not prime numbers. I know you find it convenient to ignore all of that (because you did ignore it), but that is just dishonest.
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@andrehashimoto8056 No, that's an incredibly stupid comment. For one, most people don't consider ads on TV particularly annoying, because for over 40 years, it was literally the norm and everyone was used to it. Also, nowadays, you can literally record shows being streamed within the TV service itself, so you can watch the recording, fast forward through the ads in the recording, and it's completely legal since it's literally part of the service. That's how my parents watch TV: they go to work, but they set their service to record all the shows being streamed live, and then watch the recordings when they get home. This also is more convenient since schedules are not an issue.
Besides, when you pirate shows, you still encounter an ungodly amount of ads, and those are actually infinitely more annoying than the ones on TV. These are worse in that they actually contain NSFW content and viruses and annoyinv pop ups. I'd rather watch ads on TV than deal with ads on KissAnime or other pirating websites. But we still pirate anime anyway. And the real reason we do it is because Western TV channels don't actually broadcast anime (other than a few Adult Swim series and some Toonami stuff, like Dragon Ball). Most anime series can't legally be accessed, though, since Western TV channels don't allow it. That's the real reason behind piracy. It has nothing to do with the ads. Streaming services were created to combat this problem, and in principle, it's a good solution. But because Sony is a garbage corporation, the execution of the principle is just bad. Hence this problem we have now. All of you are saying it's the same problem we used to have in the 90s, and it really isn't. This is just a problem of corporations being crappy, not an inherent flaw of streaming services. See, in the early days, streaming services were perfectly good, hence why they became popular. The reason they are crappy now is because companies got greedy.
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@dlevi67 I may have misunderstood your comment. I was trying to put your comment into the context of the comments that preceded yours, all of which seem to heavily imply or carry the connotation of an argument that theoretical physics is illegitimate somehow. I apologize. I tried to accomplish multiple roles with a single comment, but I should have just split my replies accordingly.
I have yet to see, 50+ years on from Gell-Mann/Veneziano's first insights, any experimental verification or even a realistic proposal for experimental verification of any independent prediction of string theory.
You're probably right in this regard, but I was merely using string theory as an example because it is one of the dominant areas of research and most of the unfair criticisms that I have heard are ultimately presented as criticisms toward theoretical physics when in reality they are only trying to argue against string theory, a very specific branch thereof. What I said applies to theoretical physics more generally, and I should have been more careful in making the distinction there. Also, given how long it can take for scientific breakthroughs to happen with such ambitious ideas, 50 years is relatively short in comparison. But I can definitely understand why the little progress for the field in this amount of time can be considered very unsatisfactory by reasonabke standards. Again, though, my point isn't about only string theory, but theoretical physics in general. String theory is just the more controversial aspect of it, and that merits a discussion of its own because of that.
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@didierborne166 Give me one good explanation why light looses speed in a medium and this within the first few molecules.
The group velocity of light decreases in a medium due to the presence of dipoles that cause an in-phase oscillation of the electromagnetic field in combination with photon scattering and inverse scattering. This is quite easy.
And then also this needs to explain the evanescent light experiments of Prof Nimtz.
There are hundreds of studies that and explanations that have already shown that the evanescent light experiments by Nimtz are in complete agreement not only with GRT and QFT, but even with CET. However, I am busy and lack the time to give you an entire crash course of CET and QFT, so instead, here is a list of articles that will do that job for me.
https://arstechnica.com/uncategorized/2007/08/faster-than-the-speed-of-light-no-i-dont-think-so/
https://arxiv.org/pdf/0709.2736.pdf
https://web.archive.org/web/20111218061131/http://sitemaker.umich.edu/herbert.winful/files/physics_reports_review_article__2006_.pdf
https://pdfs.semanticscholar.org/59ed/0d4abdf7a0a55ef0be1738d99ae71b65a41b.pdf?_ga=2.57485865.2044838613.1605987711-1591890454.1605987711
You will quickly see that GRT and QM fail on something as simple as this phenomena.
There is no failure of either on this phenomena. The only thing I quickly saw is that your understanding of physics is that of a middle schooler, but you have the arrogance of a Ph.D graduate. In a nutshell, this is the Dunning-Kruger effect.
Probably a lot more than you.
Somebody that does not understand why the group speed of light beams decreases in a medium does not know more about QFT and GRT than my sister who is still in highschool, let alone more than me.
GR needs continuum and QM needs discontinuum.
No, you have it backwards. GR needs a non-singular second-quantization, and QFT just needs a better standard model.
This challenge of yours has revealed to me that you are either a troll, a buffoon, or someone who needs serious help, and it has also revealed that your ignorance is much worse than I expected. In spite of that, your arrogance is clearly unending, so I am not going to bother wasting my time trying to talk with you. I will leave you to your self-delusions of grandeur and mute you because I have more important things to do with my life. Farewell.
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@elijahbedinger1222 For two sets X, Y, if an invertible function f : X —> Y exists, then X and Y have the same amount of elements, and are said to be equinumerous. What is the number of elements of a set S? Find a natural number n, and treat it as a set. If there is an invertible function g : n —> S, then n is the number of elements of S. If S is infinite, then just replace n by a cardinal number λ. If there is an invertible function g : λ —> S, then the cardinal number λ (which may be a natural number, or it may be infinite) is the number of elements of S. However, –1 is not a cardinal number. 0 is the empty set, {}, and –1 is the predecessor of 0. Therefore, if –1 is a cardinal number, then the union of –1 and {–1} is 0 = {}. However, by the axiom of union, if the union of U and V is {}, then U = V = {}. Therefore, what you are claiming is that –1 = {} = 0, and {–1} = {} = 0. However, this is impossible, because if {} = {–1}, then –1 is an element of {}, which is false. Therefore, there is no cardinal number that is a predecessor of 0. Therefore, there is no set with –1 elements.
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@user-fb2jb3gz1d I like the constant digs at my character...
I take digs at you, because you have been acting in a way that is very intellectually dishonest this entire time. You seem to not have any care for details or for anything that you are not willing to accept as true, so you barely even listen to what people tell you. Not unlike someone else from earlier in this thread.
What I'm saying is, that to prove their experimental verified predictions are correct, they have to compare that to the real thing. And the really thing happened a long time ago. Thus, no one can truly verify if their experimental verified predictions are correct.
No, comparing it to "the real thing" is not at all necessary to verify the data. This is a classical misconception about how the scientific method works. Besides, you ignored the part where I mentioned that the conditions have been reproduced in particle accelerators. See? This is why I take digs at you: because whenever I say something inconvenient for you, you simply ignore it. I am not even sure why I am even bothering to talk to you at this point.
Who is to say that tomorrow, they will not discover something that puts the 370,000 number further or closer away?
The experiments have been conducted so many times that, by now, the probability that an experiment could find completely different data is close to 0. And, even if a new study did find such data, this would not actually accomplish much of anything. I mean, in that case, the most likely scenario is that were was an error in the methdology. You seem to be under the impression that if a single study finds data contradicting all previous studies, that this automatically proves all the previous studies wrong. That is not how it works.
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0:55 Multiplication is just glorified addition
This is widely taught in schools, but it is not correct. For example, you cannot use repeated addition to calculate the value of sqrt(2)·π. It can be represented as repeated addition only if one of the factors in the product is a natural number, meaning any integer that is 0, or bigger. However, this just a coincidence, and importantly, it is not the definition of multiplication. If it were, then sqrt(2)·π, as it is understood, would be undefined.
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1:30 - 1:45 The problem with this argument is that, as I already explained, this is not how division is defined, so it does not provide you with a valid reason for why you cannot divide by 0. In fact, if you attempt the same algorithm to divide 20 by -1, then you also have a problem, because 20 – (-1) = 20 + 1 = 21, and 21 – (-1) = 22, etc., and the algorithm is non-terminating. Yet it is common knowledge that 20/(-1) = -20, so it is not undefined.
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7:19 - 7:28 It is not equally insane. There is no insanity in doing something most mathematicians already do. I frankly have no idea why teachers and professors in high schools and colleges say that 0^0 is undefined, when this is not the consensus mathematicians have. As for the fact that people in the comments argued 0^0 should be 0, I can bet that many of these people had no valid justifications for it and simply were operating on some aesthetic worldview or some misunderstanding of some mathematical construction. Meanwhile, there are legitimate mathematical reasons for why 0^0 = 1. For example, n^m in the case that n and m are natural numbers is an expression that counts that the number of times n appears as a factor in the product is m. 0 is a natural number, so there is no reason why this definition cannot be simply applied in this case. n^0 is thus the product in which n appears 0 times and no other factor. A product with 0 factors is called an empty product, and in the real numbers, there is only one such product, namely the number 1. Therefore, n^0 = 1. Notice how n being zero or nonzero was absolutely not relevant at all. The argument is still valid in the case that n = 0. Another reason is because if a is the cardinality of A and b is the cardinality of B, then b^a is the number of functions with domain A and codomain B. If A = {} and B = {}, then a = b = 0, and 0^0 is the number of functions from the set {} to the {}. There is only one such function: the identity function. Therefore, 0^0 = 1. Yet another reason is because the Maclaurin series of any function with nonzero radius of convergence always has the term 0^0/0! in it at x = 0, and such functions are continuous at x = 0, which is only possible if 0^0 is defined, and in every single case, the consequence is that 0^0 = 1.
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9:16 - 9:21 This is false. lim z^z (z —> 0) = 1 even in the complex plane. What you need understand is that z^z is actually just technically defined as exp[z·log(z)] wherever log(z) is defined, which is everywhere in the complex plane, except 0. So lim z^z (z —> 0) := lim exp[z·log(z)] (z —> 0), where exp(z) is defined via its familiar Taylor expansion, and log(z) is defined as ln(|z|) + arg(z)·i, where ln(x) is the familiar natural logarithm defined on the positive real numbers as the integral of 1/t on the interval (1, x). Anyhow, exp(z) is analytic by definition, so lim exp[z·log(z)] (z —> 0) = exp[lim z·log(z) (z —> 0)], and ultimately, proving lim z^z (z —> 0) = 1 reduces to proving lim z·log(z) (z —> 0) = 0. log(z) = ln(|z|) + arg(z)·i, so lim z·log(z) (z —> 0) = lim z·ln(|z|) (z —> 0) + i·lim z·arg(z) (z —> 0). arg(z) is multivalued, but it satisfies the condition that (2n – 1)π < arg(z) =< (2n + 1)π for any given branch and for all z, so |(2n – 1)πz| < |z·arg(z)| =< |(2n + 1)πz|. lim (2n – 1)πz (z —> 0) = 0, and lim (2n + 1)πz (z —> 0) = 0, so by the squeeze theorem, lim |z·arg(z)| (z —> 0) = 0. lim |z·arg(z)| (z —> 0) = |lim z·arg(z) (z —> 0)| = 0, hene lim z·arg(z) (z —> 0) = 0. Finally, we must prove that lim z·ln(|z|) (z —> 0) = 0. z·ln(|z|) = |z|·ln(|z|)·exp[arg(z)·i]. |exp[arg(z)·i]| = 1 for all z and independent of the fact that arg(z) is multivalued. Therefore, ||z|·ln(|z|)·exp[arg(z)·i]| = |z|·ln(|z|). Hence |lim z·ln(|z|) (z —> 0)| = lim |z·ln(|z|)| (z —> 0) = lim |z|·ln(|z|) (z —> 0). Let |z| = r, so that if |z| —> 0, then r —> 0+. Hence |lim z·ln(|z|) (z —> 0)| = lim r·ln(r) (r —> 0+). But this limit is well known to be 0. Hence |lim z·ln(|z|) (z —> 0)| = 0, which implies lim z·ln(|z|) (z —> 0) = 0. Q. E. D.
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9:35 - 9:44 But mathematicians don't get emotional when people say 0^0 = 1. Quite to the contrary, many mathematicians use 0^0 = 1 quite often in their papers. It is not as controversial of a claim as it is being taught to be. If you had told me that 20 years ago, this was controversial, then I would agree, but mathematicians have mostly moved past this. There are still a significant chunk who show resistance, but not enough to really break the consensus. Also, as I explained, the limit does not change depending on the angle. Even if it did, though, this would be irrelevant, because I already talked about how limits are not relevant when defining a function at a point. I can choose to define 0^0 = 1, and this is not going to change the fact that lim x^y (x —> 0, y —> 0) still does not exist.
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Ultimately, the true reason division by 0 cannot be done in standard arithemetic is because of our fundamentally axiomatic understand of division. The division of two real numbers a and b is just a notational shortcut for a·b^(-1), the product of a and the multiplicative inverse of b. Therefore, division by 0 is equivalent, and honestly, defined, as multiplication by the multiplicative inverse of 0. Thus the question is reduced to simply asking "why does 0 not have a multiplicative inverse?" The answer is because there is no real number x such that 0·x = 1, because in reality, 0·x = 0 for all real numbers x. Okay, but why can we not axiomatically create some nonreal, noncomplex number q such that 0·q = 1 in the same way that we axiomatically created some nonreal number i sucg that i^2 = -1? Well, let us actually try it.
Let q exist. So, theorem 1: 0·q = 1. Trivial. By definition. Theorem 2: 0·(0·q) = 0·1. Okay. But 0·1 = 0, and multiplication is associative, so 0·(0·q) = (0·0)·q = 0·q. So theorem 2 can be equivalently expressed as 0·q = 0. Uh oh! This is a problem. Because, by theorem 1, 0·q = 1, but by theorem 2, 0·q = 0. And equality is a transitive relation, so the conjunction of both theorems implies 0 = 1. Yikes! Okay, but this can be fixed. How do we fix it? By declaring that 0·(0·q) is not equal to (0·0)·q. Okay, cool, coolnowaitamoment what? No. We just lost associativity. Multiplication is no longer an associative operation.... breathes This is just becoming pointless. But fine. I'm a mathematician. I'm willing to give up multiplicative associativity for the sake of mathematics. So let's continue. So I cannot conclude that 0·(0·q) is 0·q. .... ...There are still problems. Why? Because 0 = 0 + 0. Surely, 0·q = (0 + 0)·q. Multiplication distributes over addition. This is just axiomatically true. So I can say that 0·q = (0 + 0)·q = 0·q + 0·q = 1 + 1 = 2. Oh no. We really just concluded 1 = 2 even after patching our arithmetic by removing associative multiplication, huh? The problem this time was that multiplication distributes over addition. So... we remove the distribution of multiplication over addition...
As you can see, you can continue playing this game and removing axioms and patching this new arithmetic with a 0 with multiplicative inverse and no multiplicative associativity and no multiplicative distributivity over addition. As you continue going, the system becomes more and more bizarre, less intelligible, and overall, a mess. I suppose you can imagine that someone with sufficient patience may find a working system within all this chaos, but the mathematics are unrecognizable and do not resemble anything useful or comprehensible. So mathematicians prefer to say division by 0 is undefined in this context.
Of course, there are other ways of enabling division by 0 if you get rid of the concept of multiplicative inverse and invent a new unary operator / instead. If you introduce the elements /0 and 0·/0 to your set, and modify your rules of distributivity slightly, you can get a working system without sacrificing anything else. This is what is done in wheel theory. Another route you can take, which is far more obscure, is by working with an arithmetical meadow, where 0 = 0^(-1), among other axioms. But these routes do give up the concept of multiplicative inverse and basically redefine division so that it is different altogether.
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but questioning fundamental assumptions is an important part of science
No, not quite. There is very important difference between mindless skepticism and science. Science is not about zombie-like questioning of claims. Science is about testing the claims, which is different than mindlessly questioning. You are right that we are not supposed to take any assumptions for granted, but the whole point is that if a claim has been tested over and it has never been falsified, then it likely is a true claim. Does rain exist? Well, we can never for sure. There's always the tiniest of chances that all of humanity has been hallucinating and rain doesn't actually exist. But we've tested our claim, and it is super likely rain exists. So we can claim beyond the shadow of a doubt that rain exists. I don't see you or anyone in the world questioning the existence of rain. Why? Because the experimental evidence suggesting rain exists is so statistically overwhelming, that you'd have to be actually insane to still think rain doesn't exist after seeing said evidence.
This doesn't mean you can't acknowledge the fact that nothing is set in stone and that any claim can always be overturned. But contrary to what mindless skeptics think, not all claims are created equal, and not all evidence is created equal either. So while you can and should question a claim when presented, you shouldn't do it in a fashion in which you just ignore the entirety of the work of the scientific community that has gone through for millennia. You're absolutely supposed to take that into account. Why do you think "review the literature" is one of the steps of the scientific method that is taught to children in middle school?
Einstein questioned the assumption that gravity was a force, and in doing so discovered the Euclidean geometry of space-time
No, he did not discover spacetime is Euclidean. He discovered spacetime is Minkowski (in the absence of gravity), which if anything, is the opposite of Euclidean. The metric signature of spacetime is (+, -, -, -), which is the signature Einstein and Minkowski discovered, but Euclidean spacetime woukd have a metric signature (+, +, +, +).
Also, you are right in that he did question these assumptions, but he didn't do so "for the sake of questioning," he didn't do it mindlessly. He did it rather systematically instead. Einstein's questioning was motivated by (1) his special relativity theory (2) observational anomalies with Mercury's orbital precession. He began his questioning of the assumptions by taking into account those two things and using that as a stepping stone for his deductions. Notice how he didn't randomly start questioning every basic principle he didn't understand. Instead, he questioned the principles he did understand and were well understood by everyone, and he did it by starting from the anomalies, not from any random starting point like mindless skeptics do. And he took into account the already existing data very carefully, he didn't choose to just ignore it and be like "okay but what if." What Einstein did is science. It isn't mindless questioning. Einstein did it because (1) he understood how it works (2) he understood what parts about it don't work. That's very different than, for example, a flat Earther coming to me and denying every scientific discovery ever on the basis of (1) not understanding basic principles (2) not understanding what about our current models works and doesn't work. What a flat Earther does isn't science, it's just mindless denial of reality. Do you see the difference? Science isn't a philosophically simple thing you can describe as simply being skepticism. That's why there exists a whole method to it. That's why there is an entire branch of philosophy, called philosophy of science, that is solely dedicated to try to understand what science is.
But now we're a bit stuck because, although the maths works on the whole, it treats gravity as essentially falling through space-time, rather than a force, which is why we need quantum gravity as far as I can tell.
That actually has nothing to do with why we need quantum gravity. Treating gravity as the effect of "falling" through spacetime is perfectly legitimate and correlates with data better than ever. The reason we need quantum gravity is because the Einstein field equations cannot undergo second quantization like every other physics equation can. If you try to quantize the spacetime metric into an operator with the conventional methods, it doesn't work: it breaks mathematically, leading to things like division by 0, contradictions, and other things. This is why we need a theory of quantum gravity. The whole point is to be able to quantize spacetime.
how to create the effect of gravity in space? Centripetal force via rotation. Positive kinetic energy. Einstein taught us, if it quacks like a duck...
I'm not sure I follow.
It would be more accurate yo say gravity has an "effect" of negative kinetic energy rather than actually "being" it
No, this is still totally inaccurate. Gravity has the effect of negative potential energy, not kinetic energy. There is no kinetic energy component to gravity at all, not in classical physics at all, and it doesn't work that way either in general relativity.
I've just extended Einstein's own logic
Nothing that you have said here matches Einstein's logic. Again: don't make claims about things you don't understand. Understand them first, then you can question them. I told you that you should study general relativity and at least become familiar with all of its implications. If you don't, then you're in no position to be questioning general relativity, testing it, or extending it. You do know scientists have to go to college for a reason, right?
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@Lincoln_Bio especially when you consider that you measure both things with the same device.
Uh, no, you do not. They are different devices. I also already explained this.
If you can't explain it, it's just blind acceptance.
Well, I already did explain it. Whether you want to read that explanation and acknowledge its existence or continue ignoring it is unfortunately not up to me.
My skepticism is mindful.
Saying it alone doesn't make it true.
I didn't say "it's wrong" I said the maths seems off 'to me' and I'd like to do some investigation.
You said you think I'm wrong at the very beginning of your previous comment. You can try to lie all you want, but I have screenshots to prove it.
I just want to advance my own understanding.
Good for you, but you're not going to accomplish that by ignoring the words of people who speak to you, especially when those people have significantly more education on the subject than you do.
I've repeatedly stated I'm still learning this stuff, so repeatedly telling me I don't know stuff is rather redundant, however.
I'm not exactly happy to remind you of these things. But when you tell me you learned something wrong, and I explain to you the concept that you admitted were taught wrong, and you respond by either ignoring entirely or by telling me I'm wrong, then it is only reasonable for me to assume that you have already forgotten the fact that you admitted to not know about the particular thing. Hence why I remind you.
Ohhhh of course, this is the internet isn't it, you're supposed to prove everyone wrong as condescendingly as possible [rolls eyes]
This goes both ways. If your genuine impression of me is that I'm here solely to prove you wrong and not help you like I explained multiple times earlier, then you've ironically proven to me that talking to you is a waste of my time. If you think I'm a troll, then the feeling is mutual. I have no reason to think otherwise, after all.
I sincerely hope you're not suggesting one requires a phd in physics to type a comment on a youtube video, or have an idea or an opinion about it.
I never said anything that even remotely implies this. But, again, if you think this is exactly what I believe, then I'm not going to bother proving you wrong, because I'd be wasting my time.
I'm just screaming into the void like everyone else here, my dude.
And you think I'm not?
In my original reply I was just throwing some fun silly science ideas around, but you had to get technical
Oh, cry me a river. You're complaining because someone presented criticism of those ideas on a comments section that is public and essentially free domain? So that is how you justify acting like a troll? Wow, I didn't know you had the emotional maturity of a kindergardener. Good to know.
you've only got yourself to blame ya smart-arse
Wow, you can't even say the word "ass" correctly, confirming my suspicion about having the maturity of a child. And then you pin the blame on me? This is some rapist-logic right here. This is almost turning out to be comedic. Suddenly, my view of you has completely changed. It is so funny how once you get called out, your pseudo-intellectuallist front falls apart and your immaturity gets exposed so easily.
Well, given that you have the maturity of a child, I'm fairly certain you're going to want to have the last word, though quite frankly, I might not even bother reading what you have to say, because that is just how much of a waste of time it would be. And I certainly will not bother replying to it either. Have a good life and a fun journey. Get some academic help and some psychological help too. You will certainly need it. さようなら。
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@lonestarr1490 The claim that the universe has no concept of prime numbers, in the way that you seemingly meant it, is false. The universe does have such a concept, and it is concept not unique to the integers, but which exists for all commutative rings. The general concept is that of an irreducible element, and it is a concept that holds even in rings where the unique factorization theorem does not hold, because the concept is more fundamental than the theorem. In all rings, the objects can be classified into exactly 4 categories: (a) zero divisors, (b) units, (c) irreducible elements, (d) composite elements. A zero divisor is an element x such there exists some nonzero y such that x•y = 0. 0 is trivially a zero divisor, but in some commutative rings, there are other quantities with this property too. Those are called nontrivial zero divisors. One important characteristic of zero divisors is that arbitrary multiples of zero divisors are also zero divisors. Not all commutative rings have nontrivial zero divisors, and those that do not are called integral domains. The integers are an example of this. Rings of polynomials are also typically examples of this. A unit is just an invertible element: an element x such that there exists some y such that x•y = 1. 1 itself is a unit in all rings, and –1 is a unit in all commutative rings (all rings, in fact). Some rings have other units too, such as the Gaussian integers, where the units are –1, 1, i, –i, and in the rational numbers, all nonzero rational numbers are units. As such, they all have the same algebraic properties. One key feature of units in rings is that they are just incapable of generating the rest of the ring. To put it more precisely: they are multiplicatively closed. They form a group. The product of two units is always another unit. Because of this, a unit always divides every element in the ring. In the integers, for example, –1 and 1 divide all integers. Also, the only divisor of a unit are the units themselves. Units are fundamentally different from irreducible elements. The irreducible elements of a ring are not multiplicatively closed: if p and q are irreducible, then p•q is composite. In fact, the irreducible elements generate the rest of the ring, which the units do not do. The irreducible elements do not divide all integers. Also, irreducible elements have divisors besides the units. In the integers, for a prime number p, these additional divisors are –p and p. In the Gaussian integers, they are –p, i•p, –i•p, and p. What is an irreducible element? There are various definitions which are equivalent. They are exactly the elements which are not zero divisors and whose only proper divisors are the units of the ring. The units do not satisfy this definition, because the units have no proper divisors, since they are divisible only by themselves. For context, a proper divisor is divisor where the divisibility relation only works one way. For example, –3 divides 6, but in the integers, 6 does not divide –3. So, –3 is a proper divisor of 6. –6 divides 6, but 6 also divides –6, so –6 is a divisor of 6, but not a proper divisor of 6. An alternatively definition of an irreducible element p is that if p = q•r and p is not a zero divisor or a unit, then q = u•p or r = u•p, where u is a unit. The entire point of the definition is that irreducible elements can only be written as themselves, or as themselves multiplied by units, but there are no non-trivial products that can generate them, unlike the composites, which are generated as non-trivial products of irreducible elements.
Earlier, I mentioned integral domains. If an integral domain satisfies the unique factorization theorem (as do the ring of integers, the ring of Gaussian integers, fields, rings of polynomials, etc.), then it is called a unique factorization domain. Not all integral domains fall under this category, but they all still have a fundamental distinction between 0, units, irreducible elements, and composite elements. Also, in the rational numbers, the numbers such as 2, –3, 5, 7, etc., all lose their properties as prime numbers. They are no longer capable of actually generating any composite elements. The problem is that because all nonzero rational numbers are units, they all have the same algebraic properties as 1 and –1. They divide all rational numbers, and are divisible by all rational numbers. Since they are all units, there are no irreducible elements at all, and so also no composite elements. And the only zero divisor is 0. Integral domains like this are called fields. This comparison matters, because it shows how –1 and 1 were never irreducible elements. The move from integers to rationals does not change their properties, but it does change the prime numbers into units. This distinction is completely fundamental in concept.
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I have a few pet peeves with how some of the information was delivered in the video.
That numbers such as sqrt(2) or π are irrational is not because we have checked their decimal expansions and saw no repetition. They are irrational because we can definitely prove they are irrational: we do not need to check their decimal expansions, because we can definitively prove their decimal expansions are not periodic and do not terminate without having to check them: all we need to now is only the definition of the numbers to prove it.
Also, the way "integers," "whole numbers," and "natural numbers" were presented was sort of confusing. For one, the set of integers is alos called "the set of whole numbers" in most places outside North America, even in places that use the English language, while no analogous distinction between "natural numbers without 0" and "whole numbers (natural numbers with 0)" exists in countries that do not use English as a language. On the other hand, most mathematicians today, and even authoratitative sources on mathematical notation, such as the ISO 80000-2, use the symbol N to refer to the natural numbers with 0, which in mathematics are just called "natural numbers." The set of natural number without 0 is not really considered any more fundamental than, say, the sequence of Fermat numbers, which is why mathematicians do this. So presenting a distinction between the two sets, especially with this very particular naming convention that is not even used by mathematicians, is definitely going to confuse people, and it could even be considered misleading, although I understand this obviously was not intentional.
I also think it would have been informative to include an additional layer within the set of real numbers, talking about the algebraic numbers and nonalgebraic numbers, since their definition, while a little more complicated, is still intuitive enough to be understood from a simple presentation, and there is merit in doing so due to the historical importance of algebraic numbers and the types of implications it has in, for example, architecture. Even though irrational algebraic numbers cannot be written using finite decimal expansions, they still have decidedly nicer properties than transcendental numbers, making them much more convenient to work with. This also gives context as to why e and π are such important constants, but sqrt(3) is not, for example.
Regardless, I understand why the decisions made in the video were made, and I think this was still informative and relatively good.
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@petevenuti7355 Is there such a thing as a ternary operation that can't be broken down into binary operations.
There is, surprisingly. These are called irreducible n-ary operations. They exist for all n > 2.
What is an operation then? It must involve action, yes?
An operarion is a function, but this raises the question of what is a function, does it not? So, what is a function? We intuitively tend to think of a function f as being fed by an input x, and spitting the output f(x). This makes it sound like a function has to refer to an algorithm, a physical procedure. However, a function is actually just an abstract relationship. f relates x and f(x) in an abstract way. The reason teachers present it as an algorithm is because it makes the axiom that defines what a function is easy to visualize, but at the cost of being misleading.
Consider two sets X and Y. In mathematics, we typically consider all objects to be sets, but for the sake of explanation, we can allow the members of X and Y to be arbitrary objects, they do not necessarily have to be sets themselves. Given X and Y, you can form a third set, the Cartesian product of X and Y. The Cartesian product of X and Y is the set of ordered pairs (x, y), where x is in X, and y is in Y. Now, there is a special class of subsets of this Cartesian product. These subsets G satisfy the following property: for all x in X, there is exactly one y in Y (always one, and only one), such that (x, y) is in G. This property is the property that teachers are ultimately alluding to when they talk about inputs and outputs of a function. The unique y such that (x, y) is in G is called the image of x under G, but in school mathematics, the teachers just call it "the output." As you can see, there is an abstract relationship between x and y that defines what the set G is, but there is no physical procedure involved. You can say y exists, but actually finding what y is, that is not required in order for G to be a valid "special subset" of the Cartesian product.
I should mention that this is not the complete definition of a function, but the technical details that I have omitted are not important for the point I am making. The point I am making is that for every function f, there is an associated set G that satisfy the above property, that for all x in X, there is exactly one y in Y such that the ordered pair (x, y) is in G. And that is all there is to it. These sets exist as abstract objects, not as physical procedures. Now, in a given model of computability, there are some functions, for which you can prescribe an algorithm that explicitly constructs or produces what the corresponding y is for a given x. In doing this, not only have you shown y exists, but also, you know what y is, you can give a finite description of how you obtained y. However, y could exist without being able to construct an algorithm to determine what it is. If this is the case, then such a function is called uncomputable (within that particular model). The most famous example of this is the busy beaver function, which I will not define here because I am not confident I understand the definition well enough to explain it.
The only way you can limit yourself to computable functions, realistically, is by saying that the axiom of infinity is false (meaning there are no infinite sets, as far as the axioms are concerned). However, this means that you are saying that there is no such a thing as the set of natural numbers. And by doing this, you give up Peano arithmetic and a bunch of other things. Building a mathematical system that is actually useful from this is very tedious, and not really worth the trouble of denying the axiom of infinity, especially because this axiom does so much for mathematicians and physicists. You cannot do any science without accepting the existence of infinite sets.
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@petevenuti7355 I was just pointing out how significant it is that there can be such widely different meanings of the same word between people even speaking the same language.
Well, I think the point is that we are not actually speaking the same language. Computer science and mathematics are very, very different, and the conventions are very, very different as well.
I meant linear to denot the one to one or many to one relationship defining a function, you meant linear essemtially as a line in a coordinate system, and Angel I believe originally meant it in the strictest sense of linear algebra.
rms is using the word linear in the same sense I am using it, only that they are essentially presenting the definition in a bit of a simplified fashion. I imahine that the thing you are calling a "function" is actually just a program, rather than a function in the mathematical sense, though I could be wrong. Strictly speaking, a program, in terms of mathematics, is just a computable relation. A function is a relation that is left-total (in terms of programs, it means an output exists for all inputs, but without the requirement of computability).
You should take a look at a video called "What Does a Diagonal Argument Look Like?" or something like that, you will recognize the thumbnail as it talks a bit about The One Trick to Them All. In the video, the distinction between a program and a function is expanded upon a bit more.
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@nics4967 I accept that there is evidence that is interpreted by some to mean as you say. The problem is, did they interpret it properly?
If you are going to initiate this discussion by already dismissing their findings on the basis that you disagree with the interpretation, without having even looked at the evidence, then I have no reason to engage on the topic seriously. See, this is exactly the thing I was warning you about in my previous comment.
It may show levels of devoutness, not lack of doctrine.
How are you making this assessment when I have not even cited my sources yet? This is the lack of intellectual honesty I was talking about earlier. You are not actually interested in engaging with the evidence. You have an apologetics script prepared beforehand, ready to pull out some bullet point in response to anything anyone says, rather than engaging with integrity and a sound epistemology.
Biden goes to Mass. If he didn't, does that show there is not an expectation to go on Sunday.
Why are you comparing 1 human being to an entire civilization of thousands of human beings? Biden is only one person out of 300 million people in the United States of America. What Biden does or does not do has no relevance, as far as archaeology is concerned, when describing the practices of modern Usayite civilization as a whole.
Your argument is pretty obviously fallacious, but this is the issue: you do seem to not really care about it being fallacious. You seem to have put no effort into it at all. This sounds like something you regurgitated, something you pulled out of a script. Again, I have not even cited my sources, and you are already trying to debunk them. There is saying that goes "Don't judge a book by its cover." Dude, you have not even looked at the cover of the book at all, and you are already judging it!
What seems needed is evidence of doctrinal development that can not be explained by laxity.
This is an entirely baseless assertion.
Some experts think there is evidence in biology for I.D.
No. There is one "expert" in the world who alleges that I.D. is true, but he has never presented a single piece of evidence to support his claim, he has not conducted a single experiment testing his hypothesis. This is completely different from the situation in archaeology I am referencing here.
I'm interested.
Your comments demonstrate otherwise. Are you lying, or are you just extremely lacking in self-awareness?
You dismissed the evidence without even waiting for me to cite my sources. No who is genuinely interested would do such a thing.
Why can't I start with such when there isn't anyway to know if the person is interested in honest dialogue?
I do not understand what this particular sentence is saying, but I suspect that you are criticizing me for taking the approach I took for initiating the conversation. I could be wrong, because your sentence lacks coherence. If I am right, though, then let me say this: if you want other people to be serious with you, then you need to demonstrate that you are serious with them. You did not do this. I had to warn you, because I needed to avoid miscommunication, and make sure you understand that I will not waste my time. You had the opportunity of making good use of the warning and replying to me with intellectual honesty. Dismissing the evidence prior to looking at it, obviously, does not satisfy the criterion of intellectual honesty. Why are you complaining in spite of this? I have no idea.
I would hold on philosophy on theology, I am morally obligated, to be honest.
...or so you say, but you have not demonstrated any honesty whatsoever in this conversation. Are religious people unaware of what intellectual honesty actually looks like? Because you continue acting as if you really believe you are honest, but your comments demonstrate such a seamless intellectual dishonesty, I find it impossible to believe you.
I will stand by what I said earlier. Whether you truly are sincere or not, your standards for what count as "honesty" are so clearly different from what reasonable individuals hold, there is no point in me trying to have this discussion with you.
I wish you a good life.
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Why would I need to rephrase a definition? Do you not understand the definition? If you do not understand the definition, then you are not qualified to be having this conversation with anyone at all. If you do understand the definition, then there is no need for me to rephrase. If I were your teacher, then I absolutely would rephrase. However, you came into this conversation correcting everyone else, as equals, pretending you already knew enough about the topic to actually present arguments and issue corrections from it.
So, you have two options here: (A) you can continue pretending to understand the topic, and we can have a conversation as equals, but in the process, I will continue exposing your ignorance and your lies until you have walked yourself into a deadly corner, because regardless of how much you are pretending, you are still not knowledgeable enough to present well-substantiated argument. Option (B) is, you can admit that you were pretending to know about the topic when you actually know little to nothing, and stop issuing corrections you are not qualified to make. In that case, we can still have a conversation as equals, and one in which we are both being intellectually honest. In that case, I would gladly rephrase the definition, and then we can openly have a discussion as to how both the Christian narrative and the Greek narrative are examples of mythology.
It is your decision to make, and you are the one at odds here to miss out.
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Wikipedia is not a scholarly source, though. Neither are documentaries. Actually, to be honest, Wikipedia is hundreds of times more reliable than a documentary. Also, it should be mentioned that George Lemaître was explicit in telling everyone, including the Pope, to not misinterpret his theory as being about the book of Genesis at all. He came up with the theory by analyzing Einstein's field equations, and realizing that the equations necessarily predict the expansion of spacetime. Meanwhile, most theologians were against the idea, because they believed a steady-state universe better reflected God's perfections. As for cosmologists, they only accepted the steady-state model, because that was exactly what the evidence suggested, prior to Lemaître's work. But it was not secular cosmologists who opposed the changes. It was people like Fred Hoyle who did.
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Well, you are right, but this problem is fixed as soon as you replace "infinity" with "Aleph(0)", and then there is a legitimate question to pose: if the Earth orbits 30 times as fast as Saturn, and both planets have been orbiting for Aleph(0) years, then they must have both completed Aleph(0) orbits. How can this be possible?
Thankfully, mathematics have an answer to this. The reason this seems like an absurdity is due to a misconception that we have, regarding the way sizes of sets work. Intuitively, we think that if X is a proper subset of Y, then Y must have a larger size than X, because it has every element X has, and then some other elements, so intuitively, the number must be larger. However, if there happens to be a bijection from X to Y, then X and Y are the same size, because they have the same cardinality, and every element can be matched correspondingly, even if X is a proper subset of Y. This is counterintuitive. This phenomenon is called Cantor's property, and it is a property that only sets that are infinite satisfy: they can be the same size as proper subsets of themselves. This property is the property, so unintuitive, that mathematicians prior to Cantor simply could not accept, and which is why the idea of infinite objects was not accepted mathematically until Cantor developed his set theory. By rejecting the property, you are required to accept certain implications, and therefore, certain contradictions and absurdities. Rather than doing that, mathematicians dispensed with the idea of infinite sets altogether. Cantor gave them serious treatment, and his discovery is what led to realizing that our intuition was wrong, and that Cantor's property needs to be taken seriously. At the core of this, lies a profound and unexpected revelation: that for infinite sets, there actually do exist two distinct, incompatible notions of size. These are order type and cardinality. As it happens, these two notions are equivalent when dealing with finite sets, but distinct when dealing with infinite sets. This explains why Cantor's property is so counterintuitive to us: order type concerns membership of elements in a set, and it concerns properties of subsets of a set. So proper subset of Y has a different order type than Y does, even though the two sets may have the same cardinality, because there may exist a bijection between the two sets. What this also reveals to us is that addition with infinite quantities works differently than it does with natural numbers. Actually, to be more concrete, there are two different kinds of addition for infinite numbers: one with regards to cardinality, and one with regards to order type, and these are called cardinal addition and ordinal addition. This explains the inherent weirdness behind the concept of "infinity + 1 = infinity": because when considering cardinal addition, ω ++ 1 = ω, but when considering ordinal addition, ω < ω + 1. This is because, introducing one new element to your infinite set does not change its cardinality, but it does change its order type. For natural numbers and finite sets, both notions are indistinguishable: a set A with 8 elements and a set B with 5 elements has a larger cardinality, because there is an injection f : B —> A, but there is no injection g : A —> B. On the other hand, it is also true that 8 comes after 5 in the sense of an order relation: I need to keep counting further to get to 8 than I do to get to 5. So 5 < 8. These notions are equivalent for natural numbers, but not for infinite sets. 5 + 3 = 8, regardless of whether I consider cardinal addition or ordinal addition, but not so for ω + 1.
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@Melesniannon Substituting one linguistic absurdity for another absurdity still affects nothing in reality.
Nothing that I have said here is a matter of linguistics. It is a matter of mathematics. None of the things I have mentioned here are absurdities, either. Just because a theologian many centuries ago called it an absurdity, due to the counterintuitive nature of the phenomenon. For it to be an absurdity, it has to be a logical contradiction, which it is not, in this case. The claim Aleph(0) = 2·Aleph(0) is not a logical contradiction.
If the cardinality of a set is defined as infinity, and infinity is undefined, then the cardinality is undefined and thus can't be said to equal anything, least of all another undefined infinity.
There are various misconceptions in this argument that need to be addressed.
0. Infinity is not a cardinality, so the cardinality of a set cannot be defined as infinity, this is nonsensical. It is however, sensical to say that a set is infinite and that the set has a given cardinality. They idea to understand here is that two sets can be infinite, but have different cardinalities. As such, different infinite cardinalities exist, and so it is nonsensical to define infinity as being a cardinality itself.
1. Infinity is not undefined. Infinity is a property of sets. It is, however, accurate to say that infinity is not an object, and so you cannot say that "infinity = infinity". However, I at no point have made the claim that "infinity = infinity". My claim is that Aleph(0) = 2·Aleph(0). Yes, Aleph(0) is an infinite cardinality, but it certainly is not the only infinite cardinality, and so it itself is not "infinity", which refers to a property of sets. I know you have said earlier that you are not a mathematician, but my claim that Aleph(0) = 2·Aleph(0) is not particularly technical and can be understood by mathematicians. I do not want to be accused of saying "infinity = infinity", because I have never made such a claim.
It's the turtle and hare "paradox" where the hare can never catch up to the turtle, as long as you regress the time interval infinitely. Yet in reality, that turtle eats the hare's dust.
The issue with these arguments from paradox is that these paradoxes never constitute an actual contradiction, they only always constitute apparent contradictions, originated from a fault in deduction that is not detected by our intuition. This is because our intuitions are not logical in nature. The idea that, as long as you regress the hare's time interval infinitely, the hare can never pass the turtle, is flawed, but the flaw does not lie in the assumption of infinite regress, the flaw lies in the assumption that infinite intervals cannot be traversed.
While you can validly state that infinity = infinity, when two potential infinities are contingent upon the same constant which affects them differently, stating they are identical is nonsense.
I have no idea what you just said here, but I will say that 0. potential infinities are not a thing in mathematics, and so, not a thing in reality, they are just an outdated concept invented by ancient philosophers from a time before we understood how infinity works, 1. I never claimed infinity = infinity.
You yourself pointed this out when talking about order addition, which is fundamentally the same as what I do: thinking about infinity in multiple dimensions,...
No, ordinal addition is not fundamentally what you are talking about, and it also has nothing to do with thinking about infinity "in different dimensions", whatever that means.
...which in my simple example I was referring to a WIDTH and not a LENGTH.
Oh, for fuck's sake, seriously? Width and length are literally the same thing, mathemagically. In the English language, we use different words for them to account for the direction of the line segment we are interested in, but as far as measurement and size goes, they are literally identical concepts: the distance from an end A of an object to the other end B of the object in a certain direction.
At any point on its length, the surface area of the 2 cm wide line is twice that of the 1 cm wide line. It doesn't matter that both are potentially infinite in length, this will be always true.
Again, potential infinity is a meaningless concept. As for your actual claim, yes, it is true, only if the point at its length being considered is finite.
However, if you do that the Kalam cosmological argument breaks instantly, because you no longer have causality,...
Dude, I never even said the Kalam cosmological argument works. I am an atheist. I thought this was very clear from my very first reply. sigh Well, I suppose trying to have a conversation with you was a waste of my time. I have no idea why I ever hoped to be understood by you. Good bye.
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@radicalfamily No, what he wrote is correct. If you let y = f(x), you just recover the equation [f^(-1)]'[f(x)] = 1/f'(x), which is correct, or [f^(-1)]'(y) = 1/f'([f^(-1)](y)).
Also, you are wrong. There is a general formula for the antiderivative of f^(-1) with respect to f, as long as f is differentiable. For example, consider [f^(-1)][f(x)]·f'(x) on the interval (a, b). On the one hand, this is just x·f'(x). On the other hand, let f(x) |—> x. Hence f'(x)·dx |—> dx, and the interval of integration transfoms into (f(a), f(b)). So the integral of [f^(-1)](x) on (f(a), f(b)) is equal to the integral of x·f'(x) on (a, b). With integration by parts, this is b·f(b) – a·f(a), and subtract the integral of f(x) on (a, b). Change the dummy variable to t, and let b = x the variable of interest. This way, the integral of [f^(-1)](x) on (f(a), f(x)) is just equal to x·f(x) – a·f(a) – F(x) + F(a), where F is the antiderivative of f. If we denote G as the antiderivative of f^(-1), then we have G[f(x)] – G[f(a)] = x·f(x) – a·f(a) – F(x) – F(a). Keeping in mind that G[f(a)], a·f(a), and F(a) are all constants, they can be combined into a constant of integration C so that G[f(x)] = x·f(x) – F(x) + C. Finally, compose with f^(-1), so that G(x) = x·[f^(-1)](x) – F{[f^(-1)](x)} + C. Done. G expressed solely and cleanly in terms of f, f^(-1), and F. I told you the general formula existed.
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@charlesmadison1384 From Wikipedia, "...the theory describes an increasingly concentrated cosmos preceded by a singularity in which space and time lose meaning (typically named "the Big Bang singularity")."
I suggest that, next time you try to quote an article, you actually bother to provide the quote in-context, rather than taking it out of context and giving it your own unwarranted spin. The article on the Big Bang starts with "The Big Bang theory is the prevailing cosmological model explaining the existence of the observable universe from the earliest known periods through its subsequent large-scale evolution.[1][2][3] The model describes how the universe expanded from an initial state of high density and temperature,[4] and offers a comprehensive explanation for a broad range of observed phenomena, including the abundance of light elements, the cosmic microwave background (CMB) radiation, and large-scale structure." This is how the Big Bang theory, at its most basic level, is defined by the scientific community. What follows afterward is not a definition, but merely a pointing of relevant facts: for example, "Crucially, the theory is compatible with Hubble–Lemaître law—the observation that the farther away a galaxy is, the faster it is moving away from Earth. Extrapolating this cosmic expansion backwards in time using the known laws of physics, the theory describes an increasingly concentrated cosmos preceded by a singularity in which space and time lose meaning (typically named "the Big Bang singularity").[5] Detailed measurements of the expansion rate of the universe place the Big Bang singularity at around 13.8 billion years ago, which is thus considered the age of the universe.[6]" This is why context is important. Your out-of-context quote paints this non-scientific source as presenting said quote as the defining feature of the Big Bang. An in-context analysis reveals instead that this idea of the singularity is merely one model of the theory historically arrived at by a rather simple extrapolation, which is compatible with other well-evidenced phenomena.
In the "Features of the Model" section of the article, there is more detail as to why your initial comment on the Big Bang theory is inaccurate. Specifically, in the "Expansion of Space" subsection, we have, "The expansion of the Universe was inferred from early twentieth century astronomical observations and is an essential ingredient of the Big Bang theory. Mathematically, general relativity describes spacetime by a metric, which determines the distances that separate nearby points. The points, which can be galaxies, stars, or other objects, are specified using a coordinate chart or "grid" that is laid down over all spacetime. The cosmological principle implies that the metric should be homogeneous and isotropic on large scales, which uniquely singles out the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. This metric contains a scale factor, which describes how the size of the universe changes with time. This enables a convenient choice of a coordinate system to be made, called comoving coordinates. In this coordinate system, the grid expands along with the universe, and objects that are moving only because of the expansion of the universe, remain at fixed points on the grid. While their coordinate distance (comoving distance) remains constant, the physical distance between two such co-moving points expands proportionally with the scale factor of the universe.[16]". So in summary, part of what the Big Bang Theory comprises is the fact that the spacetime universe we are familiar with has the FLWR metric, which results in a homogeneous, isotropic expansion of spacetime. This explains the precise mechanism by which the universe went from its earliest states to the current state. Nowhere in this expansion is there any causation to account for. Furthermore, "The Big Bang is not an explosion of matter moving outward to fill an empty universe. Instead, space itself expands with time everywhere and increases the physical distances between comoving points. In other words, the Big Bang is not an explosion in space, but rather an expansion of space.[4] Because the FLRW metric assumes a uniform distribution of mass and energy, it applies to our universe only on large scales—local concentrations of matter such as our galaxy do not necessarily expand with the same speed as the whole Universe.[17]". In other words, there was not something that went "bang", because there was no "bang" to begin with, despite the misleading name of the theory, which is actually the result of Fred Hoyle mocking the theory and misunderstanding it.
Do you take note of the keyword "precede"?
"Precede" is not a keyword here, and there is nothing here to take note of, as you are misunderstanding what the out-of-context quote is saying in-context. As I said, there was no object that went "bang". The so-called singularity refers to the fact that for time 0 of the stages of the universe, the mathematics of general relativity result in nonsensical results. As Wikipedia itself put it in the first paragraph of the subsection "Singularity" in the "Timeline" section, "Extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past.[20] This irregular behavior, known as the gravitational singularity, indicates that general relativity is not an adequate description of the laws of physics in this regime. Models based on general relativity alone can not extrapolate toward the singularity—before the end of the so-called Planck epoch.[5]" In other words, there is no event called "singularity" that "preceded" the Big Bang. The singularity simply refers to a point of spacetime in the universe where the laws of physics are not well-understood and require a new theory to be well-understood. This singularity just so happens to take place during the Planck epoch. Look, the article even expands on this later on, in the "Inflation and Baryogenesis" subsection, saying "The period from 0 to 10−43 seconds into the expansion, the Planck epoch, was a phase in which the four fundamental forces — the electromagnetic force, the strong nuclear force, the weak nuclear force, and the gravitational force, were unified as one.[25] In this stage, the characteristic scale length of the universe was the Planck length, 1.6×10−35 m, and consequently had a temperature of approximately 1032 degrees Celsius. Even the very concept of a particle breaks down in these conditions. A proper understanding of this period awaits the development of a theory of quantum gravity.[26][27] The Planck epoch was succeeded by the grand unification epoch beginning at 10−43 seconds, where gravitation separated from the other forces as the universe's temperature fell.[25]". So, for all we know, the events of the Planck epoch may not obey the same type of causal relations that events in posterior epochs obey in the first place, and so there may not need to be anything that precedes time 0, or anything, of the sort.
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@emiledin2183 No, it was not. The order of operations allows a + (b + c) = (a + b) + c. This is called associativity. In fact, the order of operations exists only because of associativity being true.
Also, I should point out that you are not actually obligated to follow the order of operations, since it is just a convention. Mathematicians could have chosen a different convention altogether, and it would have been just as valid. In fact, you can rewrite all of mathematics using Polish notation, where no order of operations is even needed, as the notation is unambiguous, unlike the infix notation most mathematicians use when publishing, which is the same notation we use. The ambiguity of infix notation is why the order of operations exists. However, with that being said, the order of operations was followed in the video. Anyone saying otherwise does not understand the order of operations.
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Zeetee Pippi In the hyperreal numbers, the reciprocal of the infinite quantity ω is ε > 0. However, ω is not equal to lim x (x —> ♾). In fact, lim x (x —> ♾) is not a number, but rather, in surreal number theory as well as in class theory, it is called a gap. You can define inverses for gaps, and we do that in undergraduate-level calculus, although we never use the name. The inverse of said expression is 0. ♾ is just abuse of notation to represent that object, because no one would want to waste their time writing that expression more than once.
In measure theory, we never work with the hyperreal numbers. This is because it just does not work well with the category of sets that we work with in the theory, and they are ill-suited for probability theory. That is, unless we work specifically with hyperreal measure theory, which we do not use for situations such as the dartboard.
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@JelloBeanzer This is a sin in itself.
No, it is not. Nowhere in the books is it stated to be.
Someone who actually reads their Bible and tries to comprehend it without bias would know that the verses used to support slavery were about the relationship between man and god, not man and man.
No, this is false. The verses very explicitly talk about interhuman interactions, and never mention YHWH as being involved in the interaction.
Not to mention that there's not a single white person in the Bible, making all racial arguments immediately void, because there's no mention of whites being more deserving.
You are ignoring the historical context behind the arguments they presented back then, as well as their actual content. To start with, the ancient Greeks and ancient Romans were very much classified as "white" by the so-called "racial scientists." Since they featured prominently in the Bibles, the statement that there are no "white" peoples in the Bibles are objectionable. As for the historical context, the backdrop for these arguments was an ideology known as Manifest Destiny, a worldview that stated that the Anglo-Saxon race was chosen by God Himself to bring Christianity and the Gospel to all the inferior races of the world, and to destroy the enemies of God, and that the United States of America were going to set the stage for the Second Coming of Jesus Christ. There were entire books and doctrines written on the ideology, this was not some whimsical concept thrown around only to preserve slavery. This concept had already been around for centuries before the Civil War.
In religion, it's the same as studying a scientific study, and oversimplifying the results of it to prove an opinion.
The problem is that religious thinking encourages this. The very spine-and-backbone of religions' existence is this type of rhetorical strategy. Christianity was built almost entirely by cherry-picking concepts from Hellenism, Second Temple Judaism, Zoroastrianism, and the various Roman religions, and syncretizing them together into one belief system, going so far as to distort the meaning of texts in the Tanakh, and re-interpret it in far-fetched ways to claim that Christianity is the correct religion over Judaism. As far as religion is concerned, this is a form of divine revelation, not a form of intellectual dishonesty.
There is no "what if," simply understanding the logic behind anything can be unbiased and objective (to an extent).
The "to an extent" caveat makes the argument self-defeating.
Slavery in the Bible typically in the context of indentured servitude,...
No, it is not. There are plenty of verses about owning people as property for life.
...and not the typical view of slavery.
Biblical slavery is not identical to chattel slavery, and is slightly less brutal, but it definitely is a form of slavery, and not merely indentured servitude.
And racism is not supported in the Bible.
This is just false. The Bible very explicitly condones ethic cleansing (an extreme form of racism), and it also portrays clear prejudice against the Canaanites and against Egyptians, as well as many other ethnic groups.
Again, making the biblical justifications for slavery immediately void in the whole context of the Bible.
If your assertions were true, then maybe, but they are not true. I honestly do question whether you have read any of the Bibles at all.
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@suntzu7727 I think you are the one who missed the point. The statement is by no means tautological, as you are actually misrepresenting what the host commenter said, and thus the analogy with the corpse does not hold.
The host commenter said if [the universe] was slightly different, we could have still been around but in a form that is unrecognizable to us in this universe, and still make the same dumb argument that THAT universe was created for them [us] to exist that way. This is in no way a tautological claim. This is a version of the anthropic principle. To restate it more concretely, the idea is that, if the universe were different, but life intelligent life arose in said different universe, then the intelligent life in that universe could still come to the conclusion that the universe was finely tuned for them to exist. Whether they would come to the conclusion, that is a different question to answer, but it is conceivable that they could.
Why does it matter? Because it is conceivable that, if multiple universes exist with intelligent life in them, that intelligent life in those universes could each postulate that the specific universe they inhabit was specifically fine tuned for intelligent life to exist, and as in this conceivable, scenario, fine-tuning would obviously be false, it suggests that the starting premise of the argument is unsound.
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@suntzu7727 my criticism is of the anthropic principle in general which is just a truism
A truism is different than a tautology, which is what you claimed it was in your criticism. Also, you are going to have to explain how it actually constitutes a truism, given that I showed it has a nontrivial implication, making it not a truism.
How would we be we if the entities that existed would be of an unrecognizable form?
0. If I knew how, then "unrecognizable" would be a definitionally inaccurate description.
1. This a semantic fallacy, confusing the signified with the signifier. It also does not help that my restatement of the argument presented above does not use any personal pronouns whatsoever, so this objection is of absolutely no relevance. I am talking about intelligent life forms in general, not us, and the generally accepted definition of "intelligent life form" makes no mentioning and has no relation to the specificity of a recognizable form or to a particular universe in general.
You wouldn't be your if you were a squirrel, you're a human.
I addressed this in point 1 above.
And saying, that the reason that the constants were such that allowed us and not nobody or the unrecognizable ones to have existed is because if that weren't the case we wouldn't exist is just an empty truism.
No. This is wrong in more than one way.
0. Nobody here has stated that the reason the constants are such is because if they were different, we would not exist to observe that they are what they are now. You are misrepresenting the host comment, my argument, AND the anthropic principle as a whole. Triple strawman fallacy, yay! No, the anthropic principle makes absolutely no claims regarding the reason the constants are what they are. The anthropic principle is simply an observation that indicates that our observation that the constants are what they are is a non-argument about anything concerning intelligent life forms, because an intelligent life form could never observe itself inhabiting a universe it cannot inhabit. The probability that we observe that we inhabit a universe which we can inhabit, assuming no deities, is 1. As such, there is nothing to be concluded from said observation, and in particular, there is nothing to be concluded about the specialness of life forms from said observation.
1. If your representation of the anthropic principle as being about "the reason" the constants are what they are was any accurate, then it would not be a truism at all, because it would in fact be making a statement that implies the existence of a reason, which is not only non-trivial, but a very boldly unsubstantiated claim. There is no reason to think that there would be a reason for why the constants are what they are, so claiming that the reason is anything in particular at all is bold, and far from a truism. Good thing the anthropic principle makes no such bold unsubstantiated claims!
That's not different from you getting shot at from 50 people for all angles, within a reasonable distance, every bullet missing, and when you wonder how that happened, someone comes and says "Well, it's not strange, if they hadn't missed you wouldn't be alive to observe it".
This is a false equivalence. You are presenting your example as if the thing that needed explaining was the reason it happened, but the anthropic objection to the fine tuning argument is not at all an argument for why the constants are what they are. The anthropic objection is an argument for why the observation that we live in a somewhat-habitable universe is unimpressive and not sufficient information to conclude that there is any fine tuning at all.
But the anthropic principle itself doesn't say anything interesting.
What you think the anthropic principle says is not interesting, but what the anthropic principle does actually say has nontrivial implications concerning whether we should care about the fact that we know that we live in a habitable universe. Whether those implications are "interesting" is a matter of taste, and not relevant to the discussion.
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That is the entire point of conspiratorial thinking. It is unfalsifiable. It is akin to solipsism or Last Thurdayism. Unfalsifiable things are bliss for those who have a fragile ego: they are unfalsifiable, so they cannot be proven or convinced to be wrong by even the smartest of people, and because they cannot be convinced that they are wrong, their feelings and ego are safe from any danger. Everyone has some degree of fragility within their ego. It is the truth. Being wrong doesn't feel good. It lowkey does suck sometimes when you realize you believed in a lie for 20 years. Often, these lies are insignificant facts. Often, not so much. However, plenty of people are capable of still accepting that, yes, they were wrong, even if it temporarily hurts a little bit or a lot of bit, and learn from the mistake, to some capacity or another. But flat Earth believers have an ego so unbelievably fragile that they are willing to be ostracized by the entirety of society if it means not accepting that they are wrong.
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@SlimThrull I've looked other places, no one wants to discuss it let alone answer a fairly simple question.
Interesting. Thank you for confirming my hypothesis. So this was the case after all. Also, if you want to insist that the question is simple, despite it obviously not veing so, then you do not deserve to have it answered.
I have no doubt you do question my sincerity, as you think I'm "demanding" an answer.
I have no idea why you think I would expect you to doubt my claim that I think you are insincere. Saying "I have no doubt..." as if that adds anything to the conversation is weird, suspicious, and only reinforces my point about your insincerity, ironically.
No reasonable person could possibly come to that conclusion with what I posted.
You would be right, if your standards for "reasonable" were not so skewed and disingenuous.
There was no demand anywhere. You're assuming things that simply aren't true. Naturally, this leads to bad conclusions on your part.
Right, which is why you were able to refute my claims so easily— oh, wait. You refuted nothing.
Well, anyway, I am not going to entertain having a full blown discussion with you, especially because I know this will just boil down to ad hominems. I merely commented to provide with you constructive criticism regarding your bad attitude. You proved my point, and chose to reject the criticism. This is fine by me, as it is ultimately none of my business. My job is done here. Ciao.
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@forbidden-cyrillic-handle Obviously, some mathematicians had different opinions, and routinely used it in the past.
Yes, a definition that was used over a thousand years ago. Do you think I care? No, I do not care. The correct definitions are the ones which are used today, until such a time comes when those definitions are changed, if they ever do get changed. Some definitions were thought to have been correct in the past, yes. This is fine, but today we know them to be incorrect, so the past is irrelevant. We study mathematical history to learn from the mistakes we made in the past, not to continue making them by continuing to use definitions that no mathematicians today use.
What is routinely used by some mathematicians is not what the definition is.
You are wrong. The definition I proposed is used by all mathematicians, not just "some." You will not find any mathematicians from the 20th or 21st century who use any other definitions. You can try searching all you want, but you will not find it.
Until it officially changes, and becomes something more than routinely used by some, I prefer to keep the current definition.
The current definition is the one I provided. It has been the current definition since the early-mid 1800s. This is the definition that originated from rigorous research in ring theory. It is the definition every mathematician, without fail, has used since the late 1800s.
You need a big conference to vote the new definition,...
This was done more than over a century ago. You are behind the times by millennnia. This is ignorant.
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@nuggetoftruth865 The OP wasn't saying that no regular shapes exist.
I know OP never said this, but the logic that they are using in their comment does absolutely imply this, to whatever extent "shape" even has meaning in that worldview. They may not be aware that their argument has these logical consequences, which is I am making the remarks I am making.
To your point, it might be the case that in reality, shapes don't even exist because of quantum physics. However, that doesn't help us answer the question in practicality.
The topic of "practicality" is completely irrelevant to the argument OP presents, because OP has been explicit in that they only specifically care about "reality," not "abstraction" or "practicality." The reason I mentioned quantum physics is because quantum physics, it being quite literally the most accurate description of "reality" as he defines it, as of today, completely contradicts his argument. If you so want to insist that we talk about practicality instead, then here is my answer: the question has no valid answer, and in practice, the question is both nonsensical and irrelevant, because it does not actually matter how many sides a circle has. It makes a difference whether a stop sign has 8 sides or 18 sides, but it makes no difference whether a circle has 1, 2, or infinitely many.
For the OP to make the argument that he does, he has to make the assumption that molecules have defined shapes and that a physical disk would be perfectly circular to measure those sides, both of which put aside reality for the sake of practicality.
They both put aside both reality and practicality. There is nothing practically valid about the concern OP is expressing, and it is also completely wrong in reality. If the argument OP wants to make requires an assumption that is false, then this is definitive prove that OP should not be making this argument at all, whatsoever, which is precisely the whole point of my counterargument.
OP is wrong in more than one way. Why? Because (1) shapes do not actually exist. More importantly, (2) because, even if assume they do exist, the assumption that such shapes exist AND that shapes are necessarily physical objects as opposed to abstract constructs, which is what OP claims, then said assumption implies, by logical and physical necessity, that regular polygons do not exist. Maybe OP was not aware of this, but this is true nonetheless. By this argument, squares and regular octagons and whatever shapes you like do not exist, because every shape formed by any set of molecules is either a triangle or just some other no-name polygon that is not regular. Even if we consider molecules as infinitesimal dots and the edges connecting molecules as the sides of a shape, this remains true. Again, I am fairly certain OP would disagree with such an idea, which is why they never said, but nothing can change the fact that said idea is literally an inevitable logical and physical consequence of the argument presented.
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Steve Mattero The reason is fairly simple. You have an unordered pair of numbers (a, b), and you want to square them and add them. Well, there are only three possible distinct pairs: (even, even), (even, odd), and (odd, odd). Keep in mind that (odd, even) is the same as (even, odd) because addition is commutative. That is why I specified unordered pairs. Anyhow, let's test all three pairs
(2n)^2 + (2m)^2 = 4n^2 + 4m^2 = 4(n^2 + m^2) == 0 (mod 4)
(2n + 1)^2 + (2m)^2 = 4n^2 + 4n + 1 + 4m^2 = 4(n^2 + m^2 + n) + 1 == 1 (mod 4)
(2n + 1)^2 + (2m + 1)^2 = 4n^2 + 4n + 1 + 4m^2 + 4m + 1 = 4(n^2 + m^2 + n + m) + 2 == 2 (mod 4).
I tested all three pairs, but notice how 3 (mod 4) is never a possible result in any of those pairs. This means that natural numbers of the form 4k + 3 cannot be decomposed as the sum of two square numbers.
As for why factorization matters, if you multiply out 4k + 3 and 4l + 3, you get 1 (mod 4), which is decomposable into a sum of two squares. Meanwhile, if you try to do (4l + 1)(4k + 3), you get 3 (mod 4) again, which means these numbers cannot be decomposed either. This implies that (4k + 3)(4l + 3)(4r + 3) results in 3 (mod 4), and from here, you can use induction to show that an odd number of prime factors that are 3 (mod 4) result in 3 (mod 4) again, while an even number results in 1 (mod 4). This is how you prove the theorem stated in the video, and this is why the form 4k + 3 is what makes it all matter.
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@caiomateus4194 Philosophy is not at all grounded in common sense. Quite the opposite, really: philosophy is all about going against common sense. Hence you have things such as solipsism and what not. And besides, 2 + 2 = 4 is legitimately false is you are working in a finite field of characeristic 3. Also, I should let you know that most of science defies common sense, and a lot of mathematics as well. For example, in mathematics, we have a theorem that say that there are true statements that cannot be proven to be true. This is completely contrary to common sense, and yet it is a theorem, a fact. The entirety of quantum theory is just so flabberghasting that "defying common sense" is a huge understatement. I say this as someone with a degree in physics who specializes in quantum physics. Common sense is stupidly overrated. Which is all the more reason why the Kalam argument is so stupid. The Kalam argument is not good philosophy. It is just sophistry. No self-respecting philosopher takes the Kalam seriously: only apologists, and philosophers who are specifically dedicated to countering especially bad arguments, take the argument seriously, though the latter take the argument seriously because it needs debunking to a bunch of people.
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@nahometesfay1112 Actually, I partially take back what I said. The Wikipedia article on the Pirahã does not even outright state what you quoted. I just checked, and here is the quote I found under the 'language' section. This is a full paragraph:
Curiously, although not unprecedentedly,[11] the language has no cardinal or ordinal numbers. Some researchers, such as Peter Gordon of Columbia University, claim that the Pirahã are incapable of learning numeracy. His colleague, Daniel L. Everett, on the other hand, argues that the Pirahã are cognitively capable of counting; they simply choose not to do so. They believe that their culture is complete and does not need anything from outside cultures. Everett says, "The crucial thing is that the Pirahã have not borrowed any numbers—and they want to learn to count. They asked me to give them classes in Brazilian numbers, so for eight months I spent an hour every night trying to teach them how to count. And it never got anywhere, except for a few of the children. Some of the children learned to do reasonably well, but as soon as anybody started to perform well, they were sent away from the classes. It was just a fun time to eat popcorn and watch me write things on the board."[6]
It doesn't seem to me as though they're incapable of arithmetic, based on this. Reading from the context, what I'm getting is that (1) some of the researchers have a poor understanding of the Pirahã (2) those who do seem to have investigated them more carefully agree that it's not an inability to learn arithmetic, but an unwillingness to incorporate it into their culture. And those have very different implications.
Perhaps you can tell me which Wikipedia article has the quote you used? Even so, that quote alone doesn't seem to imply much. It sounds to me that, as Taxtro said, the culture is rather poorly understand by researchers and that there is a ton of exaggeration occurring, which is super common in the study of isolated cultures like this one. This why anthropological studies should be taken with a grain of salt (well, all science should be taken with a grain of salt, but especially anthropology).
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@vladislavanikin3398 It is the most powerful argument, and I would say, the only actual argument that really succeeds. All other arguments rely on logic that is fallacious, uncompelling. For example, this whole "it makes the fundamental theorem of arithmetic easier to state" nonsense is highly specious outside of the context of unique factorization domains. In most other areas of mathematics, mathematicians are completely unbothered when theorems have exceptions built into them. Heck, even for other theorems about prime numbers, there are many, many theorems which have 2, 3, and sometimes even 5 as the exception, and yet no one bats eye in calling these prime numbers anyway. The double standard makes no sense. Besides, the validity of a theorem should not depend so much on the precise details of how its phrased. Otherwise, you could prove literally anything by choosing "the adequate phrasing." This makes it obvious that the actual answer to the question "why is 1 not a prime number?" has nothing to do with the fundamental theorem. The question is answered, as you said, by the fact that 1 is a unit, and that units are conceptually and fundamentally different from irreducible elements.
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3:52 - 3:59 Yes, that is precisely where we should start. I am glad this is the approach Rationality Rules is taking here, because I always find that other channels analyzing the argument fail to explain what exactly is it that is problematic with the premises, and it makes it confusing for both theists and nontheists.
4:04 - 4:30 To elaborate on this, we work with formal theories to discuss existence in any given context. We start with first-order logic, or possibly, second-order logic, and this enables us to use the existential quantifier "There is some" and the universal quantifier "For all." We then find a set of axioms telling us about what we say must or must not exist, along with some proof-theoretic criteria to derive other existence propositions from the axioms. These criteria, ideally, will be based on verificationist epistemology. This also requires having some primitive notions. The most prominent example of such a formal ontology is axiomatic set theory: especifically, Zermelo-Fraenkel set theory, where the primitive notions are sets. The ontology is about which sets exist and which sets do not exist. Something like this is mere abstraction, though, and is not an ontology that has immediate consequences for discussing physics, for example.
4:40 - 4:46 A more careful phrasing of this definition is that "There exists some t such that for all t' > t, X exists at t' AND such that for all t'' < t, X does not exist at t''." This is what I would take as the definition for the notational abbreviation "X begins to exist (at t)." This much precision is needed if one wants to effectively demonstrate the fallacy in the old cosmological Kalam. But I applaud Rationality Rules for getting this right in spirit, because it is important to understand this definition and its implications.
5:35 - 5:50 I hope this video delves deeper into the tensed theory of time and explains why it is unscientific, since this is a key task in debunking the old Kalam, and even the new Kalam.
6:02 - 6:07 Immediately, this is a problematic definition. Christian apologists like to do this thing where they define some undefined terminology in terms of other terminology that is also undefined, and like to pretend that this somehow solves the problem. What does it mean for something to "come into being"? Defining "begins to exist" in terms of "come into being" achieves nothing, since "come into being" itself requires defining.
6:08 - 6:45 The problem with this definition lies, not with the definition itself, but with how WLC interprets it. He claims that this definition merely formalizes the notion of past finitude, but it does not. Because it is not sufficient that the object has past finitude. It is necessary that x exists at t AND there be some t' < t such that x does not exist. This is what his requirement (ii) is in the definition, and translating the definition as formalizing past finitude ignores this requirement altogether. Why? Because it is possible that the length of time interval of existence of x is finite, but that there is no t' < t, where t is x's earliest instance of existence, such that x does not exist. Such a situation would violate criterion (ii) of the definition, yet WLC would still insist x has a beginning.
6:54 - 7:07 I am not convinced this is a coherent definition. What does it mean for x to bring about y? All this does is rename the object "cause" into the action "bring about," but no unique characterization of the action with this name is being given. Thus, it actually defines nothing. And, so far, all it describes is a relationship between x and y, but the parameters of the relationship remain unstated.
7:18 - 7:35 I acknowledge that the controversy is there, and that defining what "causation" is metaphysically is so difficult that there is no consensus. But that is still a problem we cannot just let slide. By going with the "intuitive" understanding, we are walking right down the path WLC wants us to, and that is itself part of the fallacy in WLC's argument: it lies in the fact that his terms for causation are all ill-defined. This is where the appeal to intuition fallacy kicks in. So we really should not just grant him an intuitive understanding, and commit to an actual definition of causation, regardless of how much controversy it may cause. Besides, in my view, I disagree that this should be so controversial at all. I think the only reason behind any essential disagreements on the definition is mere pettiness.
To be continued in the replies...
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This is not the definition of a prime number. This is the definition that is often taught in grade school, but it is incorrect. The definition of a prime number is an integer which is nonzero, not invertible, and which, when written as a product of two integers, must always contain a factor of –1 or 1. Since 1 and –1 are invertible, they are not prime numbers. The grade school definition is meant to be a simplification of the true definition, to keep the concept intuitive, but it is an incorrect simplification that does not lead to the correct intuition captured by the true definition. A composite number is a nonzero, non-invertible integer which is not a prime number.
Alternatively, one can define a prime number as an integer which has exactly 4 divisors (–p, –1, 1, p). –1 and 1 are integers which have only 2 divisors (–1, 1), so they are not prime numbers. 0 has infinitely many divisors, and so it also is not a prime number. Or, you can define a prime number as an integer which has exactly 2 positive divisors. The integer 1 only has 1 positive divisor, so it is not a prime number. However, these alternative definitions, although strictly "correct" as far as semantics are concerned, are bad definitions conceptually. The definition provided in the previous paragraph is the one that actually has genuine mathematical meaning.
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@Happy_Abe No, that does not use the definition of a 2-tuple. A 2-tuple is itself a function, a function from 2 to the target set, where 2 = {0, 1}. The object {{x}, {x, y}} is actually called a Kuratowski pair. The corresponding two-tuple is instead the set {{{0}, {0, x}}, {{1}, {1, y}}}, which is built from the Kuratowski pairs {{0}, {0, x}} and {{1}, {1, y}}. The distinction is subtle, but it exists, because there is no 3-element analogue for the Kuratowski pair construction, while there is such a thing as a 3-tuple: a function from 3 to the target set, where 3 = {0, 1, 2}. In this case, the three tuple would look like {{0}, {0, x}}, {{1}, {1, y}}, {{2}, {2, z}}}.
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@MrElmattias They did not "blow their own theory," you simply misunderstood the entirety of the contents of the video, and then rushed to comment without carefully processing the fact that the video explicitly acknowledged that they way modern Tetris is played works nothing like what was stated in the video. You not having the composure to process the disclaimer carefully enough is not an oversight on the video's part. And, by the way, this analysis is not "the video's," it is an analysis that is well-established in the professional mathematical study of Tetris. If you look at the description of the video, which I know you never bothered to look at, you will actually find the published paper "How to Lose at Tetris" by Heidi Burgiel. What the video is doing is just relaying the contents of the paper in a fashion that the lay audience can understand, while also clarifying that you are not supposed to think that these results apply to games in practice.
Anyway, I realize this conversation is going to be a waste of my time, so I will stop replying. I said what needed to be said, so I am done here.
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This entire video is a non-starter. It grants WLC's claim that the causal principle is confirmed by our experience, but even this is not true. In fact, the causal principle, as stated by WLC, is completely misconceived: it is conceptually mistaken. Why? Because WLC's metaphysics of causation relies on Aristotelian physics, and an Aristotelian understanding of causation, where (0) causes can be discretely categorized into formal cause, final cause, material cause, and efficient cause; (1) objects and phenomena can be discretely categorized into causes and effects. Our modern understanding of physics and the scientific method reveals that Aristotelianism is false, and as such, WLC's principle of causation could not possibly be true. It is not compatible with the scientific understanding of causation that we have today. Causation delineates spatio-temporal relationships between objects and phenomena across spacetime, and so it exists as a spectrum. Therefore, there is no coherent notion of discrete categorization into causes and effects. There is also no coherent notion of causes being discretely categorized into types, as such. Instead, any statement of scientific causation must include a discussion of geodesics and worldlines, and an ontology that accounts for locally Minkowski spacetime. Any hypothesis that you present as the explanation of some body of evidence must meet these criteria before even being considered a coherent hypothesis. On that note, any ontological notion of beginning to exist must also be defined in such a way that the definition is congruent with these ontological considerations. This should be the starting point for any honest individual discussing causation today.
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@logicalliberty132 No, those are definitely mistakes of the video, and your response is completely incorrect too.
20:20 - 20:41 "Two things can be said in response to this example. First, we know that some kinds of quarks, which are, as far as we currently know, not composed of simpler things, can turn into other kinds of quarks. This doesn't seem to be a case of already existent things being re-arranged to form a new composite; instead, it's a case of a fundamental particle beginning to exist."
This is exactly what the video claimed, and it is simply incorrect. Quark decay is in fact an example of a rearrangement: it is an interaction where energy in the quark field is lowered and redistributed into the neutrino field. Nothing here is beginning to exist in a sense other than as a re-arrangement of energy states of quantum fields.
20:42 - 20:51 "Many composite objects aren't merely arrangements of pre-existing things. For instance, I exist, but I am not identical to an arrangement of a particular group of particles."
This is incorrect. True, I am not a particular arrangement of a group of particles, but I never said I was. I am still a composite object that is merely an arrangement of wordlines of particles throughout spacetime.
Also, you claimed I said that the video claimed that I am composed of atoms, but that is actually not what I said. Read my previous comment again.
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contingent things by definition don't exist in some possible worlds
I know that, and I never said otherwise, but the issue is the notion of what counts as a possible world. I believe the concept of possible world is ill-defined.
in our world cars exist, but in some other possible world humans maybe did not invent cars
Except you do not know that it could have been possible. Conceivability and possibility are not the same thing.
thus the essence of a car is combines with it's existence to form the whole
No. Again, that is not how that works. You can try to convince me otherwise, but as long as you appeal to false concepts, I will keep pointing it out.
nothing in the definition of causal loops of causality implies they are time dependent
Then you do not understand the definition of causality.
Alexander Pruss' work which you have not refuted demonstrates this with Paradoxes that are time independent.
I never said anything about time-dependent paradoxes. You are getting different parts of my response mixed up together. I do not have the time to provide a thorough refutation of everything that Alexander Pruss has written, it would take dozens of pages, and it would be impossible to type in a YouTube conversation, not to mention that constructing a well-written response would literally take weeks. That being said, many philosophers have written rebuttals of Pruss' works. So I have no obligation to provide my own refutations. I am justified in simply rejecting his works on the basis of those refutations. You can insist otherwise, but you are wasting your time.
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You can't take the negation of "for all" if there is no "all", i.e, no members of φ, to begin with; by definition of logical negation and empty set.
No, this is nonsensical. You should really sit down one of these days and read an introductory-level book in formal logic. You absolutely can negate quantifiers over an empty domain. Logical negation of the universal quantifier is actually equal to to the existential quantifier of the negation, and this remains true in the empty domain.
But also, this objection is genuinely stupid: he never used universal quantifiers in his proof.
The contradiction is that φ (the empty set) is not a subset of A, φ is empty to begin with.
0. φ does not denote the empty set. Stop calling it "phi". We do have notation for the empty set, namely ø and {}. φ is not one such valid notation.
1. {} is empty, which is why it is a subset of A. There is no contradiction here. Since the empty set has no elements, every element this empty has (a.k.a none) is an element of A, and this much is true.
If you assume {} has members to begin with, the contradiction doesn't change that.
It absolutely does. If {} actually has elements, then at least one of those elements is not an element of A. So {} is not a subset of A.
The contradiction applies to subset, not membership.
No, it applies to membership, because the contradiction occurs on the existential quantifier. Also, this is a silly objection: the subset relation is defined exclusively in terms of the membership relation.
Otherwise, your argument runs: Assume {} has members...
No, that is definitely not at all how his argument would run.
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This is an excellent objection, but the real problem for Benny's argument is simply that the universe is not random. The universe is deterministic. When we throw a die, since it is a chaotic system, under usual circumstances, we lack sufficiently precise information about the initial conditions of the throw to determine accurately the trajectory of the die, and thus, to predict the outcome of the throw. This is why we use probability theory to model a throw of a die as a stochastic process. However, in principle, if you can program a dice-throwing machine with sufficiently precise control of the motion of the throw, you would be able to accurately predict the outcome of every throw, despite the throw still looking like a random motion to a human not equipped with the mathematics to make this prediction. Therefore, making an appeal to randomness, which does not actually occur in nature, as an analogy to demonstrate that the universe is designed, is hopelessly fallacious.
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@rmsgrey The problem is that when you are working with real numbers or the algebraic numbers, it becomes impractical and conceptually useless to work with explicit constructions and encodings. To properly give a formal introduction to the algebraic numbers, you need ring theory, and to properly give a formal introduction to the real numbers, you need lattice theory on top of ring theory. Set-theoretic constructions are not appropriate when dealing with these higher-level mathematical objects.
For instance, axiomatically, it is very easy to write a list of simple axioms that uniquely define what the real numbers are. Talking about the algebraic numbers is even easier: the field of algebraic numbers is the algebraic closure of the field of rational numbers. However, while it is very easy to understand the axioms, actually constructing these objects using nothing but sets is complicated, and to be honest, a waste of time. That is not to say that it cannot be done, but rather, that it should not be done.
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@DoofusChungus I don't even know what Answers in Genesis, but I'm just saying that it not only makes sense in a religious context, it makes sense in a science context.
I have already demonstrated how fine-tuning is false if classical theism is true, but fine-tuning makes even less sense in a scientific context. There is no fine-tuning, scientifically speaking. The claim that the constants of the universe could have been different at all is unfalsifiable. The claim that life could not have existed in a universe with different values of the constants is unfalsifiable. The claim that the universe would have even had any constants at all, if it had been different, is unfalsifiable. The claim that the constants taking on the values that they did is highly improbable, if random, is also unfalsifiable, since we do not know how these variables are distributed: they could be uniformly distributed, which is what Christian apologists assume, but this assumption is unjustified, and unfalsifiable. They could also be normally distributed, Poisson distributed, discretely distributed, or any other probability distribution imaginable. In short: it is scientifically impossible (or otherwise) to know what the universe could have been like if it were any different than what it is.
My point is that these 1 in a gazillion chances happen over and over.
These events do not have a 1 in a Gazillion chance. The universe is, by all acounts of the available evidence, deterministic (quantum deterministic). So,... no, these events do not have a probability of 1 in a Gazillion. They have a probability of 1.... in whatever way it is even meaningful to actually say that. This, of course, assumes that all events can be meaningfully assigned a probability, which we know is mathematically false.
Do you believe aliens are real?
I choose to withhold my judgment about the existence of aliens, as I do not think we have sufficient evidence to make a conclusion in favor, or against, their existence. Some form of extraterrestrial life probably does exist somewhere in the universe, but it being intelligent enough for us to call it "aliens" is an entirely different subject. We do not even have a sufficiently rigorous understanding of what it means for life to be intelligent here on Earth, so we are definitely not ready to make those judgments for non-Earthly life.
The possibilities and outcomes are virtually infinite, no?
If you mean possibilities in the sense of randomness, then, no, since the universe is definitely not random. If you mean possibilities in the sense of possible states of the universe, that depends entirely on knowledge we can never have: things such as the size of the total universe, its topology, the initial conditions of the universe, if there even exists such a thing as initial conditions of the universe, etc. Anyhow, there is no scenario where we are justified in concluding that the space of possibilities is infinite.
Not only the perfect set of events, but so improbable that this sequence of events will never happen again.
This assumes there is no multiverse, which no one knows to be true (or false). Also, again: the universe is not random. Events have probability of 1. Events happened, because the way the universe behaves means they had to happen. It would have been impossible for the universe to behave the way our current laws of physics say it does behave as, and yet for those events to not have happened.
When there are 100 zillion events that are required to occur just so life on Earth can exist at all,...
The events are not random. You do not believe they are random, because you believe God is guiding these events. We do not believe they are random, because we believe these events are a consequence of the physical and deterministic nature of the universe. Thus, no one in this conversation believes these events are random. So, why do you keep throwing this strawman as an objection?
...then it didn't just randomly happen by pure chance!
I agree! Which makes me all the more confused as to why you keep saying the events are random. Virtually no biologist thinks life emerged from random events. No, we think life emerged via organic chemistry. And, I do not know if you have ever taken a course in chemistry, but let me just say this: chemistry is not random. If it were random, then there would not exist such a thing as "the laws of chemistry." If you throw a sodium coin into a fountain of water, what is the probability that it will react with the water, release energy, and form hydroge gas with sodium hydroxide? The probability is exactly 1. It will happen (well, in Earthly conditions, anyway). It is impossible for it to not happen, and we have verified this experimentally so many times, the number of experiments is probably in the millions by now. Not only can we say for certain it will happen, we can predict the speed of the reactions, the amount of particles that will interact, the amount of energy released, and the amount of hydrogen gas molecules that will be produced, and more: and we can do all of this with such high accuracy, it would make you cry tears of joy. Chemistry is not random, and neither is biology. Therefore, unless there exists such a thing as the soul, all life can be reduced to the chemical reactions it undergoes. Therefore, the origin of life is described by some sequence of chemical reactions, by all accounts of the evidence. Do we know which sequence of chemical reactions? Not yet, no, but we are getting closed every year.
Anyway, in conclusion: (A) since God is all-powerful, there is no fine-tuning, because there being fine-tuning, by definition, implies God is limited by physical constraints, and simply making the creative choice to make a physical compatible universe/life combination is not an example of fine-tuning, because that is not how the concept of fine-tuning has ever been defined. (B) Science does not claim that events in the universe are random, so you need to stop insisting that they are.
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@DoofusChungus How do you know he wouldn't?
I never claimed to know that. And the point is, neither should you, or anyone else, since such a claim is unknowable. You are the one who made the claim that you know God would, so the burden is on you to provide evidence.
Yet my argument isn't baseless, as if he creates the universe with rules, he created those rules for a reason.
How do you know God had some reason in mind when creating the universe? Did you ask Them if They had a reason to do so?
So, what would be the point in going back on those rules?
Oh my Siesta. Are you illiterate? No one is arguing that God should go back! That is not the point of the argument. Stop making this strawman already.
"Even if theism is true" yet "I never even argued that God doesn't exist."
Again, are you illiterate? Saying "Even if theism is true" is not the same as arguing against theism.
You also even called yourself a non-theist.
I did. I am a non-theist. But that does not mean I argued against the existence of God. The only thing I have actually argued here is that the fine-tuning argument, as presented in favor of classical monotheism, is unsound. That does not prove God does not exist, nor does it approve all other apologetic arguments are unsound. We can have a discussion about those other apologetic arguments, but I doubt you wany to. Also, lacking a belief in God is not the same as having a belief in the nonexistence of God. Those are very different things.
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@DoofusChungus Brain waves basically refer to oscillations in the electromagnetic activity fields in the brain, these fields emerging from the fact that there is a net electron flow between neurons to exchange information. Electromagnetic fields are physical. They are a primary topic of study in physics, and we have characterized them mathematically.
Physicalism, in a nutshell, makes the claim that everything is physical: in the sense that, for whatever systems we can conclude exist via the study of physical theory, the systems they are causally related to are also physical, meaning that they too can be subjected studied via a physical theory. Basically, physicalism is a revamped version of materialism, but more modern, and meant to account for how modern physics describes the universe, rather than just naively asserting "everything is matter," since we know this to not be true.
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@DoofusChungus But you call yourself a physicalist, which is a belief that everything is in the physical (at least from what I've read online).
Yes.
So you very much have a belief in the nonexistence of God,...
I do, but I have never argued in favor of this belief, and yet, you are pretending that I did. And, what do we call it someone pretends that someone else did something, and yet they did not do it? Want to take a guess?
...and you're also arguing against many things about God, not just finely tuned.
No, I really have not said anything about God, other than that, if God is omnipotent, then the fine-tuning argument's premises are false. I have not made any other arguments about God. I do not know where you get that from. Maybe you are confusing my claims with the claims of someone else.
Also, side note, I don't know if there's an actual definition for the term finely tuned religion-wise, but what I mean by it, is that the universe has specific scenarios that had to play out for us to exist.
If you assume a godless universe, then yes, it is true that a very specific sequence of events had to happen for us to exist. This is not true if God exists, though.
So it's "finely tuned" so that those scenarios did play out, and here we are.
No, that is now fine-tuning works. Yes, IF God does not exist, or if God is not omnipotent, THEN, the fact that we are here necessitates that a certain sequence of events have happened. This is true, simply because if God is not omnipotent, then physical constraints matter. However, if God is omnipotent, then there is no sequence of events that "had to happen" for us to exist.
My point for that first one is that we both can go back and forth about "you don't know that" and "how do you know?"
Yes, and my point is that the fact that neither you nor I can know whether God would have wanted a particular outcome or not defeats your argument.
Everything played out as it did. Everything happened to such a degree that here we are, so obviously, there's a plan going on.
No. The fact that things happened the way they happened does not at all demonstrate that there is a plan.
And being all knowing, even if he didn't create the universe with a reason in mind, being an omnipotent God, he knows what's going to happen, or I guess in his case, how to make it happen.
Yes, per classical monotheism, God does know what will happen. This does not mean that God has a plan. It also does not mean God actually cares about what will happen. Look, I could, hypothetically, grant you the existence of God, and you still would not be able to prove anything else about God, at all, much less prove that Christianity is true. And if I grant you that God exists, I can still debunk the notion of fine-tuning, because the notion of fine-tuning in direct, definitional contradiction, with the omnipotence of God.
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@southali If the room has no vacuums, how can the vacuum be a subset of anything since it doesn't exist?
As I pointed out already, it does exist. But also, this argument is fundamentally fallacious, since this is inadequate analogy for sets. The problem here is that objects in space do not merely form sets. Sets are not compatible with the idea that elements appearing a different number of times changes the set, yet in this room, changing the number of particles of any given molecule type definitionally changes the mereological sum of the particles. Also, because the existence of any given configuration is also dependent on the amount of space, and not just the number of elements, there is no well-defined notion of a subset. For example, the configuration where we only consider the oxygen atoms is also not a valid subset, because it is not true, at least according to you, that the room only has oxygen atoms. This is the issue. So the vacuum is not the issue here: the scenario itself is incapable of working with a coherent notion of subset. Again, this is because the scenario you are describing is not at all analogous to a set. What you are describing is a differentiable manifold with topological deformities of a given type. This has nothing to do with sets, really.
Also, I should point out that it is impossible to refute a valid formal proof using analogies from intuition, regardless of how much sense you think those analogies make. Analogies, by their very nature, are necessarily flawed. This is why we use logic, rather than analogies, for evaluating the validity of proofs.
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This is incorrect. The domain can actually be any set, including the empty set. In particular, the domain can be N, Z, Q, or C. It need not be R. In fact, the equation f(x) = 5 does not specify a function. The domain of a function cannot be uniquely determined by an equation, and neither can be the codomain or the range. To the contrary: for f to be a well-defined function, the domain must be specified in and of itself, and the same is true for the codomain. I could have chosen the domain to be Z, the codomain to be Q, so f : Z —> Q, such that f(x) = 5 everywhere, and this would be a well-defined function. If instead, g : Q —> C, such that g(x) = 5 everywhere, then this still would be a well-defined function, and it is a different function from f. In both cases, the range is {5}, but the domain and codomain are different. In fact, the meaning of the symbol 5 is technically different in both cases. For f, f(x) = 5 is a rational number, whereas g(x) = 5 is a complex number. Despite the fact that we use the same symbol to denote them, these are different, unequal mathematical objects.
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Your Friend I don't think it's an issue about patience and effort to understand something. You require effort to understand quantum physics, set theory, cosmology, organic chemistry, evolution, epigenetics, music theory, languages, and so on. Those require effort and patience. But understanding the evidence and the basic physics behind the globe Earth does not require such effort and patience. These things can be understood by 7-year old children with cognitive disabilities. And I genuinely don't mean that offensively, I mean it in that having a cognitive disability can often make certain concepts difficult to understand. But this isn't the case even among people with cognitive disabilities. The globe Earth is one of the easiest things to understand that there are. So I can't honestly be convinced that they're not ready to put in the effort, because there is very little effort that you have to put in to understand it. Rather, it's an issue of fragility. The ego of a person who believes in the flat Earth is often so big that when they realize that their worldview is in danger, rather than accepting that they have been wrong this entire time, they choose to be stupid. Because this is how fragile their ego is. It's a variant of "I'd rather not know the truth, because learning the truth is too painful."
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@x-popone6817 because nothing would be able to happen if it was eternal.
False. "Eternal" is not synonymous with "timeless" or "changeless".
So you continue with your unfounded accusations that I am dishonest?
The accusations are not unfounded. I explained the accusations, and other did before me too.
It's your religion,... to call someone "dishonest".
The fact that you think a frequent retort makes a stance a religion proves that you do not understand what a religion is.
I literally just did explain it.
No, you did not, you simply provided a false claim that someone else debunked before Ant even replied. The fact that you pretend otherwise actually further proves your dishonesty.
God isn't a set of events, which is what the problem is with an infinite regress.
This does not actually explain what is it that makes an infinite regress problematic.
Scientists are biased as well...
Here, you demonstrate your ignorance of science. Scientists are biased, but the scientific method is not.
An infinite regress isn't possible because then we would never reach this point,...
No, the latter does not follow from the former. ω is an ordinal number infinitely larger than 0, but yet it is well-defined.
unless, of course, the past, present, and future are all equally really, and we just live in some type of block in time at different "locations", but that seems counter-intuitive.
It is counter intuitive, but that is how time actually works. We have known this for 106 years now, thanks to Einstein's theory of general relativity.
No one casually thinks that's how time works.
False. That is precisely how physicists and cosmologists think time works.
Faith is not, by definition, belief in the absence of evidence.
Yes, it is. Your Bible literally says so.
The Biblical definition of faith is trust,...
No, it is not, and you will not be able to find a Biblical verse that defines it as such.
then the implications of that conclusion lead to a mind.
They do not.
Yes, you don't know what dishonest means.
Projection much.
An infinite regress is impossible.
An assumption with no evidence, not a fact.
Something from nothing is impossible.
An assumption with no evidence, not a fact.
Conclusion: the universe had a beginning.
This literally does not follow from the previous two sentences, and depending on how you define "beginning", it can be the case that this conclusion is contradicted by the previous two sentences.
This leads to a mind behind it.
An assumption with no evidence, not a fact.
refute this.
There is nothing to refute, as your claims are all baseless assertions.
A natural explanation wouldn't work because how can an impersonal force suddenly, randomly, create the universe?
This is an example of the argument from ignorance fallacy. Your ignorant self is unable to imagine or understand how quantum physics, which are themselves beyond your own understanding, can lead to the universe existing as it is. Therefore, you deny the possibility altogether, but it does nothing to actually disprove said possibility. Things can be possible, regardless of whether you understand how they can be possible, or not.
The problem with an infinite regress is that we wouldn't be able to reach the present.
You have not explained how is it that we would not be able to reach the present.
Time exists outside of the physical world
No, it is not. Time is one of the four axis of spacetime, and spacetime is part of the physical world.
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@x-popone6817 No, it's not an argument from ignorance. Quantum mechanics does not show that something can come from nothing, nor does it should that the effect of an impersonal force, shouldn't be permanent.
Unbelievable. You went as far as to misrepresent your own argument in order to also misrepresent mine. Yet you complain when we accuse you of dishonesty. At this point, I find it hard to believe that you are not just trolling me.
The paragraph in which you mentioned the point to which I replied by appealing to quantum physics is completely different from the paragraph in which you mentioned your "something cannot come from nothing" claim, which I also replied to separately. So I have no idea why you are pretending that my appeal to quantum physics was in response to your "something cannot come from nothing" claim, and at this point, I do not care to know why. When you have to lie this much in a conversation, it just demonstrates without the shadow of a doubt that your position is indefensible, and that you are just desperate, and that there is no point in trying to continue have a conversation with you on the subject. Frankly, you have defeated yourself.
My argument was philosophical, that it isn't possible for an eternal cause to suddenly have an effect that hasn't been permanent.
No, it was not. That was an entirely different paragraph altogether, which I had already addressed. Your argument was, and I quote your exact words: "A natural explanation wouldn't work because how can an impersonal force suddenly, randomly, create the universe?" I find it amusing that you are lying about what your argument is, despite the fact that your actual written argument is still not only part of your comment, but part of the exact words I quoted in my previous response. Here I am quoting them again. Those exact words constitute an argument from ignorance, whether you want to admit it or not. Why do they constitute an argument from ignorance? Because you asked "how could (this) be possible?", (this) referring to an impersonal force suddenly and randomly creating the universe. The fact that you asked this is precisely what makes it, definitionally, an argument from ignorance. There are more than a dozen of plausible naturalistic explanations for the hypothetical beginning of the universe, in the assumption that such a beginning did exist. You are apparently, personally not acquainted with them, because they are highly technical explanations that require more than just a degree in quantum physics to suitably understand. As such, you are unable to imagine a plausible naturalistic explanation, so you rhetorically ask, "how can that be possible?", as if trying to drive home the point that, "of course it is not possible, for if it were possible, I would be able to imagine it, and then I would not need to ask how". In other words: it is an argument from incredulity, and an argument from incredulity is a special case of an argument from ignorance, as incredulity is a consequence of ignorance. So, yes, it is an argument from ignorance, and no, it has nothing to do with your false claim that "the eternal impersonal cause caused a non-eternal effect" is a logical contradiction, which I already addressed elsewhere.
Science can't disprove logic. Science presupposes logic.
I never stated otherwise here. The issue here is that you have not been using logic at all. You have not provided me with a sound syllogism. You have provided me with claims that you have not proven, terminology with no definitions, and then you expect me to either take your word on those claims, or you simple reiterate the claims, saying "well, this is obviously true, how can you not see that?" rather than, well, actually explaining the damn claim and proving it. You arguments have all been non sequiturs, misrepresentations of my responses, or an argument from incredulity. Nothing about that is logical.
Finally, your comments seem kind of like ad hominem. You say quantum mechanics is beyond my understanding. In other words, that I couldn't understand it even if I tried and studied it, that I don't have the mental capacity.
Firstly, let me go ahead and admit that I made a mistake. The phrasing "beyond your understanding" was very poor and careless. I apologize. I definitely intended to communicate the point that it was beyond your current understanding, but by trying to be concise in my words, I just made that into an insult. Secondly, now onto actually addressing the argument at hand. No, saying quantum theory is beyond your understanding is not an ad hominem. At worst, it is just rude. It would have been an ad hominem if I had said "your ignorance in quantum mechanics falsifies your conclusion", but I never actually said such a thing. Besides, in saying that it is an ad hominem by putting the claim out of context, not only do you misrepresent the claim, but you also miss the point of the argument.
You also said I am dishonest, without any proper basis.
No, I definitely provided more than just a proper basis. Not only did I explain exactly in what ways have you been dishonest and how exactly those things qualify you as dishonest, I also have provided explicitly examples along my commentary of such dishonest shenanigans, and have made sure to individually call those moments out in my responses. I quoted exact words, too. I am completely justified in calling you dishonest. Of course, you say otherwise, but guess what: a dishonest person would never admit to being dishonest. So you are only making your case worse here.
Speaking of dishonesty, I spent way more time in this comment correcting your misrepresentations of my arguments than I spent actually discussing any logic or philosophy, which is honestly disappointing. Conversations like that are not productive, frustrating, and a waste of time. They are not even entertaining. Now that I know exactly what you are all about in this conversation, I am going to be prudent and stop reading your replies, and stop replying to your replies. There is no point in discussing anything else further with you, and I have more important things to do with my energy than continuing to get baffled by this tomfoolery. Farewell.
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@Fred-tz7hs No one is seriously gonna read, let alone understand that,...
The comment has literally over 70 likes.
...and it's not even obvious to me how this is relevant to the comment you are replying to.
...then read the comment. It will become obvious. How are you criticizing a comment without even reading it?
For some people, it makes more sense to "see" the distance, and why it's useful,...
Tom literally explained why it is useful in the video. The comment that you chose to apparently not read by Fred also explained with even more detail, and your response was to literally complain and dismiss the explanation by saying "no one is gonna read it." This is like when a child ask their parents for a toy for months on end, and when the parents buy the toy to the child, the child immediately changes their mind and says "actually, I no longer want this toy."
I'm very sensitive.
...I would say you are just antagonizing solely for the sake of antagonizing, but go ahead.
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@joda7697 f(x) = e^x is a function, so is f(x) = 0
No, f(x) = e^x and f(x) = 0 are equations. A function, by definition, must consist of a specified domain. exp : R —> R, exp(x) = e^x is a function. Exp : C —> C, Exp(x) = e^x is also a function, but a different function, because the domain is different, even though Exp and exp agree on their outputs for every input in common. f(x) = e^x is an equation would form part of the definition of f, but if no domain is specified, then f is not a function. Also, f(x) is the output of the function, not the function itself. This seems like a pedantic distinction in semantics, but it is not: it is a meaningful distinction. If I tell you to solve the equation f(x) = 0, then I am asking you, "what number x in the domain of f is such that the real number f(x) is equal to 0?", but if I ask you f = 0, then I am asking "what function is equal to the zero function, a.k.a what function f satisfies f(x) = 0 for every x?".
It is also the unique solution, even though there are multiple ways of expressing it.
I know it is the unique solution, that is literally what my previous comment was about. You are just restating what I already said, and presenting it as if it were a different claim.
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@angelbrother1238 I would say that fine-tuning is a real problem.
If so, then you should cite several peer-reviewed studies that prove the claims being made by Ben Shapiro here. The problem is that I know you cannot do it, because such studies do not exist, because Ben Shapiro's claim is false.
This is why the physicists that disagree with this have to posit a multiverse.
No, this is false. To start with, the idea of the existence of a multiverse in physics is much, much older than the idea of fine-tuning in theistic apologetics, dating back to the early days of quantum mechanics. Also, the existence of the multiverse is not a hypothesis endorsed by most physicists or cosmologists, it is a minority position, and not the consensus. In particular, the multiverse hypothesis is only considered as an actual prospect in string theoretic research, which is itself just the study of a particular hypothesis that we are not yet able to test experimentally.
The problem is that even with that, you need to deal with an ultimate beginning.
This assumes that the universe had a beginning, which is an assumption I have no reason to grant.
Then you have the problem with explaining consciousness, and with that, near death experiences.
Explaining consciousness itself? Yes, but near death experiences are fairly well-explained now in the neuroscience research. It is to the extent that we can actually induce the sensation of a near-death experience by applying certain stimuli to the brain, all without any of the near-death stuff actually happening to the person. Also, there are many studies that have served as strong evidence that the no actual out-of-body experiences are happening with patients who feel them happen. Take for example Parnia, S.; Waller, D. G.; Yeates, R.; Fenwick, P. (2001-02-01). "A qualitative and quantitative study of the incidence, features and aetiology of near death experiences in cardiac arrest survivors". Resuscitation. 48 (2): 149–156. or French, Christopher C. (2005-01-01). "Near-death experiences in cardiac arrest survivors". The Boundaries of Consciousness: Neurobiology and Neuropathology. Progress in Brain Research. Vol. 150. pp. 351–367. or UK Clinical Trials Gateway. Primary Trial ID Number 17129, entitled "AWARE II (AWAreness during REsuscitation) A Multi-Centre Observational Study of the Relationship between the Quality of Brain Resuscitation and Consciousness, Neurological, Functional and Cognitive Outcomes following Cardiac Arrest". or Greyson, Bruce (2014). "Chapter 12: Near-Death Experiences". In Cardeña, Etzel; Lynn, Steven Jay; Krippner, Stanley (eds.). Varieties of anomalous experience : examining the scientific evidence (Second ed.). Washington, D.C.: American Psychological Association. pp. 333–367.
As for the stimuli that induce near-death experiences, consider Van Gordon, William; Shonin, Edo; Dunn, Thomas J.; Sheffield, David; Garcia-Campayo, Javier; Griffiths, Mark D. (2018-12-01). "Meditation-Induced Near-Death Experiences: a 3-Year Longitudinal Study". Mindfulness. 9 (6): 1794–1806. or Vincent, Jean-Louis (2009). "Towards a Neuro-scientific Explanation of Near-death Experiences?". Intensive Care Medicine. [S.l.]: Springer New York. pp. 961–968. or Judson, I. R; Wiltshaw, E. (1983). "A near-death experience". Lancet. 322 (8349): 561–562. or Martial, C; Cassol, H; Charland-Verville, V; Pallavicini, C; Sanz, C; Zamberlan, F; Vivot, RM; Erowid, F; Erowid, E; Laureys, S; Greyson, B; Tagliazucchi, E (March 2019). "Neurochemical models of near-death experiences: A large-scale study based on the semantic similarity of written reports". Consciousness and Cognition. Consider also “There is nothing paranormal about near-death experiences: how neuroscience can explain seeing bright lights, meeting the dead, or being convinced you are one of them” by Dean Mobbs and Caroline Watt, 17 August 2011, Trends of Cognitive Sciences. At the end of the day, we do not know everything that there is to know, and the research will continue for all of the foreseeable future regardless, but the scientific explanations are there.
As for explaining consciousness, we are actually much closer to explaining the fundamental aspects than you realize. However, I also want to point out that this is irrelevant. The fact is, a worldview has no obligation to explain the origin of consciousness, or the origin of anything, for that matter. Having explanations for the phenomena that we observe is desirable, yes, but the only obligation a worldview has is that the assertion that it makes actually be sufficiently justified and correspond to reality. Not having an explanation to a particular phenomenon is perfectly acceptable. Saying "I don't know" as the answer to a question is perfectly acceptable, and in fact, there will never be a point in existence when we will know everything that could possibly be known. Saying "I don't know" is not giving up. Saying "I don't know" is literally the first step in acquiring the knowledge and in solving problems with that knowledge.
Like I said, atheists are entertaining. I should know, I used to be an atheist...
Atheists are indeed entertaining. This is why I have atheist friends I enjoy spending time with. If they were not entertaining, then I would not spend time with them.
...until a huge miracle happened in my life.
I would love to hear your story.
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@angelbrother1238 Again, you are trying to fit your emotionally based opinion into the argument.
This is just a baseless assertion.
You can also say that God designing life this way makes it even more rare and precious.
There are two things to say to this:
(A) In the assumption that what you asserted is true, that still would not mean the universe is designed for life. It is entirely possible that God designed the universe to create black holes - something that the universe is actually really good at doing - and that the existence of life is only a by-product that God nevertheless found acceptable, as it does nothing to interfere with God's holy plan about black holes. In fact, even if God did design the universe, it still follows logically that the universe is not finely-tuned, and it does not have to be in order for life to be valuable. Most physicists and cosmologists who are religious will tell you this, an most religious people have no idea what "fine-tuning" is anyway, nor is it relevant to their beliefs.
(B) I have no reason to think that this makes life more valuable. To the contrary: most "theories of value" would render life less valuable in this fashion. The theory of value that the real world seems to operate by is the supply-demand theory: the more accessible a supply with fixed demand is, the less valuable it is, and the less accessible a supply with fixed demand is, the more valuable it is. As it currently stands, life exists on only a finite supply. Even in the assumption that life exists elsewhere in the universe, it is clear that life is extremely, extremely rare in the universe: less than 0.001% is inhabitable. However, if God is invested in creating life, then life becomes an infinite supply. Yes, it is still rare for life to exist in the universe, but only because God decided this would be true during these times. In actuality, the access to the existence of life becomes trivial if God can create life at will and is willing to do so. Hence, in whatever form there exists any demand for life, the supply has increased infinitely, so its value has decreased likewise. Besides, if you postulate the existence of an eternal afterlife, this also decreases the value of life. This is because life is valuable almost entirely due to the fact that death is an inevitable part of it. This means that every second we spend alive is precious and counts. Every second you spend with someone counts, every second you devote to the improvement of humanity counts. If an eternal afterlife exists, then being alive here on Earth no longer has any value of any kind. Well, maybe it has value to God, but most definitely not to us. The reason humans appreciate life so much is precisely because it is delicate, feeble, can easily evanesce.
The better question is what emotional event caused you to not want God to exist.
There was no such event. In fact, even for a few years after I deconstructed my Christian faith, I still felt anguish, because I did not want to abandon the religion. I wanted to believe, but found myself in pain when I realized that, if I take a close look at the evidence, the beliefs become completely untenable. To be clear, I no longer feel this way. I am completely at peace with my lack of religiousity, and the quality of my life has improved significantly since then. All I am saying is that, even after I stopped believing God exists, I still had continued wanting to believe God exists. There was never a point when I said "I wish God does not exist." As for why I stopped believing, there was a myriad of factors. I cannot deny that there were no emotional events that happened that affected my ability to believe, but ultimately, what had the most impact in my ability to believe was obtaining an education in science, philosophy, and religion. I have become acquainted with many religions, and I have started investigating all the ones I studied more deeply, including Christianity, my own religion, and the more I investigated, the more problems there were. Getting a better education in science and philosophy also helped me stop taking many deeply-ingrained ideas that Christian spokespeople inject into you for granted, and started questioning more deeply.
Irrationality rules is someone that would bend his beliefs and convictions when he is socially ostracized.
This is a baseless accusation, and the only thing it proves is that you lack any arguments to present against his points, so you have to resort to insulting his character instead. How extremely emotionally mature of you.
We both know this
Nope, not at all.
This guy is basically making money off his sheep viewers.
Sheep viewers? I have a degree in physics, and another in philosophy. And there are many things I disagree with, when it comes to Rationality Rules. In fact, in one of his videos in against the Kalam series, I heavily criticized the video, having an entire thread dedicated to that. I still ultimately agree with his overall conclusion that the Kalam argument is a bad argument, but that particular video was not a good one.
But again, you seem to have to resort to insulting his viewers, because you actually have no arguments to present against the points being presented. This is pretty pathetic, if you ask me. Every sentence in your comment decreases my opinion of your emotional maturity. So, if you actually have any arguments to present, then I suggest you lead your next reply with those, and omit the insults. Otherwise, I will just dismiss you as an Internet troll.
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Quaternions not only are useful in quantum theory, but also in general relativity, describing four-vectors. It is better to see this if one describes quaternions as paravectors in 3-dimensional space.
Let V be a vector space and F be the scalar field under V. Let P be the direct sum of F and V, such that if p is in P, s is in F, and v is in V, then p = s + v. Now define an inner product on V such that v•v = ||v||^2, and v•w = 0 if w is orthogonal to v. In other words, the inner product is Cartesian. Then the multiplication operator of P is defined by p*q = s.t + s(v) + t(w) + v•w, where p, q are paravectors, s,t are scalar elements, and v,w are vector elements, with * being the paravector multiplication operator, and . denoting the multiplication of scalar elements in F. Define the magnitude operator by ||p||^2 = p’*p with v’ = -v and s’ = s being the conjugation operator, such that p’ = s - v if p = s + v. Then p’*p = s^2 - v•v =
||p||^2. One has v = x(i) + y(j) + z(k), where i, j, and k are unit vectors in V, and x, y, and z are elements of F.
Now it is evident that the algebra of P is equivalent to the algebra of H, the set of quaternions. Further, we can define division too. Given some p, p^(-) = p’/||p||^2, where ^(-) denotes the multiplicative inverse, or otherwise, the denominator operator, if one wishes to call it this. Then for some v in V, v^(-) = - v/(v•v). This is still equivalent to quaternions division.
In relativity, the 4-position is a parevector of the form (ct, r), with r = x(i) + y(j) + z(k), as a function of proper time τ, and ct being the time coordinate multiplied by the speed of light. The spacetime element, which is usually described as the square of the Lorentz invariant magnitude of the 4-position, is equal to the quaternions square (or paravector square) of the 4-position with itself. Namely, ||X||^2 = (c.t)^2 - ||r||^2.
In this respect, four-vectors, which are a special type of paravectors, can also be treated as quaternions.
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@timothyhicks3643 Some Orthodox Tewahedo Christians consider the Kebra Nagast to be sacred scripture on par with their version of the Bible (which itself also contains scriptures not accepted in any other Christian denomination).
The Ethiopian Orthodox Tewahedo Church does not consider the Kebra Nagast to be sacred at all. It certainly is an important book to them, but nowhere in the official stance of the church, or any associated theologians publishing, it is stated that the Kebra Nagast is sacred scripture. There could very well be groups of believers who treat the Kebra Nagast as sacred, but such groups are by no means representative of the population that belong to the church.
Before the 4th century, the earliest Christians used many different gospels (or none at all, in the case of very early followers such as Peter and Paul) and Jewish texts as sacred scripture, but it would be silly to deny that they were Christians because of this.
This is another terrible argument. Why are you comparing a religion to itself separated by millennia, when this discussion revolves around discussing the religions that exist today, comparing how they differ, and how and why thet began to differ? Obviously, Christianity originated from somewhere, but what it originated from is not itself Christianity. It originated from Second Temple Judaism, but Christianity is not a sect of Second Temple Judaism. In fact, there is no such a thing as Second Temple Judaism today. Whatever sects of Judaism did not become Manichaeism, Islam, Montanism, Christianity, or some other minor religion, became Rabbinic Judaism, which essentially is a daughter religion of Second Temple Judaism. In a not-too dissimilar fashion, Mormonism is essentially a daughter religion of Christianity, and as I noted earlier, it has its origins in Christianity. But we are not comparing Mormonism today to Mormonism's origins, and asking if they are the same religion, now, are we? Just as we are not comparing Islam from the 7th century and asking if it is the same religion as Islam today. Such questions are completely not analogous to what we are discussing here, so I have no idea why you think this is somehow a good point.
Many different people with massively contradictory beliefs call themselves Christians around the world.
This is discussion is about classifying worldviews as either religions in their own right, or as sects of a religion. It has nothing to do with the beliefs of any individual person. No two Christians in the world have the exact same beliefs. No two people of any worldview, for that matter, have the exact same beliefs. And a person can call themselves whatever they want to call themselves. This is fine, but this has absolutely nothing to do with what we are talking about. This entire comment of yours has just been red herring after red herring. Besides, the extent to which two sects of Christianity disagree is usually exaggerated by the extremists, who are usually far more vocal than most Christians, who, for the most part, are just minding their business, and have no real investment in getting caught up in denominational wars. The extremists feel common, but hey, this is to be expected when 2+ billion people follow the religion.
To single out the beliefs of a portion of them to use as a model by which to judge whether everyone else deserves the title is worthless.
I have done no such thing, and no self-respecting scholar is doing such a thing either. And what is this nonsense about "deserving the title"? That is not at all how any of this works. Informal projects like this video, as well as formal research, when it comes to the topic, are all about one thing only: finding empirically-supported and conceptually-sound ways to usefully classify worldviews. This ultimately has no bearing on how a church chooses to identify itself or name itself. If a church names itself by a name, then I will call by said name. I have no problems with this. In fact, the only reason I have not named the Church of Jesus Christ of the Latter Day Saints by name is because... well... I was never talking about that church in particular. I was talking about the entirety of Mormonism and the Latter Day Saints movement, which, as I said earlier, has multiple sects in its own right. That being said, how a church names itself ultimately has no bearing on how we classify it as a worldview, just as how the Democratic People's Republic of Korea names itself has no bearing on its actual structure of governance.
And by the way, while we are talking about names, you mentioned how the name Jesus Christ is mentioned in the name of the church, while completely neglecting the emphasis the name makes on the Latter Day Saints concept, which is actually a fundamental part of the soteriology of the movement in question. After all, the movement's soteriology is centered around the exaltation of humanity, and in the idea that by entering into the Christian convenant of baptism, they, the latter-day saints being named in the name of their church, can become gods in their own right, and become coequal with Elohim. This is the entire point, and your entire argument so far has simply underplayed the importance of this concept of the church for the sake of denying their uniqueness as a religious tradition.
In my view, the only useful way to draw the line of which religious traditions are Christian religious traditions and which are not is the single factor that all self-professed Christians have in common: their faith revolves around Jesus Christ.
And this is an extremely naïve, myopic way of doing things, which is why professional scholars, who have put in centuries of research into this, do not agree with you at all on this point. And I have already cited plenty of scholarly sources to prove it, so there is really nothing else for me to say here.
Anyway, I am going to stop engaging with this conversation, because I am fairly certain this is going to nowhere, and I think I have more than sufficiently supported my point, and I have nothing to gain by continuing this.
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No, 0*0 = 0. However, this does not imply 0^0 = 1 is false. We take this as a definition for the following reason: if you take a and you multiply by 5 two times, what do we have? 25a. So 5^2*a = 125a, proving 25 = 5^2. If I multiply a by 0 exactly 3 times, what do we have? We have 0a, so 0^3 = 0. Then if multiply a by 0 exactly 0 times, what do I have? a, because multiplying by 0 exactly 0 times is the same as not performing any operation on a at all, which implies 0^0 = 1. And this also fits perfectly with the notion of the empty product. If I multiply 0 factors together, I always should get 1 regardless of what those factors are. It is the only way to make products consistent.
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It is completely valid within the context of mathematical reason to claim that the sum of all natural integers does equal to -1/12 simply because it is sensical to say that Riemann-Zeta(-1) does equal said sum. In the video, it is mentioned that the function is defined at s=-1 not by the original sum, but by the analytical continuation. However, the analytical continuation simply extends the definition of the sum so that it is also defined at -1. Remember, the exponential function f(x) = exp(x) is only originally defined for real arguments. Notwithstanding, we know that exponentiation is now an operation defined for all complex numbers, and we know that e^πi=-1. Mathematicians do not say that the analytic continuation of the exponential function for x=πi is evaluated to be -1. Instead, they use the continuation to extend the definition of exponentiation itself and then claim that exp(x)=-1. The definition of infinite sums is not defined in principle for divergent sums. However, we can extend the definition so that infinite sums can be defined also for divergent sums, and rather than saying the evaluation of the analytic continuation of the function that expresses the sum equals a certain value, we prefer to say that the extended definition of the sum allows to say that such divergent sum simply equals such value by said extended definition. Therefore, -1/12 = the sum of all natural numbers because Riemann-Zeta(-1) = -1/12 by its extended definition. The definition itself is extended by the analytical continuation, not just the mapping of the function, in the same way that exponentiation has an extended definition rather than simply an extended mapping from continuation. Because, to say otherwise, it would mean that exponentiation at complex numbers is still not meaningful, yet we can assign numbers to the analytic continuation of exponentiation for all numbers. And that is actually very different and very painful. That would mean that the identity e^πi=-1 is false since exponentials for complex numbers are undefined. For this type of reasoning to be consistent, we either need to give that and many other identities, or we have to declare that the Riemann-Zeta Function has an extended definition which makes divergent sums meaningful within the context of finite complex numbers. The latter is simply preferable due to mathematical convenience.
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Raphael Schmidpeter 1. Adding a number to the sum of the Riemann-Zeta function does not need to produce another sum which is also in the form of the Riemann-Zeta function. The property f(x+y)=f(x)*f(y) is a property that by definition only the exponential function f(n)=e^n needs to obey. So I see not why does that bother you.
2. Convergent sums are already defined in terms of a limit, which should constitute a problem for you according to your argument. Let f(x) be Σ (a,x) of g(n), where (a,x) is the summation interval. Convergent infinite sums are defined as the following: f(Infinity) = lim (x approaches infinity) of Σ (a,x) g(n). Infinity is not a number, and as such, it should never be an input for which there is a defined output. Therefore, convergent sums should also be undefined because, according to the notion of a function, the idea of infinite sums being defined contradicts the rules of algebra. However, the notion is not only defined, but also defined as the limit of x approaching infinity, which itself should constitute a second problem. The second problem, there are functions for which f(c) does not equal the limit of f(n) as n approaches c, and for which the limit is defined, but not the function at that value. Thus defining f(c) for an arbitrary value (which happens to be infinity) as its limit should not be regarded as valid. Yet it is. See, in Calculus, the limit as x approaches 0 of 1/x^2 is infinity, but the expression 1/x^2 remains undefined. If the approach of defining an infinite sum as its limit is valid, why is assigning a solution to 1/x^2=y, x=0 not? There is a problem of inconsistency there with our definitions. But that axiom is still accepted in mathematical disciplines. Then, it follows that there is no reason to reject the axiom that at infinity, continuity is no longer meaningful, and that, hence, divergent sums can be defined to result in finite numbers via the Riemann-Zeta function. In fact, the indeterminate form 1^Infinity yields discontinuity at infinity for many exponential functions, but that does not stop mathematicians from evaluating those limits for those functions. Infinity is not a value, it is not an interval, it is not defined as anything that can be the input of a function. Therefore, the concept of continuity at infinity should be meaningless, and it should not present a problem regarding Riemann-Zeta sums and divergent sums. So, if you regard the definition of convergent sums to be valid, there is no reason I should be restricted from giving divergent sums a finite definition. Sure, you can choose to reject the axiom, but that does not give any justification to say the axioms itself is far-fetched. The axiom is completely valid and accepting or rejecting it is entirely a matter of choice.
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Allow me to rephrase what I've said. A priori propositions are not opinions, but if they must be assumed axiomatically, and if they are arbitrary, then they are subjective, because such a choice as true or false is dependent upon the subject claiming the proposition. The fact that you assume that "Truth is knowledge of things as they are, as they were, and as they are to come" to be true a priori, arbitrarily, and axiomatically, implies it is a subjective claim, and it is subjective to the stream of epistemological thought that one subscribes to. The fact that God knows that K implies a truth-value assignment for P(0) and P(1) we assume a priori, but any attempt to justify renders infinite regress, and axiomatizing this renders subjectivity.
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mackdmara Such an accusation is invalid for the reason that your description of my argument is inaccurate. My argument concerns not the reliability of science and math and whether they are real or not, my argument concerns the axiomatic nature of every set of premises and the impossibility of complete a posteriori justification, which is very different. Reality is itself a construct, because the definition of reality already depends itself on the assumed set of axioms. That cannot be taken lightly. That implies not that by any means, anything can be true if you assume it to be true. The process of choosing which axioms to assume is a very difficult, precise, and sophisticated one, and we base it on practicality. Skeptics care not about practicality and challenge all axiom sets equally. I am not challenging all axiom sets equally, I am merely postulating such axiom sets do exist, but must be assumed a priori according to arbitrary standards. However, I do base my choice of axioms on efficiency and practicality. That is why my argument is not skepticism. My argument postulates that any claim about truth is in itself a truth claim, and that is the only reason there is an infinite regress, this is not a matter of whether science and mathematics are reliable constructs. Mathematics and science simply cannot be concluded from an objective standpoint and must be assumed axiomatically, but from a pragmatic worldview this is entirely valid.
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mackdmara No, you're still misunderstanding.
You claim that, according to my logical worldview, no conclusion can be true, and because I'm claiming such a circumstance to be true, my argument is self-defeating. This isn't true, and I already addressed this a dozen of comments ago. I clarified that nothing is meaningful UNLESS you allow axiomatization. There's nothing wrong with axiomatization: most philosophers accept it. But the one thing some of these philosophers need to accept is that axiomatization is equivalent to subjectivity. You haven't been able to accept subjectivity because you ignore the existence of axiomatization. My proof is axiomatized: it's not objective, but it's still meaningful because I've admitted I had to assume a set of axioms, called logic, and I've admitted everyone else in this thread, except for you and kws is assuming that set of axioms. So that's fine.
No, God doesn't need to be omniscient in order to exist. See, the omniscience of God is a logically derived property, not a definitional one. Jewish theology states that the definition of God is that God exists by logical necessity: virtue of God's own nature. All other properties can be derived from this, but they're not part of the definition and thus unnecessary. The idea that God must be omniscient stems from the intuition that existing supernaturally should logically allow omniscience, but it doesn't imply it by necessity.
Now, if you do wish to assume axiomatically that God must be omniscient, then yes, I would in that case be claiming God can't exist. Why? Because the infinite regress exists by logical necessity, regardless of whether God can logically be omniscient or not.
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mackdmara I never claimed subjectivity is practical outside of philosophical circles, but that by no means implies subjectivity is false. And no, if I'm right, it does not mean I'm not right, it merely means I'm right because of the axioms everyone has chosen, and choosing other axioms is impractical. Again, if you don't understand the argument, it's best to not critique, you made a wise decision of stepping out of a discussion you admitted having little credentials for, and it would do you good to keep things that way, unless you prefer to make a fool of yourself. I already explained why my argument is consistent. Logic renders my argument correct. You can assume a set of axioms different than logic to make my argument false, but realistically speaking no will do that. So, no, it isn't absurd.
Also, I already explained why the statement "existing before time" is self-defeating.
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Maximilian Kircher I think you misunderstood the proof.
Truth is a quality of a proposition. But what does that quality represent is subjective. Why? Because you can have different standards and there isn't anyway to choose any of them objective.
K: corresponds to physical reality - empiricism
M: corresponds to mental reality - anti-empiricism, or Cartesian rationalism
I provided two standards of truth as examples, there are many more though. But the point would still apply.
Let's say that a statement is true if physically observable via evidence or directly. So,
P(0) is True IFF P satisfies K.
But you realize this definition itself needs to be proven true, because, how do we know empirical adequacy implies truth? If we merely decide that as a convention, such is subjective. So we actually do need a justification for our choice, no matter how intuitive that choice is. Otherwise, it just becomes logically inconsistent. So the notion of empirical adequacy implying truth must itself be true in order for the above definition to be correct. Therefore:
[P(0) is True IFF P(0) satisfies K] is True, which by logical necessity implies [P(0) is True IFF P(0) satisfies K] satisfies K... but this is circular reasoning. It is circular reasoning because in order for satisfying K to imply truth, the latter needs to itself meet K AND imply truth as a consequence. And undoing the circularity yields infinite regress, because the proposition given above becomes
[P(1) is True IFF P(1) satisfies K]=P(2)
Where P(1)=[P(0) is True IFF P(0) satisfies K].
So we see that the only way P(0) can be assessed as either true or false (rather than as indeterminate or unknown) if P(1) is already known to be true, but P(1) can only be true if P(2) is true, but P(2) can only be true if P(3) is true.... P(N) can only be true if P(N+1) is true... ad infinitum.
Hence infinite regress. There is nothing anchored in P(0) as you state
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William Brown They aren't false dichotomies though, you haven't proven they, and what is worse, you can't prove they are.
Lack of free will DOES imply lack of moral sense. It is already implicit in the definition of morality itself, so that statement actually is true by definition.
Also, if it is part of your character to act a certain why, then that also by definition implies lack of free will. I think you lack understanding of how free will operates. If God is choosing an act, then there is a justification behind the choice, and because there is a justification, that means it was not by any means influenced by any personal characteristics. That is how choice simply is defined, it isn't even really a debatable idea.
Omniscience means all knowledge. But knowledge implies absolute certainty, lack of certainty implies belief and not knowledge. However, both by the Occam razor and the burden of proof, every claim requires a justification. This is because if a claim is true, that means the claim satisfies all the properties of truth that are determined by the definition of truth itself, but that definition of truth can't be proven, because proving it implies a demonstration of it being true. And demonstrating that the definition of truth is itself a truth requires circular reasoning. One can only undo the circular reasoning by using an infinite chain of justification. Justify the definition with a claim, but that claim itself will require justification, and that justification also needs justification, ad infinitum. Hence an infinite regress.
To give an example:
2+2=4 is true.
Well, why? What makes any true statement true? Well, they meet a definition. That definition is, the conclusion can be implied from the premises. So, 2+2=4 is true iff it follows from the premises AND if following from the premises makes any statement true. But this last part cannot be proven. Proving the definition of truth is impossible.
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William Brown William Brown William Brown William Brown William Brown William Brown William Brown William Brown William Brown William Brown William Brown William Brown William Brown William Brown "You are the one making the claim omniscience requires infinite regress, so explain."
I already did. I'm not sure you're even trying to read my comments or that you understand them then.
"It seems to assume past events have occurred in order for it to be an infinite regression."
No, that isn't what an infinite regression is. Take a course on logic. I already explained it, an infinite chain of justification.
"Morality: principles concerning the distinction between right and wrong behavior."
Not a valid definition, as it is circular and hence a fallacy. You must define morality in terms that aren't itself.
"Free will isn't necessary. If someone cannot carry out an action, it doesn't meant they can't comprehend that it would be."
Actually, it does mean that, yes. Empirically shown. That is why... parenting is a thing, you know. And it is impossible to derive a non-arbitrary correct assertion about an action hasn't been experienced by the actor.
"Free will is the act of choosing, acting and thinking voluntarily."
Exactly. But if your choice is driven by a personal characteristic, it isn't voluntary, by definition. It follows from the definition of both voluntary and of driven.
"Where do you get justification is opposed to personal characteristics?..."
See above.
"So if God makes a choice to take a life or not take a life, it's no longer a choice no?"
I never claimed that. Stop using straw men fallacies, and stop twisting my words. What I did say is that if actions are driven by a personal characteristic or property, they're not a choice, it has nothing to do with the action itself. You need to learn how to read and/or be intellectually honest in a debate.
"You're assuming falsely that you need a justification for your justification."
No, that isn't a false assumption, it's a basic law of logic. It's called the burden of proof. If a claim has no justification, then the claim is arbitrary and as such is dismissed as a non-sequitur fallacy.
"Something doesn't need to satisfy your opinion in order to be true."
Straw man fallacy AND red herring fallacy. I never claimed this. This is intellectual dishonesty, and it shows you have no grounds to support your position. Sad.
"You defeat your entire position if you claim the definition of truth requires circular reasoning."
No, it doesn't. It defeats your argument, because you argue truth is objective. I'm not arguing that. It is only self-defeating for an objectivist position.
"Where is your justification."
Above: the burden of proof.
"Where is the justification for the justification."
The Occam razor. Although, in all technicality, I'm not required to have one, because I'm not assuming logic is objective, so I can just assume the burden of proof to be true and then match on.
"Truth: that which is true in accordance to fact or reality."
...a circular definition. Those are valid linguistically, but not semantically nor epistemologically. A definition that references itself is a circular statement, which is a fallacy. Philosophical definitions don't make self-reference for that reason, but then it follows by the burden of proof that they necessitate a justification.
"Axioms exist, both logically and mathematically."
1. Mathematics is a discipline of logic.
2. Axioms yield objectivity as false, which would destroy the idea of omniscience.
"They are things that don't require justification because they are self evident."
Quite ironically, that isn't the definition of an axiom. I can show this with a very simple example: Euclidean geometry vs non-Euclidean geometry. Both are true because both use axioms, yet both axioms are mutually exclusive. Which means one of them isn't self evident. But that isn't a problem, because:
1. Objectivity is false.
2. That isn't the definition of an axiom.
"I don't need to prove my existence, it's an axiom."
...in your worldview. However, many other world views have non-existence as an axiom.
Two notes:
1. Self-evidence isn't real. If the burden of proof is self-evident, then there is a contradiction. Plus, self-evident assessments are subjective.
3. The idea that something is self evident is not only itself not self evident, but the idea that self-evidence means anything is also not self evident. You're on several layers of self-contradiction here.
To clarify, axioms are a sign of subjectivity. Subjectivity renders omniscience impossible by the simple virtue that omniscience obviously implies objectivity.
"This fails to honestly tackle the contradiction between omniscience and perfection."
I never claimed there was a contradiction between the two. More straw manning, more word twisting, more intellectual dishonesty. You honestly don't understand what contradiction means. Listen mate: a claim is contradictory if it contradicts itself, but two claims are mutually exclusive if they contradict one another. I never claimed they're mutually exclusive. In fact, I literally can quote myself saying "Perfection is impossible BECAUSE omniscience is impossible." I literally wrote that in one of my previous responses. I'm tired of your dishonesty. If you can't debate me honestly, then don't: admit defeat. But you're wasting both of our times here with this word twisting.
...and I already showed why omniscience is impossible. SMH.
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William Brown "To be honest I was typing the response as I was reading. Its obvious that I read your comments and explained my objection as to how they are flawed."
You clearly failed at it, in any case.
"Circular? The statement doesn't even use the word "morality" to describe itself."
No, but it does reference right and wrong, whose definitions indirectly reference morality, eventually, or at least in the case of the definitions relevant to ethics.
"I guess Meriam Webster, Cambridge, and Oxford are just dullards then because they all practically use the same words to describe morality."
They're indeed dullards. Check out the Merriam Webster's definition of the word happiness.
"We both now that is false as you can know whats moral and not without actually committing the act itself or be capable of it. A wheelchair bound person can know its an immoral act to kick someone in the face for no good reason by just being told. They don't even have to see someone kick another person, they can see the action of kicking and derive, "It wouldn't be good to do that too a person".
My argument was concerned with comprehending a moral act, although I somehow deviated from that. Regardless, I will concede to this. However, there still is an impossibility in being moral without free will. Why? Because free will is tied to consciousness. Notice, for example, that organisms whom we have discovered to have a consciousness have the capacity to actually make legitimate choices in various circumstances. The higher the order of consciousness, the higher the order of choice. This is an empirical observation, but the concept of consciousness, if generalized to ontology along with its correspondent aspect of free will, then we can reach similar conclusions, which many if not most philosophers do. Total lack of free will stems from total lack of consciousness, which means absolutely one choice, and this is because if only one choice is possible, then there is a causal relationship between the cause and effect, and there is no choice, and causal relationships are blind. This implies, because they're guaranteed, there is either no cognition of any external circumstances or no concern for them. This is different from your examples, in which one is unable to do on particular action, but still able to do many other.
"All choices are driven by characteristics,"
Only in a deterministic universe.
"...but we still choose which to exhibit and when. If God could choose between showing justice or showing mercy, both which are good things, why does that then not count as a free will choice? Hes not forced to choose one or the other."
Funny that you speak of God as being male. Regardless, this argument is flawed. Both may be good, but God is maximally good, a.k.a benevolent, in theological terms. God won't merely make a good choice, but actually necessarily make the best choice, simply by the nature God is characterized by. Now, if showing mercy and showing justice in terms of a simple quality are both the maximally best actions, then both would need to be executed because this would imply they are both elements of a composite action that would represent the absolute maximal good. This is an argument similar to the reason why only one perfect entity can exist at once. Moreover, the two qualities cannot exist in a situation such that separately they're both maximally good but combined not because this would imply mutual opposition or exclusivity, in which case both could not be maximally good to start with.
"You seem to be taking free will as, "acting outside of your nature" which has never been the definition of free will, yet seems to be what you are implying (note I said implying in case you try to accuse me again of something I didn't do)."
Not at all. God can only necessarily make one action simply by the nature of benevolence, so this actual does mean lack of free will.
" I never said you said that I'm using an example so calm yourself down. If I was setting up a straw man, I would claim what you believe and tearing it down."
Then it wouldn't be a straw man. Tsk.
"I'm asking a question, which obviously wasn't a statement of what you believed. But instead of looking at it honestly you get all angry and throw around insults."
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William Brown I can show that you were straw manning, but it is a waste of time to do so since it is irrelevant. But I will say that the fact that you claim I threw insults at you even though I literally never did does prove my point to begin with. And yes, I did not insult you. Just read my comment again. And yes, you did claim I insulted you. The claim is right in the quotation marks.
"I'm showing an example. The actual definition of free will doesn't lessen if its, "driven by personal characteristics" If you have no choice but to do one action out of a myriad of others, then you don't have free will. Lets go by the actual definition."
1. We're going by the actual definition.
2. You're committing the fallacy of assuming that every true assumption about a particular concept stems from the arbitrary dictionary definitions of the word representing the concepts. Even if you can't see or aren't willing to admit it, your arguments are phrased in such a way.
"I didn't say anything about a claim having NO justification, I'm talking about a justification FOR a justification."
Yes. I know. What you fail to understand is that justifications themselves are claims.
"If we go along that line of reasoning we get an infinite regress. If something is faulty, then it has no justification to begin with."
Or it does have a justification, which is in itself faulty, but those who accept aren't aware of it until later. Which showcases my point about justifications being claims. They are.
"You don't need a burden of proof for your proof and so on, thats all i'm saying."
Except you do. That literally is his both inductive arguments and deductive arguments operate.
"Even if I constantly said, "Where is your justification for that?" The burden of proof ends SOMEWHERE..."
Really? Where is the proof for that? No, I'm serious. So there is a certain point at which I'm no longer required to justify my claims? Then show me where, it'd be very helpful at ending several debates I'm involved in, including this one.
"Unless now you want to argue how your whole premise is not objective."
I do want to argue this, and there is nothing wrong with that.
"Are you not being the least bit introspective and apply the same standard to yourself?"
I am.
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William Brown Truth is objective otherwise the entirety of your arguments could be false."
They could be false, but if you assume the basic axioms of logic, then they're true. And sure, you could reject the basic axioms of logic, but according to you they're self evident so I think you and I would both understand the problem with that. So I'm not concerned.
"You assume both Occam's razor and burden of proof are objective standards to define truth yet argue against objective truth."
No, I'm not assuming they're objective. They're subjective alright. But if we're going to use any axioms to try to build logic and our entire framework of knowledge, then those axioms will've to be the ones of basic logic. For our purposes in particular (not necessarily for any other purposes, certainly not for all purposes), they're the best axioms to use. A.k.a, in relation to our purpose, accepting them as true is the best choice.
"Its entirely nonsensical. Its like using logic to prove logic doesn't exist."
False equivalence. Claiming logic is subjective is not even close to the idea of declaring logic to be false. I'm claiming logic to be subjective, but I still assume it a priori to be true.
"Again, I guess Oxford and Webster don't know what they are talking about when they describe what an axiom is?"
1. They disagree on the definitions in some aspects, which means: even if they do know what they're talking about to some extent, each of them is wrong about something.
2. They're English language dictionaries, not philosophy dictionaries, so for all we know, their definitions could be completely irrelevant.
"Webster: a statement accepted as true as the basis for argument or inference : "
Yes, a statement ACCEPTED as true. Not because something is accepted as true does it mean it is objectively true on an ontological, metaphysical sense. Axioms are different from any other claim in that we assume them to be true a priori, a.k.a they're definitions constructed by us, also known as analytic statements. Any other claim requires justification and must be a posteriori true, these are also known as synthetic claims by epistemologists. So, that definition only helps my argument. Because this is the very thing I was arguing.
"...postulate 1 one of the axioms of the theory of evolution 2: an established rule or principle or a self-evident truth cites the axiom “no one gives what he does not have” "
Self-evident is subjective. The fact that there are no claims on which exactly 100% of the population agrees with goes to support this. Although it isn't much of a support.
" 3: a maxim widely accepted on its intrinsic merit the axioms of wisdom"
Same as for #1. These axioms do have intrinsic merit, and that intrinsic merit relies in that they are natural to us because we construct their truth based on our instincts and what appears obvious due to our mechanisms of survival. That doesn't make them true on an ontological aspect though.
Oxford: "A statement or proposition which is regarded as being established, accepted, or self-evidently true."
Yes. This actually helps my arguments. Axioms are precisely statements that we establish to be true. We decide that they're true. But they can't be proven to be true or false. We can choose whether they're true or not. A.k.a: subjectivity holds. That is what subjectivity fundamentally is.
"But all make a distinction between a mathematical axiom and a philosophical axiom."
Mathematical axioms are special forms of philosophical axioms, but they're underlying nature and how they operate within mathematical systems is essentially the same with some differences corresponding to the sub-field itself.
"So I made a mistake when assuming a logical axiom is the same as a mathematical axiom, my bad."
You actually assumed the opposite in a way, or so did your claims imply. But that wouldn't have been a mistake, in any case.
"The axioms i'm referring are indeed self-evident,even the translation from the greek word is, "that which commends itself as evident" "
1. This only means they were self-evident to the Greeks specifically, not universally or ontologically.
2. The etymology of the word is irrelevant. The word logic comes from the Greek work logos which literally translates to speech. Yet this is clearly not what logic today is. The word physics comes from the word physis in Greek, which means nature. But that isn't what the word refers to today, that isn't even the definition you'll find in most dictionaries. This is a fallacy.
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William Brown "So 1. Objectivity is not false (otherwise you couldn't say is it false without it being objectively true. So that statement defeats itself)"
It doesn't defeat itself. It isn't objectively true, but that doesn't detract from my argument because the argument follows from the laws of logic as I mentioned above, which once again I'm assuming to be true. And that isn't a problem, because you agree that these laws are self-evident and thus the more "correct" or more natural to accept as true.
" "...in your worldview. However, many other world views have non-existence as an axiom. " Then that worldview logically fails if non-existence was an axiom, they wouldn't be there to ponder the question. "
Not at all. These worldviews have already come up with ways to debunk this rebuttal. See, there is a flaw in assuming that existence is a requirement in order to be able to ponder. In fact, it is even fallacious to actually pretend the action of pondering behaves anything like it does in our worldviews.
"1. Self-evidence is real. Otherwise you would have no argument to make."
I never self-evidence is real, I said it is subjective, which makes it totally meaningless and irrelevant.
" We all make our arguments..atleast logical ones..on self evident truths."
Correction: on OUR self-evident truths. See, to some people, the idea that the Earth is round is self-evident. To many others, however, it isn't. Same with the existence of God.
"Even you disagreeing with me, you don't go by anything that is an "irreducible primary"?"
I assume the basic laws of logic, but I don't claim they're necessarily true by virtue of themselves.
"Because by that logical you could be wrong and have no argument to stand on."
So could all claims be. This isn't a rebuttal of anything and I'm not sure it contributes to your argument at all. I COULD be wrong, but I've yet to be proven so, and since I'm assuming the axioms of logic, people either need to show to me that the axioms objectively are false or that my argument doesn't follow from the axioms of logic.
"Axioms, by philosophical definition are not subjective, you only contradict yourself by trying to contradict an axiom."
They literally are though. And you haven't provided proof your claim: the dictionary definitions you provided actually help my argument LOL.
"Just try using your line of reasoning on your own arguments, you will hopefully see how it easily falls apart."
No, I've used my line of reasoning on my own arguments for years, in hundreds, perhaps thousands of debates. It's not for nothing I'm arguing this.
"Did you not say this: (unless you go and edit your own post), "Perfection is impossible because omniscience is impossible" ..."
Yes. I did claim this. And the word BECAUSE is NOT A CONTRADICTION, it is a causal-justification relationship. I hope that was merely a brain fart, because this really is a stupid claim you make here.
"You hinged one upon the other, so why not stop presuming straw men, apply your same "logical" reasoning to yourself, and stop lying on me?"
I'm not lying on you. I'm not the one who claimed I'm being insulted when I'm not.
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Ananta Sesa das 1. I never claimed individuals have no free will, I merely claimed they would not in the hypothetical case of an omniscient god existing, which is logically impossible, implying this is not the actual case.
2. Adequacy is different from perfection. Individuals misuse the term perfection to mean adequacy. Perfection is, once again, a concept of objectivity.
3. Omnipotence is impossible because it is self-inconsistent, hence, a contradiction which can be dismissed as a fallacy and thus an impossibility.
4. Omniscience is also self-inconsistent, so omniscience is impossible regardless of the impossibility or possibility of omnipotence.
Omnipotence is impossible because an entity cannot choose to be able to choose to be omnipotent due to the logical necessity of being able to choose to be omnipotent in order for the property of omnipotence to be satisfied.
Omniscience is impossible because it can be proven that within any randomly given collection of propositions, at least one of these must necessarily be unprovable due to infinite regress, and I explained by showing that truth requires a subjective definition. Because such claims are not provable, there is no certainty in asserting their truth, thus there is no knowledge but belief, with shows that at least one proposition must be believed by the omniscient entity, showing the entity is in fact not omniscient. This is obviously a contradiction, so omniscience is impossible.
Thus demonstrated (Q.E.D).
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"That is not what the record shows."
This is evidently false. I have looked at the record, and you provide no evidence of your claim, precisely because it is false.
"There were many people that came up with a counter that you used jargon to answer."
No, no one did. Show me 1 single quote of someone besides yourself who did more than once and I will concede. Of course, you won't find one.
"Also many of these people did not understand you either. Hence them not pressing their case."
No, they understood me, and this is evident because everyone offered a counter argument at least twice before giving up. They stop pressing their case because I showed they were wrong, not because they did not understand. They would not have tried counter-arguing had they not understood.
"Also many of those that did press it you deflected with jargon, until someone who Knew the jargon disproved you handily. Did you concede? Nope."
Show me the quote in which someone disproved and I will concede. But i have read the conversation time and again, and I know for a fact no one has disproved me. Again, here is your chance to show me the quote where I was disproved and I will gladly become Christian again.
"Once that happened you engaged me directly even though I had made no comment."
Not true. I engaged with you because you responded to my argument. The evidence is right in the thread, actually. Stop lying.
"You just wanted to fight."
First ad hominem.
"It is not about what is right or logic or spreading reason."
For me, it is. For you, it isn't, because you demonstrate you don't care about logic or reasoning all that much. You're not willing to let go of your faith and open your mind to facts. But this has been about logic all along. My argument literally is based on formal logic.
"Your just trying to prove to yourself that you are the intellectual superior to those around you."
Second ad hominem.
"Your just trying to prove to yourself that you are the intellectual superior to those around you."
Third ad hominem.
"In summation, I told you why I do not want to argue with you,..."
...you did, and your reasoning is wrong and full of feces. Simple.
"...you do not respect me."
...because you do not respect me either, nor do you respect anyone commenting on this thread. I have freedom of speech. By telling me to not respond to your comments, you infringe on my freedom of expression, hence disrespecting me. So I have no need or obligation to respect you.
"I then offer to cordially debate, you basically call me deficient, you do not respect me."
I didn't call you anything. You made that lie up. All I said is that it isn't my fault you didn't understand my argument, and you accused of not speaking on plain English even though I literally am speaking plain English. On top of that, you blatantly lie about "what the record says" and deny that you started this. Also, fourth ad hominem.
"It is called ad hominem & it is your favorite fallacy."
You've used more ad hominems within this one comment and the previous than I've used within my last few responses to you. If anything, it is your favorite fallacy. In fact, declaring that it is my favorite fallacy is both an ad hominem of itself and a straw man as well.
"You are not entering into this with good faith...."
Sixth ad hominem, or maybe seventh. Also, I did enter here with good faith. Hence why I engaged with everyone here. You're the one who refuses to engage with me. So, I think it may be you who came in good faith, but it is not my position to declare this.
"You are not entering into this in an intellectually honest way."
Neither are you.
"You are educated, but it has not given you wisdom."
Eighth or ninth ad hominem. Also, irrelevant. You don't have any more wisdom than I do. Wisdom isn't real.
"You claim seventeen years of faithful service to God, is that one to seventeen years old?"
I claim seventeen years of faithful service to God. i was a pastor. The age range here is irrelevant though. Straw man.
"What part of writing to you is productive?"
All of it, because I listen to people. Maybe you're not willing to listen, but personally, I am.
"All you have to offer me is hate & all I have for you is pity"
I have no hate to offer. I don't hate anyone. I just want to teach you the reasoning behind why omniscience is impossible, but you refuse to listen either out of fear or for some reason you hide from me. Trying to teach you something doesn't make me hateful. Nor does it make you hateful either.
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"I looked over it. You are wrong & it is all there if you want to read it."
I read it all, and I know I'm not wrong.
"Odd that you missed the part where I did not want to argue with you because, I am not a philosophy major."
You only chose to stop arguing after you failed to persuade me with your flawed arguments, but rather than admitting this was the reason, you claimed you lacked sufficient scholarly education in philosophy to argue with me. Which is okay, but it is not an excuse. Most people commenting in YouTube are not scholars in philosophy either. So your point is lame.
"I do not want to argue with you but you insist on doing so."
If you truly wanted to not argue with me, you'd have stopped engaging and responding a long time ago. You've not chosen to stop though.
"This is proof that your last two posts to me, after I told you I do not want to argue with you, are at least against my wishes if not with intent to harm."
Ad hominem. I also have no intent of harming anyone. You can stop creating false assumptions.
"If that was true, would you not seek people that can understand what you said or make it accessible to any reader by teaching the principle?"
This is precisely what I've done. Once again, everyone in this comment section has understood everything I've said, and my arguments are fairly accessible. Perhaps you can't understand them, but it is irrelevant whether you understand or not: if everyone else understands, then this is proof it is accessible.
"Instead you choose to come after someone who told you they cannot understand that level of philosophical discussion."
No, I came at you because you actually told me to shut up and stop responding to your comments, which you have no right to do.
"Then you claimed the thread as yours."
I did not.
"Saying I could not comment to anyone on it (not verbatim) without engaging you."
Not what I said, and the fact that you refuse to provide verbatim quotes for the sake of twisting my words makes you dishonest. You're a real hypocrite, and yes this is an ad hominem, but still a true claim.
"This is the thread started by SirBagicious, not you."
It wasn't started by you either. I came to this comment section before you were here, and you responded to me and to the entire thread even though no one was talking to you.You use double standards.
"Make your own comment thread, then you might have a claim."
Make your own comment thread, then you might have the right to not engage with everyone at once.
"Otherwise it is clear who I was speaking to, & that is not you."
No, you were talking to EVERYONE including me. That is literally how comment sections work. Everyone has a right to respond. It's called free speech. Maybe that doesn't exist in your country, but it exists in mine. Learn to respect that.
"Let me state this plainly, your a bully. Your need to seek out a clearly weaker target in your area of expertise. This shows your inability to compete within it or a perceived inadequacy within yourself to your peers in your field."
And this claim shows you're an idiot who likes to make personal assumptions about people without any logical justification only so that you can feel better about yourself. I'd give you respect if you respected me, but you clearly fail to do so. But regardless, I do have the capacity to compete within my own field of expertise, and I do so everyday.
"If you did, you would choose to make what you said accessible or you would only speak with similarly skilled individuals who could fully appreciate your endeavors."
I made it accessible. It isn't my fault you didn't understand it if everyone else did.
"For your own good seek help of a certified clinician."
You are a person who hallucinates with a man in the sky who does not exist, and who somehow has magic powers and has infinite knowledge, all of which can be proven to be impossible, and whom by the way you have never seen or heard because he literally is immaterial. You're essentially comparable to a moderate schizophrenic. You need the clinician much more than I do.
Have a nice day .
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Christopher Sewell Citing a link of a source site which can be edited by any person is not particularly reliable nor is it intellectually honest.
Equally, declaring that the only thing which can disprove your argument is another, even more powerful link which states something in particular is a logical fallacy and therefore an invalid argument. It is called an appeal to authority,
I can play the same game though: find any written research paper by a group of mathematicians which states that Ramanujan summation is a function and not a summation, and which explains why is it called a summation despite not being such. Unless you can find any, I will not take your argument seriously.
Again, you did nothing to actually address my arguments explaining why it is NOT a function. You simply cited a link which, again, is questionable, which by itself didn’t address the arguments either.
Is this off-topic? Of course it is: your invalid arguments are dragging the topic off hand. I’m simply responding to what you’re saying and explaining why it is wrong. Summation is an operation you perform over a function, it is not a function itself. There is a reason summation is not written as S(x), but rather as an operator over a function which given boundaries. Since you didn’t address my argument at all, I can safely rest my case without saying much else.
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This is just wrong, though. The 24-hour day was not chosen due to the number of natural divisors of 24. The 24-hour day was chosen, because this implies that 1-hour is equivalent to a 15° = π/12 rotation of the Earth, and 15° is a very important number of degrees in computational trigonometry, which made it an important constant for the ancient civilizations, especially for astronomy purposes, and likely for navigation as well. Every important trigonometric constant can be expressed in terms of sin(15°), for example.
Also, in dividing day and night into two parts, this means you get cycles of 12-hours in a day, with 12 being a divisor of 60, and since the ancient pre-Babylonian civilizations used base 60, this was also important.
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@tomdekler9280 No, it has nothing to do with that. If anything, the number of importance here is 180, since it is half the number of degrees in a circle. There exists a theorem that some sin(180°/q) is constructible if and only if q = 2^m·3^n0·5^n1·17^n2·257^n3, where m = 0, 1, 2, ..., ad infinitum, and n0, n1, n2, n3 = 0, 1. Here is the interesting part, though: this is because an angle can be halved-arbitrarily, and still remain constructible, but trisection, for example, is not constructible. 180/q = 15 means that q = 12. Furthermore, sin(15°) easily generates sin(30°) and sin(45°). This not something you can do with sin(18°), for instance.
The only other generator is sin(3°), but using this division would make hours extremely short, only 12 minites, and it would make too many hours in a day, 120 of them. Besides, sin(3°) is a much more complicated quantity to express in radicals. Thus the natural generator is sin(15°).
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"morality's subjective nature has an objective qualitative factor, (i.e.
reality). In other words, human behavior (objective reality) sets the
tone for morality."
There is no objective qualitative factor in morality. There is universal moral agreement which is practical, hence allowing us to ignore the subjectivity of morality, because humans only adopt the pragmatic set of assumptions which make morality practical for our anthropocentric purposes of survival. However, this still is arbitrary and subjective to our own discretion.
"I think the notion that "one has to know they're omniscient in order to
be omniscient" is problematic when applying it to a eternal being."
No, it isn't problematic, because being eternal and being omniscient are completely unrelated and irrelevant to one another.
"That being we call God has a rational mind, and we know this in part
given our understanding of our material environment or creation."
1. Humans define rationality.
2. It is impossible to conclude any properties about an immaterial being based on an immaterial world. Non sequitur fallacy.
"But from a purely creative material standpoint, God knows everything relative to that."
This argument suffers from a flaw, since it assumes that God created the material universe, which is false. God could not have possibly created the universe because due to the very nature of the universe itself, it can't have a cause.
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"Intelligence - the kind were displaying in this mere discussion - is the product of an another intelligent agent(s)"
Intelligence is the product of coding, which can be done simply by storing chemical and thermal information on particles and waves. Life is not a requirement for intelligence to begin to exist, and by extension neither is intelligence itself. We know this from the Stanley Miller experiment.
"We can't apply these material attributes to a being, I believe, is immaterial and eternal."
It is special pleading because both the immaterial claim and the eternal claim are unjustified. In fact, it may be possible to argue that both are actually impossible, but I have yet to verify this with an argument myself.
"Unfortunately we have no way of pinpointing or measuring when such a phenomena may occur."
Then there is no reason to believe such a God exists, nor is there reason to think God is the cause of anything. You're not making much sense here.
"We can't anthropomorphize the universe."
Claiming the universe is everything that is not an anthropomorphization of the universe, it's an ontological claim, and a semantic one. It's true by definition.
"By contrast human civilization ties God with ethical constructs."
This is ad hoc and unjustified. There is no reason to think God is moral.
"Thus we conclude God is a being with a mind. "
No, in that case we define God as having a mind. Words are labels. If something doesn't have a mind then it possibly cannot be God. That's part of the definition of the label itself.
"It can't be predetermined if humans exercise accountability."
If humans exercise accountability, then God is not omniscient.
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"Dennis summation of opinionated desirability / undesirability is correct."
We agree on this much.
"His point is that, if the proposition is true, it is merely a emotional appeal w/ out any real inherit substance and or moral force to apply or agree upon."
Not true. Be careful. While it is true that this would imply it is an emotional appeal, it does not imply there is no real substance to it. We do have reason to agree on a set of collective morality. The only implication stemming from emotional appeal is that this collective morality is not objective and therefore highly arbitrary and non-deductive. It says nothing about whether it is meaningful or not.
"Also, this isn't how the real world operates."
It need not be. Objectivity has nothing to do with practicality. Something can be established subjectively while remaining practical. Please review the definition of objectivity: nowhere in it is there mention of practicality.
"Nobody in their right mind believes there isn't something inherently objectively wrong in certain human behaviors."
This is true, but a straw-man argument. Humans are animals, which means that we have an instinct to survive and use our abilities for this purpose. Hence, we have an inherent bias into believing that certain things are simply moral imperatives, but only because these things serve our own purposes, and purpose is also subjective. If we had been robots instead who hadn't an instinct for biological survival, but instead had some other purpose, then our moral biases would still be very different. Which goes to show that morality is still subjective, and this isn't negated by the fact that we have inherent biases towards moralities of survival.
"We need not hold to an unconscious unintelligent standard - a ludicrous standard IMO - when applied to rational autonomous agents."
I agree. We need no objective morality in order to operate on a functional society. We are rational beings, so we are intelligent enough to agree on subjective moral standards to serve our purposes, which is completely fine. However, keep in mind that by no means does this imply morality is objective. You seem to think it does.
"Logically only conscious / intelligent beings can set the objective standards for conscious / intelligent beings behavior."
False by definition. Revise what objectivity is. Keep in mind that objectivity and practicality are very much unrelated. I'm claiming morality is completely subjective, and that it does indeed boil down to a debate of what is desirable versus what isn't. However, this has nothing to do with the fact that, despite morality being subjective, there are multiple arbitrary, subjective, yet practical and intuitive reasons to lean towards our moral biases and instincts and socially agree that murder is immoral. You keep confusing practicality with objectivity, which are not at all equal.
"History has shown that societies can conclude the murdering of others can be highly desirable regardless of the dangers and or rationale."
I agree. This isn't relevant though.
"Individuals or societal conclusions opting for moral behavior being nothing other than opinionated desirability has no business declaring anything on anyone."
It does have a business, actually, because the lack of objectivity debunks the notion that God must necessarily exist, which is super relevant for society. It is relevant because whether God exists or not is incredibly important and our futures literally depend on figuring it out.
"The objectivity you speak of isn't going to happen."
I know. It is impossible by definition. Objectivity is self-contradicting. Again, I never said it was going to happen. You keep going on tangents and making comments that are either irrelevant or that make me think you are confused as to what this debate is even about.
"Morality set by our creator and or God would still be insufficient for you in establishing objectivity? Incredible."
Yes, because by definition, it LITERALLY cannot be objective. It isn't difficult to understand and I'm tired of repeating. Any morality established by God must by definition serve God's own purposes, REGARDLESS of whether God is omniscient and eternal, or not (and God is in fact NOT omniscient nor eternal, both of these are self-contradicting and therefore impossible, and I already discussed this). It isn't incredible: it's called logic.
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christosvoskresye Basically, in order to simplify the statement of the theorem ever so slightly, a more complicated definition of product is adopted.
No, this definition of a product was not invented to simplify the theorem. This definition of product was invented for multiple other reasons. Have you never heard of exponentiation? Exponentiation is a binary operation ^ which, when denoted n^m, is equal to the value of the product in which n is a factor m times, and no other factor appears. n^1 = n is a well-known fact that I think everyone in this comments section accepts. Yet this is just a product of one factor by definition! Similarly, n^0 is also a product, one with 0 factors, indicated by the fact that m = 0 in the exponent. n^0 = 1 is a valid product everyone already accepts. The same applies for the cases with the factorial, which is also a product. 1! = 1 and 0! = 1 are examples of products of only 1 factor and 0 factors respectively. This is no more strange or no more foreign than the existence of the empty set {}. These products are not more complicated: to the contrary, they are completely natural, and throughout high school, you used them all the time.
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grant kohler For me, it's not that 1 isn't a prime, it's just not a useful multiplication prime.
No, that is incorrect. 1 is simply not a prime. It has nothing to do with it being "useful." 1 never satisfied any rigorous definitions of primality to begin with. As for the claim that "but 1 is only divisible by 1 and itself," that is irrelevant, because that is not a real definition of primality. The problem is that teachers often do not know how to teach the definitions with precise wording, and this is worsened by the fact that most people would never remember the exact wording, because, let us be honest with ourselves: the vast majority of people are bad at learning and doing mathematics. So under those circumstances, it is, of course, only natural that people be confused when mathematicians say 1 is not a prime. However, this is not a problem with the definition of a prime, it is just a problem with how mathematics is taught. Trust me, this is not the only area of mathematics that is taught poorly on a worldwide level. I could write an entire book of all the misconceptions that even school mathematics teachers themselves do not realize are false. Maybe one day I will make a YouTube channel about it.
but the theorem isn't stated for addition, which is why I don't care for using 'to the power of' as a form of multiplication.
Exponentiation IS multiplication. This is not a question of whether you visualize it as multiplication or not. Exponentiation is defined as multiplication. If it is not defined as iterative multiplication, then it is defined as a multiplication homomorphism. In both cases, it is still multiplication by definition. Even if you do not visualize it as such in your head, it still is multiplication. This is the beauty of mathematics. Definitions in mathematics do not depend on subjective visualizations or methods. They just are. Whether you accept those mathematics or not is irrelevant, but the truth in mathematics does not depend on whether you care to visualize something one way or not. Also, the fact that the theorem is not stated for addition is completely irrelevant. To me, your argument sounds like you are saying "This theorem does not agree with my visualization of multiplication ==> exponentiation is not multiplication." This is non sequitur.
Mr Grimes pointed out that the professors tired of always having to say "excluding 1" so some group of mathematicians changed the 1 out of the sequence of primes and into its own category of number.
No, this is inaccurate. It has nothing to do with professors being tired. Mathematicians stopped considering 1 a prime long before education even became commonplace on a worldwide scale.
Basically, they just want to exclude the number 1 from the list of primes
No, it has nothing to do with wanting to exclude the number 1. Do you think mathematicians gathered together at a dinner table and started asking each other "who here hates the number 1? Should we exclude him from the list?" I am sorry, but this is most definitely not how it works. Mathematicians make decisions based on logical deduction, not person desire. Mathematics is not politics. 1 is not a prime number, so when mathematicians understood this, they decided to stop including it in the list, not because they want to, but because it literally makes no sense whatsoever to include it in the list. Your proposal to create a new list does not solve that problem, because fundamentally, 1 is not a prime number, and it has never been. This is not to mention that your proposal is just impractical and unintuitive: there is absolutely no circumstance in which it is rational to have such a pair of almost identical for ad hoc reasons if one of them will never be used or consulted.
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grant kohler Now the product of 1 can be written using at least two different expressions that utilize only prime numbers
No, that is false. 3^0 and 2^0 are the same thing. In fact, the fundamental theorem of arithmetic never mentions the word "expression" or any conjugations thereof. You are misunderstanding the theorem, and you are misunderstanding how the empty product works. n^0 = 1 for all n, so it follows that 1 = 2^0·3^0·••• ad infintum is the unique product of primes for 1. This is not cheating. This is just how it works. You can omit writing any of the given powers, but that does not turn it into a different product. You must not confuse a product with the symbolic representation of a product. It would still be the exact same empty product. This concept is no different than that of the empty set. There are multiple operations that produce the empty set, but the empty set is unique: there is only one empty set. So the empty set as a union of sets is unique too.
I still have not seen 1 written as the product of unique primes that are on the current list.
You are extremely confused. There is no "list" that _ needs to be a part of. This is not a supermarket. 1 is a product of 0 prime numbers. Therefore, it satisfies the theorem. The theorem does not state anywhere that it is a product of 1 or more primes, it merely states it is a product of primes. A product of 0 primes is still, whether you like it or not, a product of primes. The empty product is a product of primes. It does not need to be in the set of prime numbers to be a product of prime numbers, and there is no "list" it needs to be part of. None of this is cheating.
Here is a different explanation. There exists a set of numbers called the prime numbers. You can think of it as a list, although strictly speaking, this is incorrect. This set has many subsets. The subset {2, 3} is a subset if the set of prime numbers, for example. So is {71}. The empty set is a subset of every set, so {} is a subset of the set of prime numbers. With this, you can restate the fundamental theorem of arithmetic as follows: every n in N\{0} is equal to a product of elements of exactly one finite subset S of the set P. This is completely equivalent to its usual formulation: both statements mean literally the same thing. Anyhow, with this formulation, it should be perfectly clear how 1 satisfies the theorem.
they don't mind saying 1 = _ somehow fits the theorem even though _ is not listed as a prime number
It does not need to be "listed" as a prime number. The fundamental theorem of arithmetic, in its usual formulation, literally says nothing about lists, and it definitely does not say anything about "being listed as a prime number." All it says is that numbers are unique products of prime numbers. The empty product is a product of prime numbers. Period. That is all there is to it. You can either accept that as the truth, or you can deny it, but denying it will not make it false.
defining the primes that 1 is a product of requires breaking the theorem's definition
No, it does not. The empty set is still a subset of the set of primes, even if it contains no primes.
It really is cheating
It really is not. You cannot claim that a theorem is cheating if you clearly do not understand said theorem.
I know I'm not wrong
You can be in denial all you want, but you are wrong. I am sorry to be blunt like this, but it is what it is.
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flyingmadpakke But I could remove just one of those and have;...
And? Omitting one of the primes in the writing does not change the product itself, it merely changes its representation, so the product itself is still unique.
That means the product isn't unique, and the theorem wouldn't apply.
No, it does not mean that. You are confusing the product with its representation. Nobody goes around saying that 2.0 and 2.000 are different numbers, and nobody goes around saying 2/5 and 4/10 are different numbers either, but the representations are completely different. Similarly, 2^0 and 3^0 use different symbols, but the product they represent is the same, so it is unique by definition.
But this is probably one of those things which we have just defined, i.e, an axiom.
(1) An axiom and a definition are not synomymous, though they are closely related.
(2) x^0 = 1 is neither a definition nor an axiom, it is a theorem.
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@aldotemi3405 You claim that you are fairly certain that a theorem exists where any arbitrarily long repdigit can be found in π.
Yes, correct. I am glad you at least know how to read. This is better than 90% of the people on YouTube already.
What I said is that there's no such theorem.
No, you did not. What you did say is that a claim significantly stronger than the claim I made has not been proven, but that has absolutely no bearing on whether the claim I made has been proven or not. Hence, your argument is a strawman fallacy.
There is, however, a conjecture that states every single finite digit string (including your arbitrarily long repdigit case) is in π, but it is not proven. Really all I said was a more generalized version of your statement.
Yes, I know what conjecture you are talking about, and yes, I know that this conjecture is a stronger than the statement I made, and this fact is precisely why your argument is a strawman: that a statement is mere conjecture does not imply a weaker related statement is also conjecture. This is why, for example, the Riemann hypothesis, and the generalized Riemann hypothesis, are looked at separately. There are many experts mathematicians who have looked and concluded that the generalized Riemann hypothesis may be undecidable, even if the Riemann hypothesis itself as a special case is not. If we ever prove the Riemann hypothesis, it will likely not come equipped with a proof for the generalized Riemann hypothesis. In that case, the generalized Riemann hypothesis would remain conjecture, but the Riemann hypothesis would not be. So your claim is that the "π is a normal real number (in base 10)" conjecture is, well, a conjecture, and therefore, my much weaker claim is also a conjecture, but unfortunately, your argument is fallacious. That inference just does not work.
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Bob Thing “...ordinary addition (for which the sum of a divergent series is infinite)...”
No, that is not it, chief. Infinite summations are not ordinary and they do not follow the rules of ordinary summation, even when the series are convergent. Ordinary summation is concerned with 1 + 2 = 3 and 9 - 4 = 5. Infinite series are completely unrelated to ordinary addition, and how we define infinite summation is arbitrary. In particular, it is possible to evaluate infinite sums with some definition such that divergent series have a finite value. The limit of the sequence of partial sums over an ordered set is fundamentally different and unrelated from the complete ordered summation over all elements of the set, and the choice of creating an equivalence between the two of them is 100% arbitrary, unnecessary, and in many cases even, can be insufficient or anomalous. Changing the metric is not needed to sum divergent series. Changing the metric simply yields a different algebra, and this is done for other reasons unrelated to infinite summation. In the first place, talking about infinite summation is nonsensical without axioms of infinity, which we do not have in Peano arithmetic.
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KC Sutherland No, π(x) is only defined for whole numbers, because this function is, by definition, an arithmetic function. You can define a function with similar properties where it is total for the domain of real numbers, but it would not be the same function. Regardless, such a function would NOT be continuous as it necessarily would be discontinuous at the prime numbers, and therefore, the intermediate value theorem does not apply. It is not sufficient for the function to be piecewise continuous for the theorem to apply if it has step discontinuities, because for such functions, a sign flip can happen without there ever being equality.
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Raphael Schmidpeter Sorry, but no, that is wrong. Have you ever read the basic definition of exponentiation? It is an iteration of the multiplication hyper-operator, and by definition, iterations are only defined for natural numbers. The exponentiation function was defined not in terms of the complex numbers, but the natural. Roots and logarithms are mere inverses of it. Exponentiation does tie back to arithmetic, and exponentiation is not closer under the reals for a completely distinct reason than you mention. Otherwise, you are claiming that the absolute value function is not defined in the reals simply because the functional definition that accounts for complex number requires the structure of the complex plane to begin with. But that is quite ridiculous since the absolute value function which gives an output for complex inputs is nothing but an extension of the pre-existing defined function, in the same way that complex exponentiation and complex logarithmic operations are nothing more than extensions of the pre-existing definitions for those functions on the reals.
And no, it is not unfair to say that the complex numbers are not closed under log simply because log is undefined, because that would imply that one could not claim the reals are not closed under even fractional exponents (I.e., square roots) to begin with. That is how the concept of closure works in mathematics. Log 0 could easily be defined as a new number and one could create axioms such that this number preserves the common properties of algebra and arithmetic we all know. But that does not mean the complex numbers are closed under logarithms: whatever log 0 is, it isn't a complex finite number, so the complex finite number system is not closed under log. That's the end of it.
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1. The exponential function can be defined for the reals without necessity of analysis. The limit definition of e^x can be expanded by the binomial theorem into its infinite series without any form of analytic continuation, and the approach suffices to define the function for the real numbers without the existence of complex numbers in them. The problem only truly arises when defining e^ix, which requires trigonometric functions already defined for the complex plane.
2. The trivial set {0=1} allows 1/0 to exist while following the commutativity, associativity, and distributivity of algebra. I agree. Where we disagree is that there exists a non-trivial set where this is also possible, and this set is invoked by wheel algebra theory. The involution operator, "/", is a modified form of division such that division is a unary operation, and in the general case, x^(-1) is not equivalent to /x. However, commutativity, associativity and distributivity are all preserved, albeit with a tiny bit of modification, and whenever 0x=0 holds for any x, the identities all simplify down to identities that we use in non-wheel algebras, thus proving the wheel as an extension of a field. The extended Riemann sphere, which is the best projected model for this structure, is the set C in union with {/0, 0/0}. This is a mathematically sound structure, and you should read on it. My point is, there exists a non-trivial set for which it works. But my conjecture is there is no non-trivial set which is closed under all operators.
3. You ask what is that I define as an operation, and furthermore, what do I imply by all operations. That is a very good question. For our purposes, we can speak of all the integral hyper-operators and their respective inverses.
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Steven Van Hulle No, that is incorrect. The reality is that 0 to the anything-th is NOT 0. For instance, 0^i is undefined. 0^-2 is undefined, not 0. And 0^0 = 1, not 0. This is how most working mathematicians define it. Your teacher might tell you otherwise, but that just means your teacher, who is not a mathematician, is wrong. 0^0 is equal to the cardinality of the set of all maps from the empty set to the empty set. How many such maps are there? Exactly 1, and it is called the identity map. Therefore, 0^0 = 1. e^x = x^0/0! + x^2/1! + ••• for all x. If x = 0, then e^0 = 1. But for all n > 0, 0^n. Therefore, 0^0/0! = 1. 0! = 1, so 0^0 = 1. 0^0 is also equal to the zeroth iteration of the function f(x) = 0x. If I multipy x by something 0 times, then what am I doing to x? Nothing: leaving it unchanged. Thus 0^0 = 1. Consider the map f: C —> C f(z) = z^z. Consider the limit as z —> 0. What is it? 1. If we want f to be everywhere continuous, then f(0) = 1. Therefore, 0^0 = 1.
Basically, most of mathematics are secretly build around the convention that 0^0 = 1. Mathematicians know this too. The only reason some teachers prefer to tell you it's undefined is because it's too complicated to explain the truth. It's for the same reason they tell you can't square root negative numbers, even though, clearly, you'll learn a few years later that you can.
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Giannhs Polychronopoulos ah, I don’t the know the specifics, to be honest. I watched John Conway’s lecture about games and how his theory of games led to his discovery of the surreal numbers. However, the class of games is a super-class of the surreal numbers and is said to be “bigger” in some sense. In this lecture, he did say specifically the cardinal number describing the class of surreal numbers is that of the set theoretic universe. He did not specify an explanation of proof of this, which I imagine he intended for us to take for granted in the lecture, since the lecture is not about classes and cardinalities in general. I have read online that the surreal numbers form a class, not a set, but I don’t know of any specific books to read. Actually, I’m looking for books of my own myself to read about it too. I can link you his lecture, though, if it helps
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Onechill Braj The overall (net) Gaussian curvature of a torus is zero. The outer circumference has a local positive Gaussian curvature, in which case it is logically homeomorphic to the sphere. However, the inner circumference, a.k.a, the tiny hole of the donut, has local negative curvature, it is homeomorphic to a hyperbolic shape. The rate of change of the Gaussian curvature is constant throughout the entire torus, and the circumference which is perpendicular to the height or vertical axis of the tori has local flatness. By use of an integral, one can deduce that the overall curvature of the entire torus is zero, flat as a square, which is why a square can be embedded into a torus. The sphere has positive curvature everywhere, thus some of the original conditions are broken or lost in the process of transforming a square into a sphere, by producing the net curvature.
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This is just all jumbo not supported by evidence. Historically, authoritative parents have never been successful and have never been considered an ideal type of parent by well-informed psychologists. This so called “expert” provided an example with a teacher, and providing an example is fine and all, but the problem is that what actually happens in real life shows that authoritative teachers are not successful. Children are not really supposed to simply obey everything and period. This is not how raising children works, and children do not like authority. Children are curious. Historically, children of authoritative parents are actually not obedient at all, but their parents are unaware because they’re also secretive about it on top of that. It just completely destroys the parenting. Children want explanations. You have to provide these explanations to them for then to learn and understand and comply to an order. If they get argumentative and are able to pick apart your explanations, then that means your explanations are objectively bad and you’re a bad parent. That’s what it comes down to. If you tell them “because I said so”, that isn’t going to convince them at all. Sure, it may instill fear in them, and some children will react to fear by applying an obedient stance. But most of the other children won’t. If they’re not convinced and you tell them that, they’ll simply lose your trust and keep disobeying your orders, but in secret. And you know what we call it when we use fear as method of governing and ruling? Terrorism. Think about this for a second.
Also, you act as though obedient children are rare in this generation, and as though children are intrinsically little devils that need quarantine, but neither of these claims has evidence to support it. Children have always been disobedient.
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@Dhen Phu Limits are not "best-we-can-do" approximations. Limits are a well-defined operators that just so happen to be misunderstood by everyone because no one teaches them correctly. However misunderstood they are, though, they are perfectly well-defined, and nothing about how they are defined is problematic, nor does it require that you be able to accept the existence of infinite processes, although there is nothing actually problematic with the existence of infinite processes anyway.
Suppose I have some function f and some limit L. How do I know that L is actually indeed the limit? If I take the distance between the image of f and of L, written like |f(x) – L|, then no matter how small I make this distance, I can always find some positive real number that bounds the correoonding distance of the preimage, written |x – c|. If you tell me that you want to find a region in the preimage of f centered around c such that |f(x) – L| < 0.5, for example, and I cannot find such a region, then that means L is not the limit.
Often, the above explanation is a little complicated and difficult to understand, because humans have not evolved to ordinarily be able to think in terms of existential and universal quantifiers and use them correctly. So, teachers always instead teach students that, what the above means, it means that "as x gets closer to c, f(x) gets closer to L." And this is a useful visual that is much easier to understand. Unfortunately.... it is also totally misleading, and it actually only worsens student's understanding of limits, it does not improve it. The only reason it seems to improve their understanding is because they no longer feel confused. But even though they do not feel confused, they now have the wrong understanding of what a limit is. When you tell students the above explanation, it plants the notion that limits are an algorithm, a process, and that this process is supposed to be infinite. But this is totally false. Limits are not processes, and there is no algorithm you need to carry out for the limit to be well-defined. And if you tell them that this is how they should think of a limit, then I guarantee you with 1000% confidence they will never be able to write a proof regarding the existence of the limit of some arbitrary function. Why? Because their understanding of limit is false.
Teachers need to try to create an intuition for the ε-δ definition of a limit instead of giving visual analogies that are not accurate. Because ultimately, limits are perfectly well-defined.
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@Dhen Phu Let me go into more detail about the definition of a limit itself. First, it should be noted that there are two types of limits: proper limits and improper limit.
Proper limit is the type of limit where teachers say "as x approaches c, f(x) approaches L," with c being some real number. I have no clue what your background in calculus is, but the reason I want to give you the definition is only so you can recall the symbolic structure of the definition, not the actual details of said definition.
lim f(x) (x —> c) = L is defined as "for every ε > 0, there exists some δ > 0, such that |x – c| < δ implies |f(x) – L| < ε." Now, notice how there is nothing here about "approaching." The definition has two quantifiers, one in ε, one in δ, and then a proposition that needs to be true in accordance with those quantifiers. If you can find that, for a particular value of ε, there is no δ that can make the above implication true, then L is not the limit. Otherwise, it is the limit. Structurally, it is this simple. This is about existence of bounds of distances between numbers, not about processes. Notice how this definition does not say anything about "approaching" or anything. Notice how this definition does not have any notion of "process" or "algorithim" built into it. Sure, you can think of it as a process if you need a visual, and you can even use this idea of approaching for numerical approximations. But these are just that: visualizations. The actual definition has none of that. No processes, no algorithms, no nothing. The definition just asks "for every ε > 0, does a number δ satisfying certain conditions exist?" and this is the take away of my argument.
So, if you understand the take away, then now it should be obvious that the definition of a limit is not at all problematic.
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despite the Earth's significantly quicker than Saturn, both have made the same number of orbits
This is not an absurdity, but in reality, mathematically correct, and justified by transfinite set theory, which was developed by Georg Cantor in the late 19th century. Since an orbit is a discrete object, of which there can be an exact integer amount, it is a counter, and therefore, in an infinite period, there can only be Aleph(0) orbits. Aleph(0) is a cardinal number that satisfies the property that that Aleph(0) = n·Aleph(0), where n is any natural number. So yes, Saturn would have made Aleph(0) orbits, and Earth would have made 30·Aleph(0) orbits, but both quantities are equal. This is because if I consider the sets N and 30·N, there actually exists a (trivial) bijection between the sets. This is all, just to say that, if al-Ghazali had been born after Georg Cantor, al-Ghazali would have been acquainted with set theory, and would have never created the Kalam cosmological argument to begin with. The entire premise behind the argument, the so-called "absurdity", is false, because there is no absurdity.
The use of the causal principle is not to be found here, for Kindi's arguments are based simply on the notion of the succession of temporal segments.
The fact that WLC wrote this betrays his lack of understanding of the scientific ideas of time and causation. For starters, his argument would need to be rephrased, as it is well-established today that it is not time that passes, but that our time coordinate changes based on our innate spacetime motion. Even if we applied this rephrasing, though, there is still the conceptual problem that time and causation are inherently linked. In fact, causation in science today is understood in full-rigor by way of equations in the calculus of time scales.
But the problems raised by the illustrations are real ones, for they raise the question of whether an infinite number or number of things can actually exist in reality.
This is an entirely valid question to ask, but the problem lies in the approach to answering the question, not the question itself. It is in the approach that al-Ghazali is wrong.
Ghazali argues that this results in all sorts of absurdities; therefore, the series of temporal phenomena cannot regress infinitely.
Ghazali did argue this, but he presented these arguments during a time when mathematics did not stand on a rigorous foundation, and the concept of infinity was very poorly understood and controversial. His assertion that, the Earth cannot have orbited the same amount of times as Saturn, yet also have orbited 30 times as many orbits as Saturn has, is an absurdity... such an assertion is completely unsubstantiated, and stemmed from a very naïve and incomplete understanding of infinite quantities. There is no absurdity here. If anyone in the year 2021 is trying to use the concept of finitude and infinitude to present a syllogism, then their concepts better be grounded on axiomatic set theory, or on type theory, not on intuition.
Infinite quantities or magnitudes are those that are measurable but have no finite measure.
This is a fine definition, although it does have the problem that the definition is only coherent if you also define what "finite measure" means. Later, you mention that an infinite measure is greater than the measure of any finite number, but this still requires having to define "finite" in some way or other. This is not a big issue for the video series as a whole, but I certainly think many people would find it very helpful for "finite" to be defined in a way that is unambiguous, precise, and not overly reliant on intuition, since it would help parse out justifications for arguments concerning the finite and the infinite more easily.
For instance, there are just as many natural numbers as there are [nonnegative] even numbers, despite the fact that the [nonnegative] even numbers are a proper subset of the natural numbers.
This is true, and this is exactly what Ghazali failed to understand when he presented his arguments about the infinitude of time... which is understandable, since Cantor's development of set theory did not exist during his time. There was no possible way for Ghazali, or any of his contemporaries, for that matter, to understand that he was wrong, or why he was wrong. Nonetheless, this does not change the fact that he was indeed wrong, and theologians today need to start acknowledging this.
*A potential infinite is, strictly speaking, not an infinite at all. It is a quantity that 1. is increasing 2. has no finite limit 3. is always finite.
I find this to be a really strange and confusing way of defining a potential infinite, especially when followed by the central caption on screen that talks about lack of definiteness, itself not a well-defined concept. Now, I imagine that when you speak of a quantity that is increasing, you therefore talk about a quantity that we call a variable. So the potential infinite refers not to a set of objects, but simply to a variable, and this variable has the property of always increasing, and the property of always being finite. However, what makes this confusing is the second property: that it has no finite limit. The word limit seems to be used here with a rather loose and intuitive definition, which is not adequate for the explanation, since we are dealing with a precise mathematical idea and making appeals to set theory. When I hear the word "limit", what I think of is the topological concept of a limit point, or equivalently, the idea of limits that is taught in an introductory calculus course. In other words, a potential infinite is a quantity that can take on various values, and is monotonically increasing, but while the set of values the variable takes on is infinite, the values themselves are finite, and bounded. So in other words, this definition, at least to me, communicates the idea that a potential infinite is a variable that converges as its argument increases without bounds. If this is what you intended to communicate with your definition, then all is good, but otherwise, clarification is certainly needed here. I say this, because the following caption only makes this worse.
A potential infinite collection is one "in which the members are not definite in number but may be increased without limit".
This is WLC's definition. I find this to be, at the very least, nonsensical. Ignoring the fact that I have never seen any mathematical text that even makes reference to this kind of "infinity", the fact that "the members are not definite in number but may be increased without limit" is problematic. For starters, in axiomatic set theory, there does not exist any notion of a set not having a definite, fixed cardinality. Perhaps what WLC is talking about here is a sequence of sets, in which the next set in the sequence has a cardinality bigger than the last, yet every set in the sequence has finite cardinality, and the limit point of the sequence is a set with infinite cardinality. However, if that is what he is referring to, then the definition is useless and redundant, because it makes no meaningful, non-trivial distinction between actual infinities and potential infinities: in this case, every potential infinity necessarily gives rise to an actual infinity, so every infinity is both actual and potential. This is why I said clarification is needed. Honestly, what this proves to me is that the theologians' uninformed attempt at trying to make such a distinction in the first place is mathematically unsound and philosophically misguided. I have never seen a definition of potential infinity that is not redundant or nonsensical.
After having looked at the video's coverage of stage 1 of the New Kalam, I also must say that it does disappoint that none of the resources that theologians have provided that I could find on the subject have bothered to provide a scientifically rigorous definition of a "cause" and an "effect". This does make any arguments that attempt to use such a notion fall completely flat on their face, since, if there is no coherent causation to discuss, then there is no cause of the universe that needs to be discussed. In modern science, the universe is studied by way of understanding it states and how those states evolve. These evolutions are described by time-scale equations, and so the idea of causation is not even present: scientifically speaking, there is no such a thing as a cause or an effect, there is only a special kind of mutual dependence between states and variables, called functional dependence.
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Also, talking about the actual definitions:
Defining 'existence': existence is easy to define if you choose the appropriate framework with which to define it. Existence is properly defined by axioms in free logic. All ontology, when formalized, should be presented in free logic.
Defining 'begins to exist': the definition provided on screen is problematic for many reasons. It fails to define what "x exists at time t" means, and while I know that existence itself is already presuably defined here, "at time t" is not defined, because formal ontologies, as defined in free logic, do not use any notion of temporal dependence. Temporal logic, as an extension of free logic, is needed here, but the best way to achieve this, actually, is instead to formalize the concept of a parametric family of ontologies: for each t, one has an ontology O(t), and "x exists at t" if and only if "x exists," in the sense of free logic, is true in O(t). However, this also requires defining what time is, and for this, we need to recur to mathematical foundations of the general theory of relativity, which I contend, no expert philosopher alive is currently equipped to do, undermining the entire argument. This brings me to the other objection to this definition: the idea of "earliest temporal boundary," which is simply ill-defined here, even in cases for objects which have a finite age. You see, the time axis can only be coherently conceptualized (that we know of) as a connected subset of the real numbers, i.e., an interval of real numbers. However, there exists open intervals. For instance, the open interval (0, 1) has a finite length, but there is no smallest real number in this interval: for all s in (0, 1), there exists some t in (0, 1), such that t < s. In the sense of topology, this interval has a boundary, but it does not contain said boundary, and if we are using this to model the physical world, there is nothing indicating that such a boundary actually exists. So, even for objects of finite age, talking about "begins to exist" is problematic using this definition. One amendment I can suggest is to say that "x begins to exist at t" if and only if "x exists at t, and for all t' < t, x does (did) not exist at t'." However, this definition would present problems for WLC's worldview, and I am certain he would not be able to accept this definition, despite his complete inability to propose an alternative. I myself cannot think of a better alternative, though.
Defining 'universe': I think it is simpler to define "universe" as the totality of all the different spacetime manifolds that may exist, all quantum fields that may exist, and all mereological sums thereof, as well as possibly strings, if they do exist. Dark matter and dark energy would be included in the above.
There is an ambiguity as to what "whatever" or "everything" means in premise A of the argument. When it says "Everything which begins to exist..." does it refer to the universal quantifier, "For all x,...," or is it referring to a specific kind of thing, like mereological sums of states of quantum fields?
Defining 'cause': the definition of 'cause' in the video is wholly inadequate for anything: it is just not even really a definition. "x causes y just in case x produces or brings about y" is not a definition, as the relation "x produces y" is itself undefined. All the people writing this did was rename the relation, rather than actually define it. Also, we absolutely DO need to worry about the questions dismissed in the video, as those are essential for having an appropriate definition of 'cause.'
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@vernonvouga5869 I don't know I would consider anything that "looks like an edge" an edge.
I don't think you'd even be able to know whether a thing "looks like an edge" or not, especially because "looks like an edge" is an undefined predicate.
Specifically because science has made itself so untrustworthy in the past couple of decades.
This is just false. There is absolutely no proof that the scientific method is less reliable today than it was 60 years ago. Saying that there is amounts to nothing more than misinformation. Besides, I have my question as to what you think the scientific method actually is.
I'd rather believe what I see, and what I read about from the few honest scientists that are out there.
You say this as if (a) you were qualified to understand the conclusions presented by the scientists in question (b) you had a sound methodology capable of telling apart scientists who're honest, and scientists who're dishonest. This'd, by the way, necessarily include proof of some significant portion of scientists lying, which you simply don't have. If the scientific community wanted to participate in a global conspiracy to lie to every single layperson and science student on Earth, as you insinuate that they would by calling them "dishonest," which is a bold accusation to make in the lack of any evidence, then it'd be 100% impossible for someone else to know that they're lying to the world. Why would they make it so easy to find out that many of them are lying, if the goal is in fact to lie? The answer is simple: they aren't lying to begin with, and they aren't trying to, so they aren't hiding anything.
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@erikziak1249 Contradictory axioms. First I learn that there cannot be a square root of a negative number, since every number squared is a positive number. Then I am told that it is not true. And that that it sort of still is true, but I have to imagine that there exists such a thing.
These are not contradictory axioms. The real numbers form a mathematical structure called an "ordered field." The fact that they are ordered is actually very important, it is part of how real numbers are defined. To put it simply, the real numbers being ordered just means that there exists a well-defined notion of positive real numbers, and negative real numbers, and a well-defined notion of comparison. I can compare two real numbers 3 and 5, and conclude that 3 is less than 5. This is what the concept of order refers to.
In an ordered field, it is true that the square of every quantity is nonnegative. However, not all fields need to be ordered. If the field is not ordered, then there is no well-defined notion of positive or negative quantities in this field. In such a field, it is entirely permissible for all quantities to have a square root, but this just means the field cannot be ordered. As I said, the real numbers form an ordered field. However, we can choose to get rid of the ordering altogether, and just forget about there being such a thing as positive numbers or negative numbers. Now, even after you get rid of the ordering, the fact is, some numbers still have no square root. This is because you have not actually changed the multiplication at all. But, that being said, now that there is no ordering restricting you, you can just extend these numbers to a larger class of numbers where everything has a square root. This is all fine, because you got rid of the ordering. Notice that there is no actual contradiction here: it still remains a fact that if you want to keep the ordering, then negative quantities cannot have a square root. There is no "well, actually..." caveat here, this actually is just what it is. The extension is only possible if you get rid of the ordering. This is not a contradiction: by getting rid of the ordering, you are legitimately changing the type of mathematical object you are working with.
What will be next? We can divide by zero?
Despite what many misleading videos on YouTube claim, we cannot divide by 0. This is not because we choose to not define division by 0. No, this is actually a theorem. The axioms of arithmetic imply that 0•x = x•0 = 0, and this already just makes division by 0 impossible. There is nothing anyone can do about it.
I am pretty much aware of limits, when something approaches zero, but what if it IS zero?
Limits are actually irrelevant to the discussion, and they have no implications on the topic of division by 0. Discussing x —> 0 is very different from discussing x = 0. If a person tells you that limits are relevant, then you should immediately conclude that they do not understand how limits work at all.
Expecting me to think about a number as having a "real" part and an "imaginary" part is also quite stupid.
It is not a stupid at all. The concepts of the real part function and the imaginary part function are very essential in complex analysis. Also, they are important in the vectorial/geometric understanding of complex numbers.
What is a "real" number?
The real numbers have a very precise mathematical definition. To put it in simplest words, they are the unique field of numbers that form a continuum. This idea of "continuum" is important, because it enables you to do geometry and calculus. The rational numbers, for example, do not form a continuum. Instead, they are discrete points with gaps in between. The real numbers are an extension that fill in those gaps, and no other extensions exist that actually succeed in filling those gaps.
No numbers are real! They are just a mental concept.
Numbers being a mental concept does not mean they are not real. No, numbers are not physical, if that is what you mean, but 'physical' and 'real' are not synonymous. That being I said, I do think that the name "real number" should be replaced by an actual descriptive name. But, this is also your mistake. You are just placing an unhealthy and unnecessary amount of importance on mere names, to the point that it has become an obsession, and are not even willing to actually look at the concepts behind the name, which is where you should be looking. To put into perspective why this is a problem, just consider this: my legal name, which is also my birth name, is Ángel. Do you think I am actually a literal, true-to-the-Bible angel? No, of course I am not. But, you have absolutely no qualms with seeing my name on the YT username, you think nothing of it. You understand that the word "angel" is just a name when it comes to people, and take no issue with it being used to describe humans who clearly are not angels in the biblical sense. It has no meaning beyond this. Well, names for mathematical objects are no different at all. There is no reason you should even be paying attention to the names much beyond just the convenience of being able to communicate with people. If you are trying to learn mathematics, then what you should be studying are the concepts hidden behind the names, the actual definitions. The definitions may be confusing, but there are actual explanations that you can find for them.
Look, this is not exclusive to mathematics either. It applies to all areas of life. There exist many more concepts than there exist English words. So, necessarily, some words we have to recycle, and use in two completely different ways, having completely different definitions that are unrelated. We do this in mathematics, we do this in history, science, politics, economics, engineering, law, etc. Every career that exists has this conundrum. Yet, I am sure that in most other areas of life, you actually do overlook the fact that the same word is used in two different ways, and you just adapt to it. Everyone does. There is no reason why mathematics should even be the exception. In fact, even within mathematics, you already do this, and you have not even noticed it. For example, it is statistically likely you never noticed that in mathematics, there exist two completely different definitions of the word 'division.' You just adapted to the fact, and your brain processed it like it does anything else. What it all comes down to is that the names really do not matter outside the context of communication. They ultimately have no actual bearing on the mathematics. If I use the name of an object in mathematics, what you should be doing is asking yourself if you have already learned how this object is defined specifically within mathematics. If you suspect you have not, then you should ask for the definition, and the definition will be given to you. If you do not understand the definition, then that is perfectly fine! Someone will explain the definition to you, that is what education is for, and that is what we have YT videos now for. This is how one approaches learning mathematics. Focusing on the name itself is not how one learns mathematics. In fact, this focusing on the name thing is not an effective approach to learning mathematics even if we improve the names to more descriptive ones. Atthe end of the day, you are not going to learn anything if you are not focusing on the definitions, regardless of how "good" the names are. The names, ideally, should be a helpful bonus. But they are definitely not meant to be the core of it.
This is very, very bad. Maybe it is being taught at schools differently today, I do not know, but the stupid name "imaginary numbers" is still used.
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@kiraPh1234k It quite literally is counting.
The pigeonhole principle is the following sentence in first-order set theory: for all sets X, Y, if there exists an injective function f : X —> Y, and there exists no surjective function g : X —> Y, then there exists no injective function h : Y —> X. In Zermelo-Fraenkel set theory, this is a direct consequence of the Schröder-Bernstein theorem. This is not counting. The mathematics of counting are founded on first-order set theory, but set theory is not synonymous with counting.
That's even being more polite than the truth, because truth is that it's a useless tautology.
Can you demonstrate how the sentence "for all sets X, Y, if there exists an injective function f : X —> Y, and there exists no surjective function g : X —> Y, then there exists no injective function h : Y —> X" is a tautology?
It states if n > m, then n > m.
No, it does not. In the theory of orders, the sentence "for all m, n, if n > m, then n > m," is indeed a tautology, but this statement is not the pigeonhole principle.
We use the underlying mathematics to solve the problems, not these useless tautologies.
I agree, but the pigeonhole principle is not a tautology, and it is nontrivial.
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@kiraPh1234k I think you are just misunderstanding the video, quite frankly. It is true to say that, for example, if m, n are natural numbers, and n < m, then there is an injective function from n to m, and no surjective function from n to m, and therefore, no injective from m to n, but this is only a special case, which I think is fine for illustrative purposes.
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Jeffrey Black “Induction is a form of logical reasoning.”
Source?
“But if you want to restrict it to mathematical induction...”
No, it does not work if I restrict it to mathematical induction, precisely because mathematical induction, by definition, only encompasses natural numbers. You cannot prove something holds for infinity using mathematical induction. All mathematical induction can do is prove that something holds for all natural numbers, not for infinity.
“If you think my proof doesn’t hold, then prove it.”
I don’t need to prove that the principle of mathematical induction (which actually isn’t true induction at all because induction is an epistemically empiricist concept, the principle is still deduction) only proves statements about finite natural numbers because that is just the definition of the principle.
“I love how you demand I prove things, but you just assert things are proven.”
I don’t need to prove a fact to cite it. I only need to prove what has never been proven, and I’m not asserting any such claims. The one misquoting induction isn’t me.
I’m not responding the your long rant about straw men and lying as that in itself is a straw man. You spend too much effort lying about your own quotes. You know people can read, right?
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Jeffrey Black I’m not baselessly asserting anything. Now it is your job that what I am asserting is indeed baseless, because to my knowledge and everyone else’s, it is not. Also, you cannot cite the claim that the sum of all positive numbers is positive without proof precisely because this has not been proven. This is the difference between your claims and my claims. My claims exist in the mathematical literature with proofs. Yours do not, and it is clear that they could not exist with the proof since you cannot use induction to prove it. There is no double standard.
The reason I quoted you verbatim is not for the reader, but for you to see a point, which you then showed you are incapable of seeing.
I already told you what is wrong with your proof. I have no need to repeat myself, you not understanding the explanation is your problem. You’re misusing induction, and that is that. I do not have to prove a DEFINITION. That is the whole point of a definition.
I am not using my degree to prop up my claims. This is clear from context. Once again, you failing to understand the context is your problem. In any case, you saying “if you use your degree to prop up your claims then you don’t deserve your degree” is a claim you should prove.
I have still yet to see proof that “induction” is a form of logical reasoning. Or, I should say, a reliable source.
You have yet to present a single valid argument.
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Jeffrey Black I never admitted I did. I cannot admit to that which I provably never did. Once again, you put words in my mouth. And in saying you never did, you are in fact pretending, so you disproved your own stance.
My dismissal of your proof is not baseless as it literally cites the definition of mathematical induction. No sane rational person would call this baseless. Hence I must conclude you are insane or irrational.
No, there are not “two forms of logical” reasoning. You obviously have no understanding of what logic is. In logic, you start with a set of axioms and a set of inference rules, in which you use the connectives and semantics of the language in which you formulate the axioms. Only once you have established the set of inference rules can you ever prove any claims from the axioms. This is called deduction. This is all logic is. There is no induction in logic. What is known as mathematical induction is just deduction, as it reasons that one prove a premise, and then one proves a conditional that will later serve as a new inference rule from which we can derive universally quantified propositions which will be our conclusions, but universal quantification only happens over the natural numbers. True induction is not a form of logic and you encounter it in empiricism and science. Induction is the claim that our observations of events we have observed will be alike in the future as in the past, such that we can assert a general statement about the entire population of things we observe. David Hume has written many books on this.
The statement that the sum of TWO positive numbers is positive does not imply that the sum of INFINITELY many positive numbers is negative, and the only way to show otherwise is by proving it.
There is no double standard. My claims are facts. Yours are not. Any reader will understand this. Any reader knows it is a fact that mathematical induction only quantities over natural numbers, hence making your so-called proof invalid. Again, definitions are by definition not needed to prove, so I have no epistemic responsibility to prove them. The only person making assertive non-skeptic claims here is you.
And you denying it does not mean anything and it does not change the truth. So can stop being irrational, or you can stop having this conversation, as you will always continue to fail to present a valid argument, and as such, will continue to fail to convince me of anything that you say. You see, you cannot easily convince many people of false claims. It could work on other people, but it will not work on me. Simple as that. So keep wasting all of the time in your world, pal. I will continue to keep promoting the truth and I will continue to keep doing research and solving problems while you get stuck here in a debate you objectively cannot win, because your claims are objectively, provably false, and I already gave the dismissal from definitions.
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@jeffreyblack666 "So you are still pretending you haven't made one?"
I cannot pretend I made one because I never did, and I cannot pretend to have done that which I never did: only those with a talent for acting can do this, and I have no such talent.
"I'm not putting words in your mouth, you are just outright lying about me. I am not pretending. I didn't make a strawman. You did."
Exactly what a person who defended their defenseless argument with a straw man and lies would say.
"Trying to cite a definition to dismiss an argument which doesn't even need that definition amounts to a baseless dismissal."
Correct, but this is irrelevant since your argument did require the definition of mathematical induction, because you used mathematical induction, and you can only use it if the definition corresponds to the claim you want to prove. In this case, it does not.
"You also never cited any definition. So no, sane people wouldn't be on your side."
I did cite it, and you denying I did will not make it false.
"Why don't you try sticking to simple math and leave logic for rational people."
This is a question I would like to ask you, since you are the one behaving irrationally here. In fact, i would like to ask, why do you not try stciking ti simple math and leave logic to people who have a degree on the subject and understand it better than you do? Of course, such a question is futile, because you are going to continue not trying to pretend you understand induction and other topics.
"Go look up inductive reasoning and deductive reasoning."
I already did years ago, and I continued by reading many books and then taking several courses on the topic. You are the one who should look it up, considering you were not even aware that mathematical induction is not induction, and that you still are not acquainted with the definition of mathematical induction.
"They are 2 forms of logical reasoning."
They are not.
"Yes, mathematical induction uses deductive reasoning."
Ah, thank you for proving my point. Because mathematical induction is deductive, this means that citing it as a premise means the argument is either invalid or valid and sound or unsound, but I already showed it is invalid because your claim is trying to prove a proposition about infinities, not natural numbers, yet you can only infer universally quantified conclusions about natural numbers from mathematical induction, not otherwise.
"That doesn't magically mean inductive reasoning doesn't exist."
I never said it does not exist. I said it is not a form of logic. Big distinction. There you went again and lied and misrepresented my argument for the nth time, as I should have expected.
"It is the basis for science, you know, useful things where you don't magically start with all the answers."
I am aware, since I am the one who told you this in a previous comment.
"You not wanting to call it logic doesn't magically mean it isn't."
Correct, but me not calling it logic because by definition it is not logic does mean it is not logic.
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@jeffreyblack666
"Again, I have proven it. The sum of any number of numbers can be changed into multiple sums. For example sum(a,b,c) is the same as sum(sum(a,b),c).
This means the summation only ever has a sum of 2 numbers, which will always return a positive number."
This claim only is true for finitely many numbers, and the reason it can be shown true is due to the principle of mathematical induction, but this cannot be proven for the infinite case because mathematical induction cannot prove statements about infinity.
"The fact that there are infinitely many sums doesn't magically change that."
Yes, it does. That is the entire problem with infinities in arithmetic. This is why the real numbers are constructed specifically to be Archimedean. Mathematical induction cannot prove statements about infinity.
"Again, if you wish to assert this magically breaks and somehow you can then add up positive numbers and get a negative number the burden of proof rests upon you."
You were the one to make the assertions first, you were in this discussion first. I simple came later to call you out on making a baseless claim. The burden of proof states that a claim with no evidence can be dismmissed without any evidence, and only if I make an assertion that is not skeptic of another assertion or dismissive of another do I need to give proof. Since you were the first one to assert and you refuse to give proof, I have no epistemic responsibility to give any proof of my own.
"You really don't understand what a double standard is do you?"
Given the sentence you wrote before this one, you obviously do not understand double standards any better than I do.
"You asserting that your claims are facts while mine are not is simply another double standard. You wish to be able to assert whatever garbage you want without having to back it up, while dismissing whatever anyone else says."
Yet you were the first one to do exactly just this: you were the first one of us to decide to make unsupported claims only to ask me to support my dismissal of your claim, which is based on a basic, common-knowledge definition. If any person is promoting a double standard, it is you.
"Sorry, I'm not stupid enough to fall for that BS."
Whatever floats your imaginary boat.
"My statements are facts, as such I shouldn't need to back them up. Either accept or back up your own."
I refuse to accept an order from a hypocrite who is unable to practice what they preach.
"No appealing to pathetic definitions, either back up your argument or shut up."
The definition of mathematical induction is not pathetic, it is rigorous, and you calling it pathetic is an ad hominem-like fallacy that does not debunk my objection.
"You have made it clear that absolutely nothing will convince as you will simply dismiss it with baseless garbage and pretend you have no burden of proof."
Correction: I have made it clear that absolutely nothing that YOU say will convince me, as you simply refuse to verify the definition of induction and instead decide to launch personal attacks and red herrings, all to distract from the fact that your argument has been shown invalid and baseless.
"I am not the one being irrational here. You are. You are not promoting the truth, you are dismissing it."
Exactly what an irrational person would respond when they are losing control of the debate THEY initiated.
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@Alex-02 If I say all green animals should be considered plants because they have the same color, the current definition of “plant” is irrelevant because we are discussing the definition itself.
If you say all green animals should be considered plants because they have the same color, then this does not address the current definition of "plant" at all, and as such, it does not replace said definition. Also, you chose a poor example for an analogy, because not all plants are green. In any case, if you want to define categories of objects by their color, then all you have defined is the color itself. This has nothing to do with biology. You can even define a subclass of the category of living beings based on color. Again, though, this has nothing to do with biology: this is simply about the color of the living beings. You are still including green fungi, green protozoa, green bacteria, etc., all in this category. There is no biological property that green animals share with plants, such that no other entities share that property, living or non-living. Therefore, it is ontologically inadequate to insist that they are in the same category.
Definitions are non-universal, made up by humans and always changing.
Well, this is just a false belief. Definitions are not mere labels you come up with arbitrarily on the basis of preference for labeling objects of your preference. Definitions have to be consistent with reality. To apply definitions to a set of properties, you first have to ensure that these properties actually are well-defined, and that there exists at least one object which has those properties. Then, when you apply the label to this set of properties, it follows that the label applies to all objects satisfying these properties. If you want to single out certain such objects as being qualified, and excluding the rest, then you need to specify properties which are satisfied exactly only by those objects you are trying to single out. Also, for this to be justified, you cannot do this on an ad hoc whim. Otherwise, that renders all definitions as completely redundant and useless. What makes definitions useful is precisely their non ad hoc nature. Finally, there is the problem of decidability. A definition has to be decidable, because if there exists no algorithm by which anyone would ever be able to tell if a class of objects satisfies the given definition or not, then it is completely pointless, and as such, the 'definition' does not actually define anything.
How precise you want to get with all of this, it all depends on what academic discipline of study you are a part of, or whether the definition is just intended for colloquial nonsense. In mathematics, though, the highest level of achievable precision with these definitions is required.
On the note of universality, mathematics actually are universal. Mathematics are independent of culture, nation, religious belief, etc. Yes, the types of notation you use to talk about mathematical concepts vary from language to language and culture to culture. The mathematical concepts themselves, though, are universal. A quasigroup is defined in Norway in exactly the same way it is defined in South Africa. A topological space is defined in Malaysia in exactly the same as it is defined in the United States of America. The works that you see published by mathematicians from New Zealand are not in disagreement regarding the concepts and their properties with the works published by mathematicians in China. You seem to insinuate that mathematical definitions are on the same caliber as definitions of words you find in the Urban Dictionary. Yes, those words do vary in definition from location to location, and even just from person to person, and really, those words are not defined in any particular way at all, they are used arbitrarily, because they are not used to discuss concepts, they are just used to communicate basic bits of information about a particular ill-defined thing. This is not how mathematical definitions work at all, though.
Therefore you can discuss hypothetical scenarios were the current definition does not exist.
I can guarantee you that there is no hypothetical scenario in which the current definition would not exist and in which people have the knowledge of ring theory that we do today.
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@Alex-02 The point of this exercise was to ignore the current definition.
Then such an exercise is itself futile and pointless. It is not worth engaging in. It is sophistry. When in a discussion in a topic where definitions already exist, those definitions must be acknowledged before discussing if them existing is, in fact, justified. Otherwise, you may as well be speaking gibberish.
I know, I agree that that would be a terrible definition!
Yes, and my point is, so is any definition where 1 ends up being a prime number. Hence, why I have said in the past, that saying 1 is a prime number is like saying that my Toyota is an animal.
That is kinda irrelevant for my point tho, all I'm saying is a discussion can be had about what the definition should be, and that's a matter of opinion.
We can have the discussion, but if you are going to ignore the current definition, then said discussion cannot be had. If you want to change the definition, then you need start by (a) acknowledging that the definition exists, then (b) explaining how or why this definition fails in any way. Also, appealing to extraneous ideas like that of "factors" without even acknowledging how those are related to the current definition, and therefore, to whether there could be any merit in changing the definition, is entirely pointless.
I agree with the general point you are making in this section, yes, the mathematical concepts are universal. But that's different from mathematical definitions.
It is not. Definitions are merely how we formally classify and compartmentalize those concepts by using language. The actual strings of letters used for naming the compartments vary, of course, but that falls under notation, which I already addressed. The substance, the structure of the definition, does not vary.
The example of prime numbers in this video is proof of that. The definition has changed throughout history, so that is a counterexample to definitions not being universal.
No, it is not, because the universality of a definition has nothing to do with whether it has existed throughout all of history. Sorry, but that is literally not how the word "universal" works, and it never has been how the word works. Besides, the definition of primality used in the past according to the video, if we account for restrictions to positive integers only, is actually completely in agreement with the modern definition of primality. Therefore, your claim is false. Let me explain: the definition used in antiquity and in medieval times, according to the video, when applied only to positive integers, is that an integer is prime if and only if the only (positive) integer that measures it is the integer 1. This definition is completely equivalent to the modern definition, and in fact, the integer 1 does not fit this definition, hence even by the old definition, 1 is not a prime number. 1 does not fit the definition, because 1 does not measure 1. This is because integers do not measure themselves, as the video itself explained. Mathematicians in the past never really cared for numbers being divisible by themselves: they only cared about the smaller divisors, what we today call the proper divisors. The only positive integer that are proper divisors of a prime number is the integer 1. 1, on the other hand, has no proper divisors. The set of positive integers smaller than 1 that divide it is the empty set. Notice how this contrasts with prime numbers, which by definition, do have proper divisors. This is completely equivalent to the modern definition. However, in the same way that there are people today who do not actually understand the modern definition of prime numbers, back then, there were also many people that did not understand the definition of prime numbers that was used back then. This was, as the video explained, because of philosophical debates surrounding the nature of the number 1, and whether it actually was a number. Mathematicians began to argue that we should actually consider 1 to be measurable by itself, despite not providing any sound reasoning for it. This led to this weird inconsistency where 1 did not fit the definition of primality, but for philosophical reasons, it was included in lists of prime numbers anyway, as if it was supposed to be "an exception" to the definition, in a very ad hoc, for reasons that had nothing to do with mathematics. Such inconsistencies were common even during the medieval period, because mathematics lacked rigor. We say these same inconsistencies in older formulations of concepts in calculus, and even in concepts in algebra. Until a few centuries ago, for example, it was widely believed, without any reason at all, that an integer divided by 0 must have been 0. There was never any mathematical reasoning for this, not even heuristically. It was grounded on philosophical biases. Even so, in practice, no one actually ever evaluated divisions by 0, because it led to results that they knew were incorrect. Hence the inconsistency.
But... you just described your own hypothetical scenario, then claimed it doesn't exists when it clearly does because we can talk about it.
No, that is not how that works at all. Me being able to give a verbal description of an impossible situation does not mean the situation is actually possible and actually exists. This is an astronomical leap in logic.
The whole point of a hypothetical scenario is it doesn't have to be likely or even possible to happen in the real world.
No, that is not the whole point of a hypothetical scenario, not even close. A hypothetical scenario just refers to a coherent scenario which has not yet been known to happen, and which may or may not happen, but which is possible in principle (hence "coherent"). The fact that said scenario can be talked about does not make it real. If it were real, then, it would not be hypothetical, it would be actual.
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@Alex-02 Anyhow, you seem to be insisting on wasting time with banal platitudes, rather than actually getting to the point of the discussion of whether 1 is a prime number or not.
The utility of the definition of prime numbers comes in considering the divisibility relation. For any two integers m, n, we are concerned with whether m divides n or not, and we want to classify the integers in such a useful way that it facilitates that study. The integer 1 is characterized by the unique property that for all integers m, m•1 = 1•m = m. This means that 1 divides all integers. –1 also divides all integers, because (–1)•(–1) = 1, which means that –1 divides 1, and 1 divides all integers, and the divisibility relation is transitive. –1 and 1 divide all integers. They are the only integers with this property. We can prove this. The only integer between –1 and 1 exclusive is 0, but 0 only divides 0 and no other integers. All other integers m satisfy m < –1 and m > 1. But, if m divides –1 or 1, and m is not equal to –1 or 1, then m > –1 or m < 1, but this is a contradiction. Therefore, for all other integers m, there exists n such that m does not divide n. The only divisors of –1 and 1 are, in both cases, –1 and 1, and –1 and 1 are the only integers that divide all integers. As such, –1 and 1 are structurally isolated from the other integers in this fashion. They form a multiplicatively-closed structure, called a group. This already implies that, regardless of how we structurally classify the other integers in terms of their properties with respect to the divisibility relation, they belong in a different class than –1 and 1 do. 0 is also structurally isolated, because for all integers m, n, if m•n = 0, then m = 0 or n = 0, and because for all integers m, 0•m = m•0 = 0. All integers divide 0, and 0 only divides 0. All the integers that are not equal to –1, 0, 1, thus, when classified by their properties with respect to the divisibility relation, are demonstrably in a different class than either the class of 0 by itself, or the group that –1 and 1 are in. Earlier, I mentioned that for all integers m, 1•m = m•1 = m, and this is the defining property of 1. This means that for all integers m, m divides m. This also means that for all integers m, –m divides m, and m divides –m. This is possible, because –1 divides 1, and 1 divides –1. This means, that for all integers m, –m, m, –1, 1 divide m and –m. However, remember that only –1 and 1 divide –1 and 1. This means that the only integers that divide all their own divisors are –1 and 1. Again, this is why –1 and 1 are multiplicatively isolated, and form a multiplicative group. All other integers are divisible by –1 and 1, but do not themselves divide –1 and 1. Thus, for all other integers, we can define a concept of a divisors which is not divisible by the dividend. Such divisors are called proper divisors. –1 and 1 are proper divisors of –m and m when –m and m are not equal to –1 and 1. –1 and 1 have no proper divisors. –m and m always divide m and –m, which are themselves divisors of –m and m, so –m and m are not proper divisors of –m and m, and this is true for all m. This raises the question, are there integers whose only proper divisors are –1 and 1? Yes, and those are precisely the prime numbers (if you include their negative versions too). All the other integers, besides them, and besides –1, 0, 1, can be written as products of proper divisors which are not –1 or 1. In fact, the prime numbers are characterized, precisely by the idea that the product of two prime numbers is never a prime number. This means that, in this classification, they must be categorically distinct from the remaining integers not considered, which we call the composite integers.
This is how you motivate the definitions: you need to focus on the analyzing the structure the definitions are meant to intuitively capture. Furthermore, this classification of the integers into four classes with defining properties extends to all commutative rings, making it an extremely robust and general classification. Functions which preserve the structure of a ring (ring homomorphisms) also preserve these classes and divisibility relations: such functions will never map prime numbers from one ring to composite numbers from another, or will never map –1 or 1 (or some other unit) to a prime number. This is how we know this conceptual classification is the correct one. The way the classification is done for arbitrary rings is as follows: we consider all the zero divisors. A zero divisor is an element m such that there exists some nonzero n such that m•n = n•m = 0. In the integers, the only zero divisor is 0, but in other commutative rings, there may be nontrivial zero divisors. We know this works, because all multiples of zero divisors are also zero divisors, and zero divisors can only divide other zero divisors. Once we have considered this, we consider all the units. The units are those elements m such that there exists some n such that m•n = n•m = 1. In all rings, –1 and 1 are units, but some rings have other units. Units are only divisible by units, and units divide all objects in a ring. Also, the product of two units is always a unit. Of the remaining objects in a ring, we consider their proper divisors. If their only proper divisors are units (remember, the units have no proper divisors), then they are called irreducible elements, and they are the equivalent of prime numbers in that ring. The product of two irreducible element is never irreducible, but composite, and composite elements always have some proper divisors which are irreducible.
One last important idea is the way these classifications actually make the divisibility relation into an order relation, where the units are the minimal elements, and the zero divisors are the maximal elements (well, strictly speaking, 0 is always the greatest element). Irreducible elements are always the "next" elements after the units. The way you make this work is you say that two elements m, n are equivalent (m ~ n) if and only if m divides n and n divides m. This partitions the ring into equivalence classes, which we can denote [m]. We say that [m] divides [n] if and only if every element in m divides every element in n. Now, if [m] and [n] are distinct classes, then one is a proper divisor of the other. The class [1] is the smallest element in this ordering, because for all m, [1] divides [m], and if [n] divides [1], then [1] = [n]. The class [0] is the greatest element, and the class of nontrivial zero divisors is the class "right before" [0]. Well, the zero divisors are more complicated, since some zero divisors are proper divisors of other zero divisors, but the general idea stands.
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@buycraft911miner2 Your statement is a naive one. All of number theory is one academic endeavor centered around the study of prime numbers, so I have no idea of what you mean by "almost no math problems require diving deep into prime numbers." This, to me, sounds like a case of wilful ignorance. Not only are not willing to read my comments carefully, you also have done nothing to address them, or even understand them, and it is clear that you actually have no interest in learning anything about the mathematics, or in becoming better informed about how mathematicians come to the conclusions they do. I am starting to lose respect for you, and I get the impression that there is no line of reasoning, no matter how sound, that is capable of making you realize you are wrong. I am afraid interacting with you and listening to you was a waste of my time. Then again, this is often the case when I interact with laypeople. Most people nowadays would rather protect their own feelings than listen to the facts.
History is proof of this, as even while being uncertain about the nature of 1, math as a whole went on.
Yeah, this does not prove your point at all, for the very simple reason that "the nature of 1" is ultimately completely irrelevant to mathematics. It is entirely a philosophical discussion. You see, the "nature" of objects in mathematics is ultimately irrelevant. Mathematics are only concerned with mathematical structures, how objects in the structures are related, and how the various structures interact. The actual nature of those objects is irrelevant: what matters is that the objects interact according to the axioms that define the structure. Is 1 a "number"? Well, what even is the definition of a "number"? There is no definition that is widely accepted by mathematicians, because again, it does not matter. Even if we say 1 is not a number, it has no effect on the mathematics. After all, we can do arithmetic just fine with all sorts of objects which are not numbers. We do arithmetic with functions, vectors, tensors, matrices (well, strictly speaking, matrices are special cases of functions), polynomials, etc. We can even perform arithmetic on mathematical structures themselves. So, the nature of these objects is just a philosophical concern that does not matter to mathematicians. The only defining property of 1 that any mathematician cares about is that 1•x = x•1 = x for all x in the structure.
Anyway, I replied to you simply because I think it would have been rude on my part to not let you know, but I know better than to be a fool and continue engaging with you. If, one day, you decide to be intellectually honest and take the work of mathematicians seriousky and be willing to be open to having your mind changed by mathematical reasoning, then I will be willing to interact with you again, but in the meantime, you will not be hearing back from me, and I will not be listening to you any further. I hope you have a nice day, though.
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@PureExile m = 4•k, n == 7 (mod 4•k), n = λ^2, λ^2 == 7 (mod 4•k). But, λ^2 == 0 (mod 4) or λ^2 == 1 (mod 4), so given λ^2 == 0, 1, 4, 5, ..., 4·(k – 1), 4•(k – 1) + 1 (mod 4•k). Hence, for some integer ν = 0, 1, ..., k – 1, we have that 4•ν = 7, or 4•ν + 1 = 7, so ν = 7/4, or ν = 3/2. 3/2 and 7/4 are not integers, so this is a contradiction. Therefore, λ does not exist. Therefore, n is not a perfect square. Q. E. D.
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13:08 - 13:19 It actually does not fit the definition. Earlier in the video, it was clarified that what mathematicians meant by "measured" was a notion of divisibility which only considersd proper divisors. In other words, the number of which the divisors are being considered does not measure itself, because it is not a proper divisor of itself, since it is not smaller than itself. This is included in the definition of a prime number. "A number is prime if and only if the only number that measures it is 1." Prime numbers are divisible by 1, AND by themselves, as well. This confirms that "measured by" and "divisible by" are not synonymous, because of the distinction between divisors and proper divisors. Prime numbers are divisible by themselves, but are not measured by themselves, they are only measured by 1. In modern terminology, what this means is that the only proper positive divisor of a prime number is 1. Now, it is clear that 1 does not fit the definition of a prime number: 1 is not measured by 1. It cannot be, because 1 is not a proper divisor of itself, by definition. In fact, 1 has no proper divisors. In the older terminology, this means that there no numbers that measure the number 1. Prime numbers are measured by the number 1, and only the number 1, but the number 1 is measured by no numbers at all. Therefore, it does not satisfy the definition given by medieval mathematicians of a prime number, even if you argue that 1 is actually a number at all. In other words, what this tells me is that medieval mathematicians were just inconsistent, and were incapable of detecting that treating 1 as a prime number was inconsistent with the definition of prime number they themselves used, since the definition of "measured by" that they used necessarily meant 1 was not measured by any numbers. They still came to the conclusion that 1 was measured by 1, because they were applying their own definitions inconsistently. This is something that was not uncommon back then, and it is the reason why rigor became a necessity later on.
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I think it's a good thing that there's no objective morality. We as a society decide what's moral and what's not.
No, society does not decide that. This whole "morality is objective" vs "morality is not objective" discussion is stupid, and also, dangerous. Ultimately, when it all comes down to it, the way people evaluate whether an action is moral or not is based on what they believe the consequences of those actions to be, regardless of whether those beliefs are scientifically accurate or not. We can measure suffering, scientifically. However, this measurement necessarily must respect the confessions of the people being affected by those actions. If the person claims to be suffering, then they probably are (unless you can categorically demonstrate that they are lying). No one else gets to make the decision on whether that given individual is suffering or not. Society as a whole does not get make this decision. Society does not decide whether some particular action causes suffering or not, to those affected by it. The "society decides" thinking is extremely dangerous, and it is how we got things like chattel slavery, colonization, and the Crusades.
This involves us as individual parts of the society and thus makes each of us responsible to hold up these moral values (as in a democracy).
Nope. The "democracy" idea is very dangerous. Again, this is how we got chattel slavery in the U.S.A. Democracy might be great for doing politics, but certainly not for making judgments of moral value.
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These questions are not alike at all. Also, if you are thinking of mathematics conceptually, then not considering 0 to be a natural number is never useful, and considering 1 to be a prime number is also never useful. In fact, while you still can find peer-reviewed sources today of authors still excluding 0 from the natural numbers, albeit as a minority, you cannot find any sources, peer-reviewed or otherwise, where 1 is considered a prime number. The number 1 has not been considered a prime number by any mathematicians at all since the mid 1800s. There are absolutely no applications where the idea is useful, and it is also not in accordance with the formal mathematical theories that serve as the foundations for arithmetic. I have no idea how this myth that "considering 1 a prime number is useful in some circumstances" originated, but I hope said myth will die off some day.
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Regarding Galileo's paradox, you said that a function which matches every possible integer which is an even number with itself is injective. Fair enough, more integers than even numbers. But what about the follow-up: you have a function where each integer is assigned to its double, and it is a bijective function. How is that possible?
I do not understand your question, but hopefully, what I will say is still helpful. The set of integers Z has a proper subset that we call the set of even integers, which we denote as 2Z. There are two functions that were mentioned in the video from 2Z to Z. The former function, f, is the function such that f(n) = n for all even integers n. f is an injection, but not a surjection, so it is not a bijection. The latter function, g, is the function such that g(n) = 2·n = n + n. g is an injection AND a surjection, so it is a bijection. The question that is important here is, do there exist any bijections from 2Z to Z at all? If the answer is no, then 2Z is smaller than Z, since an injection between 2Z to Z does exist. If the answer is yes, then 2Z is the same size as Z. So, what is the answer? The answer is yes: there exists at least one bijection from 2Z to Z, and g is one of those bijections (there are infinitely many such bijections, actually, g is just the simplest one to talk about). Actually, this gives us a hint as to why we denote the set of even integers as 2Z to begin with.
If there are more integers than even integers, how is the second case showing a bijection and not an injection as well.
It is not the case that there are more integers than even integers. I believe you are misunderstanding what the video is saying. Intuitively, one may think there are more integers than even integers, based on the fact that f is not a bijection, but g is a bijection, so despite what your intuition would say, 2Z and Z have the same size.
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B Tread You're the only idiot here.
1. Half of the cities that you mentioned don't actually have liberals in positions of power. Quite the contrary. Especially Atlanta and Chicago. These are republican cities.
2. no elected official was telling me what I could and could not do and no curfews were ever even suggested. And because of that, the U.S.A is the country with the highest rates of COVID-19 infections and COVID-19 mortality, and it has led to American citizens being travel-banned from the European Union. Surely, your elected officials were just as idiotic as you are. The rest of the world has already recovered.
3. There are no mindless vandals destroying property and no hoards of thieves pilfering any of our minority owned businesses. None of that is happening in Chicago or Atlanta either. Do you live in either city to confirm otherwise? No, you don't.
4. I'm not sure who generates the statistics to which you are referring The Guardian, Forbes, and the Bureau of Statistics, among other non-journalistic sources. None of these are liberal pieces of media, but I suppose you're too much of an idiot to know that.
Black Lives Matter is one example of a group with espoused Marxist leanings.
Thank you for confirming your idiocy.
Their title is just a cover and their agenda is anarchy.
Sorry, but I don't buy into tinfoil conspiracy theories that have no scientific evidence. You're no better than a flat Earther or an anti-vaccine numbskull.
Anyhow, I'm not going to waste my time paying attention to nonsense from a person who believes that things that aren't happening in Chicago are happening with no evidence whatsoever. You're also too much of an idiot to realize that statistical data trumps personal experience from one single person. Lastly, I don't want to waste time with someone as dishonest as yourself, who decided to open up the discussion with an ad hominem. I won't be hearing from you again, because I won't be receiving message notifications from you again, because as I said, talking to willfully ignorant people like you is a waste of time. Bye.
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@maxaafbackname5562 Yes, there does exist such a number. I can construct it very easily too. Let Z be the set of integers, where + and · denote addition and multiplication, respectively. Since · is commutative, I can form the set of polynomials with integer coefficients, Z[X]. Every polynomial can be written in the form R(X)·(X^2 + 1) + A·X + B. Two polynomials P(X) and Q(X) are equivalent if and only if the difference P(X) – Q(X) is divisible by the polynomial X^2 + 1. In other words, A·X + B is equivalent to R(X)·(X^2 + 1) + A·X + B for all polynomials R(X). If two polynomials P(X) and Q(X) are equivalent, then we write P(X) ~ Q(X). For a given polynomial P(X), the set of all polynomials Q(X) such that P(X) ~ Q(X) is denoted [P(X)]. In other words, [P(X)] = [Q(X)] if and only if P(X) ~ Q(X). [P(X)] is called the equivalence class of P(X). We can find the set of all the possible equivalence classes, {[P(X)] : P(X) in Z[X]}. This set is denoted Z[X]/~, it also denoted Z[X]/(X^2 + 1). We can define addition +' for these equivalence classes, by letting [P(X)] +' [Q(X)] := [P(X) + Q(X)]. Similarly, we can define multiplication ·' by letting [P(X)]·'[Q(X)] := [P(X)·Q(X)]. Remember, P(X) = R(X)·(X^2 + 1) + A·X + B for some polynomial R(X) and integers A, B. As such, [P(X)] +' [Q(X)] = [R(X)·(X^2 + 1) + A·X + B] +' [S(X)·(X^2 + 1) + C·X + D] = [R(X)·(X^2 + 1) + A·X + B + S(X)·(X^2 + 1) + C·X + D] = [(R(X) + S(X))·(X^2 + 1) + (A + C)·X + (B + D)] = [(A + C)·X + (B + D)], while [P(X)]·'[Q(X)] = [R(X)·(X^2 + 1) + A·X + B]·'[S(X)·(X^2 + 1) + C·X + D] = [(R(X)·(X^2 + 1) + A·X + B)·(S(X)·(X^2 + 1) + C·X + D)] = [R(X)·S(X)·(X^2 + 1)·(X^2 + 1) + R(X)·(C·X + D)·(X^2 + 1) + S(X)·(A·X + B)·(X^2 + 1) + (A·X + B)·(C·X + D)] = [(A·X + B)·(C·X + D)] = [A·C·X^2 + A·D·X + B·C·X + B·D] = [A·C·(X^2 + 1) + (A·D + B·C)·X + (B·D – A·C)] = [(A·D + B·C)·X + (B·D – A·C)]. In other words, +' and ·' are well-defined. This means [A + B·X] +' [C + D·X] = [(A + C) + (B + D)·X], and [A + B·X]·'[C + D·X] = [(A·C – B·D) + (A·D + B·C)·X].
With this construction in place, I can prove that [X]·'[X] = –1. [X]·'[X] = [0 + 1·X]·'[0 + 1·X] = [(0·0 – 1·1) + (0·1 + 1·0)·X] = [–1 + 0·X] = [–1]. We typically denote [A + B·X] as A + B·i, with [X] = i, so i·i = –1. There, I have constructed a number, an element of Z[X]/~, such that when squared, it is equal to –1. The set Z[X]/~ is called the set of Gaussian whole numbers, or the set of Gaussian integers. This set contains the integers. A Gaussian integer A + B·i is an integer if and only if B = 0.
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@barnaclejones822 The problem, in my eyes, is trying to appeal to an authority that already excludes what you're positing.
It does not exclude such a thing, and I am not even sure you actually know what it is am positing. Actually, you definitely do not know, because I never even clarified what it is I am positing. All you are doing right now is playing games, pretending you can read my mind, when the only thing I did was respond to an erroneous argument you presented.
You just have to concede that America & its history is not some bastion of liberal democracy.
Oh, I agree completely, but not at all for the reasons you think I would agree. I am the opposite end of the spectrum of where you are.
I never claimed it was an inherently religious document, my claim was that it was written under religious assumptions with the intent to be applied only to a moral and religious people.
And this claim is factually incorrect, as has been pointed out to you by multiple people. Also, what the intent was is completely irrelevant to this discussion, and completely irrelevant, in particular, to what OP said.
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Is time more like the naturals or the rationals?
If we go by our current understanding of the physics, it is definitely more like the rational numbers, but a reliable answer to this question requires that we develop a quantum theory of gravity. A quantum relativity, if you will.
The answer has significant philosophical implications that you won't be able to detect by just considering cardinality.
You are correct. This is why we consider the order types of sets, rather than just their cardinality. (N, <•) is a poset, and so is (Q, <••), and there exists an order-embedding from N to Q. However, there is no order-embedding from Q to N. Therefore, the order type of (Q, <••) is larger than the order type of (N, <•). To be precise, the order type of the latter is the ordinal ω, while the order type of the former is ω^2. Time in our instantiation of spacetime probably operates with a set with order type ω(1) or larger, but ω^2 is closer to ω(1) than ω is, which is why I said time is more like the set of rational numbers. Though, if Wolfram's program is accurate, then time is actually of order type ω.
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@nobodyspecial9 Let's transpose your question to the point of transgender identity. So, how exactly is claiming they have a certain gender when they factually do not, not "lying"? It's not as if they do not know what their actual gender is. That would be impossible, except for the case in which they were extremely cognitively deficient, which is not clinically or legally the case for them.
This makes the assumption that it is actually possible to know what someone's gender is without being told by the person what it is to begin with.
Do you see the problem with that flavour of argument?
I do not, partially because you are strawmanning me here. My claim about Rachel lying is not merely in her claim about her racial identity, but in her claim about what her heritage is. I know I did not make it clear in my comment, but I thought we were discussing common knowledge here. I supoose we are not.
Identity refers to how you perceive yourself, not to what is apparent externally.
This is wrong. Identity has many facets and is significantly more complicated than a matter or self-perception. Of course, self-perception plays a mjor role, but it is by no means the sole defining factor.
You can look, dress and act as a member of a certain gender and still perceive yourself as belonging to another.
Yes, because how you dress and look is much more an issue of gender roles and not a matter of gender identity. Even a person who denies the validity of transgender identity would tell you this much. But the picture that people are missing is that the concept of gender identity is incoherent without gender roles and a cultural context to talk about it. Functionally, how one's gender is perceived, self-declarations aside, is via those gender roles. And gender roles are very much not about self-perception, but societal perception. Think about this: what does it even mean to "dress according to a gender"? This is an incoherent notion if gender truly is just another name for biological sex. One cannot dress according to a biological sex. This is nonsensical, as biological sex does not have a form of clothing intrinsically tied to it. There is nothing inherently female about skirts. The only reason they are "female" clothing is because society said so. And so, if one sees someone on the street with a skirt, they would assume the skirt-wearing person is a woman, and they would be ready to die on the hill of that assumption, regardless of what the truth is.
With race, this is even worse, as race inherently does carry external components to it, and in particular, a very historical context that cannot be changed by one's self-perception. But the difference is, that outside certain specific contexts in the medical field, and outside the specific context of having chidren, one's biological sex is functionally irrelevant, and so is one's gender. Race is not.
As such it makes no sense rationally to exclude racial perception while lauding gender perception, because they both depend on what your perception of yourself is.
But that is the thing: racial perception does not determine racial identity. Because as I said, real, tangible, external component to this exists. Meanwhile, how society perceives gender is almost solely a function of how one chooses to present oneself.
I'm not saying Dolezal wasn't "lying", but to dismiss their claim outright based on your perception of their condition smacks of the same type of disingenuity that many people in the comments accuse Dawkins of having.
Again, I thought we were discussing common knowledge. The reason Rachel has been accused of lying is because anytime she has been asked to explain herself, she has literally lied, regarding how she justifies her claim of being transracial. Lying about your own family still counts as lying. And besides, this objection ignores the fact that race and gender are functionally different in how they relate to one's identity. And gender dysphoria functions very differently from any other kind of dysphoria, which is why the DSM and scientific organizations classify gender dysphoria as an entirely different kind of mental condition altogether. You are not even comparing apples to oranges. You are comparing apples to vegetables.
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@nobodyspecial9 Gender roles and gender identity are two parts of the same coin, wherein one is the internal perception and the other is the external perception. And they are both arguably based but not necessarily in alignment with the tangible and physical component of gender, which is sex. Sex, in the absence of self-declaration to the contrary, is considered to be the basis of gender identity precisely because gender is often derived from and is compatible with sex, however because they are not mutually assured they do not necessarily follow.
This fails to present a full picture of the situation. The reality is more complicated than this. Gender identity, traditionally, has been devoid of self-perception, and has been based entirely on the biological sex of an individual. However, there are major problems with this.
One problem is that our understanding of biological sex today is much more refined than the understanding we had 80 years ago, and we realize that biological sex in humans is not a simple discrete binary, and not uniquely determined by chromosomes either. This also means that while, in the past, the understanding was that biological sex had functional relevance societally, which is what motivated gender roles in the first place, it turns out that we know today that biological sex has no functional relevance societally, and as for an individual, it only has relevance for the purposes of medical necessities, and for the purposes of sexual interaction, both of which are completely private matters. If gender identity is based on biological sex, then it too has no functional relevance. This makes gender as a classification very much arbitrary and meaningless. Effectively, we say gender exists only because we have traditionally believed it exists, not because it has any objective existence to it.
And on that note, another problem is that this makes the traditionally held connection between gender identity and gender roles fallacious and fictitious. If what determines an individual's gender is which reproductive organs the individual has, then masculinity and feminity are equally arbitrary and meaningless distinctions. There is no intrinsic relationship between how one dresses, how one talks, or how one does the vast majority of things in social interactions that do not directly involve having sexual intercourse, and one's reproductive organs. Masculinity and feminity are perceived to be real only by virtue of societal fiat.
But how one interacts socially is functionally relevant and does carry non-arbitrary, meaningful distinctions. So if gender roles were real, in other words, if how one manifested social interactions actually did have an intrinsic relationship to one's reproductive organ, then biological sex-based gender identity would have functional relevance. Gender identity has functional relevance only if gender roles do. The coin with two sides that you describe metaphorically in your paragraph is thus not a symmetric coin, or to say, not truly a coin. That being said, while gender roles are fictitious, they still carry subcultures with them, those being types of masculinity and feminity, which are themselves dependent on the broader context of one's national culture, religion, politics, ethnicity, and yes, race, among other things. This is where self-perception becomes relevant. Self-perception operates on one's sense of belonging which is based entirely on this arbitrary, meaningless distinction between masculinity and feminity as subcultures, and unlike the distinctions themselves, self-perception is not arbitrary nor meaningless, and has functional relevance. Gender identity acquires some level of functional relevance, albeit still artifical, only as a function of how one's self-perception interacts with these gender subcultures. And otherwise, it is completely devoid of objective existence. In fact, hypothetically speaking, if society were to realize that gender is objectively unreal, then it may get rid of the distinction altogether, yet choose to keep the gender subcultures as part of one's identity, albeit choosing to disconnect the notion of those subcultures from any notions of biologica sex, which is the sane thing to do, and would simply treat them as precisely that: subcultures. This is why it is nonsensical to consider gender as anything but a function of self-perception.
Similarly, race has an external perception and an internal perception, based on a tangible and physical component of genetic race. Racial perception can also differ from its genetic roots as it is also based on identity and, to borrow your phrase, "how one chooses to present oneself", and as such distinct from their genetic race.
No. This is demonstrably not the case. Race functions very differently from how gender functions. The distinctions between races are based on external features that, unlike with gender, are very much functionally relevant in society, and are intrinsically tied to socioecionomic history and ethnicity. Race, unlike gender, is hereditary. The existence of race is also very closely tied to colorism. And unlike one's reproductive organs, which very few people know what they are unless they are explicitly peeping on you, or they have had sexual intercourse with you, or unless they are your parents, or unless they are your medic, and thus have no functional relevance aside from an arbitrarily assigned and fictitious importance, one's color of skin is very much visible to everyone, and can singlehandedly affect your life in drastic ways, for the better or for the worse, even if this is not necessarily true for every single person out there.
Race is also inextricably linked to one's geographic location, and to one's cultural history. These are aspects that race does not share with gender. And this means that, unlike gender, race is defined and determined by things that have an objective, unarbitrary, meaningful reality of their own. This much is true, despite the fact that race is not a legitimate biological distinction, contrary to what racist people many centuries ago used to claim. Also, for what is worth, unlike with gender, there is no historical-sociological precedent for transracial identity or race dysphoria that can legitimize self-perception as not only being a genuine component of one's race, but also one which decisively overrides all of the above.
The very fact the Dolezal chose and existed for years as a "black" person before being outed as "white" is proof that racial identity both internal and external can be distinct from genetic race.
This is a very uninformed and ignorant opinion to have on the subject matter. Dolezal was not "outed" as white after many years of having existed as black. It has always been known by everyone who has known her that she is white. And she was merely black-passing, which is not the same as being black. People knew she was white. For context, she was born in 1977, and she only began her charade in 2009. By the time, she was not particularly well-known, but the people who did know of her did criticize her. Yes, the controversy only exploded in 2015, but that does not mean that plenty of critiques of her behavior did not exist from years prior. And the reason the controversy began was not because she was "outed", but because a major article was written about her in which her past hate crime allegations and on her lying, and this was enough to draw attention from the news media, and naturally, the news media exacerbated the issue to the point of irritation, because that is what they do. The reality is, she had already been outed even before this, as her parents had already made public statements as early as 2011. But the news media brought this to the attention of people who were completely unaware of her existence previously, hence why controversy began. Portraying it as "we only found about her being white after she was outed" is not only ignorant, but disingenuous. So, no. This does not prove your thesis about how race functions.
In my opinion the point of contention between us is in that you seem to adhere to a kind of special pleading fallacy wherein gender identity is allowed to be solely determined by internal perception while racial identity has to be determined by external perception, or worse a tangentially related physical characteristic.
This is far from special pleading, as both of these claims have actual justification. If anything, I would say your argument is a false equivalence: pretending that race and gender function the same without any good reason to treat them as such, and even when one should expect them, even outside this context, to function differently.
So she lied about her genetic links to the race of her identity. So what? What could that possibly have to do with her identity?
Everything, as I explained above.
That's like outing that a trans person has sex organs that are not in line with their identified gender and then accusing them of lying when they refute it. It might be true, it might not, but more to the point it doesn't matter.
And again, this is a false equivalence, as explained above.
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@nobodyspecial9 I dislike your tendency to cut up my arguments into random sections,... ...so I will simply refer to them in the order your rebuttals appear. -Paragraph 0
I am not dissecting your arguments into random sections. I am dissecting your arguments according to how you yourself separated them on your paragraph structure, each paragraph being its own claim. I acknowledge that you are trying to build a narrative, which is precisely why I make sure to address every single individual claim you make, rather than only focusing on one claim and then pretending that your stance has been fully debunked. I also acknowledge precisely the nnarrative in question at the end of each segment of my reply, after explaining why a particular claim being made is inaccurate. Hence the length of my previous reply. This is also the reason why I copypaste your arguments and quote them in boldface: it makes sure that both you and I know which of my claims addresses which of your claims, and it ensures that in me addressing your claims, I am addressing exactly what you said, and not some strawman of the claim. I put as much effort into being intellectually honest as I expect my opponents to be, even though I am aware that most people I will have discussions with are not intelectually honest, as is the nature of humanity. This is because I care about it. So, despite your qualms, I am not going to stop using this format. If it is a dealbreaker for you, then feel free to stop replying to me. No one is forcing you to jave a conversation you do not wish to have, and if you did, it would only be a waste of my time, not to mention of your time. Besides, contrary to your claim, the topic of my reply is precisely the subject of discussion. To be clear, your thesis here is that race and gender have an equivalence, in such a way that you think transgender identity is valid if and only if transracial identity is. The very topic of my reply is in explaining why that is not the case.
Part 1. I completely agree that biological sex exists on a spectrum, and I have never claimed otherwise... ...as gender roles are an external construct that has its foundations in societal norms and perceptions, an ever changing paradigm. -Paragraph 1
I never said you claimed anything about biological sex. My explanation was there to provide further context for my argument, and without that context, the claims made in the argument would have appeared to any reader simply as complete baseless assertions. I am not conflating anything. Everything I said in that particular section is specifically reliant on the distinction between the three, and I even explained in painful detail how the three are related. However, I never said they are the same thing. The reason I assert gender identity was historically and traditionally solely determined by one's reproductive organs is because this precisely what was considered correct in the majority of developing societies. In the language of SJWs, the traditional view on gender identity is the trans/phobic view (the things one has to do get around censor bots these days), not the modern view we have access to today. The existence of transgender identity is by no means recent, but it certainly has never been societally recognized until modern times, hence why it is not the traditional view. This has absolutely nothing to do with me conflating gender identity with gender roles. It has to do with the historical fact that it has been traditionally inaccurate and even absurd to say that one's gender identity is determined by one's self-perception of it, outside of some minority ethnicities, especially in the West-European values, which are the values that eventually were forced on most of the world many centuries ago.
Traditionally, gender roles are determined by gender identity. But this is not what my reply is contesting or objecting to. The determination of gender roles from gender identity is fallacious for the reasons I explained. Since the only thing that even has the potential to have functional relevance is gender roles, it should be the case that gender identity should instead be determined by gender roles. As I explained, this is where self-perception comes in, and this explains phenomenologically why self-perception must be the determining factor in gender identity for it to be a coherent and meaningful distinction between individuals in society.
It may be unrealized, but it is fallacious to claim... So when you base your argument on that foundation, your entire argument falls flat. -Paragraph 1
No. The claim "traditionally, gender identity was devoid of self-perpcetion, and so was based solely on one's biological sex (which was traditionally understood to just be one's reproductive organs)" does not imply that I believe that gender is a choice or that gender dysphoria is a novel phenomenon. This is not even a syllogism, so it is impossible for this to be a material implication. When one speaks of past tradition, one speaks about what was held to be true on a societal level, regardless of the actual accuracy of the belief in question. Talking about traditions is not fallacious. I know the distinction between traditional beliefs and what is understood in modernity.
Your claim that biological sex has no functional relevance is proven wrong... They do not cease to exist just because certain individuals do not experience them. -Paragraph 2
This is disingenuous, and a complete misrepresentation of what I said. I never stated, nor implied, that sex drive is non-existent. Many individuals do indeed experience sex drive. Sex drive is the reason why the industry of p**nography exists. It is also one of a few reasons why sexual crimes occur, though not by any means the primary reason. I never denied any of this, and nothing that I said regarding functional relevance of categorization of individuals in-context is in opposition to this. The fact that you are stating this, and then proceeding to assume my motivations for making the claims I never made but you said I made, is just a complete misrepresentation of the concept, and it is also intellectually dishonest.
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@nobodyspecial9 You also argue that gender identity only gains function due to... As such my point about them being two sides of the same coin stands. -Paragraph 3
No. The gender identity of someone else is not determined by how you perceive them. Self-perception is the perception of oneself, not the perception of others. Regardless of your perception of them, their self-perception as it interacts with the gender norms of society determines their own gender identity. Gender norms are also not determined by your perception of them either, since the perception of any one individual person is incapable of dictating gender norms: gender norms are necessarily a sum of perceptions, as they are a societal phenomenon. But besides, as I already explained, gender norms are completely fictitious, since the pre-established norms one's self-perception interacts with for gender identity to occur to begin with are themselves completely arbitrary, meaningless, and based on a tradition we now know to be false. The masculinity and femininity subcultures historically emerged from how society traditionally connected gender roles with one's gender identity. However, as I pointed out, these can conceivably exist independetly of the existence of gender, given that what their existence is also determined by several other major factors I mentioned in my reply. And these subcultures are themselves entirely subjective anyway: their existence is contingent on several factors independent of gender, but their determination for any given individual is not, as there is not even an objective metric that is capable of distinguishing the two. Society is able to dictate that masculinity and femininity exist, but it is unable to dictate how or where one fits in that distinction.
Part 2 Despite your claims to the contrary, race is not based solely on external features. Race is also a social construct that has gained relevance due to subcultures in the same way that gender has. -Paragraph 4
Race is indeed a social construct, because race has no basis in the laws of nature, it is a completely made up form of categorization of individuals that is arbitrary and meaningless in origin, and made up solely for the purpose of justifying oppression against one group of people or another. I never disputed this. This does not imply that determination of one's race is not societal and that there is no objective metric by which one determines how one fits into the social construct. I already mentioned that race is indeed tied to culture in more than way, but how it is tied to culture is very different from how gender is tied to culture.
You are conflating colorism, and racism based on color,... This statement smacks of ignorance. -Paragraph 4
The statement does indeed smack of ignorance. Thankfully, I never actually made this claim. You are twisting my words, presenting them to mean something completely different from what I actually said. So much for representing my argument fairly. I am not conflated race with colorism. The fact that I used the language that the two are inextricably connected implies that I acknowledge a distinction between them in the first place, and I furthermore explicitly stated that race is also inextricably linked to one's cultural history, geographic location, heritage, ethnicity, among other things. You are completely ignoring this. I am not so dishonest to assume whether this is due to a lapse of judgment on your part, or out of malicious intent, but you can stop pretending you present my arguments more fairly than you think I represent yours, at any rate.
If one’s racial identity is not... ...into classifications similar to gender identity. -Paragraph 4
No, such reasons do exist. I provided them in my reply, and rather than addressing them head on, you decided to ignore them, and pretend that the only factor I considered when discussing racial identity was colorism. This is dishonest on your part.
And if race exists... ...unarbitrary and meaningful reality of race fails to hold. -Paragraph 4
Yes,... if.
Part 3 I believe that I have already answered this claim. Your biased language aside, you have yet to prove that she didn’t truly believe her claim. You have also failed to prove why her supposed lack of black ancestors matters. -Paragraph 5
I never claimed that she did not truly believe her claim. This is completely tangential to the point I actually made. I have no obligation to prove to you a claim I never made. I have briefly overviewed why ancestry matters in the segment of my reply on race. You completely have misrepresented that segment of my reply, and pretended I did not connect race to anything else besides colorism. So, no, you have not answered anything.
Parts 4, 5, and 6... ...which is the very definition of special pleading. -Paragraph 6
The only thing you did was misrepresent my arguments more than once and twist my words to conclude that I made claims I never made, and then address those claims, rather than addressing the claims I actually did make. So, no, your thesis does not stand.
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@BlarglemanTheSkeptic2 From a fraction of a second, until 370 kya, the entire universe was like the inside of a star: super hot, super high pressure, composed of nuclei and electrons that were FAR too hot to stick together, and photons EVERYWHERE, being emitted and absorbed and electrically charged matter.
The universe was very hot, dense, and pressurized, yes, but it was nothing like the inside of stars. Nuclei did not exist: quarks were actually free, as the temperature of the universe was above the Hagedorn temperature, 10^12 K. Stars do not achieve these temperature in their interiors.
Then, as the expansion continued, it's like travelling outwards from the center of the star...
No, the cosmic expansion was nothing like this at all. The cosmic expansion is an expansion of spacetime itself: the actual matter embedded in the spacetime is not experiencing any motion.
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@BlarglemanTheSkeptic2 You are beating a dead horse, as I already acknowledged that the hadron epoch started 10^(–6) seconds after the initial moment. However, you continue to be wrong with regards to the existence of matter in the form of plasma, which did not occur until the photon epoch, 10 seconds after the initial moment. I was pretty clear about this in my previous comment. I guess reading comprehension is difficult for most people, though.
An atom is defined as nuclei comprised of baryons (nucleons, to be more precise) with electrons in a state bound to said nuclei. Lone protons do not meet this definition. They do meet the definition of an ion, though, but protons bound to antiprotons, which is what would have existed in the hadron era, are not lone protons, nor are they atoms.
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@dieSpinnt I wouldn't bother. The dude is a troll. He keeps insisting that that I have not admitted to a (minor and careless) mistake I acknowledged I made. He continues to be unnecessarily hostile, utilizing a lot of CAPS LOCK, in the style of what a teenager would do. Also, anyone who tries to cite Wikipedia as a scholarly source of discussion is not worth their salary. He also fails to address the rest of my arguments, while consistently strawmanning my claim. I have a degree in physics, and while I did get a timeframe mixed up, I think it is clear he does not have an actual grasp of what he is talking about. By the way, I have also seen him comment in other threads in this video, as well as other videos I have been watching these past two days. He always does this: antagonizes people over minor details, while being wrong about them, always being rude in the process. I have saved screenshots and sent them to some friends, to laugh over at them. He is not worth it. Your energy would be better spent on someone with intellectual honesty.
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@user-fb2jb3gz1d you keep using indefinites to say it's a fact.
I am not using indefinites.
Close to 0 is not 0. Could is not is. Mostly likely is not is.
These are not indefinites. The reality is that, in the scientific method, there is no such a thing as probability 0 or probability 1. You yourself acknowledged this in your previous reply, by noting that science is ever-changing.
Were you there? NO Was human there? NO
True, but irrelevant. A human does not need to be there in order for us to verify that it happened. Were you there when the colonization of the Americas happened? No. So, according to your logic, we should actually be rejecting the idea that such colonization happened, since according to you, it is not a fact. Is that what you want? Okay, then. That is great, actually, because now, I can hit you with "were you there to experience the resurrection of Jesus?" And I know the answer is "no," because I know you are not 2000 years old. As such, by your own logic, you have an epistemic duty to not accept the resurrection as factual.
See? Your argument only works against your worldview, not against science.
In all those times the experiment was done, how many times were the variables changed?
Plenty.
Now if new findings bring new variables, there is no possible way you can tell me that you know it won't change drastically.
I know it beyond the shadow of any reasonable doubt. There. I said it.
That's just stupidity.
No, it is not. Just because you are scientifically illiterate and you have no understanding of how the scientific method or how scientific linguistics work, it does not mean that experts who have studied physics for at least 3 decades more than you have are being stupid. In fact, you are the one who said in the other thread that simply criticizing someone's purposeful actions when they are vastly more knowledgeable than oneself is stupid. Well, that is literally what you are doing right now.
I'm pointing out that we don't know for a fact that there are no other variables because we don't know exactly what happened.
No, we do not know exactly what happened, but we know more than sufficient information to know that it did happen.
To say it's a fact is BS. To say it's most likely or most probable..........that's correct.
You contradicted yourself in the same sentence. After all, a fact is nothing more than a statement whose probability of being false is so small that it is unreasonable to take seriously. There are no statements whose probability is exactly 0. No such a thing exists.
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@user-fb2jb3gz1d I mean, I do have a degree in physics, but this is not about me. The scientific method is a collective, even global effort. There are millions of physicists around the world that have been contributing to our knowledge of the Big Bang theory for almost a century. It is laughable that you think that all several million of them are utterly stupid, just because you have no understanding of basic science and epistemology.
I agree that 99.999% is not equal to 100%. This is irrelevant. In science, there are no statements which have a 100% probability of being true. It is impossible for such statements to exist, due to the unfalsifiability problem. For example, solipsism is unfalsifiable. There is no method by which you, or anyone else, can prove that solipsism is false. Another example is the gargantuan reincarnation conjecture. What is that, you may ask? It is a conjecture that I literally just made up right now, and here is what it claims: it claims that every 1 second, the universe flashes out of existence, and then comes back into existence an infinitesimal amount of time later, such that the discontinuities in the timeline are unmeasurable. This is unfalsifiable, as well: there exists no method by which you can ever establish that the probability of it being false is 100%, because any objection you throw at it can be addressed trivially, simply by appealing to flashing out and into existence.
None of the statements you consider "facts" have a 100% probability of being true. The Earth being round? That does not have a 100% probability of being true. That the rotation of the Earth takes 24 hours? That does not have a 100% probability of being true. That plant leafs are green dureng springtime? That does not have a 100% probability of being true. That your phone will not crash right now, as you are reading this comment? This does not have a 100% probability of being true. Absolute epistemic certainty is impossible, and the scenarios presented demonstrate this pretty convincgly.
This is why, rather than adopting a dumb definition of "fact," which you propose we should do, which would result in facts not existing at all, scientists opt to not work with the concept of epistemic certainty, and instead use Bayesian epistemology supplemented by empiricist methods. This has always worked fine for us, and it is how the technology you are using to type these comments have been working so well.
I read that we can't see past the cosmic dark ages, a period that lasted from 370 000 to 1 billion years after the Big Bang.
Uh, no. It would be about 150 million years, and no, it is false to say we could not see past that. What is true is that there were no stars during those dark ages, but light still definitely traveled around the universe. That is how see the cosmic microwave background.
So because of this, we can't see when the first stars were born, 100 to 500 millions years after the big bang.
The numbeds you gave now literally contradict the ones you gave in your previous sentence. Are you so bad at this that you cannot paraphrase a number correctly from a website without contradicting yourself?
Read that on phys.org
I have read articles from them before. I guarantee you, they are nowhere near so badly informed that they would make mistakes this dumb. So, I have no idea where you actually got it from.
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Consider the set {0, 1}, and define ⊕ : {0, 1}^2 —> {0, 1} by 0 ⊕ X = X ⊕ 0 = X and 1 ⊕ 1 = 0, and define • : {0, 1}^2 —> {0, 1} by 1 • X = X • 1 = X and 0 • 0 = 0. ({0, 1}, ⊕, •) is a field, and it is the Galois field of order 2, also called the Boolean field, which is denoted F(2). Now, let n be a natural number, expresed as a set. Let C(n) be some nonempty set of functions f : n —> {0, 1}, and define + : C(n)^2 —> C(n) such that for all m in n, (f + g)(m) = f(m) ⊕ g(m), and · : {0, 1}×C(n) —> C(n) such that for all m in n, (k·f)(m) = k•f(m). (C(n), +, ·) is a vector space over the field F(2), and this vector space is the definition of a binary linear code. The functions in C(n) are the words of the code, and n is the length of all the words. This vector space can be equipped with the L1 norm, the taxicab norm, with w : C(n) —> [0, ∞), such that w(f) = Σ{|f|}. This norm is called the weight of a word. Thus, binary linear code is defined as Boolean normed space.
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@lukeriely4468 This is stupid and unreasonable. Human beings are not mind readers. If you say a sentence, then I have no reason to expect you intended to say something different than what you actually said. When I try to have conversations with people, then I expect them to have the extremely basic communication skill of knowing how to say exactly what they mean to say. This is something 8-year olds learn how to do in grade school. If you failed to communicate the point you wanted to communicate because you made the choice to use a sentence which does not mean said point, then this is your fault, not anyone else's.
Saying, please, before nailing someone to a wall is still nailing someone to a wall.
This is completely irrelevant to what we are talking about here. In fact, there is nothing analogous between this conversation we are having, and this scenario you are describing. However, let me add this: in an scenario like this, literally no one in their sane mind would care about the intentions of the person doing the nailing. No, they would care about what their actions. Communication is no different, as I explained above.
Simple, really.
Calling it "simple" when you are the one failing to understand requires a certain type of graceless audacity.
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Let (Q, 0, 1, +, ·) be the field of rational numbers. Hence (Q\{0}, 1, ·) is an Abelian group. Let f : Q\{0} —> Q\{0} be such that f(x) = x·x everywhere, and let f[Q\{0}] denote the range of f. Since f(1) = 1, and since f(x·y) = f(x)·f(y) for all x, y in Q\{0}, f is a group homomorphism. Therefore, if • is · restricted to f[Q\{0}]^2, then (f[Q\{0}], 1, •) is an Abelian group. Q. E. D.
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Hepad No, the equation x^3 + y^3 = (x + y)(x^2 - xy + y^2) does not restrict x and y to anything. I do not understand where you get the idea that x ~ y simply because this equation is true. The only restriction that this equation has is that, if x^3 + y^3 > 0, then sgn(x + y) = sgn(x^2 - xy + y^2, and if x^3 + y^3 < 0, then sgn(x + y) = –sgn(x^2 - xy + y^2). Those equations probably mean nothing to you. Truthfully, they mean nothing to anyone, because as restrictions, they do not pose much of a restriction at all. There is nothing that forces the solutions of x^3 + y^3 = 113 to be close in order of magnitude to the solutions of x^3 + y^3 = 115. In fact, generally speaking, this is false.
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Hepad And usually, if you have an equation of the form x(1)^n + x(2)^n + x(3)^n + ••• + x(n)^n = c, this equation is easier to solve computationally than x(1)^n + ••• + x(n - 1)^n = d, not the other way around. The more x(i) there are, the easier it is to solve. This can be proven, but should also be intuitive. For example, there are many c for which x^3 + y^3 = c has solutions in N^2 or Z^2, but for which x^3 = c has no solutions or less solutions, meaning that the solution density for most numbers increases as the number of variables increases. And the other thing that should make this intuitive is that the equation w^2 + x^2 + y^2 + z^3 = c is guaranteed to have solutions for any integer c, making it much easier to solve than x^3 + y^3 + z^3 = c.
Also, I am not sure why you mention N + i in the original comment, since (-1)^3 - i^3 is not 0.
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By analogy, consider quadratic equations instead of cubic equations. Take a look at x^2 + y^2 = c. If we take a number of the form 2n + m, then when we square it, we get 4n^2 + 4nm + m^2 = 4k + m^2. If m = 0, then m^2 = 0 = 0 mod 4. If m = 1, then m^2 = 1 = 1 mod 4. If m = 2, then m^2 = 4 = 0 mod 4. If m = 3, then m^2 = 9 = 1 mod 4. Therefore, any integer squared is 0 mod 4 or 1 mod 4. Therefore, the sum of squares of two integers must be 0 + 0 mod 4, 0 + 1 mod 4, or 1 + 1 mod 4. This implies there are no integer solutions to the equation if c = 3 mod 4. However, many c that do have an integer solution here do not have integer solutions for x^2 = c. Likewise, x^2 + y^2 + z^2 always has solutions for all natural numbers c, because the sum of three squares is 0 mod 4, 1 mod 4, 2 mod 4, or 3 mod 4, meaning all natural numbers are covered. w^2 + x^2 + y^2 + z^2 = c always has a solution for any c, but this time the amount of solutions for any given c increases, sometimes even becoming infinite. As you can see, the solvability of an equation is a monotonically increasing function of the amount of variables it has, whereas you had the wrong idea that it is the other way around.
In general, this can be proven for general n.
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Hepad And going back to x^3 + y^3 = 113, this has no integer solutions. But let's look instead at x^3 + y^3 = 115. 115 = 5*23 =
1*115 This means x + y = 5 and x^2 - xy + y^2 = 23, OR x + y = 23 and x^2 - xy + y^2 = 5, OR x + y = 1 and x^2 - xy + y^2 = 115, OR x + y = 115 and x^2 - xy + y^2 = 1, since x and y are integers. But do you see how now you have 4 systems of 2 quadratic equations to solve in 2 variables instead of solving 1 simple equation in 3 variables? Not computationally helpful at all.
And this case is, in fact, the simplest possible nontrivial case, because 115 is a semiprime, meaning there are only two possible factorizations of the number. If you had a number like 360, finding x^3 + y^3 just to find solutions to x^3 + y^3 + z^3 = 359 or 361 would be a complete nightmare, since 360 can be factorized in so many ways. Even if we exclude the case 360*1, you still get a ton of pairs to check, and for each pair, you get a system of quadratic equations to solve in two variables.
And, okay, you may say. You could simplify each system into one equation. Since each system is of the form x + y = p and x^2 - xy + y^2 = q, you could simply say let y = p - x, so one gets x^2 - x(p - x) + p^2 - 2px + x^2 = x^2 - 3px + p^2 = q. Then x^2 - 3px = q - p^2, and now we have the desired diophantine equation. But keep in mind that we have increased the number of equations vastly, based on the number of pair of proper factors, while only reducing the degree by one, and an additional cost is that these quadratics are largely non-elementary and not even depressed, whereas as the cubic x^3 + y^3 is elementary and is the simplest form of cubic involving two variables other than the trivial forms. So this method could only possibly be helpful for very few numbers, and not computationally efficient.
So, even if there were solutions to x^3 + y^3 = 115, (which there probably are), finding them only so we could solve x^3 + y^3 + z^3 = 114 is extremely impractical. It genuinely is just easier to try to solve x^3 + y^3 + z^3 = 114 directly.
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Some sentences are grammatically correct, but completely meaningless and incoherent. "Why does the Sun smell like the number 5?" is a sentence that is grammatically correct and mechanically correct in the English language. Is the question meaningful or coherent? No, it is a nonsensical question. Someone naive may think that we have an obligation to answer this question, and may even choose to call it a primordial question, and will he left unsatisfied at the fact that the question has not been answered. But the truth is that, it does not matter how much the asker protests, the question itself is nonsensical and meaningless, and so there is nothing to answer: ultimately, the question is not really asking anything, hence the question is not answerable. It is unanswerable, not because there is some fundamental flaw in the way humans reason philosophically, but rather, because the person who asked the question lacks an understanding of how meaningful questions work.
I would very much argue that "why is there something rather than nothing?" is another example of a question that is nonsensical. The question seems reasonable if you are not thinking about it at all: it is grammatically correct in the English language, and it appeals to our very flawed intuitions, so it seems like the question deserves an answer. But upon careful inspection, this question is just as nonsensical as my constructed example. How so? Because "something" and "nothing" are not well-defined metaphysical or ontological concepts. These words are merely consequences of our flawed intuition, and of the fact that abstraction is something that we have to train ourselves to do with imperfect crutches, rather than do naturally, since our brains did not really evolve to be good at abstraction. The words "something" and "nothing" can help us to transition from relying on intuition alone to relying on critical thought and abstraction, but ultimately, they serve no other purpose in the grand scheme of philosophy. And this is not for lack of trying. We have been trying to define what "something" is for millennia, and have failed so miserably, that we are not any close to solving that problem than the Ancient Greeks were. But at least now we know that maybe it just is not possible to solve the problem, because "something" does not embody any particular concept at all to begin with. Also, "why" questions tend to, very typically, be meaningless on their own right. "Who" is a question predicate that wants an individual for an answer. "Where" is a question predicate that wants a location for an answer. "Why" is a predicate that wants... some answer. But it does not tell you what kind of answer it allows. It is, in a rigorous sense, not actually a question predicate at all, which is how so many "why" questions end up being just incoherent. You can, of course, counterargue that "why" does ask for a specific type of answer: an explanation. But on its face, this is stupid, since all answers are explanations, by definition. The craftier among you will be more careful and instead counterargue that "why" asks specifically for justification. But this does still render most "why" questions meaningless, which proves my point. How does it render them meaningless? Because justification is an very specific category of answer that is only applicable if the question is being asked about the action of a sentient individual. For example, asking "Why did you cheat on me?" is very sensible, since you are demanding for a justification for an action. "Why did he cry so much last night?" is also sensible question, since again, it is asking about an action. But "why is the sun red?" is not a sensible question, because it assumes that 0) "being red" is an actuon (it is not), 1) the sun chose to be red (it did not). Of course, we do often use "why" and "how" interchangeably, but this only proves my point further: if you want to have a chance that your question is actually coherent, you should instead ask "how is it that there is something rather than nothing?". But even then, the question is still nonsense, due to the problem aforementioned concerning "something" and "nothing" not being well-defined.
So, being that the question is plainly absurd and incoherent, I am not particularly bugged by the question or interested in trying to answer it, and the fact that we still have such a primal urge to insist that the question needs an answer in spite of all the aforementioned things tells me that we have to not truly reached the age of reason. We are still in the age of instinct and intuition.
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T. Evans 11/2 = 5.5. Subtract 0.5 to get 5. 5/2 = 2.5, subtract 0.5 to get 2. 2/1 = 1. This gives the list 11, 5, 2, 1. 11 is odd, so you assign it a 1. 5 is odd, so you assign it a 1, 2 is even, so you assign it a 0. 1 is odd, so you assign it a 1. Therefore, the list is converted to 1, 1, 0, 1. The first number is 1, so we add one of 24. The second is 1, so we add one of 48. The third is 0, so we add none of 96. The fourth is 1, so we add one of 192. Therefore, 24 + 48 + 192 = 11*24 = 240 + 24 = 264.
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@Lincoln_Bio No, that is not how that works. First, you have a very fundamental misunderstanding of how speed is defined. The expansion of the universe has no speed, because it is not a movement. The rate of the expansion of the universe is measured in SI units of 1/m^2, while speed is measured in SI units of m/s. Thus, it makes no sense to talk about the expansion of the universe being faster or slower than c: it makes no more sense than saying 5 °C is hotter than 15 kg. Second, space is not being expanded. Spacetime is being expanded. The redshift tells us information about how the metric of spacetime changes with respect to proper time, but the speed at which the light is traveling is not changed, because all of spacetime is expanding, not just space by itself. Therefore, nothing is traveling faster than light. Spacetime expanding is not a form of travel, and there is no speed associated to it, because the rate at which it happens is not a speed, but an inverse length squared.
I suggest you read up some general relativity material if you want to continue having this conversation, because we will frankly get nowhere if you continue adhering to this fundamental misunderstanding of how speed works and what a frame of reference is.
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@Lincoln_Bio what I meant was how 2 points at opposite ends of the universe can appear to be moving apart ar a relative speed > c if we do not account for that expansion.
Yes, now you've stated correctly. This apparent recession shouldn't be confused with actual movement in any frame reference.
I realised gravity is therefore negative kinetic energy, which seems important and requires further investigation
Gravity is not kinetic energy. The gravitational potential of energy is, as the word states, potential energy, not kinetic. It is negative because this just follows directly from the fact that the force of gravity is a conservative force in the Newtonian paradigm. There is nothing puzzling about this. Also, this has nothing to do with the equivalence principle or with general relativity, so I'm a little confused as to what you're trying to get at.
and the gravitational constant appears to just be conversion between Newtons & kg
It is not a conversion factor between those two. If it were, then it would have SI units of m/s^2, which it does not have. It has units of m^3/(kg·s^2).
it all seems rather circular & we're accounting for mass and gravity over & over again in the maths
Uh, what?
Physics infuriated me at school because Newtons are clearly an arbitrary measure of force, I much prefer the well-defined kilogram
No, that is entirely inaccurate, because:
1. Newtons are not an arbitrary unit of force. They are a coherent unit based on the base units of the SI, which are not themselves arbitrary, since they are based on the physical constants of the universe: the kilogram is defined based on Planck's constant, the meter based on the speed of light, and the second based on the hyperfine transition frequency between states of an isotope of the Caesium atom.
2. The kilogram is not a unit of force, it is a unit of mass, and there is no fundamental equation that converts between the two in the same way that there exists conversion between temperature and energy or energy and mass. Force is a 4-vector, and mass is a Lorentz scalar.
We can only calculate mass from weight, not the other way around
What? No, this is completely false. C'mon. There are hundreds of different experiments you can do to calculate or measure the mass of something without needing to appeal to gravity. How do you think we know the mass of an electron? Do you think we put the electron on a balance isolated and weighed it? No, that is impossible.
at least without a bunch of other variables like volume, density, chemical structure, etc.
Again, this is not true. For any given system, the only thing necessary to calculate the mass is know the mass of the elementary particles comprising the system, and the potential energy of the forces binding the particles together, along with any kinetic energy of the particles. Volume and density play no role in this whatsoever.
So I'm going back to basic principles to see if I can come up with something better that uses measurable values in SI units.
Okay, but before you do that, you need to make sure you understand the concepts you're trying to debunk, because I sense that you have a lot of misunderstandings regarding the distinction between force and mass, and kinetic energy and potential energy.
Einstein came up with Relativity while he was bored at work right?
No, not quite. And also, Einstein was already a physicist when he came up with his revolutionary ideas. Einstein had a thorough understanding of the ideas he wanted to debunk and of the scientific method as a whole. You can't create a change in science of this caliber if you don't understand the science you're trying to change. This is like saying "I'm going to become a better writer than Shakespeare" but never having read the Shakespeare works. How are you going to surpass Shakespeare if you don't know what writings you're supposed to be surpassing? I'm not saying you can't become the next Einstein. But if you're this misinformed about the subject you're trying to revolutionize, then you wom't accomplish much.
Try to broaden your understanding of physics first, do it for years, as much as you can. Once you've mastered all of the basics and some of the intermediates, then maybe you can start trying to be the next Einstein.
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@Lincoln_Bio If you weigh yourself on the Moon, you will literally weigh less kilograms.
No, you will not. You cannot "weigh" kilograms, because the kilogram is not a unit of a weight, it is a unit of mass. The weight is just a nonscientific word to refer to the magnitude of gravitational force vector. Therefore, weight is a force, not a mass. You weigh less in Newtons on the Moon than you do on Earth, but your mass in kilograms is unchanged. I've already explained the distinction between mass and force three times here, and you're not paying attention.
To work out your Newtonian "mass," you'd have to take off the moon's gravity and add the earth's gravity.
No, you do not. Experiments have already shown that the mass of objects does not change with respect to the gravitational acceleration of a celestial body. We went to the Moon, and we have sent probes to Mars. The data confirms what I have stated. Also, to obtain the mass of an object, you only need to divide by the gravitational acceleration of the object on the celestial body that it is, you do not multiply by the gravitational acceleration of the Earth. This is never done. If you did this in an exam, you would get no marks, and if you did such a horrible miscalculation at a job, you would be reprimanded. We know for a fact this is not how this calculation is done, not only because of definitions, but because of experimental data.
But what has the Earth's gravity got to do with your mass when you're on the Moon, or in space?
Nothing. Earth's gravity has nothing to do with my mass even when I am on Earth. I've already explained this to you, but you're not listening to me.
Mass is not affected by gravity. It cannot be, because by definition, mass is a quantity that tells us how much matter the object has, and the content of an object is not affected by the gravity of a planet. Orange juice on Earth is still orange juice on the Moon, the chemical composition is the same, and therefore, the density is the same, so unless I spill it or drink it, the mass will be the same as well. That is literally how mass works. There is nothing to question here, since this is a definition, not a hypothesis.
Kg is only a constant of mass if you ALSO take Earth as the constant of gravity & use G-force in your calculations instead of m/s^2
Oh my God. What part of "mass does not depend on gravity" do you not understand? Do I need to say it to you in Spanish, French, or Japanese? I am teaching you Newtonian physics here, and all you're doing is spewing the same nonsense over and over instead of listening. Geez, I'm wasting my time here talking to you. I don't care about whether you disagree with Newtonian physics or not. This isn't about agreeing or disagreeing with Newtonian physics. It's about understanding what Newtonian physics claims. As I said earlier: you can't say a theory is wrong unless you actually understand what the theory is saying, which you have shown not to and even admitted it yourself. So I'm trying to help you understand the theory better, but all you keep doing is saying "mass and weight are the same thing" instead of listening. Were you like this 30 years ago when you went to school as well? If so, then I'm not surprised your education didn't go so well. How precisely do you intend to learn if you're not going to listen? sigh
If you measure it in different places, it isn't constant.
No, it is not. There is literally no experiment that supports that claim of yours. You're just pulling shit out of your ass at this point. This is precisely what I told you not to do. I told you that mindless denial of reality isn't science. But I guess I'm not going to be able to change your mind regardless of what I tell you, so I don't know why I'm still trying anymore.
I find it really interesting because there's a massive weight/mass discrepancy in all but one known galaxies last I checked, that's what dark matter is.
No, it isn't. But I'm not going to bother explaining this, since you aren't going to listen to me and just ignore me. This is like me trying to teach you how to speak Spanish and you telling me "see, I think you're speaking Spanish wrong" even though you're the one who hasn't learned Spanish. Are you also the type of person who goes to the doctor and tells them "no, your diagnosis is wrong" whenever they tell you anything? Your problem is you don't know how to listen. Unfortunately, I can't teach you that.
Rest assured if I should stumble across anything important I'll submit a paper for peer review
Oh, and are you going to listen to that peer review? Because you aren't listening to me, the person who was trying to help you. Well, good luck in life. I hope that paper you submit is a hell of a lot better than the dishonest nonsense you're pulling off here. Good bye.
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The ordered pair (2, 1) would not be {{2, 1}, {1, 2}}. The definition you provided does not support your conclusion. The only weird aspect of the definition (a, b) = {{a}, {a, b}} occurs when a = b, but even this is not actually very weird: while you do get {{a}} in that case, this is not an issue, because a is never equal to {a}.
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ohyeahyeah How are we on top of every species? First of all, bacteria have survived for billions of year more than we have, and despite all the extinctions, they still exist. Second of all, we cannot survive inside volcanoes or methane lakes. Third of all, we cannot survive in other planets or on the depths of the ocean. We cannot even fly with our own bodies. We cannot see infrared or ultraviolet light. We suffer from menstruation, which other species do not. Try fighting a bear with your bare hands. Our children take a whole 18 years to mature. Basically one fourth of our entire lifespan. Yes, other animals have significantly shorter life spans, but they're already adults by only 10% of their lifespans, on average.
Yes, we do have the intelligent use of technology to partially compensate for these things. But that is not an argument that helps support your beliefs, because it proves 1. We are such a weak species that we need to rely on technology. Other species don't have intelligence because they have so many positive adaptability traits they don't need it. It doesn't make us superior, it makes us barely on par with them. 2. There is nothing inherently superior about intelligence. And if anything, our intelligence is the very reason we are going to end up causing our own extinction. We as a species have caused more damage to the biosphere than almost any other species. Sounds like intelligence isn't such a good thing, biologically speaking. Very few species have managed to go so far as to destroy not only their own species, but damage the entire biosphere to a significant extent, to the point that we've lead to so many species going extinct. And we have damaged our own species greatly too. Not many species succeed at this. So much for intelligence making us superior, huh?
Regardless, we are a pretty pathetic biological species. We cannot even exist without the bacteria in our bodies, of which there are more cells than of our own. That's how bad of a species we are.
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It is harmful advice to tell students that the correct way to evaluate a limit is to start by "plugging in" into the function. This is incorrect, and it fails whenever the function is discontinuous, which you cannot know unless you already know the limit. Below, I present the correct method to do this exercise.
Let f(x) = x – 1, and let g(x) = x^2·(x + 2). lim f(x) (x —> –2, x > –2) = –3, and lim g(x) (x —> –2, x > –2) = 0. Therefore, lim f(x)/g(x) (x —> –2, x > –2) = lim (x – 1)/(x^2·(x + 2)) (x —> –2, x > –2) does not exist.
However, you can say that as x > –2, x —> –2, f(x)/g(x) —> –∞. You can say this, because g(x)/f(x) < 0 as x > –2, x —> –2.
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Akki B It works with every number, so you must have made a mistake. 33 in binary is 100001. You get this because 33/2 = 16.5, so you go down the list 16, 8, 4, 2, 1. You eliminate all the even numbers, leaving only the 1 and 33. Therefore, 1, 0, 0, 0, 0, 1. Concatenate and this gives 100001.
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Language is a layer on top of signaling, that both sender and receiver should understand. It works as a coder/decoder, coded from the sender, decoded from the receiver, and this language is carried on top of a materialistic medium. So "language" is not the medium itself, nor is it the signals, but it is the coded signals that we call "superimposed."
What you provided here is a description of how language physically propagates. It is not a definition of what language is, and it is certainly not how a linguist would describe language. A language is just a system of signs with an assigned meaning that can be extracted by the interpreter, together with rules or a structure that indicate how distinct signs interact together. Given that language is a tool, a sender is often required for this tool to actually be useful, but as far as what a language is, a sender doing an encoding at all is not necessary. In any case, you only need an interpreter decoding the assigned meaning, whose corresponding signs may or may noy have been put there by a sender.
So in the examples of flowers and insects, that flowers develop shiny or colorful or attractive appearance, these are signals to insects to trigger them for feedback, that's a communication system for sure.
No one disputes this, but this discussion is not about flowers signaling to insects. This discussion is about the relationship between the DNA molecule family, and language.
However, we can't say it's purely language here, as there's no coder/decoder, or superimposed messages (or there could be a basic one, I am not a biologist to identify the meaning of animals signals).
I find it absolutely hilarious whenever you theists do this. You guys love doing this thing where you acknowledge that you are completely unqualified to speak on a subject, decide to make a bunch of bold, factually incorrect statements on that subject, and then close it with "...buuuuuuut, I am just an untrained layperson in this subject, what do I know?," as if this somehow strengthens your argument at all. It is truly comical. I always find myself wondering why you guys even bother with the disclaimers anyway, when you still present your statements with all the seriousness in the world, and with all the expectation that we are supposed to accept them as true, and that it supports whatever point you are trying to make. Should you not be taking more of a humble approach, trying to learn more about the topic, before making the decision to present these arguments? I always wondered why most people fail to do this. If you know you are unqualified to make these claims, then you should not be making them at all. This is not to say you are not allowed to have opinions, but you probably should refrain from deductively coming to conclusions you are not willing to budge on without doing some very thorough self-fact-checking.
But "language" as superimpised signals on top, can't emerge.
Sure it can. Pareidolia exists.
C++, Java, etc., have never evolved by their own, but an intelligent IT developer designed them, either directly, or indirectly, and they can sure develop (mostly not on their own).
You are talking about computer languages here. Computer languages are a subclass of artificial languages. Obviously, artificial languages cannot emerge on their own, that is literally what the adjective "artificial" means. But, I hate to break to you, not all languages are artificial, and not all languages work the way computer languages specifically work.
...and accordingly, I see no reason to expect that human language just emerged as well out of nothing, if not built-in already or pre-designed with.
You mean that all of the evidence that we have for the theory of linguistic evolution (which is just a special case of the theory fo biological evolution) is not sufficient for you, somehow? You mean to say that because artificial languages cannot emerge on their own, non-artificial ones cannot do so either, despite the fact that you have absolutely no valid justification to make this strange, bold extrapolation to all languages from what is only a tiny subclass of languages?
If you rejected that last point, then we have another problem with "the coevolution between evolving entities." Here, you then assumed that these evolving entities are not intelligent, or are lifeless.
What is wrong with you? You are not even going to wait and listen for my explanation for why I disagree, and instead, you are going to decide I disagree for X reason, even if you have no way of proving it? You are so arrogant. I am sorry, but this behavior so many of you theists exhibit is the reason why many atheists have given up on trying to have reasonable conversations with you: because with this kind of dishonest, rude behavior, you are communicating to us the fact that you are NOT interested in having an open conversation, you are not interested in learning anything new, you are not willing to correct your beliefs on the basis of any information presented to you, you are not even willing to listen to what we have to say on a given subject. You have decided that we are going to say something before we even actually say it, and if we fail to meet that expectation, you are going to ignore that and move on anyway, following your script, which is why nothing we say to you actually matters. You guys do not know how to have a conversation, you only know how to evangelize and listen to yourselves talk.
So, I do apologize, but I have a difficult time taking you seriously whenever you pull rude dishonest nonsense such as this. It is a conversation-breaker. I would like to only address the actual arguments presented and not have to comment on you as a person, but that is not a reasonable stance when you behave so poorly that you are not even allowing us to say our part. I point out these behaviors so that you guys can stop doing them. And if it bothers you that I have spent this much time commenting on your character and not so much on your actual argument, then it is on you to ask yourself "why? What did I do? What should I be changing?" Reflect more upon yourself.
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In all of the levels of mentioned interaction in the video, between different cells or organisms or animals, they are all living things, with certain intelligence.
The assertion that all living things have certain intelligence is a complete baseless assertion. Here you are, again, a non-biologist, trying to pretend you know more about the topic than biologists, making a claim so bold that virtually no biologist would agree with it with as much confidence as you have asserted it. And despite your confidence, you do not make even a semblance of an effort to try to support your claim.
As IT engineers, we categorize things in layers,...
Yes, whenever it concerns something in IT, you can do that. We are not discussing something that concerns IT. We are discussing biology. IT engineers should keep quiet when it comes to discussing biology, unless you are actually going to provide all the scholarly sources to back your claims up (which I know you will not do, given your attitude thus far).
Even if we go with the hypothesis that layer 1 & 2 (till the signaling) emerged on its own, we have never observed any on-top layer being developed on its own, really never.
This is a completely baseless assertion, once again. This is you, an engineer who lacks an understanding of biology, making speculations on how biology should work based on your perspective as an engineer, and then deciding that certain ideas are facts based on these speculations. You are using what little you do know, and using that to interpret a list of hypotheses you could not possibly know is factual, and coming to a conclusion from there. To be fair, I am not insulting you. I am not a biologist either, I am just as unqualified as you are when it comes to this (though I do know many reliable sources I can consult if I wish). The difference is: I am not the one jumping to conclusions. You are.
So, it's not God of the Gaps here,...
It absolutely is. You, a competent IT engineer, are not a biologist, so you have very limited knowledge and understanding of facts and concepts from biology, as am I. We are both ignorant, when it comes to biology, but you are using this ignorance as an argument that God must be the source of intelligence and language, because you lack the sufficient understanding to make a conclusion as to how it could have happened on its own. This is, categorically, a textbook example of what the God of the Gaps fallacy us. And I am qualified to say that.
...assuming languages and coding is a self-made product is not scientific by any means, as it has never been observed, nor does the current evidence point to be self-created language.
Pareidolia is a phenomenon that has been observed repeatedly, and it is one of several ways in which language can develop without a sender intending for it to exist. Also, the gradual evolution of human languages is natural: no one programmed a piece of technology for these languages to evolve. Also, as a non-linguist, you really should stop speaking so confidently on matters of linguistics as well.
All observations leads to the same [conclusion], that any design needs a designer...
Citation needed. Also, this is a red herring, because it distracts from the fact that you have not proven that DNA (and the universe in general) is a design. If it is not a design, then it does not need a designer. You calling it a design is a completely baseless assertion.
...we have never managed to observe a self-designed system,...
Self-catalytic chemical systems would like to have a talk with you.
...but it's unproven, non-observable hypothesis.
The existence of self-catalytic systems is a fact. We have observed them repeatedly. You can do a basic Google search. I would provide sources, but I have no incentive to, seeing that you have none to provide yourself.
The problem is that this "layered" vision is missing in many biologists,...
Yes, it is missing, because it is woefully inaccurate. It is inapplicable to biological systems. It is only applicable when the concepts in IT engineering are relevant, which they are not here. What is with engineers trying to pretend that all other disciplines of scientific study have to borrow their methods to be valid? I have met way too many of you who pull this nonsense.
However, life, mind, or intelligence, are not phenomena, these are facts that we can't even agree on their proper definition...
Us not agreeing on their definition does not make them not phenomena. Your argument is not remotely close to being valid.
While, for materialistic science...
Materialistic science? This is nonsense. There is no such a thing as non-materialistic science.
material is the medium to carry data on, and material can largely impact the quality of data for sure, but it isn't data...
This is false, and we know this is false because of quantum thermodynamics and quantum information theory. Information. All physical systems, material and non-material, have an inherent data content to them, that cannot be separated from the system itself, because it is a property of the system, much like energy is.
God, and that definition can be really enough for religions...
No, it cannot be, because not all religions agree with this definition. Shinto and Daoism categorically disagree with this definition, for example, and so do some sects of Neo-Confucianism.
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@EskChan19 Slavery, for example, could be seen as morally just, in a society based on maximizing happiness, making the majority as happy as possible, even to the detriment of a few.
There are many problems with this:
0. You are conflating "just" with "moral," and treating them as synonymous. They are not synonymous. They often align, yes, but that which is just is not necessarily that which is moral. Besides, the comment you are responding to is using the word "optimal," so neither word applies here.
1. You are conflating "society believes that slavery maximizes happiness" with "slavery actually maximizes happiness (regardless of what people believe)." Only the latter would contribute towards slavery being moral. Societal beliefs can be wrong. When society used to believe that blood-letting was medically effectively and scientifically supported, their belief was wrong.
2. You are ignoring that, the way the definition is presented, an action has to both maximize happiness AND minimize suffering. Since your example does not accomplish both, it is not optimal. Now, you pointed out that accomplishing both may be impossible, but you have not actually demonstrated that the scenario being discussed is an example of an scenario with such an impossibility. There could very well exist an alternative to slavery that maintains equal levels of happiness, but reduces suffering further.
3. Even if it turns out that in a given scenario, there is no way to both maximize happiness and minimize suffering, that does not suddenly make slavery acceptable. What it does mean is that in that scenario, there simply is no optimal course or action. Dealing with a scenario with no optimal course of action is uncomfortable, but that lack of comfort does not make it a valid objection.
A society built on minimizing suffering would forbid slavery, but might allow human sacrifices.
No, probably not, since human sacrifices cause, almost assuredly, suffering comparable to that induced by slavery.
If you have enough resources to sustain 99 people, then if the 100th is born, what to do? Keep them and cause 100 people to suffer because no one gets enough? Or cast someone out who will likely just suffer for a while alone in the woods, until they get mauled by a hungry bear? Or just sacrifice them in a quick, as painless as possible way, and thus, allow 99 people to continue with enough resources, and preventing the 1 from suffering needlessly as well?
There is so much information missing from this scenario, the only rational answer is "No one knows." No real-world scenario will require you to take action with this little, tiny amount of information. In a real world scenario, you would actually know what exactly these resources are (is it food? Housing? Clothes?); you would know exactly the reasons behind why we only have enough resources for 99 people; you would figure out the circumstances behind how 1 extra person was born, even when we agreed to not have this happen; you would know the people personally, and you would be able to figure out what course of action would cause the other 99 people the most amount of happiness, and the least amount of suffering, and you would be able to map out the relationship between happiness and suffering in this case, in order to try to figure out what to do; etc. Even in that case, the answer would still not be easy to think about, which is why being a leader in a political setting is always a difficult task. This is why education became a thing.
You have essentially created a scenario with so little information, that making a determination of what action is optimal is categorically impossible due to a lack of information and a lack of realism, and then used this impossibility to... what, argue that OP is wrong about optimal actions being defined as maximizing happiness and minimizing suffering? This is essentially a God of the Gaps fallacy, except you are not actually arguing that a god exists. I am not even sure what your point is. If your point is merely that "determining what is moral is difficult in practice," then, yes, everyone agrees, and I think OP knows this. If your point is that "whether an action is optimal or not depends on many, many variables," then, again, OP probably already knows this.
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I am surprised that Matt never mentioned the fact that, while τ may be a better "circular constant," π is not merely a circular constant. It is also a triangle constant, and it is a better triangle constant than τ, but it is also a constant fundamental to other realms of mathematics that are removed from geometry. For example, π is important in analytic number theory, and it is also an important constant in Fourier analysis, which is the mathematical foundation with which we talk about oscillatory motion. In practice, it is more common and more useful to talk about half-periods in engineering and other applications that use oscillatory motion than it is to talk about periods, mainly because it allows us to talk about values at which the function is equal to the shift constant more easily. Half-rotations are also very important to talk about in physics and many other applications too.
The idea to understand here is the reason we use π is not because it is a better circle constant, since it is indeed not a better circle constant. It is just a better constant for applications that range outside elementary geometry, and it is the reason why it historically became the convention that it is today. Is it worth to change this convention to gain a slight bit more intuition that ultimate does not solve the education problem with mathematics in favor for making all applications more difficult and forcing a rewrite of textbooks globally? No, it absolutely is not.
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Is there any sense in which one set is still 'bigger' (scaled) compared to the other?
The problem here is that you are conflating "scaled" and "bigger," as if they mean the same thing. They absolutely do not mean the same thing. In fact, the set 2Z is indeed a scalar multiple of Z, in the Minkowski scalar multiplication sense. For more information on what this means, you can read the Wikipedia article on "Minkowski sum." However, it is still true that |2Z| = |Z|. The problem that you are neglecting is that if you multiply |Z| by 2, you still get |Z|.
Is this a problem with the notion of being able to 'count infinity'?
No. What this is, it is a defining characteristic of infinite sets. Your intuition tells you that if A is a proper subset of B, then A is smaller than B. This is only true if A and B are finite sets. If you informally treat the cardinality of a set as a function output, with the set being the input, then you can view this as saying that the cardinality function is monotonic with respect to subsethood, but not strictly monotonic, even though your intuition is telling you that it must be strictly monotonic. Of course, this is just an analogy: cardinality is not a proper function.
I suspect that something along this path is the break that allows the axiom of choice to lead to multiple copies of an object.
No, not at all. Even if the axiom of choice is false, all countable sets are still the same size, even one is a proper subset of another. The axiom of choice has no bearing on that. What the axiom of choice does is it guarantees that the cardinalities of sets can be well-ordered, meaning that for every size of a set, there is always a well-defined "next size" or "next cardinality," and that if you have a set of sizes, that there is always a smallest size in that set of sizes.
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@guitar-songcovers3704 ...before that, the definition was that the number should be divided only by 1 and itself.
No, that is false, and a myth. There are actually no historical mathematical sources where such a definition was given. The historical definition of a prime number was that a (positive) prime number was a number measured only by the number 1, and no other numbers. What the word "measured" means is that there exists some smaller positive integer that divides the integer being considered. For example, 6 is not smaller than 6, so although 6 divides 6, 6 does not measure 6. This is why in older writings, you find that mathematicians never considered numbers to measure themselves. But 2 is smaller than 6, and 2 does divide 6, so 2 does measure 6. 1 does not measure 1, since 1 is not smaller than 1. So, actually, 1 never actually satisfied the definition of a prime number, not even during antiquity. 1 was believed to be a prime number only between the years 1600s and 1800s, and this was because of an ongoing philosophical debate as to whether 1 was a number, and misunderstandings in the debate led to the definition of "measured" to be applied inconsistently. The inconsistencies were discovered in the 1800s, so people stopped doing it, for the most part, though a minority of people were stubborn and never admitted that they were wrong. Even so, the understanding that 1 is not a prime number became standard among mathematicians and nigh-universal in the 1900s. You will not find peer-reviewed texts that treat 1 as a prime since the 1920s. You will also not find any texts that treat 1 as a prime before the early 1600s either.
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The pigeonhole principle is a mathematical theorem, and as such, it is impossible for it to be false. The fact that it is stated solely in terms of pigeons and pigeonholes is done merely to make the principle intuitive. The actual statement of the pigeonhole principle is as follows: for all sets X, Y, if there exists an injective function f : X —> Y and there exists no surjective function g : X —> Y, then there exists no injective function h : Y —> X.
To prove this, notice that the Schröder-Bernstein theorem states that for all sets X, Y, if there exists an injective function f : X —> Y and there exists an injective function h : Y —> X, then there exists an injective and surjective function j : X —> Y. All theorems are semantically equivalent to their contrapositive statements, and the contrapositive statement of the Schröder-Bernstein theorem is that, for all sets X, Y, if for all functions f : X —> Y, f is not injective, or not surjective, then, there exists no injective function k : X —> Y, or there exists no injective function h : Y —> X. This, in conjunction with there existing an injective function f : X —> Y, implies that f is not surjective, and implies that there is no injective function h : Y —> X.
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@ianbelletti6241 every theorem has underlying assumptions.
Those would be the axioms of set theory, but beyond that, no, your statement is false.
The pigeon hole theorem, in order to always ring true mathematically relies on no limit to the number of "pigeons" each "pigeon hole" can hold.
No, it does not. In fact, as I have stated already multiple times, this is a general theorem in set theory. It actually has nothing to do with pigeons and holes. The pigeons and holes are used only for visualization purposes. You are continuing to ignore the formal statement I presented to you in my previous sentences, which is fairly irritating.
If you make the limit 1 per hole, you can never double up items per hole and the theorem falls apart very easily.
As I said, the theorem actually has nothing to do with holes at all.
Therefore, the theorem relies on not placing limits to how many "pigeons" you can fit in each hole.
It does not.
Your proof falls flat on its face because it ignores that core principle of the theorem.
No, it does not, because my proof says absolutely nothing about holes. My proof is about injective functions, sets, and surjective functions. It is very clear to me that you have not understood my proof even the slightest bit at all.
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@ianbelletti6241 You can only double up items in the smaller set if the rules of the sets allow you to. This rule cannot apply if the rules of the specific sets prohibit that proposition. Remember, set theory does have real world applications. Calling it a general theorem is fine, but the rule for it to work is that positions in the smaller set are allowed to hold multiple values or items at once. If the smaller set cannot do this, then the theorem cannot be used for that situation.
You almost have a point, but the insinuation here is that the axioms of set theory may be such that they allow for singletons, or the empty set, but no other sets. However, I do not know of any nontrivial, non ad hoc set theory, in which this is actually true. Any set theory in which the pigeonhole principle is false would also have to be a set theory in which the Schröder-Bernstein theorem is false, and even Cantor's theorem.
There are situations where the theorem is false. It's the nature of these general theorems.
There are no situations where the theorem is false. That is simply not how theorems work. You may say that there are situations where the theorem is not applicable, but by definition, a theorem is a sentence which can be proven, and the deduction rules used in mathematics are sound, meaning that it is impossible to prove false sentences from true sentences. Therefore, if a sentence is provable, then it cannot be false. Since the pigeonhole principle is provable, it cannot be false.
In many cases they work, but under certain circumstances they don't. It's like the formula P=IE for power. It works in many cases but won't give you the true power because it doesn't take into account how many phases of electricity you're using and the impedance of the circuit.
This analogy is invalid. The formula P = I•E is not a theorem, it is a scientific observation, a scientific law, and scientific laws do not work like theorems at all. You are comparing mathematics to physics here. This is like comparing apples and pineapples. Besides, no competent scientist would ever actually claim under any circumstances that the law works for all physical circuits. If someone tells ypu that it does, then they are lying to you. It is as simple as that.
The same thing is happening with this theorem.
No, it is not. Theorems are not scientific, and they do not work like scientific laws. They do not describe physical systems.
In general it works, but it doesn't apply to all sets where you're inserting a larger set into a smaller set.
I have no idea how you came to this conclusion. This is incoherent. No one is claiming you can embed the elements of a set into a smaller set. That is literally not what the pigeonhole principle states.
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@petevenuti7355 I like to see operations like adverbs, the definition being a description of the process, basically, am I wrong?
Yes, this is wrong. You are thinking of mathematics as "something a computer does," not as a genuine collection of abstract concepts, which is what it actually is. If you want to only consider a concept "mathematical" if there is an algorithm associated to it, then you will be shocked to learn that 99% of what we call mathematics would not count as "mathematical" under your definition. A perfect computer cannot do the vast majority of things we can do. In fact, under your definition, there is no such a thing as a real number or a complex number, there is no such a thing as an uncountable set. Why? Because computers can only deal with countable sets.
An arithmetic operation is not an algorithm that spits out a result. Some arithmetic operations do have an associated algorithm, these are called computable operations. The result of an operation already exists as a mathematical object, regardless of whether you or any computer can actually identify the result. This is because an operation is just a set, a function, and nothing more.
In the specific case of 0/0, would it be one, zero, or undefined?
IF 0/0 exists (and it may not exist), THEN it is equal to 1. Why? Well, what is the definition of division? In the context of semirings and generalizations thereof, x/y is an abbreviation for x·y^(–1). Here, · is the multiplication of the structure, which is well-defined, and y^(–1) is the multiplicative inverses of y: it is defined to be the unique element with the property that y·y^(–1) = y^(–1)·y = 1. The reason (x/y)·y = x is because (x·y^(–1))·y = x·(y^(–1)·y) = x·1 = x. Similarly, (x·y)/y = x, because (x·y)·y^(–1) = x·(y·y^(–1)) = x·1 = x. This is the actual definition of division. Hence, x/0 is just an abbreviation for x·0^(–1), where 0^(–1) is defined such that 0·0^(–1) = 0^(–1)·0 = 1. Here, you should ask yourself: does 0^(–1) exist? If it does exist, what is it equal to? The answer is that it does not exist (unless 0 = 1), and I can prove it, because I can prove that 0·x = x·0 = 0 is true for all x, meaning there cannot be an x such that 0·x = x·0 = 1 (again, unless 0 = 1). So, 0/0 does not exist, but if it did exist, it would be 1, because 0/0 := 0·0^(–1) = 1. On the other hand, if 0 = 1, then 0^(–1) = 1^(–1), so 0/0 = 0·0^(–1) = 1·1^(–1) = 1 = 0.
It largely depends on your definitions of division and zero...
Well, yes. ALL symbolic expressions depend on their definition. Symbols do not have definitions. Definitions are something we, sentient beings, impose on those symbols. Otherwise, the symbols are just arbitrary arrangements of matter and energy on a physical surface. That being said, there is only one definition ever used for these objects, so the dependence is irrelevant. To properly define what 0 is, you need to know about the distributive property. Consider two binary operations $ and °. $ is said to distribute over ° if a$(b°c) = (a$b)°(a$c), and (b°c)$a = (b$a)°(c$a). Normally, these operations are denoted as · and +, but I am avoiding this because I want you to realize the binary operations could be anything, they do not have to be the familiar addition and multiplication operations of natural numbers that we know. Now, say $ has an identity element e, where you have e$x = x$e = x, and say ° has an identity element z, where you have z°x = x°z = x. Then z is called 0, and e is called 1. The 0 and 1 here do not have to be interpreted as natural numbers at all, they could be any type of mathematical object, in principle, but we still denote them as 0 and 1, because regardless of the actual structure, they always play the same role. You will find that z$x = x$z = z, which means z is an absorbing element of $, motivating us to call z as 0, and e as 1. This is also why $ is generally just denoted as ·, and ° as +. If you have two binary operations, and one distributes over the other, then the identity element of the one being distributed over is called 0. This is the definition. As for division, I believe I already explained the definition. In this case, just replace · with $, and you have your definition.
On the other hand, if you considered division a multi-step process, then dividing anything by zero, even zero, would be a do-nothing function, because it would be zero steps, and what do you even return from a do-nothing function? Zero, the starting number, or it is just meaninless?
If /0 is a do-nothing function, then x/0 = x, by definition. However, your reasoning behind /0 being a do nothing function is completely wrong, since it assumes division is an algorithm, not an operation.
if division is defined as recursive subtraction...
You cannot define division as recursive subtraction. If you do, then (1/2)/(2/3) is undefined, and so is e/sqrt(π). Similarly, you cannot define multiplication as recursive addition. In the specific case of the natural numbers, you can define multiplication as recursive addition, but if you are talking about rational numbers or real numbers, then no, you definitely cannot.
if 1 is defined a set of {}, isn't the empty set just as prime as 1?
What do sets have to do with primality?
If you think of something times zero as a do-nothing function and that's why returns zero,...
You have the wrong idea of what "doing nothing" is. If I have a function f that takes the input x and maps it to the output 0, that is not a do-nothing function, because you are taking the input and doing something to it, changing it into 0. A do-nothing function is a function that maps x to x for all x. It does nothing to x, leaving it unchanged. Multiplying by 0 is not "doing nothing." Also, 0 is not "nothing." 0 is the empty set. The empty set is something. There is no such a thing as "nothing" in mathematics, and I wish teachers stopped telling people that there is.
If you think of something times zero as a do-nothing function and that's why returns zero, then zero is not in other numbers like a factor...
What are you talking about? What does 0 being the empty set have to do with the factors of an integer? This is nonsensical, to be honest, but I want to help you find where your question went wrong.
What is the concise definition of division?
x/y := x·y^(–1), where · distributes over +.
Then I can see how it works, where it breaks, and think about how the undefined can be defined without breaking the rest of math.
Division by 0 cannot be defined. This is a theorem. There is nothing you can do to "fix it" anything, and I would argue that thinking of division as being "broken" to begin with is already incorrect. 0·x = x·0 = 0 is an inevitable consequence of how · and 0 are defined.
The concept of the empty set being the building block of 1 does seem to make 0 a factor of everything,...
No, it does not. Do you understand what the word "factor" means?
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@rmsgrey For the question of 0/0, that does, indeed, come back to your definitions.
Are you implying that there are things that do not come down to definition? Because if not, then I find that pointing this out is unhelpful. Of course it comes down to definitions. OP's question is, what is that definition?
If you define division as the inverse operation of multiplication, then x = 0/0 is equivalent to 0·x = 0, and any (finite) value of x will work in that equation
This is not a coherent definition. You have not even explained what exactly it means to be the inverse of a binary operation, which is what I would argue needs defining to begin with. You said division is the inverse operation of multiplication, but then proceded to claim that the result of 0/0 is an entire set of numbers. This is nonsensical, and I would also say confusing to most people.
If the function converges on the same value from either direction, then it's convenient to assume that it's also continuous at that point, giving a value to 0/0 which can be any number.
No, this is definitely not how that works, and I strongly dislike it when professors make the choice to give an explanation this inaccurate to calculus students. To start with: no, we do not assume a function is continuous at a point. If I have two functions f, g, and I am interested in studying the behavior of f(x)/g(x), then it does not matter if f(p) = 0 and g(p) = 0. We care about the behavior of f and g near p, not their value at p. To put it symbolically, we only care about lim f(x) (x —> p) and lim g(x) (x —> p). f and g may discontinuous, they may be continuous, they may even be undefined at p, it does not matter, because what we are interested in are the limits, not the actual values of f and g at p. Also, whatever conclusions you end up drawing about lim f(x)/g(x) (x —> p) have absolutely nothing to do with what the value of 0/0 is. Absolutely nothing. Besides, the conceptual approach here is completely wrong too. What you should be looking at is to define a function h such that h(x, y) = x/y, and then look at lim h(x, y) (x —> 0, y —> 0). But, even then, whatever conclusions you end up drawing are irrelevant, and ultimately have nothing to do with the value of h(0, 0) = 0/0. h does not have to be continuous. This idea that it has to be is nonsense. Division is an arithmetic operation, so you obviously cannot define what h(0, 0) is in terms of limits. If it were defined, then you would already know what it is before ever arriving at a calculus course.
There are several ways of defining division, which pretty much all explicitly exclude division by zero,...
Are there multiple definitions? I have only ever seen one definition of division: that of multiplying by the multiplicative inverse.
One moderately standard definition of division (when working with rationals) is: (a/b)/(c/d) = (a/b)·(d/c).
This is not exclusive to the rational numbers. This is the definition of division for all mathematical structures where a concept of multiplication (defined as distributing over addition) is well-defined.
When working with integers, there are at least with two different concepts of division: there's division only when the divisor is a factor of the dividend, and there's quotient remainder division.
I have never heard a mathematician call either of those things division. The Euclidean quotient-remainder algorithm is just that: the Euclidean algorithm, and while it may be somewhat related to division, it is a different concept altogether. And you are conflating the concept of divisibility with the concept of division. Again, related, but not both concepts go by the name of "division."
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@rmsgrey How would you define division when restricted to integers? Or polynomial division?
I would not, is the answer. You can perform the Euclidean quotient-remainder in the ring of integers and in any polynomial ring, but that is not the same as division. The existence of such an algorithm is actually used to classify rings. An integral domain where such an algorithm exists is called a Euclidean domain. Polynomial rings and the ring of integers are examples of unique factorization domains, and all unique factorization domains are also Euclidean. To define division, you work with a division ring, instead.
In abstract algebra, a magma (a set with a binary operation, where the set is closed under that operation), whose binary operation can be referred to as multiplication (symbol * ) may not have an identity element, let alone multiplicative inverses, but there are still two partial functions, left division ( a \ b ) and right division ( a / b ) defined as the value x, where such a value exists and is unique, such that b = a * x and a = x * b respectively.
I have never seen a scholarly work on quasigroups where left-division and right-division are defined as partial operations. If the magma is not a quasigroup, then division is simply not well-defined. That is all there is to it. Besides, when we talk about "dividing by zero," which is the context we are in, this general formalism of quasigroups is inapplicable. Defining 0 requires having two binary operations, one distributing over the other. The structure need not be a ring (the addition and multiplication need not be associative nor commutative). The identity element of the operation being distributed over is called 0, and the operation that distributes, if it does have an identity element, is called 1. However, 0 is not a multiplicatively cancellable element in a structure satisfying these axioms. The ring theoretic definition I provided is just a special case of the quasigroup definition, because rings are a richer structure than quasigroups, where it is meaningful to talk about 0, as opposed to arbitrary quasigroups. Though, again, I remind you that I believe you have still defined division incorrectly, even for the context of quasigroups. (Right)-division, the operation, is characterized by the axioms (x/y)·y = x, (x·y)/y = x.
Where a multiplicative inverse exists, the two definitions - division as the direct inverse of multiplication, and division as multiplication by the multiplicative inverse - are equivalent (and the definition I gave for rational numbers is a special case of the latter)...
I know as much.
...except when you try to extend them to 0/0, where the multiplicative inverse approach concludes that, if you pretend you can have a multiplicative inverse of 0, you arrive at a specific value of 1 (which also breaks the general rule that 0 times anything gives 0 since 0 times the hypothetical 1/0 gives 1), while the direct inverse approach concludes that 0/0 could have any value since 0 times x is 0 whatever x is.
No. The quasigroup approach does not conclude 0/0 could have any value. The quasigroup approach concludes that if 0/0 exists, then (0/0)·0 = 0 AND (0·0)/0 = 0. In the latter, one simply has 0/0 = 0, which indeed satisfies (0/0)·0 = 0, since (0)·0 = 0. However, as stated, 0 is not actually multiplicatively cancellable, so the multiplicative magma of the bi-magma cannot be embedded in a quasigroup.
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@rmsgrey If you want, you can consider the free unital magma over S = {X}, with 0 + X = X + 0 = X (since the magma is automatically Abelian), and also consider a free non-unital magma over S, such that a·(b + c) = (a·b) + (b·c). Since 0 = 0 + 0, it follows that 0·a = (0 + 0)·a = 0·a + 0·a. 0·a = 0·a + 0, so 0·a + 0 = 0·a + 0·a. Thus, if 0·a is +-cancellable, then 0 = 0·a, so 0 is not ·-cancellable. In other words, it is impossible for both magmas to be cancellative. As such, a quasigroup approach for division cannot work regardless. However, a note to be made is that if it did work, if 0 were ·-cancellable, then the ·-quasigroup would be embeddable in a loop with identity element 1, and 0 would be ·-invertible. As such, the definitions are equivalent. The reason 0/0 = 0 in the quasigroup approach is because the built-in assumption that 0 = 1, which is the only way / can be defined at all, in this case, and in that case, 0 does indeed have a ·-inverse anyway: 0 itself.
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@rmsgrey It seems we disagree on whether the term "division" should be extended to cases where there are close, but not perfect, analogues of the field operation of division.
To start with, not all division rings are fields. A field is a commutative nontrivial division ring, but for the sake of simplicity, I do not want to boggle readers with having to assume commutativity, as it is just not relevant or important. More importantly: the Euclidean quotient-remainder algorithm is not a "close" analogue to division. It is not even in the same ballpark of ring theoretic concepts. It is something completely different. This is not to say there is no relationship at all, but that the relationship is a lot more distant than you are making it sound to be.
In any case, perhaps we can agree that division by zero is generally undefined, and that you can make plausible arguments for expressions that look like they would evaluate to 0/0 to actually take any specific value.
You can make such arguments, only with false assumptions, is my point, and in any case, I still want OP to understand what a proper, precise definition of division, not have some wishy-washy handwaving of a description of what it could be. Although, before explaining it, I think we need to rectify the superstitious misconception that the set-theoretic construction of the integers is somehow related to the concepts of primality and divisibility for integers.
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@rmsgrey It is a function when working in a quasigroup. The problem is that in a ring, the multiplicative monoid does not form a quasigroup, because the monoid is not cancellative, since 0 is not cancellable. For a division ring, division is just a shorthand for multiplication by the right or left inverse of an element, so formally speaking, it is not an operation in that case (and virtually no algebraist treats it as one, in that case). Informally, however, it makes sense for non-mathematicians to still think of it as an operation, since the exact definitions are not super important for applications.
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@petevenuti7355 Personally in my mind any linear algebra has that one input one output property and that's what linear means in that context. I don't believe you are also saying that a function has to be linear to be a function were you?
I have never seen or heard the words "linear algebra" in that fashion. This may just be my unfamiliarity with computer science, but in mathematics, "linear algebra" refers to the study of vector spaces, and of linear functions between them. A linear function is a function such that satisfies f(u + v) = f(u) + f(w) and f(a·u) = a·f(u). When it comes to finite-dimensional vector spaces, we represent those functions via matrices. Anyway, functions need not be one-to-one. Those are injective functions. Functions are in general, some-to-one, where some could be one, or many. A function cannot be one-to-many, however, or one-to-none.
I'm assuming that only needs to be the case in the description of the definition of the simple binary functions we are discussing being division multiplication and what they're built from, I believe you're only saying that those specifically have to be linear , correct?
If by linear, you mean some-to-one, then yes, that defines what a function is. However, the definition of a binary operation is more specific than that of a function. An n-ary operation on X is a function with domain X^n and codomain X. A binary operation is an n-ary operation where n = 2. However, I reiterate that I have never seen the word "linear" being used in this fashion, and in general, no, binary operations are not linear, in the mathematical sense.
What would be your opinion on giving 0/0 it's own symbol, much like the numeral " i " , (essentially making it its own object outside of the systems you guys were discussing that I don't know the vocabulary for) even if it won't allow for a conceptual definition (like i) it would at least make errors glaringly obvious.
Would it? I think 0/0 already is capable of denoting errors by itself just fine.
I think I'm seeing what you're trying to explain to me, your separating the concept of what these relationships are from the mechanism of how they are calculated.
This is correct. For example, in a mathematical proof, I may be able to prove that there exists a natural number satisfying some property. However, there are may not exist an algorithm to compute the binary digital expansion that represents such a number. For example, Rayo's number exists, and is well-defined, but it is uncomputable. There exists no algorithm that can tell me what the mth digit of Rayo's number is. And I am not saying the technology of modern computers is not advanced enough. I am saying it is logically impossible for such an algorithm to exist, regardless of how perfect or ideal this computer is. Simply a put, an idealized Turing machine could not compute the digits of such a number. An omniscient being could not compute the digits of such a number, because it is not mathematically possible for that to happen. However, the number does exist, and one can prove it satisfies a number of properties without even knowing the number "truly is."
Fun fact: Rayo's number is not the largest named finite number out there.
I never was and would never deny the concept of infinity, and I don't think you were saying I was, I think you were just saying that some of the things I was saying would point to that conclusion but in explaining that to me it sounded like you're denying the concept of "nothing" or by saying "nothing doesn't exist" or did you just mean in the sense that that's what nothing is by definition, something, everything, that doesn't exist..?
I am not sure you how you came to that conclusion, since I am not sure how that is a plausible interpretation of the words I said. The point I am making is that, in mathematics, we have certain axioms. What you are able to prove and study in mathematics depends on what those axioms are. By changing the axioms, you change the realm of possibilities of what can be proven, and what cannot be proven. The axioms most mathematicians have agreed to use for basically all intents and purposes are the Zermelo-Fraenkel axioms of set theory, which include the axiom of infinity. These axioms imply the existence of functions which cannot be computed in a computability model. However, the functions need not be computable to be well-defined. To be well-defined, you just need some well-formed formula in the language of the theory that describes the existence of such a function, and prove that the formula is satisfiable. I can give you a description that this function, and only this function, satisfies, even I cannot figure out which exact ordered pairs are the elements of G and which are not. The relationships between these properties and the properties of other objects is what allows us to do mathematics. This is why it is so easy for us to study numbers that are too big for computers to handle. A mathematician can tell you many things about the number 3^(3^(3^(3^(3^3)))), even though no computer can store its binary digital expansion (such a computer would have to be bigger than the observable universe). I can describe infinite sets, even if the elements of the set cannot all be known. I can describe uncountable sets, even though none of the elements can be known. Mathematical proofs are such that they transcend the capabilities of computation, because they deal with the properties of abstract objects, and not merely computer representations of those objects.
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@petevenuti7355 Computer science and mathematics started out quite mingled and have since diverged.
That is not remotely true. Mathematics is millennia older than computer science. Modern set theory precedes computation theory, which long precedes computer science. Very few people that are alive today were alive when computation theory as a discipline of mathematics and formal logic emerged. Mathematics goes further back than even writing does.
Like compare organic chemists, even biologist to someone who does genetic engineering, it has diverged a lot over the decades.
On that point, I would argue that genetic engineering is much more closely related to organic chemistry and biochemistry, than computer science is to mathematics.
Know of Gerald Sussman? He's old enough to remember when the fields overlapped completely and likely more familiar with both then the 3 of us, definitely me...
Computation theory emerged in the early 1920s, which is far before Dr. Sussman was even born, and mathematics had already been reformulated in terms of classical first-order formal logic & set theory by then. In fact, computation theory emerged as an offshoot of type theory, which was published as an alternative to set theory being a foundation of mathematics, it was meant to compete with what today we know as the Zermelo-Fraenkel axioms of set theory.
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@Tehom1 No. 0^0 is not undefined. lim floor(x) (x —> 1) does not exist, but this does not mean floor(1) is undefined. floor(1) = 1, by the definition of floor(x) for all x. Similarly, lim x^y (x > 0, y > 0, x —> 0, y —> 0) does not exist, but this has nothing to do with the exact value of x^y at x = 0, y = 0, which is 1, by the definition of exponentiation. The value of a function at a point is not determined by what the limit is near that point. The value of a function at a point is determined by its definition.
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@Tehom1 No, it is not. It is not reasonable for the same reason that using limits to define the floor function at the point x = 1 or at any integer point is unreasonable. Who taught you that using limits to define functions, and in particular, basic 5th grade arithmetic, is acceptable and reasonable? Because whoever was it that taught you this, I need to have a serious, long talk with them. By the very definition of what limits are, limits deal with the values of a function NEAR a point, not AT a point. What is the definition of a limit? For a function f : R^2 —> R, lim f(x, y) (x —> s, y —> t) = L iff for all real ε > 0, there exists some real δ > 0 such that 0 < ||(x, y) – (s, t)|| < δ implies |f(x, y) – L| < ε. Notice how the condition that needs to be satisfied in the definition is 0 < ||(x, y) – (s, t)|| < δ, not (x, y) = (s, t).
What is the definition of r^n for real r and natural or cardinal n? The standard definition is "r multiplied by itself n times," but this is a very unrigorous "definition," since the notion of "multiplied by itself" is very unclear. Instead, to make this idea rigorous, you would say that r^n is defined to denote the product of the elements of a tuple of length n, where the elements of the tuple are all equal to r. In this manner, if I have the tuple (x, x, y, z, z, z, z), then I can write the product of the elements of the tuple as x^2·y^1·z^4. This is also the product of the elements of the tuple (x, y, x, z, z, z, z) or of the tuple (y, z, z, x, z, x, z), because the product of two real numbers is associative and commutative, but as the product is not idempotent, the number of repetitions of an element in a tuple do affect the product. However, as multiplication is a binary operation, the meaning of r^1, for example, is unclear, because the product of the elements of the monotuple (r) is not well-defined. The solution to this problem invokes some set theory, but the idea is that it can be proven that an multi-ary operation on an n-tuple when n = 1 simply outputs the sole element contained in the n-tuple, and when n = 0, the operation outputs the identity element of the operation. The identity element of multiplication of real numbers is 1, so the product of the elements of the 0-tuple () is 1. Therefore, r^0 = 1. However, notice that the 0-tuple is unique, according to set theory, and does not depend on the value of r. So this holds true for every real number r. This includes the real number 0. In other words, 0^0 = 1.
Of course, this definition is only applicable if r is a real number, or a complex number, and n is a natural number, or more generally any cardinal number. To extend this to integral n instead, the machinery of a functional equation is needed, but this definition needs to be such that, for the special case that n is nonnegative, it agrees with the definition for n cardinal. As such, the recursion used is r^(n + m) = r^n·r^m, so that r^(-1) is equal to the unique real or complex number such that r·r^(-1) = r^(-1)·r = 1. This is why 0^(-1) is also not defined: because there is no complex number x such that 0·x = 1, as demanded by the recursion. This argument would not make sense if 0^0 = 0, as 0·x = 0 is axiomatically true. Beyond integral numbers, it does not make sense to talk about exponentiation in the ordinary sense, and instead, you must continue the function piecewise to the complex plane for arbitrary x and y by using the operation y^x := exp[x·log(y)] for nonzero y, noting that, here, log is a complex multifunction defined using a Riemann sheet, and exp(z) is defined as the Maclaurin series of x^n/n!. Notice that this definition does not disprove 0^0 = 1, because in reality, even the trivial 0^2 = 0 is not possible to establish with this definition, because log(0) does not exist. Therefore, the definition must be completed piecewise by letting 0^z := 0 when Re(z) > 0, 0^z := 1 if z = 0.
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@MuffinsAPlenty I disagree. The truth is that 1 not being a prime number has absolutely nothing to do with the convenience of how we state the fundamental theorem of arithmetic. I dislike it when people who are educated in mathematics try to present everything as if it is "a matter of convenience," because that is just not true at all. 99.99999% of things in mathematics are never about convenience. Sure, some things in mathematics are purely a matter of convention. The fact that we still use what I consider to be bad notation for derivatives, such as dy/dx, instead of using functional notation for them, such as D(y), really is entirely a matter of convention, even if there if one of the conventions is objectively superior to the other for three dozen reasons. It really is just notation. Using base 10 for our positional representation system to denote integers, instead of base, say, 60, like the Babylonians used to do, is s convention. In fact, using positional representations systems at all, rather than non-positional ones, is itself a convention. So, I am not saying there are no conventions in mathematics. My problem is that everyone on YouTube presents so many things (like 0! = 1, or 1 is not prime, or the ideas about the radical symbol being used as a function, etc.) as conventions that actually are not conventions. 1 not being a prime number is not a matter of notation, and it is not a choice we get to make. It is an irrefutable mathematical fact, that when it comes to commutative rings, and how we classify objects, there are exactly four families into which these objects can and do fall into, and these four families are exhaustive, distinct, and mutually exclusive. We can characterize these families as (a) those objects x such that there exists some y such that x•y = y•x = 0; (b) those objects x such that there exists some y such that x•y = y•x = 1; (c) those objects which are neither of the above, and whose proper divisors are exactly the objects in (b); (d) those objects which are not in (a) and have proper divisors in (c). How we choose to label these four families with four distinct labels is completely arbitrary, yes. We can even choose to have multiple distinct labels for the same individual family, yes. However, we can never choose to insist that elements from distinct families actually belong to one single family, and should be labeled as such. This is not a choice we get to make, because it is conceptually inconsistent with the mathematics above, and it leads to ill-defined terminology. Calling 1 a prime number is entirely analogous to insisting that my Toyota is a plant.
Anyway, what this comes down to is, there is an actual conceptual reason behind why 1 is not and cannot be a prime number, no matter how much we would like it to be. It has nothing to do with how easy it is to formulate the language the factorization theorem in the English language when we reject 1 as a prime number. People should be taught the actual conceptual reason behind why 1 is a prime number, not this "it's more convenient" nonsense. And no, I am not saying we need to be formal about it. Simple intuitive explanations will do.
I know that, as far as colloquial language is concerned, you can arbitrarily coin words and make them mean absolutely nothing and use them in self-contradicting fashion, or make them have useless meanings for the sake of trolling, and you can arbitrarily change how you use those labels any time you want to. But, this is not a colloquial language we are dealing with, now, or is it? We are dealing with abstract mathematics and number theory. It is a serious discipline of study. Going around telling biologists that your Toyota is a plant, because you chose to change the definition of the word "plant" in some unspecified, ad hoc way to include your Toyota in the definition, is not how science works. Similarly, simply changing the definition of terminology ad hoc so that 1 is a prime number, that is not mathematics. That is just pseudomathematical crankery. Now, I am not actually accusing anyone of having done this. That is not the point I am making. The point I am making is that educators need to stop encouraging this idea that all definitions exist only according to convenience, and that we change them willy-nilly how we want to. This is true of colloquial language, but not of language in academic disciplines of research. Educators also need to stop presenting fundamental mathematical facts that we do not get to do anything about as if they are something we choose. Perhaps you think otherwise, but that would beyond incomprehensible to me. Maybe I am out of my element. Maybe my strong advocation for the idea that people should not be taught false things makes me unreasonable, although if true, that gives me very little faith in the human species. But I remain unconvinced that this is the case.
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I would not say this is circular reasoning. While I agree with your explanation on why the fine-tuning argument is fundamentally flawed, to me, this just sounds like a problem of an unsound premise, rather than circular reasoning. Circular reasoning would be what would happen if they started with the premise that our universe is unlikely, and then concluded with the same statement in disguise. What they are doing is just starting with an undiscarded and unproven assumption to conclude a different statement, rather than proving the statement they assumed. In other words, their starting premise is not the conclusion they are trying to prove. This is why I say it is not technically circular reasoning. It just an ordinary flawed argument due to an unsound, unprovable premise. In this regard, it is very much like the ontological argument, but worse, since the ontological argument at least makes sense syntactically if you ignore the fact that the premise is unprovable.
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@kensho123456 I am not sure how you define the word "criticism," but I am not trying to criticize her argument in any way. All I am doing is pointing out that her usage of the phrase "circular reasoning" is, by the standards of a person who has taken college courses in formal logic, not exactly correct. Me making this correction has no bearing on the argument she is presenting... which is fine, because my intention was never to criticize her argument in the first place. I reiterate: I agree with her argument. That I decided to try to correct her on some little vocabulary has nothing to do with me agreeing or disagreeing with said argument.
you can't tell somebody they're wrong and ask for the mistake to be replaced with an error of your own.
sigh Leaving aside the fact that I never asked her to "replace" anything, and I am not sure what is the metaphor here, it would be nice if you, perhaps, told me what my error was. Because as much as I hate to say this, telling someone they are wrong without actually pointing out what the error is what my personal dictionary tends to call "an asshole move," and as such, I am not a big fan of it. If you could tell me where the error is, that would be much appreciated. It is not as though I want to make a big deal out of anything I or she said. It was a small and rather insignificant correction in the grand scheme of things, but I thought it would be nice of me to offer her this very secondary knowledge. If you think it was bad of me to offer such knowledge, then forgive me, I did not know it was bad. Will not do it again.
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To be fair, these mutually exclusive deities are only mutually exclusive if you include some minor petty details about the what defines the deity. It is possible to define the concept of deity such that it encompasses every deity in every currently existing religion, without also being unreasonably broad and including mundane concepts, though constructing such a definition is a different story. Religions differ not very significantly with regards to deities, just with regards to other theological and metaphysical aspects. Of course, the names given to the deity in each religion are different. For example, Judaism and Christianity and Islam all worship the same deity, categorically, even though each religion gives a different name to the same deity. But it would be unreasonable to say that the deity in all three religions are mutually exclusive to one another. In fact, the three religions differ not in who the deity is, but what the deity did. In this regard, it is unsurprising that a theological argument for deities is unable to actually distinguish between those deities.
Of course, the arguments are absurd, but their inability to distinguish petty details concerning deities is not what makes them absurd.
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The cardinality of the two planets' orbit sets being equal isn't the same as the number of orbits being equal.
It is the same. It is the same, by definition.
If the first elements of the sets are 1 and 30, respectively, and in both sets, the value of the successive elements increases at fixed but different rates, then at what point do the values of elements become equal?
Firstly, the premise of your question is wrong, because there is no such a thing as a "successive" element in a set. If you want a notion of next element in a set, or a successive element in a set, then you need not just a set, but a partial order on that set. A partial order is a binary relation on a set that is reflexive, transitive, and antisymmetric. You can induce a partial order on a set by having a sequence from the set of natural numbers to it, and the partial order is induced by making the sequence monotonic with respect to that partial order. Consider the set of natural numbers N, and consider the set 30·N := {n natural : n = 30·m, m natural}. You can take the standard partial order of N, restrict it to 30·N, and this forms a partial order on 30·N.
Secondly, by asking about different rates of increase, you must be talking about functions. However, there are problems with this consideration, as you shall later see.
The answer is that the values are never equal, the later will always be greater than the former as time trends towards the infinite.
Yes, if I have a function f of real numbers with f(x) = x, and a function g of real numbers with g(x) = 30·x, then it is true that for every 0 < x, f(x) < g(x). It also is true that the functions f and g diverge unbounded as x grows unbounded. However, this is completely irrelevant to the scenario being discussed. The question being asked is not "what happens to the number of orbits as time gets bigger?", the question being asked is "what is the number of orbits for each planet after an infinite amount of time?" The answer to the question is that, for both planets, Aleph(0) units of time have passed. This is not a contradiction or an absurdity.
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@frede1905 Surely you can vary the value of some parameter, keeping the other parameters the same, and then use the laws of physics to predict how the universe would evolve with that other value.
Mathematically, yes, you can do that, but there is no evidence that this is physically meaningful at all. There is no evidence that there exist any undetermined parameters which could have been different than what they are, and there is no evidence that, even if they could have been different than what they are, that it changing would not somehow lead to changes in all other parameters. The claim that these things are true is fundamentally unfalsifiable. Also, again, there is no reason to assume a universe different than ours has to obey the same laws than ours does, so this talk about considering different universes as being just variations of our own universe with different is, again, unfalsifiable. There is no evidence of such a thing, and there will not be such evidence in the near future, probably. Playing around with the mathematics of our current laws is physically meaningless, if we cannot confirm that a different universe could have existed, instead of our own, satisfying such mathematics.
Plug a different value for the cosmological constant than that which is observed into the Friedmann equations, for example.
There is no evidence that the cosmological constant could have been any different than what it is, or that if the universe had been any different than it is, that it would still have satisfied Friedmann's equations. These assertions are, again, unfalsifiable.
If fine tuning appears in the model, then as mentioned before, there would appear to be something suspicious in the model, and the better, revised model (which surely must exist, as our models are incomplete, as you say) should somehow resolve it.
The "if fine tuning appears" part is the problem here, since any assertion that there is fine tuning is necessarily unfalsifiable, due to things I have mentioned above. Yes, it is true to say that a model having fine tuning is a problem, I am not denying this. However, you can never actually establish that the model has such a problem, because in order to so, you have to make unfalsifiable assumptions.
That's the fine tuning argument in physics anyway.
In the assumption that there really do exist undetermined parameters in our models (as is the case with the current models), we want to find new models which do not have such undetermined parameters. It is true to say that this is a real problem that is being attempted to solve, but this has nothing to do with the notion of whether the universe could have actually been different than what it is, and how different it would be if it were. At best, this concept of "what if the value of this parameter were different?" is just an inaccurate simplification presented to laypeople in order to basically answer the question "why do we care that there are undetermined parameters?"
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One of the things that makes it confusing for people to understand why 1 is not a prime number is the fact that there is more to primality than the intuitive notion of indecomposability. There is another intuitive idea that the definition of primality should capture that people forget about, and it is the idea of non-invertibility. As we know, there are no multiplicative inverses in the integers. The only integers with a multiplicative inverse are –1 and 1. This makes –1 and 1 fundamentally different from all the other integers, be they composite, prime, or otherwise. This distinction is actually more fundamental than indecomposability, and it is what separates the integers from the rational numbers. Because –1 are multiplicatively invertible, you also get a kind of "closure," in the sense that products that contain only –1 or 1 can never be equal to any quantities other than –1 and 1. You cannot multiplicatively generate the integers with –1 and 1. The prime numbers are fundamentally different from –1 and 1, because they are not invertible. As such, you could never get the kind of multiplicative closure that –1 and 1 get. If you multiply prime numbers together, then you necessarily must produce new integers, which are not prime. It is this property that makes the indecomposability of prime numbers special.
Yes, naïvely speaking, –1 and 1 are also indecomposable, intuitively, but this indecomposability is not a mathematically meaningful property, since the only thing you can do with –1 and 1 in multiplication is just get –1 and 1. This is not so with prime numbers. The indecomposability of prime numbers actually has meaning, only because they are not multiplicatively invertible. Therefore, it makes no conceptual sense to actually think that 1 and the prime numbers should be part of the same classification system at all. Saying that 1 is a prime number is like saying that a car should be in the same classification system as animals. Sure, if you want to, you can just come up with a name for any arbitrary collection of objects you see, no matter how ridiculous it is. The collection of all animals and a car can be given its own name, say, the carnimals. You can do this if you really want to. But it is completely nonsensical. Clearly, a car does not belong in the same categorization system as animals do, at least not unless you include many other non-animal things in the classification that have the same properties as cars and animals. Well, this is completely analogous to 1 and the prime numbers. 1 is not a prime number, and that is not because "it's more convenient that way," it is because it is just simply mathematically, conceptually unsound.
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This is largely irrelevant too due to the fact that there exists literally thousands of equivalent definitions for π, many of which are not directly related to circles even if those definitions explicitly are founded on Euclidean geometry axioms instead of, for example, calculus.
Besides, even the analytical process of expressing what π is when talking about the ratios of circumference and diameter lends itself to becoming a definition that is very generalized to all sorts of applications that may not directly involve geometry. For example, even if we acknowledge that claiming that C(r) = 2πr is somewhat outlandish of a notation, this is resolved by the what becomes the analytical definition of C(r) when using calculus. What is C(r)? It is the arclength of a circle of radius r. Therefore, πr is the arclength of a circle of semicircle of radius r. The difference is that the curve for a semi-circle is a function, the curve for a circle is not. Hence the semi-circle lends itself to a manipulation with derivatives and integrals. y(x) = (r^2 – x^2)^(1/2) ==> y'(x) = -x/(r^2 – x^2)^(1/2) ==> 1 + y'(x)^2 = 1 + x^2/(r^2 – x^2) = r^2/(r^2 – x^2) ==> s(x) = [1 + y'(x)^2]^(1/2) = r/(r^2 – x^2)^(1/2). Therefore, πr is equal to the integral of r/(r^2 – x^2)^(1/2) from x = -r to x = r. This is the same as the integral of 1/[1 – (x/r)^2]^(1/2) from x = -r to x = r. Performing the variable change t = x/r implies dx = r·dt, and the interval of integration has -1 < t < 1 instead. Therefore, πr is equal to the integral from t = -1 to t = 1 of r/(1 – t^2)^(1/2). Therefore, π is equal to the integral from t = -1 to t = 1 of 1/(1 – t^2)^(1/2). In fact, from the construction of the problem, this can and should be taken as the definition of π. As it happens, this is an integral that occurs frequently in applications, justifying the usage of the constant.
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Tom H This applies to some flat Earthers, but in general, not really. Many flat Earthers, particularly in the U.S.A, do believe in the Bible, but many others are also atheists. And yes, they are uneducated in science, but they are also dumb. Here is the thing: most people are uneducated in science. This is the truth. Unless you went to undergraduate school and obtained a bacherlor's in science, your education science is effectively minimal, especially because high schools are not particularly good at teaching maths or sciences. But the difference is, most people uneducated in science aren't dumb. And they won't try to argue with you about stupid things they're not educated about because they're not dumb like this.
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@Lukasek_Grubasek Formal set-theoretic definitions for addition and multiplication for the natural numbers do exist, and they extend to the integers, the rational numbers, and other sets quite naturally. As to whether these definitions are necessary: if you want to define the integers set-theoretically, then you definitely need to define addition beforehand. Multiplication is optional, though, and 2·n would, in this case, be merely a shorthand for n + n, rather than an actual product. However, to define the rational numbers, you absolutely must define integer multiplication first.
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@yohanessaputra9274 Firstly, I didn't insult you.
Yes, you did. You told us to "read the literature", presupposing that we have not read it before this discussion, and that we are thus ignorant on it.
I was merely responding to what Alan Animus said about his dislike about the term "timeless".
He adequately had already explained that his dislike of the term has nothing to do with the neutrality of the term, which you should have known, had you bothered to carefully read the conversation before trying to butt in. Your response was to deliberately ignore that, and then tell us to "read the literature", with implications that I already explained.
Secondly, you exchange with me by Ad Hominem and you presupposed me a theist.
At no point in this conversation have I presented any ad hominem to you. I have also not presupposed you a theist. I said your comment makes you look like an ignorant theist. I never said I believe you are a theist. The fact that you do not know the difference proves my point about your arrogance.
That's a one way to have an exchange lol.
It is the way of exchange that you chose.
I feel of things we will talk is probably just insulting each other, so good luck with your life
You wanted to insult us, so I decided to reply accordingly. If you want to stop being called out for insulting people, then you need to stop insulting people. The fact that you are even pointing this out exposes you as a troll. You know what would be conducive to a productive conversation where I would glady be discussing time and philosophy, instead of your own insults? A conversation where you do not begin by insulting people perhaps. Anyway, I will not waste my time having any further interactions with you, because I have been on the Internet for a long time, and I know that feeding trolls is a bad idea. Farewell.
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@Melesniannon What you are appealing to is a different concept altogether, a concept from measure theory, which is not about determining the number of elements in the set, but measuring the length of the sets. Namely, there is a function called the Lebesgue measure, which, when applied to a set, gives you the length, area, or volume, or the set, depending on whether the set has dimension 1, 2, or 3. Since the intervals (1, 2) and (1, 3) are one-dimensional sets, they have lengths, and the Lebesgue measure, when applied to these sets, gives their lengths, namely 1 and 2. In general, if you have an interval (a, b) with a =< b, and the Lebesgue measure λ, then λ((a, b)) = b – a. So λ((1, 2)) = 1, and λ((1, 3)) = 2. However, even though the Lebesgue measure of the sets is different, their cardinality is the same.
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@Melesniannon I think you misunderstood my example, it's not about length, it's about being able to understand the set in a second dimension which leads to the understanding that even though they are infinite in one way, one infinite set will, given the same parameters, always be larger than the other.
I disagree. This absolutely is about length. You are talking about meter sticks, and comparing an interval of length 1 cm with an interval of length 2 cm, and claiming that while both intervals are infinite, the latter is larger than the former. Meter sticks are precisely about lengths of intervals, not about the number of elements that are members of the intervals. Of course, if you intended to imply something different, then your analogy does not work for your purposes.
A 2 cm wide line extended infinitely is always twice the area as a 1 cm wide line given that they are both extended towards infinity equally.
Yes, it has twice the area, but it also has exactly the same amount of area. This is because Aleph(0) = 2·Aleph(0), so if they both have Aleph(0) square units of area, then one has twice the area of the other, and they have the same area. I am unsure if you are meaning to imply they do not have the same area, but they absolutely do.
A planet that revolves 30x every time another planet revolves 1x, always makes 30x more revolutions, even when time is extended infinitely.
Mm, no, not quite. For every real number x > 0, it is true that 30·x > x, so 30·x and x are unequal. However, this does not hold for x being an infinite quantity.
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You can prove that numbers of the form 4k + 3 cannot be the sum of two squares fairly easily. There are only three possible pairs for (a, b): (even, even), (even, odd) (addition is commutative over the natural numbers), and (odd, odd).
(2m)^2 + (2n)^2 = 4(m^2 + n^2) == 0 (mod 4)
(2m)^2 + (2n + 1)^2 = 4m^2 + 4n^2 + 4n + 1 = 4(m^2 + n^2 + n) + 1 == 1 (mod 4)
(2m + 1)^2 + (2n + 1)^2 = 4m^2 + 4m + 1 + 4n^2 + 4n + 1 = 4(m^2 + n^2 + m + n) + 2 == 2 (mod 4).
Numbers that are 3 (mod 4) are completely absent from the possible outcome. Hence why they cannot be the sum of two squares.
From here, you can build the larger proof that shows that if n is divisible by (4k + 3)^(2n + 1) and 4k + 3 is prime, then n cannot be the sum of two squares.
(4k + 3)^2 = 16k^2 + 24k + 9 = 4(4k^2 + 6k + 2) + 1 == 1 (mod 4), which can result from the sum of two squares, by setting 4k^2 + 6k + 2 = m^2 + n^2 + n and solving for m and n.
However, (4k + 1)(4l + 3) = 16lk + 12k + 4l + 3 = 4(4lk + 3k + l) + 3 == 3 (mod 4), which implies that (4k + 3)^3 == 3 (mod 4). This, in combination with (4k + 3)^2 == 1 (mod 4), and by induction, implies that if n is divisible by an odd power of p = 4k + 3, then n cannot be composed into the sum of two squares, while if n is divisible by an even power of such, then it can be decomposed.
To be more general, (4k + 3)(4l + 3) = 16lk + 12k + 12l + 9 = 4(4lk + 3k + 3l + 2) + 1 == 1 (mod 4), indicating that if the number of 3 (mod 4) prime factors - these are called real-valued Gaussian primes - is even, then the product is decomposable into the sum of two squares, and if the number of factors is odd, then it is not decomposable.
One way to formally state this as a theorem is to define a function Ω(m, p) that is equal to how many times m is divisible by p. If m is not divisible by p, then Ω(m, p) = 0. If m is divisible by p^2 but not p^3, then Ω(m, p) = 2. Then, we look at the sum of Ω(m, p) over all Gaussian primes p less than or equal to m. Denote this sum as ψα(m). Then our theorem goes as follows.
There exists a pair of (a, b) such that a^2 + b^2 = m iff ψα(m) is even.
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@andresvillarreal9271 The claim that the infinite cannot be traversed is something that you have to discuss with mathematicians, not with philosophers.
Thank you! This desperately needed to be said.
In fact, there are a gazillion ways in which, through artifacts of language and not of science, mathematics or good philosophical reasoning, you end up treating infinity as a number, and not a cardinality.
This may just be unnecessary nitpicking, but as the connotation of the conjunction "and" in the English language is more ambiguous than it is in formal logic, I should remark that a cardinality is a number. It is unclear whether your comment is meant to imply otherwise or not, but just in case, I wrote this clarifying remark.
You can add an element to a set that has a cardinality of infinity.
Infinity is not itself a cardinality, though infinite cardinalities do exist. It would be more accurate to say that you can add an element to a set with infinite cardinality. This may seem like a pedantic distinction, but actually, it is quite an important one.
And the new set also has a cardinality of infinity, creating the apparent contradiction that infinity plus one is infinity. But there is no contradiction because infinity is a cardinality, not a number.
This is incorrect, and there are a few misconceptions to unpack here. As I clarified earlier, a cardinality is a number. Every cardinality is a von Neumann ordinal number, and every von Neumann ordinal number is a hyperreal number. As I also clarified earlier, infinity is not a cardinality. Infinity is the property of a set S of there existing an injective function f : N —> S, where N is the set of natural numbers. Infinite sets are said to have infinite cardinality, but different infinite sets have different infinite cardinalities, which is why infinity cannot be called a cardinality itself. The set of natural numbers has cardinality Aleph(0), which is the smallest infinite cardinality. Aleph(0) satisfies the property that Aleph(0) = Aleph(0) + 1, and Aleph(0) = 2·Aleph(0). The set of real numbers has a larger cardinality, the cardinality 2^Aleph(0), which is not equal to Aleph(0), but is larger instead. Assuming the continuum hypothesis, this means the set of real numbers has cardinality Aleph(1).
As for the equation that Aleph(0) = Aleph(0) + 1, it seems to be an apparent contradiction due to our preconceived, erroneous intuition, regarding cancellability. When we work with a commutative algebraic structure (S, +), where S is nonempty, we say that an element x of S is +-cancellable if x + a = x + b. If every element of S is +-cancellable, then we say that + is a cancellable operation. Every algebraic structure you learn about in primary schools, and even in undergraduate colleges, is a structure where + is cancellable. So by intuition, we tend to assume that every algebraic structure in mathematics must satisfy this property. So when we encounter Aleph(0) = Aleph(0) + 1, we immediately assume that cancellability holds, and so we conclude from this that 0 = 1 must be true. However, we know that 0 = 1 is false, so we believe there is a contradiction. Since most people are non-mathematicians, they lack the training to recognize that the problem is not with Aleph(0), but with our cancellability assumption. In fact, historically, it was impossible to know this was the case, because the study of non-cancellative algebraic structures did not develop until many centuries after the topic of infinity had become already controversial. This is where the apparent contradiction comes from.
When you say that the infinite cannot be traversed you are declaring that infinity is a number, and everything you do from that point onward is pure garbage.
I disagree. I would argue that the word "traversing" is not sufficiently well-defined for this conversation to hold, but even if we hold on to just intuition, saying "the infinite cannot be traversed" reasonably translates to "infinite objects do not exist", which in modern terms, is understood as simply rejecting the set-theoretic axiom of infinity. Historically, this makes sense, since axiomatic set theory did not exist until Georg Cantor came along, and it was he who provided a framework for working with infinite sets in seriousness and good faith, by way of equivalence classes and functions, and this happened centuries after the infinite was already controversial.
The answer to everything in the first minutes of your video is that those philosophers did not know much about mathematics.
This much is very true, though, in their defense, neither did anyone else, at least prior to Leonhard Euler.
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@TheWrongBrother The reason for philosophical assessment other than mathematical stems from the ambiguity of results from mathematical operations on infinity when viewed either as a cardinality or a natural number
Prior to the 19th century, you would have been correct in stating that the results of operations with the infinite would have ambiguous and with many apparent absurities. However, we live in the 21st century, and von Neumann-Bernays-Gödel set theory is a tentative foundational theory of mathematics that exists. Rigor for the infinite is well-understood now, and there is nothing ambiguous about operating with the infinite.
♾ + 1 = ♾
Here is where you are relying on an outdated idea of infinity. Infinity is a property of sets. Specifically, a set S is called infinite if and only if there exists an injective function f : N —> S, where N is the set of von Neumann natural numbers. Infinity is not a direct description of the size of a set, let alone a number. For example, the set of natural numbers has cardinality Aleph(0). The set of real numbers has cardinality 2^Aleph(0). Both sets are infinite, but it is false that Aleph(0) = 2^Aleph(0). In fact, according to Cantor's theorem, Aleph(0) < 2^Aleph(0). Aleph(0) and 2^Aleph(0) are numbers, and are infinite cardinalities, but neither of them is called "infinity", and "infinity" is not a number. So what you should have written is that Aleph(0) + 1 = Aleph(0), which is indeed correct.
since the mathematical size of the infinite quantity has grown by one, its logical size has not been impacted in any way and still remains infinity.
Wait, what? Your sentence is extremely confusing. You are making a distinction between mathematical size and logical size, but in reality, you have defined neither. Also, because you wrote "since" at the beginning of your intended sentence, you are indicating that the logical size of the infinite quantity not changing and remaining infinite is a necessary consequence of the mathematical size being increased by 1. However, the meaning of this, and therefore, its truth, is far from obvious.
If two separate events added to the infinite quantity, how would you know the size difference from two separate view points?
I am not sure I understand your question, but if there exists a bijection f between an infinite set X and an infinite set Y, then the two sets, by definition, have the same cardinality, i.e, the same number of elements. This is true, regardless of whether there are elements of Y that are not elements of X, or vice versa.
Hence philosophical assessment bodes better for this analysis...
What your comment has demonstrated seems to be the opposite of this. Philosophical assessment of the infinite is inadequate to understand the infinite, which is why no good understanding of the infinite existed historically until mathematicians took it upon themselves to set the theory of sets on top of a rigorous foundation. To put it more succintly: the ontology of infinite objects is the theory of sets. There is no other way to do it.
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@TheWrongBrother The usual assumption is that the variable V(N0) is linear and the observer assessing it is 1.
No. There are no linear variables required in this discussion. There are no observers required either.
At the end of the set...
There is no end to such a set. If a set has order type ω, then there is no such a thing as its last element.
If the V(N0) is moments in time with the assumption that there have been infinite moments in time till then, can you get to the present?
Yes, you can.
as the last element, if another moment is added, you immediately become the past and are excluded from the state of the present and by that it becomes impossible to get to the present.
This is nonsensical, since (0) time does not work that way, time is a coordinate in spacetime, (1) there is no last element in such a set.
its logically impossible to observe the last moment since new ones are added every 'time'.
No, observation is completely irrelevant here.
If you take an infinite crowd concert (non linear set with infinite elements).
There is no such a thing as a linear set. Stop saying nonsense and inventing terminology that means nothing.
how would you establish that everyone in the set has their lighters on?
Via a bijection.
Traversing a set with Aleph-Naught cardinality is logically impossible and absurd from a math/philosophy point of view.
No, it is not. Your understanding of mathematics and set theory is just REALLY bad.
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Awwkaw You really are forming strong conclusions based on the first sentence of a Wikipedia article of all things? I'm sorry, but if you want to have a serious discussion, that's not going to cut it for anyone. You must be joking if you think I can call that researching a definition. You could at least put a bit more effort, don't you think?
Going down too many dimensions just loses information. I can make a projection of a ball onto a plane to get a disk, but I cannot do the same and get a point or a line. Intuitively, this makes sense, and if you're skeptical, just try thinking about it yourself. You'll end up concluding this yourself. This alone is sufficient to show the amount of dimensions this lower-dimensional boundary has is not arbitrary and cannot be made as small as one wants it. You could do further maths to then show that for must situations, you only can go down 1 dimension, but the maths for that are rather complicated and too much to discuss on YouTube. However, the intuitive idea is there.
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@BRDRDRDAT You are correct, but all those axiomatic systems are still infinitely more useful and far less problematic than ultrafinitism will ever be. And that is ultimately the problem.
And you're right. It would be seriously funny if he thought quantum mechanics is real but that a continuum of the number line cannot exist. Honestly, if you reject infinity, then why even accept physics at all? You may as well reject part of or all of the scientific method, since so much of the scientific knowledge and inquiry is contingent on these ideas of real numbers. Although scientists don't strictly use real numbers for measurements, their system is similar enough, and procedures of taking derivatives and integration are still fundamentally present in every scientific field. I mean, if you want to reject the axiom of infinity and reject the existence of any number bigger than 10^(10^80), then fine, you can do that, nobody really cares, but at least be honest with yourself and maintain logical consistency by rejecting everything else that is implied by those ideas. The most ironic thing is that he has the gall to publish this on YouTube, a platform that only exists because the Internet exists, and the Internet exists because of quantum mechanics. I mean, his thesis is the most self-defeating thesis ever. I'd be partially okay with it if he at least accepted this much and decided to distrust every scientific achievement that is contingent on scientific theories that use real numbers. But he doesn't even go that far. It's hard to believe in the commitment and consistency of his worldview in light of those things: you'd have to give up your sanity to buy it.
I wrote a series of comments in the comments section analyzing and deconstructing the video point-by-point. I took his presentation seriously (even though it didn't deserve to be taken seriously), but after looking at it closely and carefully, I think this video is easily one of his worst. Though, in general, even his good videos don't make a particularly convincing good case for ultrafinitism. He can show that ultrafinitism works in some contexts, but he has yet to show that ultrafinitism is sufficient to replace everything we currently. And the reason is simple: he can't, because he'd be showing something that is false being true, which can't happen.
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It is not an axiom, it is a theorem. Formally, the pigeonhole principle states that, for two sets X, Y, if there exists an injective function f : X —> Y, and there exists no surjective function g : X —> Y, then there is no injective function h : Y —> X. You can prove this by appealing to the Schröder-Bernstein theorem, which states that for any two sets X, Y, if an injective function f : X —> Y exists, and an injective function h : Y —> X exists, then an injective and surjective function j : X —> Y exists.
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Zeetee Pippi Zero may be a limit, but it has no inverse.
I never said zero has an inverse, I said it is the inverse of an infinite quantity, and also of the limit to infinity of x. Not that this is relevant, since probabilities with infinite sets are not calculated with inversion, they are calculated with measures and distributions, which is what Mike addressed, and which is what you ultimately replied to.
That's one of the most commonly accepted axioms of mathematics.
No, actually. It is an axiom of a field, which is what the real numbers are. However, if not working with a field, then you can totally divide by zero. You can do this with non-division algebras, such as the sedenions, which are a well-studied type of numbers that generalize the octonions via the Cayley-Dickinson construction. You can also do this with a wheel, a type of algebraic structure which generalizes a field. For more on this, see "wheel theory" or "wheel algebra."
Mathematics is not made of one set of axioms. It is made of multiple universes, a collection of universes, if you will. Each universe has different sets of axioms, inference rules, and then some formal language with interpretation equipped to it.
1/Infinity approaches zero, but it is not 0.
It IS 0. I don't think you understand how limits work, my friend. Infinity IS a limit of some function. It is not the thing that approaches, but the thing that is approached. The question is asking what is the limit of 1/x as x approaches infinity. The answer is unanimously 0. That is synomymous with the abbreviative phrase "1/♾ is 0". They don't represent different equations. They are the same equation, and therefore, translate to the same thing in English as they do to any other language. I can say it in Spanish, if you want me to. I can even say it in Japanese, although my Japanese is rusty.
Just as Infinity is an approach to a larger and larger number.
No, infinity is not the approach, infinity is the limit of said approach. The approach isn't anything other than the linear functional whose output is precisely the limit.
A point has dimension 0, but that mostly describes how it can't be measured in any meaningful way.
No, it just describes that it has no measure. Not that it cannot be measured meaningfully. Those mean different things. The forner means "Measure = 0", the latter means "Measure = Undefined".
You get shapes by linking together points within some higher dimensional space.
Correct. This is precisely what Mike said, in fact, just in a much more formal, strict, and rigorous language. Stating that the unit square is an uncountably infinite union of measure-0 sets means exactly the same thing, except this is much less vague. So, you're only helping prove his point.
You can't link nothing together and get a line segment.
Correct, but points aren't nothing. I made this difference clear in a previous comment. Having a total area of 0 does not imply it is nothing. Points have area 0 by definition. Why? Because any object with non-zero surface area must be composed of uncountably infinite-many points. If a point has nonzero area, then you are claiming points are made of infinitely many points, and furthermore, of uncountable-infinitely many lines, which is a contradiction. So it obviously is not the case.
You are more likely than not thinking of points as they actually would be in the physical world, which is granular in its fundamental nature. Those don't represent actual points in mathematics. And using that physical analogy is unjust, since, if you wanted to turn the paradox into a question about real-world dartboards, then the paradox would cease to exist, since where a dart is going to land on the dartboard is not random, but rather predetermined by physics, making so that it has a 0 probability of hitting anywhere and 1 of probability where it will hit. Thus, obviously, we have to discuss the paradox in the context of a mathematical dartboard, not a physical one, just like how Gabriel Horn's paradox has to be discussed in the context of math, not a physical shape (because a physical shape of infinite surface area can't exist, and an infinite physical hotel can't exist either).
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Zeetee Pippi For all intents and purposes, an infinitesimal is zero.
No, it really is not. And no, it's not a question of semantics: they REALLY are different not only in definition, but in the very behavior they exhibit as quantities. The confusion is a little absurd, in my opinion. They are not as similar as you think they are.
You can't add an exact zero to get anything,...
...and that's fine. We aren't adding things in here.
...but an infinitesimal is a different type of zero that can be part of an uncountable set.
Now you are simply making no sense, and I am afraid you do not understand the terminology you are using.
1. There are not "different types of zero." No such a thing exists in mathematics. You will not find a single textbook or mathematician or written theorem supporting this notion. No.
2. The real numbers are an uncountable set. The number 0 is an element of the real numbers. Therefore, it is an element of an uncountable set. Do you know what a uncountable set is? Also, we are talking about uncountable infinitely many unions of sets, not just sets that are uncountable.
That's why I think "measure 0" is misleading when you think of adding nothing being added together.
1. We are not arithmetically adding quantities. Union is not the same type of operation as addition, although they share some properties. You could consider it the set-analogue of addition, but as sets are different from numbers, the union ultimately also has a different behavior from addition. 2. "Measure 0" is not nothing. And once again, that's not semantics: there is a fundamental difference between both things. If you don't understand the difference, then take a course on measure theory, or read a textbook. But honestly, understanding that one simple fact is really not hard and it doesn't take much understanding of measure theory in general. To give you a hint on how they are different: the set of rational numbers has measure 0. Do you still think measure 0 is equivalent to "nothing" somehow? 3. So you are telling me that the theorems of measure theory, which by the way, is 100% formulated on the "accepted axioms of mathematics" as you call them, are misleading? Okay, I get it. You're not arguing that the notion that points having measure 0 is mathematically incorrect. You are arguing mathematics in general are incorrect. Gotchu.
I was disagreeing with Mike when he said that the inverse of infinity is the real zero and not the non-real zero.
There is no non-real zero. Zero is a real number. And it IS the inverse of infinity. That's not something you can change. It's a direct consequence of the axioms. If you argue against it, then it is inherently a disagreement with those very axioms.
But if you're allowed to disagree with the axioms, then I might as well be allowed to claim centaurs are real, and you can't debate it because it's an axiom. Do you get it now? Some things, you just don't get to disagree with, even if they go against your very intuitions. So your options are 1) learn about measure theory, master it, and then if you can irrefutably, thoroughly, and rigorously prove that somehow every mathematician ever has got it wrong and you're right, then you publish the paper; or 2) learn about measure theory and realize that measure 0 does not mean "nothing", and that the inverse of infinity really is 0. And also learn some set theory and realize there exists only one zero, the real number. There are no non-real zeroes. That's all there is to it.
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In the first line only, 0 cannot be written as a sum of infinite zeroes.
Technically, there is no such a thing as an infinite sum. However, there is such a thing as the limit of a sequence. You can write 0 as the limit of the zero sequence. Consider this: z(m) = 0 everywhere. Let s[z](0) = 0, and s[z](m + 1) = z(m) + s[z](m) everywhere. Hence, s[z](m) = 0 everywhere. s[z] is the sequence of partial sums of z, so lim s[z] is the "value of the series" of z. lim s[z] = 0. This is completely valid.
As, if it is possible, then we can write the integral of a continuous function on [a, b], as the sum of the integral at each point.
No, that is actually not how that works. The integral of a function cannot be evaluated "at a single point." You can only evaluate the integral of a function over closed intervals, and finite unions or intersections of those closed intervals.
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Also, saying they are of the "same infinity", is not accurate, or rather, it is imprecise. The set of real numbers has a larger cardinality than the set of integers, while the set of integers has the same cardinality as the set of natural numbers. In other words, the cardinal number corresponding to the set of real numbers is larger than the cardinal number corresponding to the set of integers, which also corresponds to the set of natural numbers. Specifying that you are talking about cardinal numbers here is important, because a more fundamental type of infinite number are the ordinal numbers, but even though the set of integers and the set of natural numbers have the sams cardinal number, they do not have the same ordinal number. The ordinal number for the set of natural numbers is ω, while the one corresponding to the integers is ω + ω. The ordinal number corresponding to the real numbers is some ordinal number greater than or equal to 2^ω, possibly ω(1).
You also would be better of explaining why a pair of sets have the same cardinality or a different cardinality. If there exists an injective function f from a set X to a set Y, then the cardinality of X is less than or equal to the cardinality of Y. This defines a well-order on the class of cardinal numbers.
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@James Lynott The problem is deeper than that. The issue is that "infinity" is not even a mathematical object. It does not refer to anything. It is, however, a property of mathematical objects, in much the same way that "finity", or "finiteness", is. As for quantities that are infinite, they can be arithmetically well-behaved, as demonstrated by the success of the hyperreal numbers in nonstandard analysis. But the important concept is that while some hyperreal numbers are infinite, "infinity" itself is not a hyperreal number. "Infinity" is merely the property that describes infinite hyperreal numbers, in the same way that "finity" describes finite hyperreal numbers, even though "finity" is not itself an object.
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@jb76489 if you say so buddy, meanwhile in reality you (attempted) to insult me,
If you interpret it as an insult, that is YOUR problem, not mine. I merely stated a fact, that is all.
then you lied about what I said,
I did not. All I did was ask a question about what you meant, which you answered affirmatively, hence I rested my case there. Ironically, the one lying about what I said here is you, which proves that you are a troll, which is exactly what I suspected.
then asked deflecting rhetoricals .
They are not deflecting, though, because their answers literally answers your question. If you are so delusional that you refuse to see this, or if you are so moronic that you are unable to acknowledge it, then that is, once again, your problem, and not mine.
Maybe the definition of argument is different wherever you are but up here that’s a sorry excuse for one
I agree, because the definition "up there," which is different from that of "wherever I am," is objectively trash. It just implies you do not understand what an argument is, and frankly, this just indicates I should stop wasting my time with you, which I will do.
*I’m really not sure what you find so difficult to understand about “this problem, which has zero practical use, got computer time whereas other projects, which are more likely to actually help people did not. This is suboptimal” *
And I am not so sure what you find so difficult to understand about "that statement has zero valid justifications, and *you are wrong,*" but hey, I am not your professor, nor do I intend to be. I gave you the means to find the answer to your question, but you are too lazy to use those resources, so here you are berating me like the idiotic dishonest troll that you are. It's almost cute, except it's disgusting and pathetic instead.
*but please, do show how knowing this sum is going to help some with cancer or predict climate in the future or literally do anything at all to help people
*
See my sentence above.
this is what people say when they’re right, 100%
No, this is what people say when they are not so idiotic as to enter a fruitless discussion with another idiot. And with that in mind, I should point out that you were too incompetent to understand the message, so instead, I'm muting you and blocking you. You won't be hearing from me ever again (because you won't be able to), and I won't be able to hear from you (because I won't be able to.) This wouldn't be necessary if you understood what "done feeding the trolls" means. Anyhow, I know you'll want to have the last word, because this is what trolls do. So go ahead and keep typing like a maniac. Only you will be reading what you type, so if you enjoy presenting yourself as actually insane and enjoy talking to yourself, then go ahead and please yourself. I'm out.
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Actually, I disagree. I would argue that Schrödinger's cat, while popularly being weirder than E = mc^2, is easier to understand. Because as others here said: it can be very easy explained by simply talking about waves and their amplitudes, and then talking about how this translates to probabilities instead of energy. It only takes some linear algebra to understand, and you don't need to go through the entire quantum-mechanical framework to explain it.
Meanwhile, it isn't possible to talk about E = mc^2 and explore it further without going over the entirety of the theory of special relativity from scratch. I mean, yes, fundamentally, all the equation means is that a bodies energy at rest is proportional to its mass. That's very simple, but that's no deeper an understanding than the very overly simplistic misrepresentation that the mass media has given to Schrödinger's cat.
The thing about quantum mechanics is that while the math is more difficult, dissecting each concept on a qualitative level is far easier. Special relativity, on the other hand, has much simpler math, but actually dissecting the concepts qualitative is much more difficult if you can't discuss the theory in its entirety.
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@njwildberger can we spell it out completely and clearly?
Yes, of course we can, that is the entire point behind mathematical symbols. The problem is that your highly restricted and flawed definition of "completely and clearly" is too insane to allow for symbolic machinery, even though it is an entirely inconsistent worldview, considering that the decimal digital string representation of a number is also merely a symbolic tool. The decimal digital string representation of the number is just a representation of the number, not the number itself, so criticizing real analysis for using symbolic machinery is hypocritical and a fallacious form of argument. This is the basic misunderstanding you have to address before your refutations against this comment can even begin to make any rigorous sense logically. By the way, I already explained this in my own series of comments to the video, along with many other things.
Rational functions are finitely specifiable so they can be written out
This is also true of non-rational functions.
You could interpret that as giving a function on natural numbers, in the sense that given a natural number you could calculate f(n), at least if n was not too big.
That is famously not the definition of the word "function," Norman.
That turns out to be famously impossible.
It is not. Every refutation presented to your video so far presents counterexamples to this.
So the academy is not even in explicit agreement about whether "non-computable real numbers are valid objects." Most...
This is an entirely unfounded claim. You should at least bother to provide a source that presents some type of metaanalysis or a survey with a sufficiently large and diverse sample size that verifies this claim. However, you have not done this. I suspect that is because, if any such survey exists, then it would most definitely support the opposite claim, not your claim.
Most computer scientists will say: obviously not
Computer scientists are not mathematicians, so this is entirely irrelevant.
even thought it is clearly separated from reality.
All of physics begs to disagree with you, and I want you to know that the only reason you are able to use the Internet to write this very comment of yours is because of physics.
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Craig Dougan Also, if you are seriously treating Google searches and Khan Academy as serious mathematical sources to prove your point, then I cannot take your objection seriously. You will have to do that a whole lot better than that. And I said in my comment that many professors do indeed teach about the set of whole numbers, which - spoiler alert - does not actually exist in the mathematical literature. Textbooks are written by professors, not mathematicians, so naturally, textbooks will contain that type of outdated information. That is irrelevant, though. Textbooks are not part of the mathematical literature proper. It's very much how scientific textbooks are not actually scientific documents of any sort. They can be useful pedagogically, but for the sake of actually deriving knowledge, you go and read actual scientific papers, not textbooks. Scientists write scientific papers, and they research scientific papers. Textbooks are not real academic sources.
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@dennishickey7194 I point out his refusal, as all others lawyers', refusal to answer the most basic practices of the courts they presume to explain (and reconcile us to).
This is still a baseless accusation, and rephrasing it does not help your case. The usage of the word "refusal" insinuates that you communicated directly with Devin Stone, told him about your qualms, and that he replied to your communicate by saying "No." You have provided absolutely no indication that this is what happened, and I am pretty certain you and I both know this is not what happened. Also, "explaining the basic practices of the court and reconciling them to us" is not something the channel has ever stated it presumes to do.
I can appreciate that you find my comments inappropriate or "bizarre" as you said.
Do you? Honestly, I do not particularly care if you "appreciate" the comments or not. What I want from this interaction is for you to actually put some thought into what you are saying, and either understand that what you are saying is completely unreasonable, or present me with the missing justification that tells me that what you are saying is reasonable. Of course, you can refuse to, but if you are not going to provide the justification, then you are just going to look silly to everyone else.
I do hope you consider the practice of forging the transcripts in felony cases and what it means legally and morally.
I am not a lawyer, nor a law student, and I have no interest in defending any particular form of court practice, especially as I am not qualified to do so, but I also do not trust that presenting the practice as "basic" and common is accurate, nor is it presenting it as forgery.
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π is defined as the ratio between the circumference of a circle and the diameter of said circle. One can prove said ratio is constant by first, noting that this ratio does not change if a circle is centered at the origin. A circle centered at the origin with radius r has equation x^2 + y^2 = r^2. The ratio between the circumference and the diameter is equal to the ratio between the arclength of the upper semicircle and the radius of the corresponding circle. The upper semicircle is given by y = sqrt(r^2 – x^2), and the arclength is given by the integral on (–r, r) of sqrt[1 + (y')^2]. y' = –x/sqrt(r^2 – x^2), hence (y')^2 = x^2/(r^2 – x^2), implying that sqrt[1 + (y')^2] = r/sqrt(r^2 – x^2) = r/sqrt(r^2·[1 – (x/r)^2]) = r/(r·sqrt[1 – (x/r)^2]) = 1/sqrt[1 – (x/r)^2]. Let t = x/r, hence x = r·t, hence dx/dt = r, and (–r, r) |—> (–1, 1), so the above integral is equal to r multiplied by the integral of 1/sqrt(1 – t^2) on (–1, 1). The integral on (–1, 1) of 1/sqrt(1 – t^2) is independent of r, so this is a constant ratio, and so the arclength is proportional to r. Therefore, the arclength divided by the radius r is simply this constant of proportionality: the integral on (–1, 1) of 1/sqrt(1 – t^2). This integral is the definition of π.
To get a better definition that we can use to prove that π is a real number that is not rational, we can first notice that if we define g(x) as being the integral on (–x, x) of 1/sqrt(1 – t^2), then π := g(1). One can then obtain the Maclaurin series expansion of g, which converges everywhere for g, and use the Lagrange inversion theorem to prove that [g^(–1)](π) = 0, and furthermore that [g^(–1)](z) = Im[exp(i·z)]. Hence g^(–1) can be analytically continued to the entire complex plane, and it can be shown that [g^(–1)](0) = 0, and that in general, exp(2·m·π·i) = 1. This gives us a new, more useful definition of π: it is the unique real number such that it is half of the imaginary period of exp. This can be used to prove all sorts of properties of π.
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I am very glad this video was made. I do appreciate using visuals as a teaching aide in mathematics, but I absolutely despise it when exams make questions that rely entirely on the visuals with verbally giving you the information, since visuals are only that: an aide for intuition, they themselves do not communicate mathematical information, since any visual representation you make is necessarily physical and imperfect. I also despise it when people begin acting like visuals count as mathematical proofs. It completely goes against the spirit of mathematics. And those people are, ironically always pointing to this channel to excuse their pseudoscientific, pseudomathematical mindset. And while I know that was never this channel's intention, this sorely needed to be addressed and explained to people. Visual aids are learning tools, they are not mathematics in themselves. Visual aids necessarily contain flaws, and as this video shows, if mathematical proofs relied on them, then most of mathematics would be useless, because it would just be full of contradictions everywhere.
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@fokocrispis4036 If we say that existence is "something with qualities", nothingness would...
"Something" is not a well-defined metaphysical concept either, so trying to piggy back off "something" to get to a well-defined metaphysical "nothing" is not going to work. Appealing to common sense is also not going to do you any favors.
Nothingness would lack all qualities.
This statement is nonsensical, since, at least in the context of how "quality" is defined in essentially every context, "lacking all qualities" is itself a quality. What you are doing is akin to saying that there is a set of all sets, which is known to be impossible, as it is a logical contradiction. Besides, what does "lacking a quality" even mean? In a formal context, it means literally nothing, since if you "lack" a quality, then you have the opposite quality, hence you still have a quality.
If your nothingness has 0% of events happening, you are not talking about nothingness in the first place.
Exactly. Somehow, you missed the point of the comment, though, and failed to use this as the key premise to conclude that nothingness is incoherent as a concept, which is precisely what the comment set out to prove in the first place. You disagreed for no reason. And ultimately, nothing that you have described in your reply is a sensical definition of "nothing" either, so you have sort of reinforced the point. "Nothing" is not a coherent idea, it is merely a flawed intuition that humans naturally have.
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@fokocrispis4036 but the important part there is that everything in existence has a quality.
That begs the question of what "existence" is. Because, as pointed out in the video, "existence" is not really well-defined idea either. And if it was, then it would no longer really be a matter of ontology. Besides, I can make the claim that "there exists one entity such that this entity has no properties", and I do not think you can prove that this statement is false. This makes your definition of "nothing" inadequate.
Lacking all qualities is a contradiction because of language but it is by itself not a quality.
No, this has nothing to do with language. The contradiction would remain if I used formal logic to state this instead of a natural language. So it is, in fact, a quality.
Lacking a quality also doesn't mean you have the opposite. Lacking size doesn't make something big or small, it just doesn't have size.
This is a bad analogy. Lacking quality would better be compared with lacking bigness, or lacking smallness. Also, if you lack size, then you are sizeless, the opposite quality of being size-having or "sizeful". Of course you do not become big or small if you lack size: I never claimed you do, because those are not "opposites" of "sizelessness". This is a misrepresentation of my point.
I disagreed with the original comment, because I don't see that as nothingness.
The original argument does not claim to see that as "nothing" either. So you did miss the point.
The point in the original comment was nothingness negates itself, therefore it is an incoherent concept.
This is indeed the thesis of the comment, and this is a different claim than claiming that nothingness is probability.
But my point was that nothingness is a viable state, but there is no actual way to describe it as there is no actual contrast to even identify it.
If there is no identification for nothingness, then it is not a viable state. Well-definedness requires that there be precisely something that identifies the thing being defined, even if the identification cannot be put into words. Otherwise, it is not a "thing" in any meaningful sense of the word. It is a non-concept.
I don't believe there is a coherent definition of nothing, though, but that doesn't mean that reinforces the impossibility of nothing as a state,...
It absolutely does reinforce it. If there is no coherent definition of it, then it is undefined, by definition. That is just how it is. Undefined things are not concepts, let alone states. This is a lot like calling "Undefined" a number. It just does not work like that.
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This video is years old, but even today, people are still commenting on it, so let me jump in and actually explain what unique factorization means in this context, why is it important, and how to make sense of the idea of an empty product.
Consider the product 2^a(0)*3^a(1)*5^a(2)*•••, where * is the multiplication symbol and ^ is the exponentiation symbol, and a(n) is a sequence of natural numbers, a.k.a elements of the set N. For clarification, I am using the modern definition of N, which means N includes 0. Therefore, n |-> a(n) : N —> N; in other words, it is a map from N to N. Now, consider all of the different a(n) that satisfy the property that there exists some m = 0 or m > 0 such that if n = m or n > m, then a(n) = 0. Call the set of all such a(n) the set S for simplicity.
With the notation above established, I can now explain the fundamental theorem of arithmetic differently. The fundamental theorem of arithmetic states two things. The first thing it states is that if a(n) is an element of S, then the product above is an element of N\{0}, a.k.a it is a positive integer. The second thing it states, which is the important part and which is the statement that was discussed in the video, is that there exists a bijection between N\{0} and S, a.k.a the sequence of exponents in the product is unique to some particular positive integer, implying no other positive integer has that same sequence, and no positive integer has any other sequence. For example, if the product is equal to 10, then a(n) = (1, 0, 1, 0, 0, 0, ...), and viceversa. What is the unique product/factorization that corresponds to the mth prime number, then? It is the one product such that a(n) = δ(n, m), where δ(n, m) is the Kronecker delta function. Then, to answer the question of how to express 1, a positive integer, as a unique product of powers of prime numbers, simply let a(n) = 0 for all n, and this gives the unique factorization for 1. This factorization is what he called "the empty product" in the video.
What does any of this have to do with 1 being a prime number? Everything. If we were to use the product 1^a(0)*2^a(1)*3^a(3)*•••, then factorization would no longer be unique, since for any positive integer, a(0) is allowed to be any natural number, giving infinitely many factorizations, and leaving no positive integer such that its corresponding sequence is a(n) = 0, which would imply that there is no multiplicative identity, and thus, no additive identity either, which is obviously false, since 1 is the multiplicative identity, and 0 the additive identity. Why does it imply that? Because x^0 = 1 for all x in the integers, (yes, including 0, in most modern definitions and theorems. Sorry, but your professors taught you wrong if they said this is undefined.), and because ln(1) = 0, so in the first product, taking the natural logarithm would give ln(1) = 0*ln(2) + 0*ln(3) + 0*ln(5) + •••, which only makes sense when you look at this as a vector space. Also, the concept of the empty product is necessary, since it exists in every other field of mathematics, it's not something we invented to deal with factorization with primes: it pre-exists the entire concept of primes as a whole, in fact. The empty product is a fundamental concept in algebra, calculus, combinatorics, probability, set theory, etc. If you have heard that x^0 = 1, then this is exactly what the empty product means: that having 0 factors of the form x is equal to 1, or more directlt, that multiplying x 0 times exactly gives 1. There is an analogous with the empty sum, 0, which is the additive identity.
So the fact that 1 being a prime breaks both with unique factorization and with the notion of empty products means that 1 being a prime is inconsistent with the rest of mathematics. Therefore, 1 cannot be a prime.
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The construction of the Cartesian product would be far cleaner if we used the standard definition of ordered pairs, which is itself also far cleaner anyway. The Kuratowaki definition of the ordered pair, which is the best definition we have, is that (x, y) := {{x}, {x, y}}. With this definition, we can claim that x is an element of A, and y is an element of B, so both x, y are element of the union of A and B. The sets {x, y} and {x} are subsets of this union, so they are elements of the power set of the union. Now, the set {{x}, {x, y}} is a subset of the power set of the union of A and B, so {{x}, {x, y}}is an element of the power set of the power set of the union of A and B. Since this is true for all x in A and y in B, it follows that every set of sets of the form {{x}, {x, y}} is a subset of the power set of the power set of the union of A and B, and therefore, the Cartesian product of A and B is an element of the power set of the power set of the power set of the union of A and B. This also avoids the issue with the tags.
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This video is somewhat misleading. Saying 1 + 2 + 4 + 16 + ••• = -1 is perfectly valid and be made by rigorous proof. Summation does not have to obey our intuition or what we dictate is common sense: most of math does not obey intuition, and common sense is very often wrong, as both math and science have demonstrated throughout history. Whether a statement about infinite summation is true or not depends on the axioms of the theory, and there is no need to change the metric of field of numbers with which we work. For starters, the limit of the sequence of the partial sums of an ordered sequence is fundamentally distinct and unrelated to the ordered summation of all the elements of a given set. Deciding to create an equivalence class between the two is an arbitrary choice of axioms, but there is no necessity to accept this choice or exclude any other possible choice. If we can consistently and rigorously assign a value to a series, then what the limit is and whether it diverges or not is irrelevant. To illustrate the point, it can be true that x —> 0 implies f(x) —> 0, but that f(0) = 1 instead, as with a step function, or any discontinuous function. Whatever the meaning is of infinite summation, we decide it. Let Σ(from n = a to n = b, f(n) ) = Σ(from n = a + c to n = b + c, f(n - c) ) hold true for all ordinal numbers a, b, and c. Then it is trivial to demonstrate that if 1 + 2 + 4 + ••• has any arithmetic simplification, then it must simplify to the value -1. There are no constraints on whether it can have a value or not. The convergence theorem says that if the limit of the sequence of partial sums of the series converges, then the series equals this limit, but provided that the limit diverges, this tells us nothing about whether the series can have value or not, or what the value is. We can consistently find methods to evaluate the series, and these methods can be and most often are compatible with the convergence theorem, since for convergent series, the methods return the same value those series have by the convergence theorem, but for divergent series, they still return a value, much in the same fashion that the Gamma function is a domain extension of the factorial function, but it still is compatible with the factorial function whenever the factorial function is defined. In fact, one could state that both functions are equal up to a shift. Therefore, this is not even a question about redefining anything.
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@SequinBrain I'll say this, and since it's the second time I've said it, I'm repeating myself: the subject is "value." Banks have value or the account doesn't exist.
I agree, but this supports my point, not yours, and you are the one who brought banks, not me. Also, this is also me repeating myself.
The subject is value, not "things." Or if it is things, it's things that have value, and like 0, things that don't. zero is the absence of value. zero destroys everything it touches like zero oxygen destroys humans.
Well, no, that is just factually incorrect. 2^0 = 1. 0! = 1. cos(0) = 1. ζ(0) = –1/2. Saying "zero has no value" and "it destroys everything" is about the most childish mathematical viewpoint I know of.
zero multiplied by anything destroys its value and gives it none. The inconsistency is introducing a new subject when we haven't finished the first yet.
You were the one who introduced the subject of banks, not me. This is your mistake, not mine.
This is the last I have to say since I'm repeating what's already been said and apparently not comprehended yet.
No, I comprehend it just fine. I am not confident you comprehend my response, and the fact that you say this strengthens my doubt.
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@suntzu7727 How is that not trivial?
I have already explained how it is not trivial. I am not going to repeat myself.
That's like saying that being alive surviving circumstances that you couldn't survive would be unthinkable. It would.
Did you not read my explanation? I did indeed say this, and I did say it would be unthinkable, and if this unthinkable event happened, that would be impressive. Now that would need an explanation, and it would be a good argument that some supernatural force exists.
As I said, that says nothing about whether those conditions were likely, could plausibly be attributed to chance or physical necessity, which is the issue.
No, that is NOT the issue. This is precisely the crux of the explanation I have given you multiple times now, but you are unable to understand. You do NOT need to explain whether those conditions are likely, unlikely, or whether they are a consequence of physical necessity. There is no such a logical requirement. The idea that this needs explanation at all is the fallacy to begin with. Why? Because it is unimpressive. As I said, if this were a defiance of logic, such a defiance would require an explanation. Something that so thoroughly obeys logic to the extent of being unimpressive demands no explanation, and asserting that there is any justification in thinking that the basis of any argument could be formed from said lack of explanation is wrong.
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@suntzu7727 Listen, the host commenter wrote: if it [the universe] was slightly different, we [intelligent life forms] could have still been around but in a form that is unrecognizable to us in this universe, and still make the same dumb argument THAT universe was created that was specifically for them to exist that way. What does this comment do or say? It rhetorically thinks of a hypothetical scenario. Which hypothetical scenario? The scenario that in a slightly different universe, unrecognizable intelligent life forms exist. What does the comment say these life forms would do? Claim that the universe was specifically created for them to exist. Okay, and what was your response? How did you represent this comment on your arguments? You represented as if things weren't what they were they wouldn't be what they are. NOWHERE in the host comment is anything of this effect being said. The host comment talks about a hypothetical scenario about life forms different from us, and makes a claim about what those life forms could do, and the fact that those life forms could do the thing it claims they could do, very much makes the idea that the universe is fine-tuned questionable. In no way is this equivalent, or even close, to the assertion that "if not A, then not A," which is what you are trying to present it as. The fact that you so blatantly insist on trying to continue misrepresenting the argument in this fashion is outrageous and baffling. The fact that you also fail to understand that proving that "the assumption that 'the universe is finely-tuned for us' is false" constitutes a valid objection to the fine tuning argument is even more baffling, seeing that even children can understand this concept. Most baffling of all, though, is the snarky attitude accompanied by the load of logical fallacies you keep throwing in response to the explanation of these rather elementary ideas because you simply cannot accept that your petty pet peeves against the anthropic principle, or any strictly superior variants of it, are unfounded.
I have no idea of what else to tell you, but I have already done more than enough to explain this, and I have no intention of continuing wasting my time if you are just going to be in denial about it. I am sorry I am not able to help you get out of a dark place in your mind that allows you to not use cognition and judgement properly in this particular instance of discussion. Good bye.
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Theorem: No natural number n > 0 such n = 3 mod 4 can be written as a^2 + b^2, where a and b are integers.
Proof:
If a = 0 mod 2, then a = 2k for some integer k, implying a^2 = (2k)^2 = 4k^2 = 2(2k^2) = 0 mod 2 = 0 mod 4.
If a = 1 mod 2, then a = 2k + 1, implying a^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 1 mod 4.
Since a and b are arbitrary, this implies that if b = 0 mod 2, then b^2 = 0 mod 4, and if b = 1 mod 2, then b^2 = 1 mod 4.
Therefore, if a = 0 mod 2 and b = 0 mod 2, then a^2 + b^2 = 0 mod 4; and if a = 0 mod 2 and b = 1 mod 2, or vice versa, then a^2 + b^2 = 1 mod 4; and if a = 1 mod 2 and b = 1 mod 2, then a^2 + b^2 = 2 mod 4.
Since every case is covered above, no case can result in a^2 + b^2 = 3 mod 4. Q. E. D.
Note: I know the equality sign is abuse of notation when talking about modulo classes. I lack a mathematical keyboard, so you will have to simply accept the abuse of notation for now.
What this all implies is that prime integers 3 mod 4 are Gaussian primes.
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Infinite is a vague and general statement without reference points.
No, it is not. Infinity is a precisely well-defined property of sets, based on the axiom of infinity. A set S is an infinite set if and only if there exists an injective function f : N —> S, where N is the set of natural numbers.
The numbers between 1 - 2 are infinite,...
Which kind of numbers are we talking about? Because, for example, there are infinitely many real numbers r satisfying 1 < r < 2, but there are no natural numbers r that satisfy it. I will assume that you are talking about real numbers here, though, so in this case, yes, you are correct. The interval (1, 2) is an infinite set.
...but not as infinite as the numbers between 1-3, as it contains both the subsets 1-2 and 2-3.
No, this is incorrect. (1, 2) and (1, 3) are sets of the same cardinality, because there exists a bijection from (1, 2) to (1, 3). If you want a specific construction of the bijection, then consider, f0 : (1, 2) —> (0, 1), (f0)(x) = x + (–1); f1 : (0, 1) —> (0, 2), (f1)(x) = 2·x; f2 : (0, 2) —> (1, 3), (f2)(x) = x + 1. Now consider g = (f2)°(f1)°(f0) : (1, 2) —> (1, 3), such that g(x) = (f2){(f1)[(f0)(x)]} = (f1)[(f0)(x)] + 1 = 2·[(f0)(x)] + 1 = 2·[x + (– 1)] + 1 = 2·x + 2·(–1) + 1 = 2·x + (–1). It can be proven quite easily that g is a bijection, and so (1, 2) and (1, 3) have the same infinite cardinality. Yes, it is true that (1, 2) is a proper subset of (1, 3), but this only implies that they have different order type, not different cardinality. In fact, the set difference (1, 3)\(1, 2) of (1, 3) and (1, 2) is [2, 3). (1, 2) and [2, 3) have cardinality Beth(1), while the set of natural numbers has cardinality Aleph(0) = Beth(0). As it happens, the union of (1, 2) and [2, 3) is (1, 3), and that all three sets have the same cardinality reflects the fact that Beth(1) + Beth(1) = Beth(1). For the record, Aleph(0) < Beth(1).
I think this is why infinities are argued against so vehemently.
The reason infinity is argued against so vehemently is because our understanding of infinity via set theory is extremely recent for humanity, and transfinite set theory is so counterintuitive, that some people just reject it. For example, as I already clarified, if we have a set X that is a proper subset of Y, it is still possible for X and Y to have the same cardinality, because it is possible that there exists some f : X —> Y that is a bijection. This is counterintuitive, because intuition tells us that if X is a proper subset of Y, then Y should have a larger cardinality than X, and so such an f should not exist, yet we can prove that such an f can exist. Again, what this reveals is that the subset relationship only gives information about the order type of a set, not the cardinality of a set. For finite sets, order type and cardinality just so happen to align and be equivalent, which is why our intuition fails when this equivalence no longer holds if infinite sets are introduced. This is why the movement called finitism, the rejection of the axiom of infinity, has become so popular. Together with finitism, there is also the intuitionist movement and the constructionist movement.
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@stratonikisporcia8630 Wouldn't classify it as "abuse" unless it's really frequent and/or very violent.
Any amount of violence is abuse, by definition.
Sometimes, you just have no choice,...
If you honestly believe this is the case, then this precisely proves you are not ready to be a parent. I say this with as much as respect as it is possible to have. This notion that "but there is no other choice!" is a delusion: it is nothing more than a dogma not grounded in reality. There are many parents who do just fine without ever resorting to spanking, which categorically disproves the hypothesis you are presenting here.
...because your role as a parent is to punish your kids when they do bad things,...
No, this is most definitely not the role of a parent at all. The role of a parent is to care for the children, while ensuring they progressively learn essential life skills, so that they can eventually survive in the world on their own, being a functional and safe member of society, and achieve their happiness and well-being. In light of all this, punishing a child is unreasonable, because it is not conducive to any of those goals in the vast majority of circumstances, and on top of this, it is scientifically proven to be harmful.
...and the only way to punish someone, unless they understand their mistake and willingly[,] which is like 1% of people, likely lower when they're f-cking kids who barely understand language,...
Stop right there. Right here, you have already exposed your own biases and inherent disdain for children, treating them as objectively inferior and less "intelligent" than adults. There is no data to back up your claims here. I have worked in education before, and the vast majority of young children I have worked with are actually quite reasonable, and often, more reasonable than adults, and when I talk to them, it is not difficult for me to get have an "intelligent" conversation with them and get them to learn from their mistakes. If having a conversation is not effective, then there are hundreds of other methods we can apply to ensure the child stops misbehaving. Hitting them is at the very bottom of the list.
...is by using "physical constraint" which may in certain cases be assimilated to violence.
No, absolutely not. Physiological restraint is fundamentally different from violence in every conceivable way, and this is one of the early things you learn when being trained to perform such acts.
However[,] this is not abuse, as the parents' role being to make their children follow basic rules...
No, it is not. Anyone who genuinely believes this should never be a caretaker of children. Also, allow me to point out a mistake in your reasoning: you cannot logically deduce the claim "this is not abuse" from the premise "parents' role being to make their children follow basic rules." It being abuse has nothing to do with what the role of a parent is at all. The reason hitting children is abuse is for the exact same reason hitting an adult is assault.
...just for the sake of society not collapsing,...
Children breaking a few rules occasionally will not lead to a collapse of society. Do you know what can and does cause collapses of society? Prevalence of crippling mental health disorders, which are causally linked to spanking. Also, higher crime rates, which are also causally linked to spanking.
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@erlandochoa8278 That is not even true. In fact, up until the 1700s, mathematicians did not consider 1 to be a prime number. It was only towards the late 1700s and 1800s that the idea that 1 should be considered a prime number became popular, but it was discarded in the 20th century after the advent of mathematical rigor. Even so, never in history did there exist a mathematical consensus that 1 was a prime number.
Also, you talk about "the actual definition" as if there was only one definition for what a prime number is. This is not the case. In mathematics, things have multiple equivalent unique characterizations, and so can be defined differently but equivalently. For example, I can define exp(x) = lim (1 + 1/n)^n (n —> ♾). I can also define it instead as lim 1/0! + ••• + 1/n! (n —> ♾). I can also define it instead as the unique continuous function satisfying exp(x + y) = exp(x)·exp(y) for all x and y such that exp(0) = 1. These definitions are all different statements, but they can be and have been proven to be equivalent. So it does not actually matter which of the definitions you choose. The same happens with prime numbers. Prime number have multiple definitions, but they are all equivalent. One such definition is the set of all natural numbers p such that d(p) = 2. Another definition is p divides a product iff p divides at least one of its factors. 1 does not satisfy this definition, and it does not satisfy the other one either. In fact, it does not satisfy any of the definitions, because, well, it is not a prime number. It is not a matter of excluding 1 for petty reasons. It just literally does not have any of the defining properties of a prime number. Saying that 1 "was excluded" from being a prime number is akin to saying that we excluded the negative numbers from being square numbers.
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I think there is a better proof for the claim that lim 1/x (x —> c) = 1/c for all 0 < c. If 0 < δ & 0 < x & -δ < x – c < δ, then c – δ < x < c + δ. Hence, if you can find δ such that δ < c, then 0 < c – δ, and 1/x < 1/(c – δ, thus 1/(c·x) = 1/|c·x| < 1/[c·(c – δ)], and |x – c|/|c·x| = |1/x – 1/c| < |x – c|/[c·(c – δ)] < δ/[c·(c – δ)]. Therefore, |1/x – 1/c| < δ/[c·(c – δ)]. To prove |1/x – 1/c| < ε, let ε = δ/[c·(c – δ)], equivalent to δ = c^2·ε/(1 + c·ε). Thus, all that remains to be proven is that c^2·ε/(1 + c·ε) < c, in accordance to δ < c, and the proof is complete. c^2·ε/(1 + c·ε) < c is equivalent to c·ε/(1 + c·ε) < 1, equivalent to c·ε < 1 + cε, equivalent to 0 < 1, which is axiomatic. Q. E. D.
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@Korialx "Writers deserve to be underpaid because the most recently produced shows on streaming platforms are among the worst."
This is objectively one of the most pathetic arguments to make, and if you unironically believe this makes sense, then not only are you unintelligent, you are also a disgusting human being. You should be ashamed of yourself. Let me explain why:
(A) Writers have very little control over the overall production of the final show/film itself. Any of the writing that they do has to abide by the specific handpicked standards of the directors and producer, and even if the writers do just that, the directors have the final say on what the final product will be. As such, the writers have very little to do with whether the product is "good" or not.
(B) Whether the product is "good" or not is completely irrelevant to the discussion, because it bears no weight on whether the writers should get paid a livable wage or not. I swear, y'all only support the writers getting paid at all because we live in a time-period where not getting paid breaks the law. If it were legal, y'all would totally be okay with everyone being coerced to do work for free.
(C) I seriously doubt you have the qualifications to provide an objective critique of whether shows and films in streaming services are "good" or not. The most you have is a personal preference, but guess what? You are only 1 person out of 8 billion in the world. No one gives a flying pepper if the recent shows and films appearing in the streaming platforms that you use match your personal taste or not. Have you considered switching streaming platforms? Maybe you will find something you like. Have you used the advanced search options these services provide? No, of course you have not. As for me? I do not watch Western shows or films, for the most part, because I do not enjoy them. But guess what? My personal taste is irrelevant too, and despite my inability to appreciate Western media, I still support the notion that writers who work in Western media should get paid an appropriate amount, because I am not such a self-centered idiot who thinks that writers should get paid solely on the basis of how well they can satisfy me.
(D) Even if the previous points I made are somehow incorrect (and they are not, but I am speaking hypothetically), your argument is still nonsense. Your complaint is that writers get paid what they get paid because of the shows they make. You think underpaying them is somehow going to get them to do a better job? This is ridiculous. If I paid you $7.25 an hour to babysit, would you actually do a good job at it? Or would you basically let the children do literally whatever they want, and only do the bare minimum, the bare minimum being making sure the children do not die? I woukd certainly do the latter. You know why? Because that is a stupidly low wage for a babysitter. It is completely disrespectful. If you want your babysitter to do a good job, then you pay them appropriately. This idea that writers should somehow do a better job despite getting paid so little is so unbelievably stupid, that the only way I can personally conceive of anyone making this argument is if they have some cognitive disability.
(E) No, if your work flopped badly, you would not get fired in a heartbeat by the company, not if the people at the company actually want to preserve their reputation and still be able to make a profit in the future. Companies that do that go bankrupt. If your work flops, you will be reprimanded, you will be moved departments, or you will be made to work under special supervision. You may even get demoted, but no one is going to fire unless you do something exceptionally atrocious, which obviously, most workers do not do, and writers are no exception to this.
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@anghusmorgenholz1060 My previous comment was deleted, so I should say this again: I at no point presented 0 as being infinity. You did. You have poor reading comprehension skills. Your original comment asked for a definitive infinite numeral to be talked about in this discussion. I replied by saying that the video is using the numeral Aleph(0). For some reason, you are now insisting that I am instead talking about the numeral 0. I am not. The numeral Aleph(0) is a different numeral than the numeral 0, and I only talked about the former, not the latter, so you need to stop being a lying piece of garbage, and stop pretending that I ever talked about the latter as being representative of infinity.
Using the infinity in mathematical terms to prove a religious or philosophical claim without using actual math doesn't work.
Once again, you display your idiocy, and your lack of reading skills. I never presented any religious or philosophical claims here. I only answered your comment, by explaining that the video did use a definite infinite numeral in its discussion, like you wanted them to. As for using actual maths, the video also did that. It is called set theory.
Anyway, you have exposed yourself as a t-r-o-l-l. I will no longer be interacting with you. I will no longer be reading your comments.
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Actually, one could be far more careful here. dS/dr = k·S implies S = 0, or 1/S·dS/dr = k. Consider f(S) = ln(–S) + A iff S < 0 and f(S) = ln(S) + B iff S > 0. f'(S) = 1/S. Hence 1/S·dS/dr = d[f(S)]/dr = k, which implies ln(–S) = k·r – A iff S < 0, and ln(S) = k·r – B iff S > 0. With this, we have S = 0, or S = –exp(–A)·exp(k·r) iff S < 0, or S = exp(–B)·exp(k·r) iff S > 0. S = 0 = 0·exp(k·r) = λ·exp(k·r) iff λ = 0, while S = –exp(–A)·exp(k·r) = λ·exp(k·r) iff λ < 0, and S = exp(–B)·exp(k·r) = λ·exp(k·r) iff λ > 0. Therefore, S = λ·exp(k·r).
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@EskChan19 It's not quite a code in the general meaning, but it's similar to a code how it's used in computer science.
No, it is not, and this much is common knowledge among geneticists and computer scientists alike. You can even search explanations on Google for why they are only superficially similar at best, if at all.
ASCII for example is a good analogy, in that this really isn't language, it's just more a codec, a way to store and parse data.
No, it is not a good analogy. DNA does not store data anymore than any arbitrary molecule does (yes, all molecules store data, this is a basic fact of thermodynamics), and it certainly is incapable of parsing data, which requires a parsing algorithm. The fact that DNA does not store data any more than any other molecule already disproves your thesis.
And it works similar to an extent, in that certain patterns of base pair 'codes' yield certain results, and the same pattern would theoretically result in the same pattern for people.
No, it would not. Again, every geneticist knows. The reason the discipline of epigenetics exists is precisely because genes do not work this way. Gene expression has an effect on whether a given string of base pairs will give a codon for a protein strand or not. Also, there are a number of other factors that can alter the result that have nothing to do with that particular string itself. Then, there is also the issue of thermodynamic entropy changing the interactions, something that does not exist with actual code.
And just like an ASCII, if it corrupts, it can change all of it.
And unlike an ASCII pattern, a "corruption" can result in no changes at all.
DNA is not a code. It is a molecule.
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@lurch666 If you have infinite time then your past is infinite.
Yes. This is a tautology, so there is nothing to discuss here. Proceed.
So say you wait 10 billion years to do something. Since your time has no start (because you have an infinite past) there's no time to start counting your 10 billion years.
No. Having an infinite past does not imply your past did not have a start. If my time coordinate is t = ω, then because ω is infinite, my past is infinite. However, my past still did have a start, because t = 0 is the minimum time coordinate. Time starts, then ω years pass, then I start counting, until I reach ω + 10 000 000 000 000 years. So I have now counted 10 000 000 000 000 years, which is a finite amount of time, but I still have an infinite past with a beginning.
Because you have an infinite past, you can't start counting from the start like with finite time so like I said any finite amount of time is swallowed by your infinite past.
Why exactly am I obliged to count from the beginning? Just because I cannot count from the beginning, it does not mean there was no beginning. You may just be confused as to what exactly "counting" entails.
The problem with infinity is that it makes no sense.
Wrong. Infinity makes perfect sense. Zermelo-Fraenkel set theory has an axiom dedicated just to infinite sets. von Neumann-Bernays-Gödel set theory goes further by introducing the axiom of global choice, and by being able to quantify over proper classes, rather than just sets. Really, it should just be called von Neumann-Bernays-Gödel class theory. Anyway, infinity is well-studied and well-understood by mathematicians, and it works just fine. It even has applications in physics.
But if time is infinite that now needs an infinite amount of time to pass before it can happen so we never reach it.
You keep insisting that if something takes an infinite amount of time to be reached, then it cannot be reached, but you have not actually explained why this is the case. To me, this just sounds like an unnecessarily elaborate way of saying "infinite sets cannot exist".
Infinity causes paradoxes...
No, it does not. Before the 19th century, when set theory was developed, people had a really poor understanding of the concept of infinity. As such, it did cause paradoxes, though the paradoxes actually originated from wrong intuition, rather than from infinity itself. However, we live in the 21st century. There exists an entire super-discipline of mathematics called transfinite mathematics dedicated to studying infinity. We understand infinity so well today, that it is actually shocking in retrospect how bad we used to be at handling infinity as a concept in the past.
Obviously we have reached now. But if time stretches back into infinity then an infinite amount of time has to pass to get to now.
Yes, an infinite amount of time has to pass to get to now, if the past is infinite. What is the issue? Are you implying that an infinite amount of time cannot pass, so therefore, an infinite past cannot exist? This just sounds like you are saying infinite time cannot exist, which is the conclusion you are trying to prove in the first place. Why is it that infinite time cannot pass?
Because infinite time doesn't get us to now-it gets us to the future.
If it gets us to the future, then it necessarily gets us to now, because now precedes the future. So your claim is wrong. Also, infinite time does not necessarily get us to the future. So your claim is also wrong. If my time coordinate is ω, and I have a time machine, and I set my time coordinate to 0, and then I let my time run for ω units, I am back when I started, and this true, in spite of the fact that ω is infinite. No future required.
Imagine a sea with infinite depth under a sky with infinite height. The surface of the sea is now, under the sea is the past, so the deeper you go the further into the past you go and the sky is the future, the higher you go the further into the future you go.
Yes. Your scenario is reasonable so far.
Now how long of a stick would you need that could stand on the bottom of that sea so the tip of the stick just reaches the surface?
An infinite stick. Specifically, if the sea has depth ω units, then my stick needs to be ω units long.
Any stick of a finite length wouldn't reach the surface because the bottom is an infinite distance away.
Correct. The bottom is an infinite distance away. Please remember this for the next sentence.
Any stick of infinite length would reach the surface but because it's infinite, it would keep going into the sky.
No. In order for it to keep going into the sky, the stick would have to have a length larger than the length from the surface to bottom of the ocean. If both lengths are infinite, then it is not warranted for the stick to have such a larger length, and so it is not warranted that the stick would continue into the sky. If the ocean depth is ω meters, and my stick measures ω meters, then my stick will just reach the surface, and not go into the sky, but the ocean will still have infinite depth, and my stick will still have infinite length.
So there is no length of stick that can just reach the surface (now).
I have just proven otherwise. Such a length of stick does exist.
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@homotheticwren So instead of saying that if infinite time precedes an event, you will never reach the event, it might be helpful to think about infinite in a more numerical sense.
Yes, please. Are working with surreal numbers, hyperreal numbers, ordinal numbers, or just cardinal numbers?
take the set of natural numbers, for example; this is, I think, is the kind of thing you're imagining, where you could theoretically count to an arbitrarily large element in the set.
Yes, although I am not sure how counting to an arbitrarily large finite number is relevant here.
However, what about the set of real numbers? If you tried to count the set of real numbers, or even just the real numbers between 0 and 1, you would never reach even an arbitrarily low number, because where do you even start?
The set of real numbers is known as a dense set. What this means is that, for any two real numbers x and y, there exists many real numbers r such that x < r < y. Density is a property that you can have even with a countable set, such as the rational numbers. 0 is a rational number, but there is no "next" rational number, because for any rational number 0 < q, there are rational numbers r with 0 < r < q. However, the set of rational numbers is a countable set, because there exists a bijection f from the set of natural numbers N to the set of rational numbers Q. This means that every rational number can be written in a list, even though there is no "next" rational number. However, the interval of real numbers (0, 1) is not countable, because there is no bijection from N to (0, 1), so it is not possible to write the real numbers of the interval in a list. What this demonstrates is that density and uncountability are different properties that are not equivalent. I think you may be confusing one for the other.
Also, while it is true that the cardinality of (0, 1) is larger than the cardinality of N, this does not prove that an infinite amount of time cannot pass to reach a point in time.
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@lurch666 But infinity + infinity is still infinity.
Yes, if α and β are infinite numbers, then α + β is also infinite. How is this exactly relevant to what I said?
When you say the depth of the sea and the stick are the same length, then of course the stick would just reach the surface, but since infinity (depth) + infinity (sky) is still infinity then the stick would also reach the sky because the stick has a length of infinity.
No, that is not how that works. I wanted to avoid correcting you explicitly with your previous paragraph, but I have no choice now: "infinity" is not a number. Infinity is a property of sets, and therefore, of numbers, and other mathematical constructs. There are finite numbers, and there are infinite numbers, but infinity is not itself a number, infinity is simply the property of a number being infinite. When you talk about the ocean and sky having some kind of infinite depth or height, you need to specify what kind of infinite number you are working with. This is why in my arguments, I specified a size ω. ω is an infinite number, but it is a different infinite number than ε(0) or ω(1). More importantly, ω is different from ω + 1, and different from ω + ω. What you are doing here is giving the ocean a depth α (which is infinite), the sky a height β (which is infinite, and which may or may not be equal to α), and the say that the stick must have length α + β, which is not true: there is no logical necessity for the stick to have the sum of the lengths. Of course, if it does have length α + β, then of course it will reach into the sky forever and ever. But this is cheating: you are conflating various different quantities and treating them as being the same quantity solely because they all share the same property of being infinite. That is simply not how the infinite works.
That's why infinity doesn't make sense
No, that is why your incorrect understanding of infinity does not make sense. Treating infinity as a number does not make sense. Treating infinity as a property of a class of numbers, however, does make sense.
because when you think about infinity you get contradicting results and math falls apart.
Oh, yeah? Tell that do the the hundreds of thousands of mathematicians studying transfinite set theory, right now. Tell them that if they continue developing transfinite set theory like they have been doing for 150 years, their mathematics are going to fall apart. Please do that, then come back to me and tell me, how the mathematicians took that. Good luck.
Got to admit I don't know infinity from any deep learning this is just what I have figures out myself so I've make some big errors that's down to this being a difficult subject
I would very much argue set theory is not a difficult subject. Only the theorems that are counterintuitive are the difficult ones.
but an infinite past can't have a start.
It absolutely can. I just demonstrated how it can. There exists a smallest ordinal number: the empty set, also called 0. However, ω is an infinite ordinal (the smallest infinite ordinal). So if my time coordinate is ω, then I have an infinite past (because ω is infinite), but said past has a beginning, because 0 is the smallest time coordinate. Are you going to tell me that mathematicians are wrong on this?
but if you have existed for an infinite time then you would never reach the point where you can start because it would take an infinite amount of time to get there.
No, you are wrong. You saying "an infinite amount of time must pass to reach point A" does not demonstrate that "point A cannot be reached." You need to prove it. You keep insisting that it is true, but it is not.
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@homotheticwren I genuinely don't mean to be rude,...
Ah, yes, the classic "I don't mean to be X, but..." only to follow up with a remark that fits exactly the defining characteristic of X, except this time, you wrote said remark before saying. I am sorry, but I cannot take this seriously when say this immediately after saying "it's both utterly irrelevant and unnecessary". It would have been better if you had simply called it unnecessary, irrelevant, and then left it that. The feigned humility in your comment is devoid of meaning, and leaves me unimpressed.
...I don't think what you said has any bearing on the purpose of my comment.
If so, then that is your own failure, not mine. Your comment, by your own admission in your first paragraph, was an attempt to help illustrate Lurch's claim that infinity cannot be traversed or reached. My reply was a direct sentence-by-sentence response explaining why your comment does not help illustrate the point you intend to illustrate, by mentioning the fact that the phenomenon you appealed to has little to do with infinity itself, and more to do with the density property of some sets. If my response somehow has no bearing on the purpose of your comment, then neither does your comment itself. Though, I would also argue, as I indicated at the end of my reply, that much of what you said in your comment had little relevance to Lurch's claim anyway. What you and Lurch argued were effectively two completely separate topics.
Anyhow, I will not be replying to you any longer. This response of yours has made it evident than trying to correct you on your own mistakes is a waste of time, and is even unappreciated, given the rudeness of your comment. Very well, then. Stay wrong, for as long as you want. I will no longer bother you. Farewell.
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@SlimThrull I honestly question the sincerity of your "skepticism". Having questions is fair, but the Internet is a place that exists, and very in-depth discussions on the subject involving various complex topics in sociology, anthropology, physiology, and neuropsychology can be found if you just put 5 minutes of effort into it. Given this, demanding for an answer on YT of all places is suspicious. Also, the fact that you have to assume that people will bring up an objection like "oh, and before you assume that doesn't happen, Google this" as opposed to any other objection over is disingenuous. I do not think you are doing this maliciously, but with an attitude like this, you will just never get a satisfying answer, especially because I doubt many people would want to engage when you present those red flags in the first place. Saying you are open minded does not make it so.
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It seems counterintuitive that lim a(n)/b(n) (n —> ∞) = 1, but lim a(n) – b(n) (n —> ∞) does not exist, can be possible. The issue is, lim a(n) – b(n) (n —> ∞), if it exists, must be equal to lim b(n)·(a(n)/b(n) – 1) (n —> ∞). Since we know lim a(n)/b(n) (n —> ∞) = 1, it is immediately seen that lim a(n)/b(n) – 1 (n —> ∞) = 0. However, this does not guarantee that lim b(n)·(a(n)/b(n) – 1) (n —> ∞) exists, because lim b(n) (n —> ∞) clearly does not exist in this case. We can say that lim 1/b(n) (n —> ∞) = 0, so we can ask about lim (a(n)/b(n) – 1)/(1/b(n)) (n —> ∞), where both the denominator and numerator converge to 0. This is evidently not guaranteed to exist, so there is no reason for lim a(n) – b(n) (n —> ∞) to exist either.
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@forbidden-cyrillic-handle You made the initial claim about the consensus, not me. You made a claim, and you failed to provide evidence for it. I simply replied to the claim by pointing out that it is false.
At any rate, you should be checking out sources like "An Introduction to the Theory of Numbers" by G. H. Hardy, "The Development of Prime Number Theory: From Euclid to Hardy and Littlewood" by Władysław Narkiewicz, Van der Waerden's "Moderne Algebra," and H. M. Edward's "Fermat's Last Theorem" all cover the history and the mathematical theory very extensively.
Well, there you have it. I provided my sources, but I know you are too ignorant and too dishonest to actually provide any of your own. I mean, if you were actually right, then you would have provided your sources already. Instead, you are being disingenuous. Couple with your complete lack of understanding of the mathematical concepts, this tells me that continuing to have a conversation with you is just a waste of time. You are just troll, like many others on the Internet are. So, this will be my last response to you, and I will not be reading anything you have to say, because I have decided to not reduce myself to your level. Farewell.
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Of course, it all changes if you have the sum of three squares instead. This is something you can do with a 3D grid space as opposed to a 2D one. The number of unordered triples possible is 4: (even, even, even), (odd, even, even), (odd, odd, even), (odd, odd, odd). These give 0 (mod 4), 1 (mod 4), 2 (mod 4), and 3 (mod 4) respectively, and now you can decompose many more natural numbers into sums of three squares - although still not all of them, mind you.
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@DenniWintyr It is all semantics, but ultimately, the property of "being divisible by only 1 and itself" does not exist in mathematics. All integers m are divisible by m, –m, –1, 1. If you look at the actual history of the definition of prime numbers, though, we were only ever concerned with proper divisors. m properly divides n if m divides n and n does not divide m. As such, m and –m are always divisors of m, but are never proper divisors of m. m is called a prime number if its only proper divisors are –1 and 1. 1 and –1 have no proper divisors at all, so they cannot be prime numbers.
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Strictly speaking, all integers, except –1 and 1, have at least 4 divisors. This is because for all integers m, it happens that –m, m, –1, and 1, all divide m. In the case of 0, every integer divides 0, and –1 and 1 divide every integer. 1 and –1 have exactly 2 divisors: –1 and 1 themselves.
However, the actual definition of a prime number (which is also the definition that was always historically used, it was just not properly understood due to philosophical debates on whether –1 and 1 were actually numbers) is that it is an integer whose proper divisors are –1, 1, and no other integers. Under this definition, –1 and 1 have no proper divisors, and so they are not prime numbers. The divisors of –5, for example, are exactly –5, –1, 1, 5, but while –5 and 5 are divisors of 5, they are not proper divisors of 5, because 5 divides those divisors. –1 and 1 are proper divisors of 5, (because 5 does not divide them), and so are the only proper divisors of 5. Therefore, –5 and 5 are prime numbers. Incidentally, 0 is not a prime number, because every nonzero integer is a proper divisor of 0, since 0 divides no integers other than itself. So, indeed, the prime numbers are the set {..., –7, –5, –3, –2, 2, 3, 5, 7, ...}. However, in number theory, we are often only concerned with positive integers, since symmetry just makes the theorems equally applicable to negative integers anyway.
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John David There is evidence of a Creator.
There is absolutely, legitimately no evidence for the existence of a Creator whatsoever.
It's like math; it isn't seen or heard but known.
This is a terrible analogy, because (1) you can see manifestations of the consequences of the mathematics in the physical world. This is why there is such a tightly-knit connection between physics and mathematics. (2) You can prove a statement in mathematics by using some system of syntactic deduction and a set of initial axioms. There is no formal or logical equivalent for proving the existence of deities. You cannot prove the existence of a creator deity. There is no set of well-founded, indisputably proven truths from which you can validly conclude that a creator deity exists. None. 0. Nada. Nilch. Rien. If you want to believe that there exists such a creator in spite of the fact that no there is no valid argument with which you can conclude that such a creator exists, i.e, belief without justification, then you are welcome to do so, and I will not try to stop you from doing so, this is your choice, and you are free to do so. However, if you are going to claim that such evidence or proof does exist, then you are responsible for providing that evidence to us and subjecting your arguments to scrutiny. That is how it is.
It's not something I have the patience to teach.
You don't have to teach me anything. I have a degree in philsophy, and I studied epistemology and theology. I have another degree in physics, with some level of especialization in quantum theory and some especialization in astrophysics. I was a follower of Christianity for 20 years. I have also dedicated myself to reading the religious texts and studying the basics of the theologies of other religions, such as Judaism, Islam, Hinduism, Buddhism, Confucianism, Daoism, Shinto, and some shamanic religions of the Yoruba tribes in Africa and Latin America, because that is where my mom came from. I have read the Dao de Jing, several Buddhist texts, as well as Confucian texts, the Bible, some of the Quran, and what not. You can feel safe and confident that I am not just some fool who is coming into this conversation unprepared and uneducated. If you still don't want to have this conversation, then I can't force you to do so.
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PhilAOFish x I can't imagine a more miserable, pathetic, and useless view, than nihilism.
How arrogant. Why do you assume all atheists subscribe to nihilism? I am willing to bet real money on the claim that, to the contrary, most atheists are not nihilists.
To think that WE humans are the top of the pyramid is what I consider to be Hell.
Many of atheists, myself included, do not believe we are at the top of the pyramid. In fact, we are not so ignorant as to think such a pyramid exists in the first place. At least I know plenty of atheists for whom that is the case.
The best human being is still only a human being, subject to all our faults and mortality.
Yes, and there is nothing wrong with that. There is nothing wrong with the fact that we are imperfect. I can accept those imperfections and live happily with them because they make us what we are. If you cannot be happy with that, then yikes, that must suck.
Worshipping ourselves is what I believe Satanism is.
My first instinct when I read this sentence was to think "you're an idiot." I will refrain from actually consciously declaring that you are, though, because I want to give you the benefit of the doubt. With that said, though, you need to recognize how absolutely ignorant and moronic your sentence is. Atheists do not worship humans. They do not worship themselves. You are right. Satanists do worship humans. Plenty of pagan belief systems do. But atheism is not a belief system, and atheists, by the literal definition of the word, do not worship anybody, not even us humans.
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CHAD Once again, I will play devil's advocate...
You are not playimg anyone's advocate, you are simply being dishonest and disguising that as a farce of being a mediator of sorts. I can justify this accusation that I am making.
Please provide the evidence that unequivocally refutes the notion of a Universal Creator.
Infinite Monkey has no responsibility to provide you with such evidence, because they never claimed that a Universal Creator does not exist. To the contrary, in their response to your comments, they clarified that atheists do not believe a god does not exist. As such, the burden of proof falls exclusively on John David, who does actively claim that a Creator does exist.
Secondly, any evidence, or lack thereof, used to support either side of the argument, is also subjective.
It is not. Epistemology and formal logic exist.
For instance, what may be undeniable proof to some, may not suffice for others, especially when each side uses their own criterion.
Some criterion are invalid, meaningless, or simply absurd. Others are not. Whether you or anyone else agrees with this does not change that fact. If a proof that is valid does not suffice for someone, then that someone is just in denial, and they are wrong. This is why, we can say, for example, that a flat Earth conspirationist, is wrong, and in denial, regardless of what you, they, or anyone else says. If a proof is not valid, or is incomplete, and does not suffice for someone, then it does not matter if the entirety of the Earth decides that the proof that is valid, it is simply not, and it means that the entirety of the Earth is wrong, even if no one knows it, because that is how truth-value semantics work. If a proof is invalid, but it suffices for someone, that someone is still wrong, and it is their mistake for not realizing why the proof is invalid. Proofs are valid and invalid. This is not subjective. Independently of whether any human ever in existence ever acknowledges this or not, this is true.
Both sides of this argument are contingent on one's own subjective understanding of reality.
Understanding, by the very virtue of what it is, is not subjective.
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PhilAOFish x We live in a casual Universe. Every effect has a cause.
This is, unfortunately, inaccurate. This not what the current scientific understanding of a causal Universe is. Scientific, physical causality is actually understood via the theory of general relativity. "Causes" and "effects" in the classical understanding that philosophers and scientists alike had in the 1600-1800s simply do not exist. This classical understanding fails to take into account the relativity of simultaneity, and time dilation in general. It also fails to take into account non-locality and quantum theory as a whole.
Therefore, our Universe began with an external cause.
This is impossible, because according to all understandings of causality, scientifically accurate or not, include the notion that the cause and the effect are connected via temporal precession. However, anything external to the universe cannot participate in temporal precession with the universe or anything with the universe, because that would imply that they exist somewhere in spacetime, but if they do exist in spacetime, then they are, by definition, not external. In fact, the current scientific understanding of the universe is that it is impossible to interact with the universe or any parts thereof, unless that interacting entity is also contained by the universe. The universe, in its definition, includes all the set of interactions physically possible.
1A: Causality, Newton's 3rd Law, action-reaction
This is most definitely NOT what Newton's 3rd law states. What Newton's 3rd law states is that if an object A applies a force F on object B, then object B will also apply a force -F on object A. And unfortunately, Newton's laws are only an approxiation of reality. A more accurate description of reality is given by the theory of general relativity and quantum field theory.
In this case arguing the premise cause and effect requires you to argue against using science to discern knowledge.
No, it doesn't. It only requires for me to argue that your fundamental understanding of physics is completely inaccurate. And I can do that, because I am a physicist, and I have a degree in physics, and I know that the claims you're making just aren't scientifically accurate. If you want to make scientifically accurate claims, then you should be talking about metric tensors, curvature tensors, stress-energy tensors, Lagrangian densities, quantum fields, wavefunctions, and what not. You aren't doing that, so there is nothing else really to discuss here. I am sorry.
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John David I can not confirm the existence of things that must experience on your own
Personal experience is not a form of evidence. You cannot draw a valid conclusion about an unphysical entity via an experience that is contingent on the laws of physics. You can claim that you have experienced the creator, or that you experienced death, but just because you make that claim, that does not imply that you did indeed experience the creator or that you experienced death. In fact, just because you claim you understand what you experienced, that does not mean you do. Neurological studied have demonstrated that we can use electromagnetic resonance to manipulate regions of the brain, such that it induces hallucinations in patients, including sensations of a holy entity, and you can even induce consistent out-of-body experiences despite the fact that the body is not in any danger whatsoever. These can be done independently of whether there exists a creator, multiple, or none. These findings suggest that these sacred personal experiences are just stray sensations that do not have any real connection to anything preternatural, surnatural, or supernatural, whatsoever. The fact that each personal experience contradicts most or all others only further destroys the reliability of experience as evidence.
The Creator cannot be calculated anymore than a computer can surpass its programmer's knowledge.
This is factually and objectively incorrect. A computer can, and often, does, surpass the knowledge of its programmer. For example, in the decades of the 1990s, a computer, which was programmed by somebody who was not an international grandmaster at chess, defeated Garry Kasparov, who at the time, was the best chess player in the world. The knowledge the computer had qualitatively and quantitaively surpassed Kasparov's, and therefore, surpassed the programmer's, who had less knowledge than Kasparov. This is just one example, but there are potentially millions of currently existing examples in the real world today.
You must experience your own evidence.
I already debunked this above.
You cannot know what love is without experience.
This is false. You can know what love is without having experienced it. You can consult people who have experienced it and obtain a statistical consensus. You can investigate the brain chemistry and the hormonal chemistry of people who experience love and who do not experience it. You can investigate all the sociological facets of what love is by looking at different societies and looking at the different traditions centered around the topic of love. None of this requires that you experience love ever in your life. You need to experience love to know what love feels like, but not to know what it is or that it exists. Frankly, this also holds true for most things in life. You can know what a country is like without being there. You can know what a person is like without ever having seen them or met with them in person. I can know that Moscow exists without ever having needed to be there. You need not know to experience something to know that it exists, and in the same vein, that you think you experienced something does not imply that it exists.
for all we know, math is an obsolete way of keeping track of things
No, we know for a fact that math is not obsolete. It cannot be, because math is just a collection of formal theories of propositional logic constructed by humans. Math is a human invention. Mathematical statements are true because we decide that they are true in our system. For example, there exists the axiom of infinity. Zermelo-Fraenkel set theory, which is the foundation of standard mathematics as we know them, includes the axiom of infinity, which declares that one infinite set exists. This is the set of all natural numbers. Therefore, an infinite set exists, because we decided it exists. Of course, we could have chosen to reject this axiom, and work with an ultrafinitist set theory of mathematics. Sure, it is not practical to do so, but it is logically consistent, and therefore, just as valid as Zermelo-Fraenkel set theory, or any other mathematical theory. Another example is the continuum hypothesis. The continuum hypothesis not decidable in the standard mathematical set theories. You can append the continuum hypothesis as an axiom, and you maintain consistency. You can append its negation, and still maintain consistency. This is perfectly legitimate. Whether the continuum hypothesis is true or not, is genuinely and literally up to our choice. Anyhow, the point is that, because mathematics are constructed according to a set of logical criteria, and according to human necessity, it is basically, by definition, not obsolete.
An A.I. cannot surpass its programmer.
You are in dire need of consulting multiple computer scientists that will all tell you otherwise, and explain to you why you are wrong. I also gave a counterexample to this claim, so it is effectively debunked. Moving on.
If a computer is a compilation of our pre-existing knowledge...
That is not what a computer is. Once again, I STRONGLY recommend that you watch or take introductory computer science courses, and/or you consult with multipli computer scientists in-depth about the topic. I am not a computer scientist. However, I have enough of an education on the topic to at least know that this is not what a computer is.
Anyhow, none of this is evidence that a creator exists.
What do you want me to do?!?!
To give us evidence that God exists, not use some elegant-sounding rhetoric that only amounts to cute poetry. Saying "with will, anything is possible" is not an argument. If you can actually indeed prove that "with will, anything is possible," then that is an argument. Making claims without proof is not presenting an argument. Rather, it is just a waste of everyone's time. The only arguments you have presented, which were that experience is evidence, and that computers cannot surpass their programmers, were both debunked. So you need to present us a new argument and not some poetry. I'm sorry, but that is how conversation works. Poetry is not going to convince me or anyone here.
when you look at the amount of "programs" running through the human brain in comparison to the fastest super computer... let's just say it doesn't compute
Once again, this is false. The fastest neurological processes cap at a speed of no more than 125 m/s. This is due to biological limits that cannot simply be surpassed through sheer will. Surpassing them actually requires undergoing natural selection and, frankly, being lucky that the correct mutations happen in our genes. A computer has physical limitations, but not biological ones. And while a brain does run more programs than a computer at any given second, it is far less efficient than even an ordinary computer, let alone a supercomputer. Frankly, the only task at which humans surpass computers (for now) is at replicating thorough emotions. But A.I is coming closer and closer every year.
Einstein wasn't very bright. This is common knowledge.
No, it isn't, John.
The size of his brain is exaggerated.
You have to be joking. You cannot be making such blatantly ignorant claims and expect us to take you seriously. Intelligence actually does not correlate with the size of the brain. It correlates with the number and the density of the wrinkles in the brain. This is why there exist mammals who have legitimately larger brains that are considered to be less intelligent than those with smaller brains. It is because these larger mammals have smoother brains. This is why you will often find that, in the Internet, will people will call you a "smooth-brained person" as an insult: biologically, it is synonymous with "idiot."
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8:10 - 8:13 THANK YOU! There is no time at which there is no time. I think, whatever else is said in the video later on, that this is the most important sentence in the entire video. Because this one sentence, this one tautology, is what signifies defeat for the Kalam, singlehandedly. It creates problems with both premise 1 and premise 2. More on this later, but for any readers, just keep this sentence in mind. There is no time at which there is no time.
8:25 - 8:30 Exactly! This is the issue that WLC does not understand, and it is the issue that made him look completely foolish when having his live-streamed debate with Scott Cliffton. No one claims the universe popped into existence out of nothing. Such an accusation is a strawman, and all it demonstrates is the people do not understand the Kalam, not even WLC himself, apparently.
9:45 - 9:59 Yes, but the problem with this version is that premise 1, in this case, is just arbitrary baseless assertion that, whenever apologists try to justify it, they always end up having to resort to the unrestricted principle anyway. The restricted principle cannot be substantiated on its own terms. I honestly think it is not even worth discussing, until such an attempt of independent substantiation is given. The burden of proof is on WLC here.
12:08 - 12:21 But that is precisely why appeals to intuition are fallacious. Ultimately, an intuitive statement has no relationship to being a true statement or a demonstrable statement. Whether a statement can be intuited or not should be universally considered irrelevant to discourse, because it does nothing to determine whether asserting the statement is reasonable or not. It adds literally nothing to the conversation. The fact that, in metaphysics, intuition is regarded an acceptable avenue of argument by many philosophers, is part of the reason why I find it very difficult to take metaphysics as an academic field of study, seriously. In my opinion, it is philosophy done poorly.
12:30 - 12:38 I disagree. To start with, this description makes it sound like it is difficult to overturn intuition, and that is just not the case. Every form of evidence to the contrary necessarily overturns intuition, no matter how weak. Intuition is the weakest form of evidence possible, if you can even call it call that. Epistemically, the entire point behind saying that X supports Y is that, in more situations than not, X could only have occurred if Y did. But since there is no correlation between intuition and truth, intuition does not satisfy this basic minimal criterion for what constitutes evidence. Something has no correlation to truth under any circumstances cannot serve as support of anything, by definition. Intuition is as good as support as any other thing not related to truth, such a beauty, naturalness, or what have you. Epistemically, if intuition can support a statement, then literally all things can, regardless of how fallacious it is.
12:40 - 12:52 This is not true. Most mathematical truths and logical truths are not only unintuitive, but counterintuitive, and this also goes for scientific truths. The fact that we find the material implication in logic so weird and "wrong" is an excellent demonstration of this. So I fail to see how exactly these things are based on intuition.
12:52 - 13:07 Well, no, that is inaccurate. You are treating it as if "if p and p implies q, then q" is a conclusion that we make. But that is not how it is. "If p and p implies q, then q" is literally just the definition of the word "implies." There is nothing to conclude here, intuitively or otherwise. So I disagree with your explanation that logic is based on intuition. The evidence very strongly suggests it is not, and treating a definition as a conclusion does not demonstrate otherwise.
13:12 - 13:20 What? No, that is, with all due respect, utterly ridiculous. It does not "seem" to the scientist that there is any pressure gauge there. That is literally not how measurement works. The scientist simply reads what the instrument of measurement is saying to the best of their ability, and they write it down. They say the pressure gauge is 14 atm, because to the best of their sensorial-cognitive coordination, that is what the instrument is saying. There are no "seemings" here in the way that you explained them earlier.
13:29 - 13:32 No, that is just false. What makes science reliable is the scientific method, not intuition. In fact, in multiple occasions, the scientific method has had us conclude that seemings actually are unreliable. And to be honest, this entire segment on seemings felt very different from the rest of the video, very lackluster and significantly less well-substantiated than everything else in the video. It was completely unnecessary, and the way it was presented was like propaganda. The video would have been better off without this segment. Everything else so far has been on point, but this segment alone had so many inaccuracies, it made me think you had an agenda of sorts there. It was completely unneeded, as you and I both agree that something being metaphysically intuitive is not sufficient reason for accepting premise 1 of the Kalam (otherwise, this video would not exist). Hopefully, the rest of the video is better than this.
13:34 - 13:50 Okay, so I retract some of my previous statements. This here clarifies what you actually meant, but you did a poor job at explaining it. It is not the case that intuitions actually serve as support for a claim. That much is false, and I already discussed why. But if it is the case that enough people have the intuition, and there is no available information that contradicts the intuition, then it does become more reasonable to believe in the intuition, than to not believe it. This does not mean the claim is sufficiently justified, though, but in this case, a Bayesian analysis would reveal the claim is more likely to be true, based on the available information. But this all falls apart in light of the counterarguments we will encounter only a few seconds later in this video.
14:03 - 14:21 Ditto. And I think this sole response undermines the entirety of the previous segment, hence justifying my admittedly harsh criticism of it. I fail to see why one would have that segment when this objection exists. And I do not think this video will provide a good objection to this counterargument.
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14:26 - 14:52 What? No, I am sorry, but I have to become harsher with my critique. What on Earth does it mean for something to be reliable if not that it consistently proves decisively that statements are true? Now it just seems to me as though you are using the word "reliable" idiosyncratically without providing a disclaimer that you are doing that. If intuition is so often incorrect, why should we trust that it is correct in this one, isolated instance, and not be skeptical right away? All you have done is say "Intuition is not infallible, but that does not mean we should discard it" without even attempting to provide a good reason for why that is the case. I apologize, Rationality Rules, but you are just completely wrong on this point. And let us address the analogy you gave, shall we? Yes, it is true that perception also fails often. But we are talking completely different scales here. Not only is sensorial perception correct much more often than intuition is, but unlike with intuition, we can actually provide very precise and accurate qualitative and quantitative measurements of when our perceptions will fail, and when they will succeed, and we can also explain why they will or will not fail. There is no such thing for intuition that allows us to make it useful for coming to conclusions. Perceptions are reliable, as long as we limit ourselves to a certain class of circumstances that have been well-studied. There is no known class of circumstances for which we know intuition will be reliable. So we should not think of intuition as being reliable, except for the most mundane of circumstances. And as we are clearly talking about the universe beginning to exist here, we are definitely in the realm of "not mundane." So intuition definitely should be discarded in this context. So, again, Rationality Rules, with all due respect, you are super off the mark here. Everything else in the video so far has been great, but your deliberation on intuition is rife of misunderstandings.
15:02 - 15:07 And if that were true, then that would help support the above counterargument against my objection. But as pointed out earlier, this is just not true. There are very few assumptions that logic, mathematics, and the scientific method make, and some of those are not even intuitively true, for what it is worth, but are there just as a matter of formality. Also, I have no idea why you keep insisting so strongly on defending this notion of metaphysical intuition. This creates many more problems than it solves. If we ought to accept premises on metaphysical intuition alone in the absence of evidence to the contrary, then you should be deists, and not atheists. I mean, according to WLC, the existence of God is justified by metaphysical intuition. The entire worldview of reformed epistemology is reliant on this notion. What evidence to the contrary do you have that resists this intuition? To my knowledge, none. What about the kinds of mental gymnastics that people engage in, such as confirmation bias and cognitive dissonance? Those are clearly examples of intuition. And you may say, "yes. But the point of intuition, in this context, is that the evidence can overridde it." But how do you come to that conclusion? The very concept behind cognitive dissonance and confirmation bias is precisely that one does not need to accept that such evidence overrides anything. So if intuition is really to be taken as primitive, as you insist, then why should evidence override anything at all? That does not make any sense, and I know you agree that it makes no sense. So, please, for the sake of the quality of your arguments, stop trying to defend this reformed-epistemology-ridden idea of metaphysical intuitions. It creates more problems for your arguments and everything else you say in the video, than it solves. And it really is unnecessary. Again, I have no clue why you insist so strongly on this point that you have spent a full 3 minutes presenting essentially the same argument. It serves no purpose.
15:15 - 15:29 No, this is completely wrong. The law of contradiction does not rest on intuition, and as I explained, the fact that people feel comfortable in not only engaging in cognitive dissonance, but also straight up just holding mutually exclusive beliefs, demonstrates that the law of noncontradiction is not such an intuition, and perhaps it may not be true at all. This is why there is an entire discipline of formal logic dedicated to researching contradiction-tolerating logics. And the idea that an external world exists at all is not even an assumption the scientific method requires. The scientific method is all about data and making predictions. What the data is describing is solely a matter of interpretation.
15:50 - 15:57 Because that implication is built into 1 itself. It is not possible to accept premise 1 without accepting the implication that defines premise 1. This goes back to the whole "if p and p implies q, then q." You guys claimed this is an intuitive assumption, but it is not: it is literally what defines the implication in "p implies q." sigh You guys are just repeating yourselves now, and it really is just harming the quality of the video. You are trying so hard to defend the indefensible, and I do not understand why. It makes me a bit sad. I have watched many of your other videos, and I know for a fact that you do not actually think arguments from intuition are reasonable in these contexts, so why are you now pretending that you do think that?
16:00 - 16:16 What even makes you conclude that? Did you seriously not acknowledge the possibility that one possible such argument would be "...by definition"?
16:17 - 17:33 Ugh. Okay, in what domain of discourse do you think, conclusively, that intuition is reliable? I understand what you are getting at, and in the end, you are rejecting WLC's argument from intuition based on an appeal to inappropriate domain, but what was the point of that? Why use all of these bad arguments to say that there exists a domain in which intuition works, while failing to describe it, when you know that it does nothing to add nuance to the discussion and defend WLC's argument? And again, all it does is raise many more questions, and put into question everything else you have said, simply because I can argue that something is metaphysically intuitive, and that the domain of discourse in question is appropriate for the intuition. How would you ever debunk such an objection? That is why I find this entire segment problematic. By accepting that there it at least one presumably unspecifiable domain of discourse where metaphysical intuition obviously works - using very fallacious arguments to justify that - it makes anything else you have to say undermined by this. In fact, it makes the Kalam redundant, because as I explained earlier, metaphysical intuition can be and has been used by WLC as justification to believe God exists. It "proves" at least deism, if we are to accept your arguments. But I know you guys are not deists and I know you guys do not accept reformed epistemology. So I fail to see why you spend so much time defending it, when in the end, you are going to reject WLC's premise anyway. It feels like one big waste of everyone's time, and it puts you in a difficult spot. Anyway, I am moving on from this.
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17:35 - 18:39 But the problem with this is that the notion of a statement having "clarity" is inherently subjective. There is no way anyone can come up with an objective standard for this. Philosophers have tried, and all have failed. This is why formal logic formalizes tautologies in terms of truth-tables with respect to a choice of truth-functions, rather than just accepting a formal standard of obviousness.
18:40 - 19:03 There is no proposition that we can demonstrate that definitely satisfies this second criterion. Even the law of noncontradiction, upon further reflection, can seem less likely to be true. And again, this is entirely subjective.
20:17 - 20:23 Yet another example of the unreliability on intuition: the fact that it is subjective. And this brings me to the next point: which philosophers should we trust here? Is the causal principle intuitive? Or is it not? Whatever the answer is, someone is wrong here. If we are to regard intuitiveness as not a property that can be true or false of anything due to its inherent subjectivity, then how can it be evidence of anything? The more elaboration is being given here, the more thoroughly skeptical I am.
22:30 - 22:45 Oh, goodness, what is this? Metaphysically necessary truths? I thought this could not get more absurd.
24:01 - 24:29 This is exactly the problem with the entire discussion. It all boils down to "my opinion is...." "oh really? Well, MY opinion is..." who cares? Stop using intuitions, and actually start providing actual evidence. Personally, I am going to skip ahead in the video to a point in time where the interlocutors finally decide to stop discussing intuition with such unmerited philosophical seriousness, and move on to a legitimate use of logic, because I feel tired of beating the dead horse here.
25:45 - 25:50 You cannot claim that god has any properties before first determining whether god exists or not. This is a circular argument. "The cause of the universe is omnipotent." That assumes the conclusion of the Kalam is true, but that is the very thing we are questioning to begin with.
28:00 - 29:00 This principle is just false, though, and I do not need to "appeal to intuition" to tell you that. We know, for a scientific fact, that there are material things without material causes. In fact, the very notion of "material" is nothing but an emergent property, according to physics, thus making the very classification of "causes" into "material, efficient, formal, and final" deeply flawed. And this is why I complained earlier in the video when you guys gave no definition for causation.
29:55 - 29:58 There is no such a thing as "before the beginning of the universe." That would entail the existence of "before time." And that is nonsense. Remember what you guys said earlier? There is no time at which there is no time. So there is no "before time."
33:30 - 34:14 Excellent presentation of that argument. Finally, we are back to the actually high quality arguments that I am used to seeing from Rationality Rules and Joe.
34:14 - 34:51 Yes, which is why Rationality Rules and Joe's defense of metaphysical intuition as being sometimes reliable is completely untenable. There is no situation where this symmetry problem does not apply. You guys completely debunked WLC's defense, but also debunked your own defense for intuition. This is why I complained so much about. You could have simply foregone that segment altogether and just addressed the contradictions in WLC's argument directly.
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37:05 - 37:15 I am so glad you guys did decide to talk about the A and B theories of time. And yes, the previous description given of the B theory is quite accurate, and it does undermine the Kalam entirely, not only in premise 1, but premise 2. And the B-theory of time, unlike the A-theory of time, is scientifically supported.
37:15 - 37:37 Exactly.
38:20 - 38:48 This misses the point entirely. Remember, there is no time at which there is no time. Yes, things within spacetime have spatiotemporal boundaries that demand explanations in terms of other things within spacetime themselves. But as there is no spacetime outside the boundaries of spacetime, the notion that any causal explanation is required here at all is misconceived, a categorical error. The reason things with spatiotemporal boundaries require causal explanations is because they are surrounded by regions of spacetime not occupied by themselves, and so this needs to be accounted for. But spacetime is the entirety of spacetime itself. Its boundaries are just inherent properties of its intrinsic geometry, they are not causally related to any other objects, and it makes no sense to say that they could be, precisely because they are the boundaries of spacetime. A thorough understanding of this would require mathematical understanding of manifolds. This is the problem with talking about metaphysical things on their own terms without being properly defined, and is another reaosn why I find it difficult to take it seriously.
38:55 - 39:33 Yes, and this all brings me back to wondering why you guys defended arguments from intuition as legitimate, when this point undermines their legitimacy almost entirely. Again, I acknowledge there are very limited contexts where they are clearly okay, but this is not one of them, and we all agree. And this is really just the same objection to the argument from intuition as the other objections. They are all really the same objection, and they all boil down to this: "intuition sucks and is not (sufficient) for this." Which is what I was saying all along. The only objection that is different, in this case, is just that the B-theory of time undermines the Kalam.
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Overall, I think this was a good video, and some things were pointed out that needed pointing out that are typically not pointed out. My one issue with the video is the bad attempt at defending the legitimacy of intuitions, which not only was rife with false claims, but also undermines everything else that they say, and which completely distracts from the more powerful and legitimate objections that they gave that could have been made without discussing intuition. Nonetheless, the video was still ultimately effective in demonstrating the difficulties with premise 1 of the Kalam (to say nothing of premise 2, which the video did not discuss in depth).
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Since the cosine is not zero, you can cancel it out (divide by it) on both sides, and you get 0 = 2π.
This is not true. cos is nonzero, but cos could be a zero divisor, or, more simply, it may not be left-cancellable.
Divide both sides again, this time by 2, and you get 0 = π.
This is not justified. How have you discounted 2 = 0? Or 2 being not left-cancellable?
With 2 totally independent proofs, it has to be true, right?
2 is quite a small sample size, making your study statistically insignficant 😉
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Univ Univeral Let a > 0. Draw a semicircle with radius (a + 1)/2 and endpoints {(0, 0), (a + 1, 0)}. Draw a line segment perpendicular to the line connecting those endpoints, and let the segment have endpoints at the semicircle and at (a, 0). Let the endpoint at the semicircle be (a, b). Draw a line segment with endpoints {((1 + a)/2, 0), (a, b)} to form a triangle. The measure of the angle of the rays {((1 + a)/2, 0), (a, 0)} and {(a, 0), (a, b)} is t, such that (a + 1)/2·cos(t) = a - (a + 1)/2 = a - a/2 - 1/2 = (a - 1)/2. Therefore, t = arccos[(a - 1)/(a + 1)], and (a + 1)/2·sin(t) = b. For any |x| < 1, sin(arccos(x)) = sqrt(1 - x^2), this is a fundamental trigonometric identity. |(a - 1)/(a + 1)| < 1 for all real a > 0. Therefore, sin(t) = sqrt[1 - (a - 1)^2/(a + 1)^2] = sqrt[(a + 1)^2 - (a - 1)^2]/(a + 1). (a + 1)/2·sin(t) = b = (a + 1)/2·sqrt[(a + 1)^2 - (a - 1)^2]/(a + 1) = sqrt[(a + 1)^2 - (a - 1)^2]/2 = sqrt[a^2 + 2a + 1 - a^2 + 2a - 1]/2 = sqrt(4a)/2 = sqrt(a). Therefore, b = sqrt(a).
This proves the claim in the video for all real values of a with a > 0.
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Chris Sekely There is no such largest number, since arbitrary symbolic notation can be used to express arbitrarily large numbers. We can go to arbitarily high orders of logic to compress expressions as much we want.
More concretely, I can define a function F(n) such that this is equal to the smallest number not representable in x-order logic with n symbols. Then let n be the number of Planck volumes in the universe and I have expressed a large number. Normally, you would want to use the label F_x to specify the order of logic of the function. With this method, there is such a largest number. However, I can circumvent this entirely by simply creating new symbols. Symbolic logic sets no restriction on what symbols I can use, so long as they are part of the language. I can arbitrarily expand the language and add new symbols arbitrarily, which allows me to not need to label the function, but rather just use a new symbol for a higher order logic function. Then any limitations would come from the limit of possible symbols I can use. As far as I understand, though, there is no limit. For any symbol that exists, I can make a new symbol from it.
Okay, I suppose you may be able to come up with such a limitation symbols. But I'm already a step ahead of you: I can define a function such that there is a number not expressible in this type of notation, and I can do so in symbolic logic. In fact, I can define the function F(n) as the smallest number not expressible in n symbols in any lower order of logic with new symbols. And so on. You may have to consider transfinite orders of logic, and so, but you can always go one order higher and form a compression defined explicitly from the limits of the previous orders.
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Exactly! This is also my favorite reasoning. More precisely, you can think of n! as being defined as the product of (1, 2, ..., n). Hence 3! is the product of (1, 2, 3), 2! is the product of (1, 2), 1! is the product of (1), and 0! is the product of (). Since the product of () is 1, 0! = 1.
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Epic Math Time "First, you are using "=" as the relation "has an analytic continuation of..."..."
Uh... no, he is not. You make this assumption without any proof, as nowhere in his comment is this stated, and nowhere in the mathematical literature is this stated either. The symbol "=" unambiguously and unequivocally means "equals", not anything else. Also, for the record, that result can be shown with methods that are completely independent of the existence of analytic continuations and make no usage of them whatsoever. This invalidates your argument.
"Second, your 1 is obviously false."
No, it is not. Let me repeat myself. In any standard language of any model of any well-axiomatized theory in mathematics, the symbol "=" means "equals", and is actually a relation whose definition includes equation 1. There are languages witg interpretations which map the symbol to a different meaning, but those do not get used outside of metamathematics or Gödelian set theory.
" "=" is just a symbol."
Sorry, but no, it is not. This is mathematics we are talking about. Mathematics stands for a collection of theories about certain abstract objects, and these theories are not uninterpreted. Therefore, no character is only a symbol in mathematics.
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@SplendidKunoichi Uh, well... for starters, you can search "pigeonhole principle" on Wikipedia, and if you read the various sections of the article, then you will see what I mean when I say that the pigeonhole principle is a mathematical theorem. There are several formulations of it too. As for the usage of the word "theorem," a theorem is simply any mathematical sentence that can be proven to be true. A sentence not having the name "theorem" does not mean it is not a theorem. Again, you can search on Wikipedia for more details. Now, I understand Wikipedia is not a scholarly source, but as we are just discussing the introductory concepts, I think it should be fine. The pigeonhole principle is also known as Dirichlet's principle, not to be confused with Dirichlet's theorem, which is a different theorem that is unrelated (well, there are multiple theorems by that name). If you are curious for details beyond what the article states, then I can tell you this: the pigeonhole principle is a direct consequence of the Schröder-Bernstein theorem, which again, you can read about on Wikipedia. If you are familiar with set theory, then the proof is fairly straightforward.
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@Happy_Abe No, that does not use the definition of a 2-tuple. A 2-tuple is itself a function, a function from 2 to the target set, where 2 = {0, 1}. The object {{x}, {x, y}} is actually called a Kuratowski pair. The corresponding two-tuple is instead the set {{{0}, {0, x}}, {{1}, {1, y}}}, which is built from the Kuratowski pairs {{0}, {0, x}} and {{1}, {1, y}}. The distinction is subtle, but it exists, because there is no 3-element analogue for the Kuratowski pair construction, while there is such a thing as a 3-tuple: a function from 3 to the target set, where 3 = {0, 1, 2}. In this case, the three tuple would look like {{0}, {0, x}}, {{1}, {1, y}}, {{2}, {2, z}}}.
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@Happy_Abe Wikipedia, as usual, is oversimplifying things, since it is not meant to be treated as an introductory textbook to the topic. A Kuratowski pair of x, y, namely {{x}, {x, y}}, is different from a 2-tuple of x, y, namely 2 —> {x, y}.
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No, you are completely wrong. The problem is the logic. You seem to believe that the only thing that matters here is empirical evidence, but that is just absolutely not the case. I want to redirect you to mathematics. Can you provide me with an experiment that can prove that there is no rational number such that x^2 = 2? No, you cannot. By your own advice, you would thus end up dismissing the claim as being ridiculous, even though, in reality, this is an important theorem in mathematics with real life consequences in the sciences. Abstractions, by definition, cannot be empirically studied. I doubt you actually dismiss mathematics as a whole, though, especially since the scientific method is reliant on mathematics in the first place.
You say that one can construct logical arguments about a multitude of ridiculous things, like gods. This is incorrect, though. You cannot construct logical arguments to prove things that are false. That is why the entire concept of a logical fallacy exists. The very point of contention is whether gods fall under the category you named "multitude of ridiculous things" or not in the first place: that is why this discussion is happening. I find it arrogant and insulting of you to suggest that this is a waste of time simply because you fail to understand the importance of the logic of an argument.
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The implications of set theory for the Kalam are quite disastrous, because, as far as I am concerned, set theory is inconsistent with the distinction between actual infinity, and potential infinity, and set theory is consistent with the existence of sets of infinite cardinalities having different cardinalities. This to say, there exist uncountable models of any reasonable set theory you choose. Thus, the idea of the infinite being traversed being impossible is contradicted by these models, because in these models, ω < ω + 1, yet card(ω) = card(ω + 1).
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@littleredpony6868 You are talking about one of Zeno's paradoxes. The idea is that, if I want to from 0 to 1, I must first go from 0 to 1/2. Then I must go from 1/2 to 1, but to do so, I must go first from 1/2 to 3/4. Once I do, I must them go from
3/4 to 7/8..., etc. Inside the interval [0, 1] that you want to cross, there are infinitely many intervals that you have to cross individually to be able to cross the full [0, 1]. Since there are infinitely many, intuition dictates that crossing the interval [0, 1] should be impossible. However, in the real world, you cross intervals of the form [0, 1] all the time. Hence the apparent contradiction.
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@hlulanizitha9920 Craig is a well-reputed philosopher with great credentials to back it up,...
No, he is not. He is well-reputed only among Christian apologists, and even then, plenty of fundamentalists hate him too. As for his published work in philosophy, it is quite average, if not mediocre. I say this as someone with a college degree in philosophy.
he's defended his Kalaam arguments publicly many times,...
No. He has attempted to publicly defend the Kalam argument plenty of times, but in all his attempts, he fails to convince anyone who is not already a theist, and he fails to address the objections by the opponent.
This harsh criticism of his work is unfounded
No, it is not.
even the best philosophers make mistakes.
Yes, but Craig is not among the best philosophers, he is not even among the good philosophers. If we were talking about someone like Saul Kripke, then it would be a different story, but Craig? Hell no. The fact that you even attempt to compare Craig to the best of philosophers is actually insulting.
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@Hello-vz1md That was a podcast with alex not debate.
It is still a debate. It is called a debate in podcast format, sillybilly.
Theist and Atheist Academic philosophers respect Craig while disagreeing with him
"Respect" is a strong word you are using.
unlike some random people in YouTube comments section with no philosophical background.
Maybe, but I have a degree in philosophy, and I can tell that much of Craig's work is garbage.
Both Rationality Rules and Alex said Craig is an inteligent person
Well, yes, he is an intelligent person. Most people are intelligent people. This does not change the fact that he is not a good philosopher.
but I will not call him complete idiot
No one here has called him that. You need to get off your high horse and actually start reading the replies you are responding to.
philosophy is BS 😂😂😂😂😂 i have no words for your ignorance
No, I will not call him ignorant. While I agree with you that he is patently wrong in saying philosophy is BS, he is justified in incorrectly believing that, precisely because mediocre philosophers like WLC exist. It is people like him that give philosophy an extremely bad reputation where it is not deserved.
i highly recommend you to watch this introduction course on philosophy
No thanks. I already have a degree, and if I want to study more, then I will just buy some books instead.
But I can't remember when science disprove the existence of God or even study or do research about God.
You are right: science has technically never conducted an experiment to gather data concerning the existence of deities. However, a quick cursory analysis reveals that, given the nature of deities, as per theology, the fact that there is no immediate non-trivial evidence for their existence is itself evidence of their non-existence, making their existence quite unlikely.
And blind faith is complete useless and stupid but every knowledge have some level of faith if you think deep philosophically and critically
No, this is nonsense. For one, faith is, by definition, blind, so writing about "blind faith" is literally redundant, as you just said blind twice. Secondly, knowledge does not require faith. Taking something as an axiom is different than having faith.
it's completely different and philosophically, it's about the nature of reality, like do I or anything or anyone really exist, like how we see/think it exists around us.
No, this has nothing to do with faith either. Also, whether this actually constitutes a problem of knowledge depends on how you define "existence". Really, this is all just semantics.
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@Martial-Mat Aha, you're today's designated mind reader huh?
If I had the power of mind reading, that would be nice. However, you give me too much credit.
Shouldn't you be on stage conning old ladies?
No thanks. I have a moral compass.
Your arrogance is a particularly appealing trait btw.
Is it? You have not demonstrated that I am arrogant, but even if I was, how would that be appealing?
Without understanding the basis for my opinion, you are simply going to tell me why I don't understand.
No, this is not what I did, and this is not what I will do. I simply stated a fact: the available evidence contradicts your claim concerning your understanding of philosophy. I have no idea how or why you decided to twist this into me making a claim about why you do not understand, but the fact that you did that makes you quite arrogant yourself.
Anyway, this reply of yours makes you seem like a prime example of the Dunning-Kruger effect in action, so I get the impression that continuing to interact with you will be a waste of time. Have fun talking to your echo chamber.
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If a magma (M, •) has a left identity L and right identity R, then L = R, because L = L•R = R. As such, L = R = e is a two-sided identity element. The two-sided element is unique by consequence of this very same proof. If e is a two-sided identity element and f is a two-sided identity element, then e = e•f = f.
That being said, a magma can have multiple left-identity elements or multiple right-identity elements if no other-sided identity elements exist. However, if the magma has the cancellative property, which follows if the magma is a quasigroup, then the one-sided identity elements are unique. If you have associativity, cancellativity, and unique one-sided identity elements, then you have a unique two-sided identity element, and thus this is a monoid.
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Mar Tijn The series of 1/x^p converges if and only if p > 1. However, this does not count as a crossover point. This is because x^p >> ln(x) >> 1 for any p > 1, yet the series of 1/[x·ln(x)] diverges. Similarly, ln(x) >> (ln^2)(x) >> 1, yet the series of 1/[x·ln(x)·lnln(x)] diverges too. In fact, if we let (ln^0)(x) := x and [ln^(m + 1)](x) = ln[(ln^m)(x), then the product from s = 0 to s = N for any N of (ln^s)(x) will be called g(N, x). Then the series of 1/g(N, x) from any x in the domain of g to infinity will diverge for all N. Also, as N increases, the domain the of g in its second-place input approches the empty set, in which case the sum trivially converges, but this does not mean the function is a boundary.
To further worsen the problem, there is the fact that, if we extend (ln^s)(x) to fractional iterations s, then the series of 1/[x·(ln^s)(x)] diverges for all s > 0, and there is no minimum value for s satisfying this property.
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@outermarker5801 Again, I did NOT say that meditation is prayer, or even a form of it.
I understand this. I never said that you said that.
I said FOR THE CHRISTIAN it does the same thing to THEIR brain.
One of the effects prayer has is certainly similar, but this alone is not sufficient to justify treating prayer and meditation as comparable in this context.
I don't know about you, but I WAS a Christian,...
I was one as well. I never said anything that should invalidate your experience of prayer.
I'd encourage you to be like Hitchens, calm down,...
I hate to say it, but I am pretty calm. I am not convinced that you are calm at all, though. You are using CAPS LOCK every five words, and you compared me to a dogmatic Christian. These are not actions a calm person does in the beginning of a conversation, solely from a disagreement. Honestly, you seem like a person filled with anger.
...and actually analyze why religion, for all it's bullcrap, is so persistently captivating to the human mind.
I have. This was one of the many things I researched when I studied in college. I am probably more qualified to speak on the topic that you are, and frankly, I would be so bold as to say, more qualified than Hitchens. I never denied that religion is persistently captivating to the human mind, and my point was never about addressing this.
It is literally brain chemistry, specifically dopamine.
It is far more complicated than brain chemistry. Brain chemistry definitely plays a role, but it is not as major as you would claim to be.
Objective observation of the phenomenon rather 'impudent' dismissal will make you a better educated atheist.
(A) The only thing I have dismissed here is the grossly inaccurate comparison between prayer and meditation.
(B) I want to hear nothing about "objective observation" from someone who has conducted no observations of their own on the subject matter.
(C) Given your clear incompetence in holding a civil conversation, you are certainly not qualified to make any worthwhile judgments on how well-educated an atheist (or anyone else, for that matter) is.
Because right now, you remind me of some Christians - so dogmatic, you can't understand what's being said to you over the din of your own negative opinion and contempt.
This is just projection on your part. I understood your argument just fine the first time. You are the one who misunderstood my response to you. You are also the only one here being negative. You decided to proceed with this discussion by insulting me, despite my replies to you being polite. I will end my comment with an insult of my own, pointing out how this all demonstrates that you are definitely not smart enough to understand that this approach you are taking is completely counterproductive: you will not succeed at convincing anyone of your point of view, and if anything, you will only succeed in turning them away. Alas, if there is anyone here who is like dogmatic Christian, that would be you.
I am going to mute you, because there is no point in me bothering to try to have a conversation with someone so irrationally angry, that they are unable to hold a conversation.
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Lau Bjerno If we ignore the multiplicity of the prime factors in the prime factorization of the number, then no, no such number exists. In fact, the sum will always be much less than the number. This is because multiplication grows much faster than summation.
If we include the multiplicity of the factor, then the answer is yes. For example, 4 has the prime factorization 2^2, and 2 + 2 = 2*2 = 4. Other than this, though, no power of a prime satisfies this criteria. Trivially, no square-fre semi-prime composite satisfies it either. In fact, I believe 4 is the only number to satisfy this criteria.
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@Vee-Hive No, it is not logically incoherent. Corruption is one of the most well-established ways to damage one's credibility, and the I.B.A. being corrupt is well-known by now. Also, chromosomal tests themselves are not very reliable either, which is why, since the Olympic Games in Sydney at the beginning of the millennium, they were discontinued. So, unlike the corruption idea, the narrative you are promoting has no legs to stand on, beyond the very suspicious yellow-journalism going on of the trainer allegedly having verified the test, even though none of the sources actually can corroborate this independently.
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@1ToTheInfinity ...since 1, 5, and 79, are all examples of positive integers which can only be divided by 1 and itself, it makes perfect sense 1 is among the primes...
No, it does not make sense. The problem is, "only divisible by 1 and itself" is not the definition of a prime number. It never really has been the definition of a prime number. The video talked about how, in the past, different definitions of prime number were used, and the one which was used the most was this notion of being "measured" by another number. For example, "the prime numbers are numbers that are measured only by the number 1 and nothing else." This is the definition that you encounter Another Roof mentioning throughout the video, because that is the definition that was used in antiquity and in medieval times. But saying that a number is divisible by x is not the same as saying the number is measured by x. This was explicitly stated in the video too. Why are they not the same thing? Well, because (a) the mathematicians of those times did not consider numbers to be measured by themselves. In other words, 1 measures 7, but 7 does not measure 7. Again, this was explicitly stated in the video. The other reason is that (b), well, it simply is not true that 5 is divisible by only 5 and 1. 5 is also divisibly by –5 and –1. But for a long time, negative integers were never considered in number theory. This is an oversight that needs repair.
Let me go back to part (a). As mentioned, 1 measures 7, but 7 does not measure 7. That is how mathematicians used to think of it, but why? What is the difference between saying that 7 measures 7, and that 7 divides 7? The difference is that 1 is a proper divisor of 7, but 7, although it is a divisor of 7, is not a proper divisor of 7. The distinction between a divisor of x and a proper divisor of x is actually the exact distinction as the distinction between subset and proper subset. It is also completely analogous to the distinction between "equal or less than" and "less than." y is called a divisor of x if and only if y divides x. But, a proper divisor is more special. y is called a proper divisor of x if and only if y divides x AND x does not divide y. Now we get it: 1 divides 7, but 7 does not divide 1, so 1 is a proper divisor of 7. The whole point of proper divisors is that we only care about the divisors of x that are "simpler" than x: we do not at all care about the fact that x divides itself, because that is just completely useless, and trivial. Like, all numbers divide themselves anyway, so do we care about x being a divisor of x? This is why the concept of proper divisors exists. And the concept of proper divisors is the concept that the mathematicians of old were alluding to when they said "x measures y." The old language of "x measures y" translates into the modern language as "x is a proper positive divisor of y." Positive, because again, negative numbers were never considered seriously by European mathematicians prior to like the 1500s.
So, now that you know what the distinction is between divisors and proper divisors, and now that you have watched the video and you understand that prime numbers were always ultimately defined in terms of proper divisors, albeit in a different language, it should be clear why 1 is not a prime number. You see, the definition of a prime number always has been "a positive number which is only measured (positively) by the number 1." Translating this to the modern language, this means "p is prime if its only positive proper divisor is the integer 1." THIS is the definition of a prime number. This is what it has been since basically forever, in concept, even if the language used was different. But look: the number 1 has no positive proper divisors, since its only positive divisor 1 itself. So, it does not actually satisfy the definition of a prime number. The problem here is that most teachers, and most textbook authors, believe that the actual definition is too complicated for grade schoolers (i.e, children) to learn. So they simplify it down, they get rid of all the "technical details," and so they just tell the children that 'a prime number is divisible only by itself and by 1.' But what this is a mistake, because this is completely misleading: you have changed the definition itself altogether by getting rid of the technical details. You cannot get rid of the technical details, because they are the most important part of the definition, and not the least important part. Prime numbers always have exactly 4 divisors: –p, –1, 1, p. The proper divisors are –1 and 1. But –1 and 1 have no proper divisors at all! They are fundamentally different from the prime numbers, and do not satisfy the definition of a prime number, so they belong to an entirely different classification system. –1 and 1 are called "units," or unitary numbers. One of the defining properties of units is that they divide all numbers. Also, the product of two units is always a unit. Notice how this can never be true with prime numbers: the product of two prime numbers necessarily is a composite number, by definition. Also, units cannot be divided by any prime numbers at all. They can only be divided by units. 2 cannot divide 1 or –1. –7 cannot divide 1 or –1. Units are fundamentally different from prime numbers. Saying 1 is a prime number is like saying "a car is an animal." Like, no, that is just way off. Also, 0 is not a prime number either. It is also not a unit, because you cannot divide by 0. 0 is what is called a zero divisor. In the integers, 0 is the only zero divisor, but this is not true for all systems of arithmetic, actually. For example, when you work with matrices, there are some matrices not equal to the zero matrix, but are zero divisors anyway. A zero divisor is a quantity x such that x•y = 0 for some nonzero y.
By the way, this distinction between primes, units, and zero divisors, is not exclusive to the integers. It holds universally. It holds for matrices, polynomials, associative vector algebras, the rational numbers, the complex numbers, the Gaussian integers, the dual-integers, the split-complex integers, etc. It holds for all commutative systems with associative multiplication and addition. A unit is some quantity x such that for some y, x•y = 1. The integers –1 and 1 are units, and they are the only units in the integers. In the Gaussian integers, the units are 1, –1, i, –i. In the rational numbers, everything is a unit, except 0, which is a (trivial) zero divisor. The analogue for prime numbers in more general settings is called an irreducible element. An irreducible element is an element that (a) is not a zero divisor, (b) its only proper divisors are units. Remember: units have no proper divisors, so they are not irreducible elements. A composite element is just a product of two or more irreducible elements, just like in the integers. Note: how you factorize composites into irreducibles need not be unique in general. But, in the integers, it is unique.
Now that you know all this, you are probably going to object that this all just "the multiplicative perspective," and that when you look at it from "the additive perspective," it is very different. But that is not the case at all. Why? Because the definition of a prime number has absolutely nothing to do with how you write numbers as repeated sums of integers. In fact, there is nothing interesting to point out: all integers can be written as sums of two integers in infinitely many ways, and all nonzero integers can be written as sums of 1 or –1 alone. So, ultimately, the additive structure does not matter at all. My point is, there is no such a thing as "the additive perspective." Insisting that there is one is borne from a severe misunderstanding of how number theory actually works.
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You are wrong. The statement does follow from the theorem. If N is an odd perfect number, then 2N is triperfect number. This is true. No known triperfect number has the property that half of it is a perfect number. This is true. The Boolean conjunction of both statements implies that if the list of triperfect numbers is complete, then there are no odd perfect numbers, because if there are, then the list is not complete. This is true by the very definition of material implication, which itself includes the law of contrapositives. Therefore, if one finds an odd perfect number, a new triperfect number which is its double rises, in which case it is added to the list, so the old list had not been completed. Therefore, both statements in the disjunction that you wrote are valid conclusions of the theorem and the statement that finitely many triperfect numbers are listed and that none of them are the double of an odd perfect number. Therefore, you are wrong, and you simply fail to understand how material implication and the law of contrapositives work.
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@AShaif take your time not proceeding to engage
Have some patience, child. Unlike you, I have a life, and yesterday, I did not get a chance to sit down and write down a response to what you wrote. Are you one of those children who will arbitrarily alot a deadline for when I have to respond by, not tell me about the deadline, and then dishonestly claim I forfeit the conversation because I did not follow this arbitrary stupid deadline I was not told about? Geez. Talk about being aggressive: you are the only with the attitude problem here. But hey, with disingenuous, jerklash response like this, it almost seems like you want me to be aggressive you. That would actually explain why you made such a bold accusation despite having nothing to back it up.
At the first time you replied to me, you were too aggressive, arrogant and provocative, with no substantial counter-argument.
Concerning the accusations of me being aggressive, arrogant, and provocative, see my previous paragraph. Concerning there not being a substantial counter-argument: you listed about a dozen of arguments by name, and then a few others you went into more detail. You know very well that each of these arguments would require an essay-length response, followed by a discussion on that response. I know very well that writing a dozen and a half essay responses would be unrealistic, and it would become a wall of text you know you will not read nor reply to. Also, I have no responsibility to provide these counterarguments when the topic of discussion is the cosmological Kalam, a topic of discussion you seem unable to stick to. So this is yet another disingenuous remark by you. And all this does is prove my earliee snarky remark of your dishonesty.
You didn't come to me as a seeker of truth, but rather as a troller.
This is meaningless when coming from you.
However, I have time now and could spend a couple of minutes on your big discovery here :D
There is no big discovery here. Most people with the level of education that I have are at least aware of the things I have mentioned. Or do you think being educated is a rare trait?
Causality is axiomatic to science. Otherwise, no need to discover or look for illness causes, etc.
This is a non-sequitur. The need to look for cures has very little to do with causation, though we do happen to utilize causation in our quest for cures. Also, causation is not axiomatic to science. Causation is something we establish through science. The only thing that is axiomatic to science is that regulated empirical observation is necessary to establish facts about the universe, and that the value of a theory is a function of its predictive power.
Even in quantum theory, there are different interpretations of the Schrödinger equation that don't lead to non-causality, or indeterminacy, or violation of first principles.
Causation and causality are different things. Causation is the notion that things are related via causal relationships of some kind. Causality is a specific scientific principle in the theory of special relativity that restricts the possible types of causal relationships to those that are consistent with locally Minkowski geometry.
Also, I have no clue why you bring up Schrödinger's equation. Schrödinger's equation is not consistent with the theory of special relativity, with curved spacetime, with quantum spin, and is unable to account for quantum electrodynamics, among other things in quantum theory. So it is not suited for a discussion of causality, let alone causation. Also, the interpretations of quantum mechanics do not interpret Schrödinger's equation, they are just potential solutions to the measurement problem. The measurement problem is irrelevant in light of quantum field theories, though, which is how we establish the ontology of the universe in a physicalist worldview.
Bohm's quantum interpretation to give a single example.
Which one? There is an entirely family of interpretations of quantum mechanics attributed to Bohm. This smacks of ignorance, to me.
Also, even if causality is like a spectrum, it's still an uninterrupted chain of causes,...
It is an interrupted chain of causation, not of causes. Again, things cannot be discretely categorized into causes. That would literally contradict it being a spectrum.
...that can't go forever, unless you think they can, or that you adopt determinism.
I see no reason to think causal chains cannot regress forever. And sure, I would be willing to adopt determinism. The only unresolved issue here would be free will, but I fail to see how free will is relevant. There is no known physical process that is known to not be deterministic, not even within quantum theory. I know that quantum mechanics uses plenty of probabilistic calculations, but this tells us nothing about what is actually physically happening in the interactions.
But enough of the Kalam, let's speak of the worst of the bunch as you say, the contingency argument.
I think the argument from language origins is slightly worse, but sure, I am not opposed to discussing contingency.
How can you explain a universe that is made of parts and subject to addition, destruction and change, i.e. dependent, when you take the side that there is not a necessary independent existence?
This is a loaded question. The entities in the universe experience change, insofar as their world-cylinders progress through spacetime. Since the universe is the collection of all entities said to exist, the world-cylinders are themselves a feature of the universe. There is no meaningful sense in which the universe itself is changing, though, because it already contains all the histories of all the entities. There is no need for me to explain how the universes experience change, because I hold that the universe does not experience change, that is only entities within the universe that do.
My argument is as such: - Contingent existences are existences that depend on something else for their existence, and could be any other way. Necessary existence is an existence that is independent for its existence, and could not be any other way.
In what sense do contingent existences depend on other existences? What type of dependence are you talking about? There are many types of dependences, and so this makes your statement ambiguous, and so you could be referring to a wide variety of things you may not intend to refer to. I can tell you what contingency would look like in my worldview, but I doubt it matches your concept of contingency. For example, I, as a human, am a merological sum of histories of quantum field states across spacetime. I am a contingent existence, for the quantum fields did not exist as I do, or did not have the histories that they do, then I would not exist. However, the quantum fields are just... sort of there. Their existence is not ontologically correlated to the existence of any other entities.
Contingent existences could be any other way? In what sense do you mean this? Are you using counterfactuals? Are you using Kripkean semantics? Again, you could be referring to several distinct things.
A world of only a set of contingent existences is inconceivable without a necessary existence.
I am unconvinced this is true.
The set of contingent existences cannot depend on itself to bring itself into existence.
This may just be a quirk of your usage of language, but the set of contingent existences is not itself an existence. As such, there is no sense in which such a set needs to be brought into existence. The set of contingent existences may be the empty set, or it may be some other set, but it is never not a set, and there is no sense in which one can say that there is no set of contingent existences.
Therefore, there exists a necessary being that brought this set into existence , whereas it itself depended on nothing to be existent.
Again, the set does not need to be brought into existence. For the notion of "being brought into existence" to even be meaningful, one has to be able to talk about the entity not existing at some point, and then existing at some later point. Ultimately, this means that "being brought into existence" requires spacetime to be coherent. It is a spatiotemporal concept, and not fundamentally ontological. So if we are talking about some set of contingent existences without spacetime, then there is no meaningful sense in which you can say individual entities in the set can be brought into existence, much less the entire set.
prove me wrong
Before we try to prove each other wrong, we should probably try to clarify the language and understand the claims being made.
stage 2 , how many necessary beings ? stage 3, what are the features of this being ?
Okay, so listen, how about we focus only on stage 1 right now? There is no point in you bringig up these other stages if I already disagree with the argument on stage 1.
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@youwillwin7107 The teleological argument can actually be steel manned, but even then, can still be debunked. This is done using Bayes' theorem. The idea is simply this: The evidence that we observe (E) is better explained by monotheism (M) than by atheism (~M). The argument considers that P(E | ~M) << 1 and P(E | M) >> 0, and the conclusion is that given E, M is more likely than ~M, which is to say, P(M | E) >> P(~M | E). By Bayes' theorem, P(M | E) = P(M)·P(E | M)/P(E), and P(~M | E) = P(~M)·P(E | ~M)/P(E). The idea is that P(E | M) >> 0 means that, as long as P(M)/P(E) >> 0, P(M | E) >> 0, while P(E | ~M) << 1 means P(~M | E) << 1. But the problem here is the assumption P(M)/P(E) >> 0, which is impossible to justify.
So the teleological argument, as even its strongest version is not sound.
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@alveolate It could be an issue with terminology, but even if you use the most elementary English explanation, this is still confusing to people, because it is counterintuitive. What it boils down to is that humans are inherently terrible at doing mathematics. The conclusion that we tend to draw based on our intuitions are in complete disagreement with the actual logic and mathematical proofs. Our intuitions are very much anti-mathematical in the way they work, this is why we invented computers. Even the best mathematician in the world is bad at mathematics, in the sense that they really do have to fight against their own intuitions in order to be able to provide sound mathematical work. Not everyone has the ability to fight off those intuitions, however, which is why not everyone can be a mathematician.
For whatever reason, humans evolved in such a way that they have a predisposition to assume things like lim a(n)/b(n) (n —> ∞) exists implies lim a(n) – b(n) (n —> ∞) exists, even though this is false in general. What you can conclude instead is that lim a(n)/b(n) (n —> ∞) exists implies lim ln(a(n)) – ln(b(n)) (n —> ∞) exists. In particular, if lim a(n)/b(n) (n —> ∞) = L, then lim ln(a(n)) – ln(b(n)) (n —> ∞) = ln(L). This means, the difference of the logarithms of the sequences converges, but not necessarily the difference of the sequences themselves. You can go even further in the analysis: let ln(a(n)) = c(n), and let ln(b(n)) = d(n). So, lim a(n)/b(n) (n —> ∞) = lim exp(c(n))/exp(d(n)) (n —> ∞) = lim exp(c(n) – d(n)) (n —> ∞) = L, so lim c(n) – d(n) (n —> ∞) = ln(L), which means the difference of c and d converges to ln(L). However, lim a(n) – b(n) (n —> ∞) = lim exp(c(n)) – exp(d(n)) (n —> ∞) = lim exp(d(n))·(exp(c(n) – d(n)) – 1) (n —> ∞), and now it becomes obvious how the difference between a and b can diverge, since lim exp(d(n)) (n —> ∞) does not exist. The only way this limit can exist is specifically if lim c(n) – d(n) (n —> ∞) = 0, but that alone does not guarantee it anyway.
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@sidarthur8706 Why would a god have to be beyond understanding?
Labeling any system which can be understood by the scientific method, or by otherwise reliable epistemic criteria, by the name of a "god" or "deity" is redundant. The entire point of theism, and the religions for which it serves as a backbone, is that there exist supernatural beings, which, as they transcend all other forms of knowledge, they can only be understood via divine revelation. Even deism, which does not claim a personal deity, still claims that there was a 'who,' a creator, who is beyond the comprehension. That is the thesis of these worldviews. Sure, you can redefine the word "god" to mean whatever you want. You can call your cellphone "God" if you want, but we are having a philosophical discussion, we are not playing word games here. So, the discussion has to start by acknowledging the actual thesis of theism, and discuss it, and that is what I am doing.
How do you know that the natural laws apply everywhere, and not only in the bit that we're aware of?
We have very strong evidence for the homogeneity and isotropy of the universe, save for a few anomalies, and we have even stronger evidence for the general theory of relativity, which, among various things, holds that the laws of physics are covariant. Meanwhile, we have no evidence that there could even be a mechanism by which the behavior of the universe could be any different than what we observe it to be in parts where it is unobservable. So, from a scientific standpoint, it makes absolutely no sense to actually conclude that the laws of the universe are somehow not the same everywhere, and it also makes no philosophical sense, as Occam's razor applies here.
Why do you think that what God happened to do is what he was compelled to do?
I do not? I never claimed God exists. I do not hold that God exists. I am a physicalist.
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@DoofusChungus How you define physicalism depends exactly on the variant of physicalism you are considering (for instance, token physicalism is not equivalent with supervenience physicalism, the latter being the most popular variant, and the one I subscribe to, mostly), but all physicalists agree on the causal closure I described earlier, and all physicalists agree that what defines "physical" should be in reference to what the scientific method calls "physical" or "natural." A succinct way that I like to put it is this: if a conceptual entity has well-defined properties that in any way make reference to a causal process or property that characterize the entity, then there can exist a conceivable physical theory of that entity in principle. This would, therefore mean that the claim that the only way to know said entity is via something like "divine revelation" is false, unless you also think that divine revelation itself can be understood solely in terms of physical theory (and/or chemistry, biology, etc.), but the latter view is rare, and is basically something that most physicalists would say does not exist, since at that point, labeling it "divine revelation" is not really justified.
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@sidarthur8706 you're specifically discussing the thesis of some forms of christianity.
No, this is not exclusive to Christianity. Sikhist doctrines maintains classical monotheism. So do Judaism, Zoroastrianism, Islam, Mandeism, and several other major religions. Though, to be clear, the definition of the word "god" given is not exclusive to classical monotheism either. It applies to many forms of polytheism just as well, and to other forms of monotheism.
the pantheistic god isn't supernatural.
I agree, and I maintain I said earlier. Calling the "pantheistic god" by the label of a "god" is semantically unreasonable.
odin isn't supernatural.
Odin is very much supernatural. I have no idea where you get the idea that he is not.
you've argued from your conclusion back to your conclusion.
You have not demonstrated how this is the case at all. Simply making a claim does not make it true, and this is what you are doing here.
we have evidence that the universe is probably homogenous but that's not the same as knowing that it really is...
Do you not understand how the scientific works? Or even just how the concept of evidence works? Knowledge is defined in terms of evidence. I never claimed to have "absolute certainty," but that is because having "absolute certainty" about anything is categorically impossible. It is an incoherent notion, and this is why in epistemology, it is not taken seriously. Instead, we work with the idea of sufficient justification or sufficient evidence. This is how we distinguish knowledge from belief. As for the scientific method, all conclusions drawn by science are tentative. The existence of gravity is tentative. The roundness of the Earth is tentative. It is not impossible that, one day, we will find sufficient evidence that overturns everything that we know today. Yet, I do not see you doubting the roundness of the Earth. So, quite frankly, and with all due respect, your objection is very ridiculous. You are pretending to, all of the sudden, for this one very specific scenario, not understand the basics of the concept of scientific evidence, solely to try to defend a completely untenable worldview.
or that there aren't alternative laws of physics in other universes.
Why are you bringing other universes into this? There is no evidence that other universes exist. Also, I should point out that, at this stage, everything you are saying is a strawman. I never made stated assumptions about the laws of the universe prior to your replying to me. You decided to bring that up on your own, because the only thing I had said prior to you replying is that the fine-tuning is in contradiction with theism, and that a theist who presents the counterargument you had presented is missing the point. But, I suppose reading comprehension is difficult, so it is what it is.
actually from a physicalist perspective and on the anthropic principle you'd surely have to assume that besides the universe that we know there must have been failed experiments of nature with laws of physics that couldn't sustain a universe and which were outsurvived by ours
No. Neither physicalism, nor the anthropic principle, imply this conclusion. The proposal that there were "failed universes prior to ours" is unfalsifiable, and, strictly speaking, not really a scientific hypothesis. This is about as unnecessary of an assumption as assuming that a god exists.
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@DoofusChungus That's the problem I have with your argument. Much of it is based on "you don't know," which yeah, but neither do you.
You saying this proves you do not understand my argument at all. You see, my argument does not require me to be able to know anything about God's mind. Yours, however, does. Therefore, me not knowing is not a problem for me, but you not knowing categorically debunks your argument. My argument does not rely on pretending to know what God wants. Yours does.
I'm arguing that I believe that God created the universe a certain way, and the reasoning and possibility behind it.
No, you are not doing that. You have presented zero reasoning behind God creating the universe the way you claim They have created it. You made the baseless assertion of "God would have wanted the universe to make sense." Sorry, but baseless assertions are not reasoning. You made the baseless assertion of "God has a plan." Sorry, but baseless assertions are not reasoning. If you want to present this as an explanation, then you have to provide evidence, which you do not possess. This is actually pretty funny, because I never asked you to explain why God made the decisions you claim They made. I never questioned those decisions, and the fact that you keep insisting that I did question them makes me question your lacking in reading comprehension skills, and your intellectual honesty. Still, I am giving you the benefit of the doubt here. Anyhow, all I did was refute the fine-tuning argument by appealing to God's omnipotence, and you made the decision to start trying to explain God's decisions, as if that somehow addresses my refutation, which it does not.
I could assume he has a plan,...
Well, don't. If you are not going to provide evidence for you claim of God having a plan, then I am not interested in hearing said claim.
...if you "grant me the existence of God," then him having a plan for everything is almost guaranteed,...
No, it is not. The fact that God exists, created the universe, and is omnipotent, does not imply God has a plan for the universe. That is not how logic works.
...as knowing everything would mean knowing the future, and knowing the future would mean creating things a certain way for that future to come into fruition...
No, it does not. Having knowledge about the future does not mean you have a plan for the future, nor does it mean you are actually interested in the future or care about it. All of these assumptions you are making are basless conjectures. This is not "reasoning." This is not "an explanation." This is you making a bunch on unjustifiable, unknowable claims, and expecting me to be like «ah, you got me, I guess God exists now.» Sorry, but no: that is not how any of this works. You cannot just make claims about God without substantiating those claims. The only things I have granted you here are (a) God exists (b) God created the universe (c) God is omnipotent and knows the future. Having a plan does not follow from those three premises at all! Your argument is invalid.
Again, if you grant me the existence of God, the fact that these astronomically low probability events happen again and again for humanity to be where we are...
No no no no. Stop right there. We already talked about this. These events are not low probability events. They are events of probability 1, the highest possible probability. Why? Because the universe is deterministic, not random. The laws of physics are deterministic. I granted you the existence of God, but your claim about these events being random remains scientifically false.
...means he's gotta have a plan...
If it were true that the events were low probability, then maybe. But, you are wrong about the events being low probability.
Exactly, the minuscule chance that all of this stuff happened is so low...
No, it is not. You are just pulling claims out of your ass now. Somehow, whenever you make a scientific claim, you happen to be wrong, every single time.
You are arguing against my belief in God through your perspective of physicalism...
Okay, you are very confused here. My holding a belief in physicalism is not the same as making an argument for physicalism. I have not presented any arguments for physicalism in this thread, not yet, anyway. It is entirely possible to have a belief, and not actually present an argument for that belief. The topic of this discussion is not "does God exist?," nor is it "is physicalism true?" The topic of this discussion is "is the fine-tuning argument sound?" Me saying it is not sound does not actually constitute an argument against the existence of God, and I have no idea why you keep insisting that it is.
And by the way, I only brought up physicalism, explicitly to address your claim that atheists believe in random chance. We do not. I did not bring physicalism up as a way to refute the fine-tuning argument, or as a way to argue against God's existence. You would have known this if you had, you know, read my comments carefully.
which I'm course not offended by...
Are you sure? I don't know, brother, but you seem fairly offended to me. You literally lashed out at me earlier being all like "Fine, you win, God doesn't exist anymore," which is the adult equivalent of throwing a tantrum. You have also continued to repeatedly misrepresent my position, almost as if you are unable to handle facing my argument head on.
but still, you are arguing against God,...
My guy. I granted you the existence of God for the sake of discussion. I do not know in what language I should be trying to explain this to you, but apparently, it is not English.
We don't know what he had in mind when creating everything.
See, you say this, but then you are turning around and contradicting yourself by claiming that we can know for sure that God had some kind of plan, and then you are refusing to present evidence for that claim. Do you not see how the conversation cannot go anywhere when you do that? Pick a lane. Do we know for a fact that God had a plan? If so, present the evidence. If not, then, my argument against fine-tuning stands. That is all there is to it.
My original argument was simply arguing for the existence of God being needed for everything to play out the way it did.
See, now you are shifting the goalpost. OP's comment, the comment that started the thread, was about the fine-tuning argument being refuted by God's claimed omnipotence. Everyone else's comments were about that too, and you replied to those comments by defending the fine-tuning argument. Then, I replied to your replies, also by discussing the fine-tuning argument. You replied to me by, again, defending the fine-tuning argument. And now, you are trying to pretend that, all along, you were actually making an entirely different argument for God's existence, rather than defending the fine-tuning argument. Not only that, but now, you are also trying to pretend I was arguing against that argument, and against your beliefs in general, and not against the fine-tuning argument, even though that is the argument that I explicitly stated I was arguing against in the beginning of the thread. You are either being extremely careless with the way you read comments, or you are dishonest.
But, okay. Fine. Have it your way. Since you are not going to accept that this discussion is about the fine-tuning argument, even though everyone else is on the same page about that (which, by the way, makes your behavior here pretty unreasonable), let's talk about your other argument instead. Let me put it in syllogistic form for you:
(A) Many improbable events had to occur for life to exist.
(B) Since life does exist, those events either happened by random chance, despite being highly improbable, or God made them occur.
(C) It is more reasonable to believe God made them occur.
Conclusion: Therefore, more likely than not, God made those events occur.
Is this an accurate formulation of your argument?
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@DoofusChungus To disprove what I said, you gotta have proof, too, no?
If what you said is a baseless assertion, then no, I do not have to provide proof that your assertion is false. I can simply dismiss the assertion, because I have no epistemic obligation to accept an assertion for which no evidence has been presented.
You're argument requires you to assume God has no plan, or at least may not have had a plan.
My argument only requires assuming that it is not impossible that God had no plan. In this case, this would be the equivalent of the null hypothesis: as concluding it would have been impossible demands evidence.
I have said that for God to have wanted to create the universe a certain way with the outcomes it has, then there has to be a plan.
I know you have said this, but it is still just another baseless assertion. The premise "God wanted to create a universe" does not imply "God had a plan."
He didn't just create the universe and say "Well, I hope life forms."
You are assuming that God's primary goal was the existence of life, though. Maybe God created the universe because They really like black holes. After all, this universe is extremely good at producing black holes. The universe is definitely much better at producing black holes, than at harboring life. Perhaps God created the universe for some other reason. Perhaps God was just really bored, so They wanted create something. Either way, as there are multiple possibilities, and there is no evidence for any of them at all, your assumption is baseless.
And fine-tuning means having a plan, no?
Yes, but as I am arguing against fine-tuning argument, I am also arguing against the assumption that God had a plan. There is no reason to make that assumption.
It means that God created the universe a certain way, so it would work in a way he wanted it too.
Whatever plans God had, if any at all, can be accomplished in any universe. The universe does not need to be created a certain way for things to work the way God wants them to.
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@DoofusChungus You call my points invalid because I have no clue what God's intention was,...
Yes, because the validity of your argument is directly contingent on knowing God's intentions, even though you yourself admitted it is impossible to know. My refutation does not require me to know God's intentions. My refutation only requires me to, for the sake of argument, assume God is omnipotent, but you already agree with that, so you have no rational justification for not accepting my refutation of the fine-tuning argument. Then again, you have no understanding of what the fine-tuning argument even is, so I guess that expectation on my part is unreasonable.
...although everything points to life as the specific events that happened were very much only there to create life,...
No, this is a baseless assertion. Sure, the events led to the existence of life, but many events happened that had nothing to do with the existence of life, and many events that happened are actually bad for the existence of life.
...and create a world that works.
Again with this nonsense. There is no such a thing as "world that works" or "world that does not work" for an omnipotent being. Do you actually not undeestand what the word "omnipotent" means? Do you even actually believe God is omnipotent? Because I have been suspecting for a while that you do not actually believe God is omnipotent.
That would be like me saying "nope, physicalism is false because you don't have proof that there's only materialism."
No, it is nothing like that at all, because at no point have I ever claimed your assertions about God are false. I only claimed they are baseless, and since they are baseless, I can dismiss them. And here is the thing: if I were to come out and tell you "well, physicalism is true anyway, so nothing you say about God actually matters," you would be completely justified in dismissing physicalism on the basis of me claiming it to be true, despite not providing evidence for it. Because, yes, that is how the burden of proof works. Right now, the burden of proof is on you to prove your claims, and if you are unable to, then I am 100% epistemically justified in ignoring and dismissing those claims. Again, that is how it works. You may not like it. You may not be comfortable with it, but it is how it works.
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@DoofusChungus Also, I should mention that while I merely have dismissed your claims about God to be baseless, your claims about the universe are indeed false, and I have actually explained how and why they are false, I have provided you with evidence. Every single time, though, you have ignored the parts of my comments that actually provide said evidence. For example, you keep saying "these events have astronomically low probability." No, they are not. Their probability is 1. You keep saying that chemistry and the laws of physics are random. No, they are not. If they were random, then there would be no such a thing as the laws of physics. Also, it is very obvious to scientists that this universe is not "designed for life," since it is extremely bad at harboring life: almost all of it is uninhabitable. This universe is much better at making black holes, so your argument here really should be "God created the universe because They like black holes." However, there is no way you will ever actually argue for that.
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@DoofusChungus You keep saying my points are baseless because "the probability is 1".
Yes. You claim that the events have low probability. They do not. Your claim is false. Since your argument relies on this false claim, your argument is unsound.
Yes, the probability is 1 because it did happen, that's how it works.
No, that is not how probability works. The probability of an event does not depend on whether it has happened or not. People who make this claim are people who pretend to understand Bayes' theorem, but do not. The reason real-life events have probability 1 is that the universe is deterministic. It is not random. Random events do not exist. It is physically impossible, as far as the evidence points, for processes to be truly random.
Yet scientists best guesses as to what happened to lead us here, and what has been proven to lead us here, rely on such random chance and coincidence, over and over.
No, this is false, and a claim that you pulled out of your butthole. There are no reliable scientific studies suggesting that random events actually exist. Even dice throws are not random.
All I've seen you say to rebut this is "it happened tho, so you're wrong".
No. You are lying. I never used this as my response.
Please explain to me how the perfect elements that we needed were carried on meteors and just so happened to hit Earth...
The perfect elements? There are many types of aminoacids, and organic compounds, that could have served as the basis for life to form. The reason we do not know how life originated is not the lack of scientific explanations, but rather, that there are too many scientific explanations: there are too many combinations which are possible which would lead to life, and we do not know which of the possible ones exactly happened.
As for the meteor hitting the Earth... that is completely mundane. All planets are hit by a meteor at some point. Remember, though, that Earth was hit by a meteor only half a billion years after the Solar System was formed. Meteors would have been far more abundant because of this, back then.
...but only after it cooled down from being a flaming ball of fire is just luck, I guess?
Nope. It is not luck. It's called "physics." Perhaps you have never taken a physics course before, so you have no understanding of physical processes work. But I have a degree in physics.
Please explain to me how, Earth just so happened to crash into another planet randomly so we can have the moon, just luck, I guess?
Planetary collisions are not uncommon during the early stages of stellar systems. There is nothing lucky about it, just the natural order of things. Again, it's called "physics."
Please explain to me how whatever wiped out the dinosaurs was impactful enough to get rid of them, yet mammals and other animals were spared, just luck, I guess?
Are you scientifically illiterate? These are all questions answered by high school level textbooks. The only mammals that survived were marine mammals, and very small mammals, like proto-lemurs and rodents, and such. Why did they survive? Because, as they were small, they required significantly less oxygen than dinosaurs did. So, while the dinosaurs were asphyxiating from the lack of oxygen, caused by the debris-filled atmosphere from the impact, small mammals were still capable of breathing and obtaining food. In fact, small dinosaurs survived too! And they evolved into the modern species of reptiles we have today, and they also evolved into modern avians. The ones that went extinct were the large dinosaurs, due to lack of oxygen and nutrition, because the meteor impact had catastrophic effects on the global climate.
And you're right, when we say anything about God, there is no hard proof. I can't just pull out a picture of God and say, "There, proof."
Don't be ridiculous. I am not asking you to pull out a picture of God, obviously. But surely, as a Christian, you can do better than no evidence at all... right?
But yet the burden of proof doesn't only fall on me though, because you claim to have disproven my other points,...
No, it absolutely does fall on you. You are the one making the claims here, not me. I am merely responding to your claims by (a) dismissing them, or (b) showing that the science disproves them.
but until you provide proof that God doesn't exist, imma dismiss your claims.
No. That is not how the burden of proof works. You are being unreasonable. I have not made any claims that God does not exist. In fact, I have, for the sake of argument, granted the existence of God, in order to debunk your arguments. So, I am not required to provide any proof here, because I have not made that claim. You do not get to dismiss my claims, because the only claims I have made are supported by scientific evidence. The claims you have made are not.
Exactly my point: the universe is extremely bad at harboring life yet we're here.
What you are failing to realize is that the universe being bad at harboring life contradicts your argument, but is consistent with science. Life existing despite that, is also consistent with science. Your worldview is incapable of explaining why the universe is bad at harboring life. Mine is not.
Also, I can't believe you could assume that the universe is bad at harboring life.
How many planets have we discovered? Several thousands of planets. How many of them do we know of harbor life? 1, and that is planet Earth, where we live. So, less than 0.1% of planets that have been studied are capable of harboring life. That counts as "pretty bad at harboring life." If you teleport a living being to anywhere in the universe, there is 99.99999999% that they will die immediately. More likely than not, you will either end up in (a) an empty void, (b) inside a hypergiant star, (c) inside a supermassive blackhole. In the off chance that you actually land somewhere else: if you land in some other type of star, instant death. If you land in some other type of blackhole, that is almost instant death. If you actually happen to land on a planet... most of them can kill you in matters of microseconds, if not faster. Even among the planets that are actually Earth-like and apparently inhabitable, no life has been found. Furthermore, solar systems like the Sun, which do have a goldilocks zone, are pretty normal, and despite this, still no life in other places has been confirmed. This alone is conclusive to demonstrate that life in the universe is pretty rare. That does not mean there is no extraterrestrial life of any kind. It just means, the universe is sufficiently bad at harboring life, that even though life may exist elsewhere, it was still definitely not designed to harbor life.
That's baseless, as only the Milky Way might be bad at it.
Except, that argument does not work, because the Milky Way is a very typical spiral galaxy, and is actually among the few galaxy types capable of supporting a galactic habitable zone.
Where's your proof?
Where is yours? I just presented you with mine, but I know you are still not going to provide me with any evidence, because you are incapable of doing so. So, basically, your position is untenable and unreasonable. Furthermore, you are extremely lacking in basic knowledge of biochemistry, astrophysics, probability theory, Big Bang cosmology, epistemology, and basic propositional logic.
I rest my case.
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@sidarthur8706 i didn't have to show how your arguments are circular, you've done it yourself for me.
Sorry, but no, this is not how this works. You cannot claim a fallacy was committed without explaining how the argument presented is a fallacy. Ironically, doing such a thing is itself a fallacy. And dishonest.
the pantheistic god doesn't help you make your case so you've deliberately left it out of the exact definition of a god that you needed for your conclusion to follow from it.
Calling the pantheistic "god" by that name is not substantially different from calling your cellphone a god. I apologize for not considering such semantic sophistry to be a real argument worth addressing.
that's not a straw man, it really is your argument.
You are seriously lacking in reading comprehension skills, or you are just fragrantly dishonest. I did not say that you telling me I was leaving out the pantheistic god is a strawman. My accusation of you strawmanning me was specifically in reply to you bringing up the laws of physics. I even quoted the exact part of your comment I was replying to.
Well, seeing as how you are either illiterate, or dishonest (possibly both), I will not read the rest of your comment, and I will not reply to it. That would be a waste of my time. Have a nice day, and hopefully, your approach to conversations will change in the future.
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@DoofusChungus Wait, so you're telling me random events do not exist?
Yes! I have been telling you that since the very beginning of the conversation! Finally! Took you long enough to realize it.
So, then, without something pushing it to happen, how does it happen?
Can you provide an example of anything that has ever happened without something pushing it to happen? No, you cannot.
You know exactly what I mean when I say random.
I really do not, because you apparently do not know the definition of the word random.
If I'm walking down the street, and a completely random person unrelated to me in any way walks up to me and shoots me because he felt like killing someone, that's not random?
No, it is not. There is an actual cause behind why the person would have done so. Actually knowing the cause is nigh-impossible, but this is only because the number of variables one would have to know to accurate have predicted such an outcome is very large, and the amount of precision with which one would have to know them is also unmatched by current technology. Even then, there is still a lot that can be said about the causation of the sequence of events. If the person shot you, as opposed to someone else, this means that you looked easier to kill, and that it would be harder for your body to be discovered, than if they shot someone else. Of course, this assumes the shooter is actually somewhat sane. If they are insane, then... that already contributes significantly to a causal explanation for why they shot you. Much of the focus of forensic psychology is actually to understand such scenarios. You act like no research has ever been done on the topic. You have the digital era in your hands, and you do nothing with it.
Dice rolls are random.
No, they are not. You can use Newton's laws to model the trajectory of the throw of a die if you have sufficiently precise measurements of the initial conditions of the throw. In practice, this is difficult to achieve, even with machines, but it is not impossible.
They are determined by how a person throws it, how it hits the table, what the table is made out of, etc.
Yes... that makes it not random by definition. See, this right here proves that you have no friggin' clue as to what the word "random" actually means. You really do like to use words and throw them around in a conversation without knowing what they mean. If dice were truly random, then the outcome of a throw would not depend on any variables at all. You would not be able to even slightly manipulate the probabilities, regardles of how much information you had: there would be no equation that would allow you to predict the outcome, even in principle.
So, what you're arguing randomness is, sure, nothing is random.
What I am "arguing" randomness is is the actual definition of randomness, and the one that nearly everyone uses.
Stop trying to get me on scientific hypotheses and stuff like that.
No, I will not. I respect you as an individual, but I would be lying to you I told you that know anything about science. You talked about the Big Bang theory, and you got it wrong. You did not know basic facts about biology that are answered by standard high school textbooks. You did not know that brain waves and radio waves are just examples of electromagnetic phenomena. You are out here trying to pretend you know science, and you are arguing with a physicist who is correcting you on these topics, and you have the audacity to tell me to stop. I am sorry that you fail to realize how unreasonable this is.
Are YOUR literacy skills undeveloped,...
You cannot say that, and then fail to put a question mark at the end of a question.
...because what I said was the best guess' as to what led to life as we know it, such as the meteories carrying the right chemicals and elements,...
I addressed this already. A meteor with the appropriate organic molecules landing on Earth is not a random event. In fact, there is nothing abnormal about. Many meteors and comets have hit the Earth.
...all the events that led to life rely on random chance.
No, they do not. Biochemistry is not random. We can predict chemical reactions with high accuracy, much more easily than we can predict a dice roll. We can predict astrophysics with even higher accuracy.
See, my point exactly, you rely on my wording to try to prove me wrong.
So, what, you want me to rely on words you did not say, and strawman you? You want me to lie? No. Stop.
...you wanna know what there's more of than chemicals and elements that can begin life? Elements and chemicals that cannot.
And? So what? Every carbon-based molecule can begin life. There are more of those than you can count.
And we do know the ones that happened, the ones that are being used currently so we can live.
Nope, that is false. Of course, life today is still carbon-based as it was in the past, but the exact molecules and chemistry involved are not all the same. Life has evolved for billions of years.
My whole point out about the meteor is that it needed to be to such a degree that the dinosaurs were killed, but other life wasn't.
And your point is false. What happened with the dinosaurs is not specific to that meteor. Any meteor that would have hit the Earth would have caused the same effects, because that is just how meteor impacts work. Also, there are many other events that can do exactly the same thing, such as mass supervolcanic eruptions, and the Milankovich cycles of the planet Earth, combined with the greenhouse effect.
What I meant was the fact that for the elements to survive, they couldn't hit too early, because, you know, ball of fire.
And? Plenty of meteors hit the Earth during that time, and many of them probably did contain those elements. The fact that the Great Bombardment kept happening after the Earth stopped being a molten ball is not at all abnormal.
But our systems orbit patterns make it incredibly hard for planetary collisions.
The orbits of systems today, yes. Not the orbits from when the Solar System was a freaking baby. Orbits take a while to stabilize in the formation of stellar systems. What, you really thought the Solar System from 4.5 billion years ago looked identical to today's? Oh my Siesta.
Just to make sure I wasn't misremembering anything, I went to Google and it literally uses the word "uncommon."
Citation needed. I have looked at several sources, such as NASA, and Centauri Dreams, and these sources, if anything claim that they are common, not "uncommon." For example, https : // www . centauri - dreams . org / 2020 / 07 / 16 / planetary-collisions-and-their-consequences (type the URL into your browser, without the spaces) explicitly claims «Let’s hope we never share such a fate, but it’s likely that collisions are commonplace in the late stages of planet formation, and many researchers believe that Earth’s Moon was the result of the collision of our planet with a Mars-sized planet about 4.5 billion years ago.» Also, in the other thread you commented on, I demonstrated that you are not great at actually doing searches on engines for a topic anyway, so I am skeptical of your search. Incidentally, you also never cite the sources you find in your alleged search. I have no reason to think you are not lying, especially as I have already caught you lying in previous replies.
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@DoofusChungus Did you forget about the part where it had to be to a perfect degree that the dinosaurs weren't capable of living, but other animals were.
You say this as if it would have been difficult for this to happen. It was not difficult, this is literally just how meteor impacts work.
A little bit faster or bigger and everything's gone, a little slower or smaller and nothing is.
This is not substantiated by any science at all, this is just a claim you pulled out of your butt.
You have yet to debunk all these very improbable events happening, one after another.
The events are not improbable, and I already have proven this multiple times. You being unreasonable and actively choosing to ignore my explanations has nothing to do with me.
You keep saying that "they happened so, there, debunked."
I have never made this claim. You are LYING. You should be ashamed of yourself, and I would say you are lucky that the Bible does not actually ever condemn lying, but you should still be ashamed of the fact that you think lying is acceptable in a philosophical conversation. Honestly, I am not even sure you are worth any respect from me now. Perhaps I should hit the "mute" button on your name and start ignoring all your replies, as it seems that trying to have a conversation with you is a waste of time. You lack the intellectual honesty to admit that you are not able to refute my arguments, so you have to resort to lying in order to try to make it seem like there is a problem with my arguments, as if I would be too stupid to not realize that you are lying. Well, listen: perhaps the average atheist you interact with on your day-to-day life is stupid, but not me. You chose to mess with the wrong atheist. When people lie, I call them out without hesitation. If you are going to behave dishonestly, then I have no qualms scolding you whatsoever. I quote your exact words, to ensure I am not misrepresenting you, and I also go back and check my own comments, to see if people are misrepresenting me. Perhaps others let this kind of bullcrap from you slide. Not me. I hope this is clear.
You even explain how they happen in a scientific sense for me, saving the me trouble.
Saving you the trouble? No, this actually spells trouble for you. The fact that I can provide science-based deterministic explanations for these events categorically debunks the claim that they are random.
...all this stuff happening, whether you wanna admit is random or not, there's so much chance involved with.
sigh You can insist "there is chance involved" all you want, but you have no evidence for your claim. Furthermore, I have presented evidence against your claim. In response, all you have done is act along the lines of covering your ears, and going "LA LA LA LA LA I CAN'T HEAR YOU I CAN'T HEAR YOU," rather than acknowledging the evidence. I may as well be talking to a brick wall here, but that actually might be an insult to a brick wall, since a brick wall does not have the ability to listen. The brick wall also does not lie about my claims, like you have been doing. At this point, it is very hard to take you seriously, and I am very close to just deciding to stop talking to you.
That is proof...
No, it is not. A claim that has been debunked as false by deterministic (read: deterministic implies non-random) explanations cannot serve as proof anything.
...even if you don't see it as proof, then I guess we'll call it a theory,...
I want to smash a table... no, my guy, you cannot do that. Okay, so I guess you also do not know the definition of the word "theory." Let me give you the first paragraph of Wikipedia's "scientific theory,": «A scientific theory is an explanation of an aspect of the natural world and universe that has been repeatedly tested and corroborated in accordance with the scientific method, using accepted protocols of observation, measurement, and evaluation of results. Where possible, theories are tested under controlled conditions in an experiment.[1][2] In circumstances not amenable to experimental testing, theories are evaluated through principles of abductive reasoning. Established scientific theories have withstood rigorous scrutiny and embody scientific knowledge.[3]» The sources cited for this claim are the National Academy of Sciences; Winther, Rasmus G. "The Structure of Scientific Theories," and Schafersman, Steven D, "An Introduction to Science."
...as that's what Christianity mainly is. As the whole point of it is faith.
The reliance on faith makes it (a) not a form of knowledge at all, and (b) not a theory at all.
Also, you are the one making claims as well, as you claim my arguments are false.
Yes, but here is the difference: I have provided reasoning and evidence for my claims, and I can actually cite sources, as I already have. Meanwhile, you are ignoring all of that evidence, and repeating nonsense I already dismissed as having no evidence. On top of that, you are literally lying about my arguments. So much for being a Christian. Is the name of your belief system "Lying for Jesus Christ?"
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@DoofusChungus You can't respond to a point with "Well, I read a study that says you're wrong," and then not supply proof.
Yes, I can. As Christopher Hitchens once said, "That which is asserted without evidence can be dismissed without evidence." You have made many assertions, but have provided no evidence for any of them. As such, I have no epistemic obligation to actually accept them, and I can readily dismiss them. In fact, I have no epistemic obligation to even acknowledge your claims, since no evidence was presented for them. Despite that, because I actually wanted to educate you, for you to learn something about the world, rather than stay intensely ignorant on the topics of discussion, I went out of my way to actually explain to you why several of your claims are false. I have provided proof, and I have cited sources, whilst you have done neither. The fact that you chose to ignore said evidence is your problem, not mine. I already met my burden of proof for the claims I made. The only claims I have not proven are the claims that you say I made, but did not make. But, that is just you lying. I have no obligation to defend claims I never made, regardless of how much you want to insist I did make them.
And actually, I made it really easy for you to learn about the topic by citing sources aimed at laypeople, rather than trustworthy scholarly sources. I could have bombarded you instead with the hundreds of scientific peer-reviewed studies that I have read on the various topics we have discussed, and gone into detail as to how each of them proves my point. However, since everyone can tell from 1000 km away that you are not qualified to do, so much as read a published scientific study, and actually understand any of the stuff they are talking about (considering you fail to understand even basic physics and such), you would not actually learn anything. You would stay confused, and you would not have an opportunity to understand why your arguments are incorrect. At worst, you would completely misinterpret what they said in the studies, and develop some really bizarre ideas about the universe that are just not true. So, I avoided doing this instead.
You think I have read one single study, and concluded you are wrong? Let me put your arrogant self in your place then, because you need to be humbled: I did not read "one" study. I am a physicist. I have read more studies than you ever will in your entire life. I have written a thesis for my undergraduate degree in a methodology comparison between cosmology and quantum mechanics. I have a degree in physics, and another degree in philosophy. I own a dozen of textbooks in philosophy, and about two dozen in physics. I have studied mathematics much more advanced than you or anyone in your circle of friends will ever encounter. Have you ever heard of tensor calculus? I doubt it, and you probably don't even understand the definition of a "tensor." Have you ever heard of category theory? Do you know what a morphism is? No, of course not. Normally, making these assumptions about a stranger is unjustified, but I have learned quite a lot about you from these conversations. I have also studied various religions, beyond just Christianity. I probably have a better grasp of Judaism and Islam than you do. I have studied Shintoism and read the creation myth, as well as several other classics of Shinto mythology, such as the Tale of the Bamboo Cutter. I have studied some Japanese as well, though that was primarily in summer programs when I was in high school. I also own a beautiful copy of the Dao de Ching, I own a copy of the Analects, copies of several scholarly texts about Buddhism, I know the basics of Fengshui animism in China, and I even understand some of the Yoruba religions, due to my family ancestry. Now, let me ask you a question: have you ever put this much effort into understanding other people's beliefs? I doubt it. And you have definitely never put as much effort into the sciences. You think I am sitting in an armchair, just doing random Google searches, just so I can be like "ha! See? You're wrong!" Well, no. This is all knowledge I have accumulated over the years, after having had my beliefs changed several times, after having put blood, sweat, and tears into my studies.
I am not telling you all of this to brag. Look, I know absolutely nothing about theater. I know absolutely nothing about starting a business, or about making a law. I know absolutely nothing about social networking. If you gave me a car with an engine problem, and you asked me to identify the problem with car on my own, it would probably take me 4 years to do it. If you told me to make a scultpture, I would be outperformed by a toddler. So, no, I am not trying to say I am superior in any way. The reason I say all of this is so that you can understand that dismissing what experts have to say on the matter just because you have the arrogance to think you know everything, despite not being able to provide evidence for anything, is only harming you in the end. So, stop doing that. You clearly have the capacity to do better than this. You are just choosing not to. And that is infuriating, especially because you are also so dishonest about it. It honestly makes no sense, because the point of this conversation was never to disprove that God exists. In fact, for the sake of the conversation, I actually sacrificed my viewpoint entirely, and granted his existence. And I know how to do that, because I was Christian for many years in my life. I own three distinct copies of the Bible, and I have read the Bible in all the languages I am fluent in. I have even researched some of the ancient Hebrew and ancient Greek of the original text, which I doubt you have ever done. But, despite me granting you the existence of God, all you did was have tunnel vision and insist that I have to somehow prove that God does not exist, even if this meant lying about my claims. You chose this. The only thing you think about is "do they agree with my belief in God?" You don't actually listen to what they are saying to you, you do not look at the evidence, you do not bother to understand the beliefs of other people, you do not bother to learn about any of the science, even though the information is available for free online and in libraries, you do not bother to look at sources beyond the first Google result that pops up in your most primitive search. You choose to act this way, even though you do not have to, as a Christian. You think the world revolves around your relationship with God? Well, it doesn't. There are 7.8 billion people in the planet, and humans have existed for over 100 000 years. This is why you needed to be humbled, and this is why I told you everything I just said.
But, whatever. My guess is, you will not even read this comment carefully enough anyway, because I can tell you never read any of my other replies to you carefully either. And for that reason, I am going to end the conversation here, since I see no worth in continuing to interact with you. I gave you all the advice I can give you, plenty of evidence, and even tolerated your dishonest behavior. If, even after that, you cannot figure out why you are wrong, then I do not think you are actually rational enough to change your mind. Please reflect on this conversation. Anyway, I hope you have success in your endeavors.
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@user-fb2jb3gz1d where are you getting the definition of omnipotent?
The definition is very simple: we say that x is omnipotent if and only if, for all things y, x can do y.
Because having unlimited power and authority, doesn't mean if you create something, it can't have fine tuning because that would mean you're not omnipotent.
This demonstrates that you do not understand the definition of "omnipotent," nor the definituon of "fine-tuning." Let me define "fine-tuning" for you. "Fine-tuning" refers to the act of taking a free parameter of a system, and matching its value to a very narrow range of values, in order to accomplish a particular goal. For example, if you are listening to a radio station, your receiver's audio frequency has to be tuned to a very narrow range of emitted audio frequencies, in order to be able to listen to a particular radio station of your choice. This is fine-tuning. Matching a frequency to another frequency range is called tuning, but here, it is called fine-tuning, because the range of frequencies is very narrow. The argument that Ben Shapiro, and other monotheistic apologists are presenting, is that the free paramaters of the universe had to be given very specific values, in order for intelligent life to exist. In other words, they are claiming the universe is finely-tuned for the existence of intelligent life. However, if God is omnipotent, and God created the universe, then this is false: the universe cannot be finely-tuned. Why? Because, since God is omnipotent, all values of the free paramaters work equally well for producing intelligent life, since the only factor that would matter in the production of intelligent life is God's Will. Since all values of the free parameters work equally well for the existence of intelligent life in a universe created by God, such a universe is, by definition, not finely-tuned.
It's ignorance of God, that makes one think God is a god of miracles or magic. He isn't.
The Bible claims that God is a god of miracles. If you disagree, then you disagree with the Bible.
We acknowledge laws of everything around us. If God created it all, then he willingly, purposely created it as such. So he understands exactly how it works. It is us who doesn't understand.
This has nothing to do with whether the universe is finely-tuned or not. Can you stick to the topic of conversation, and not avoid my arguments, please? This is cowardly.
You acknowledge God is supposed to be omnipotent so you want him to do whatever without limits. Yet you fail to see that what he created, has limits so what you are asking him to do is to break those limits.
Yes. Is that too much to ask? Then God is not all powerful. The only reason God's creation has limits is because he put them there, no? So, he can also just remove those limitations. Yet, here you are, saying God cannot remove those limitations, resulting in a finely-tuned universe.
You know your arm can only bend so far. Let's break those limits. So now you have a broken arm.
What is the point of this analogy? God is all powerful. God is perfectly capable of doing literally whatever he wants whilst simultaneously leaving the universe unbroken and safe. Yes, if I bend my arm too much, my arm will break. And guess what? I am not all powerful. So, of course it breaks. Someone with sufficient power can bend their arm freely without breaking it, though. Such people do exist in fiction, and I would presume an all powerful being that is not fictional is not an exception to this.
What's the purpose of that? Just to see your arm can break? Why do it if YOU already know the outcome?
Why are you assuming that anything God does will break the universe? You seriously do not believe God is omnipotent.
Let's say this 4 year old does know that and he asks you to go past your arms limit. Are you going to amuse him with what you already know is not a good thing? Of course not.
No, but I would explain to the child precisely why I am refusing to engage with their request. God has never done anything analogous to that.
Does that make you stupid, less knowledgeable or weak? Of course not.
I would say that it does make you less knowledgeable or weak if you are incapable of explaining to the child why you are refusing. Such a task should be trivial for an omnipotent, omniscient being.
Yet you want God do it?
What are you talking about? I never asked God to break the universe. In fact, I am postulating that God can do literally anything he wants to without breaking the universe. You are the one who thinks that anything God does will break the universe. Not me.
You're the r year old asking God to do something he already knows the outcome.
No, I have not asked God to do anything at all. You are strawmanning me.
You're the one making God into this magician.
No, the Bible is the one making God into a magician.
Christians will say miracles, the bible will have examples of these miracles. But it's not a miracle. We just call it that to explain what we do not know.
Are you saying that everything God does can be explained by the laws of physics? Because if so, then God is not omnipotent.
Nowhere in the bible does God call himself a magician.
It does not matter. God is still a magician, even if he does not call himself by that name.
He has a reason for it. You may agree but it's a reason.
I never disagreed. God obviously has a reason to create the universe in this particular way. This does not change the fact that the universe is not finely-tuned.
You obviously don't know the gid according to the bible.
I absolute do. I think I understand the Bible a lot better than you do.
You know the god according to angel mendez-rivera
Nope. I do not have a concept of God, because I am an atheist.
It's not God according to you.
No, it is according to the Bible.
Though you can certainly think so. But that's ridiculous to presume every Christian and I should also.
No, it is not ridiculous for me to assume that you guys think God is omnipotent per the Bible. Yet, here you are, insisting that he is not omnipotent.
That's like telling a star wars fanatic that Luke has a ring and uses the Schwartz because Q says so. Or that orangutans and humans don't share a common ancestor because Q said so.
Nope, these things are not at all analogous to me saying that God is omnipotent.
That's exactly you on the character of the God of the bible.
It really is not. Your reading comprehension skills need vast improvement.
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@ruaraidh74 Atheism makes no claims about the origin of life. Atheism is not a worldview. Again, you are demonstrating that you do not know the definition of the word "atheist." An atheist is, quite literally, someone who is not a theist. That is all. Me not being a theist does not tell you what my worldviews are at all. Currently, you have no way of knowing whether I am a Daoist, Buddhist, Confucianist, legalist, naturalist/physicalist, solipsists, or something else entirely. All of the aforementioned are consistent with not being a theist. The fact that I am not a theist has absolutely no implications on a discussion about the origins of life.
Now, if you want to have a discussion about the origins of life, we can certainly do that, but just know that this has absolutely nothing to do with atheism at all.
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@user-fb2jb3gz1d I'm not saying God is not omnipotent.
Yes, you are. You have made many statements with this precise logical implication, and I have quoted you making those statements.
You have a total different definition of omnipotent than what the accepted definition is.
No, I do not. As I explained in my previous reply, Merriam-Webster, the prestigious dictionary, defines the adjective "omnipotent" as "having virtually unlimited authority or influence." This definition matches my definition.
And you obviously don't know the bible. Your statements prove that.
No, it is you who does not know the Bible.
Give your definition of omnipotent and where you got it from.
I already gave my definition. It is not my fault that you are illiterate.
And give that scripture verse that says God is a god of miracles.
I already did that. Psalm 77:14. Again, it is not my fault you are illiterate. Before accusing me of not knowing the Bible, you should read the Bible. Had you read the Bible, you would have known about Psalms.
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@ruaraidh74 Even if that were a valid argument, it wouldn't be sound, because fine-tuning is an internal critique of atheism.
It does not work as an internal critique of atheism, because the argument concludes classical monotheism is true, even though classical monotheism being true contradicts the existence of fine-tuning. You cannot present something as an internal critique, when it does not actually appeal to properties of the thing it is critiquing, and when the critique fails to be consistent with its solution.
Suppose some effect X. The cause of X can either be of type P or not P. All effects of [cause] type not P suffer from the same probability objection: [teleological argument semantics here]
The supposed "probability objection" to effects of cause type not P amounts to nothing more than a collection of unfalsifiable assertions, in this case, and the premises of the objection are, again, not implied by atheism... because atheism is not a worldview.
Therefore, all else equal, P is a more probable cause of X than not P.
This argument is valid, but not sound, since the premises are unfalsifiable.
Your objection is: "Well, if P is truly the cause of X, then [teleological argument semantics here] don't apply!"
In a sense, yes, this is my objection. If it is improbable that X has a cause of type not P, then it follows that it is more probable that X has a cause of type P, but the problem is that if X does have a cause of type P, then it is not actually the case that X is a real effect, falsifying the premises of the argument, as X not being a real effect means no causal discussion is meaningful. In other words, if the argument is valid, then it is unsound. Since it is unsound, it fails as an internal critique.
The problem is that, in order for your objection to be sound, it requires "P is truly the cause of X."
It does not. The point of my objection is that the argument in question is self-undermining: the conclusion, if it is implied by premises, actually contradicts the premises. The keypoint you are missing is that the premises of an argument can imply a conclusion without either the conclusion or the premises being true.
It supposes that "P caused X" in order to suggest that the probability of (not P) causing X is not subject to the teleological critique.
No, it does not. You are taking two completely different objections, and treating them like they actually are the same objection. You see, here, cause type P refers to the God of classical monotheism, an omnipotent, omniscient creator. But, a universe created by such an entity cannot by finely-tuned. Here, effect X being caused is the existence of a finaly-tuned universe. If the cause of the existence of a finely-tuned universe is an omniscient, omnipotent creator, then said universe is actually not finely-tuned. Hence, we have a contradiction. If you change what X is, and you claim that it is merely the existence of the universe, then there is no problem, but in that case, you cannot claim that the universe is finely-tuned. Alternatively, you can postulate P as being not an omniscient, omnipotent creator, but rather, just a generic creator god that may not be omniscient or omnipotent. Then, again, there is no problem. The specific combination of P being an omniscient, omnipotent creator, and X being a finely-tuned universe, however, is contradictory.
The above contradiction has nothing to do with the "probability objection" in the premises of the argument. If you change P and X, such that there is no contradiction, then the argument remains valid, but another problem that makes it unsound remains: namely, that X having a cause of type not P being is improbable, is an unfalsifiable, unknowable assertion. Atheism does not entail the truth of this premise, so the argument cannot even function as an internal critique. This is an entirely different counterargument from the counterargument I presented above, regarding the contradiction.
Look, let me simplify things for you, since you are still missing the point. The argument being presented is essentially of the structure A ==> B, where A is the proposition "The universe exists as it does (i.e., is finely tuned)," while B is the proposition "It was created by an omnipotent, omniscient deity." It is entirely possible for A ==> B to be true, while A and B are false. ==> denotes material implication from formal logic, here. As such, there are two possible cases to consider here. Case (i) is that A ==> B is true. Case (ii) is that A ==> B is false. My objection is the following claim: B ==> not A. Why? Because in case (i), it follows that A ==> not A, since A ==> B and B ==> not A implies A ==> not A. A ==> not A is equivalent to A being false, i.e., the universe is not finely-tuned. Meanwhile, in case (ii), A ==> B is false, so B is false, and the argument fails to establish B as its conclusion, which is exactly why I object to the truth of B. On an entirely separate note, I can choose to ignore all of the above, and still comfortably say A is an unfalsifiable proposition, so I can freely dismiss A and reject it as a premise.
Thus, your goal here is (a) to prove that B ==> not A is actually false. But you can't, because again: a universe created by an omnipotent-omniscient being cannot be finely-tuned; (b) to actually demonstrate that A is true (and not unfalsifiable).
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@ruaraidh74 That's a completely different argument from your rebuttal that "if God exists, then it's not true that fine-tuning is required for life."
Yes, I explicitly clarified that it is a different objection. You would know this if you actually carefully read my reply to you.
If I conceded that the probability objection was unfounded, then what the heck is the rigamarole about fine-tuning about?
I do not understand what you are confused about. I presented two different objections to the fine-tuning argument. One is an internal critique, whereby the conclusion being implied by the premises necessarily implies the falsehood of the premises. The other is an external critique, where I point out that, even if my internal critique were to not hold, the premises of the fine-tuning argument are unfalsifiable.
Let's not shift the goalposts around, please.
I am not shifting any goalposts here. I introduced a new objection to the discussion, while still defending the objection I had presented previously. This is not what shifting the goalpost is.
That doesn't make any sense. "Fine tuning" is another way of saying "the universe needs to be a really specific way for this to happen, given what we know about the natural world."
If it is true that the universe has to be a specific way in order for intelligent life to exist, then it cannot be true that the creator of such a universe, if there is one at all, is omniscient and omnipotent, because if such a creator is omniscient and omnipotent, then there is no specific way in which the universe must be for life to exist.
Also, there is absolutely no evidence that the universe needs to be a specific way for intelligent life to exist. Such an assertion is unfalsifiable.
If the natural world was different, then of course, the tuning would be different.
We do not know if the natural world could have been any different than what it is. We do not know whether life could exist in the natural world if it were any different than what it is. Any assertions regarding the matter are unfalsifiable. On the other hand, if the world is not natural, but the creation of an omniscient, omnipotent entity, then said world is not tuned at all for life, since life can exist in any universe created by such an entity.
The universe needs to be a specific way to permit intelligent life, given some axioms about the natural world.
This is unfalsifiable speculation. We do not even completely know what conditions a planet in our universe must satisfy for intelligent life to exist in it. Nothing that we know about the natural world can allow us to know about what would be possible in a universe different than ours.
For instance, if God decided people would be made of transcendent jelly instead of molecules and energy... then that's what they'd be made of.
Yes, and I agree, which means there is no fine-tuning. The universe could be literally anything, and the above would be true as long as God commands it.
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@user-fb2jb3gz1d Interesting. Alexandrian manuscript is the bases for modern bible translations.
Yes, this is correct, and is why I brought it up.
Apparently, you don't get into bible discussions.
I do, far more than you imagine.
When in biblical apologetics, nobody uses modern translations.
This is untrue. Cameron Bertuzzi, Frank Turek, Jay Warner Wallace, Justin Briar, William Lane Craig, etc., all use modern translations of the Bible. This also applies to New Testament scholars, such as Sean McDowell, Dale Alison, Michael R. Licona, and even Gary Habermas.
After the 1611 KJV bible and protestant reformation.............brought forth many denominations of christianity. And each takes their own translation.
Yes, and this is irrelevant. The accuracy of a translation is not based at all on how many new denominations it can spring off.
To get to the way the original teachings were, you have to use the older texts. This isn't up for a debate.
You have to use older manuscripts of the original text, not older translations.
You would know this if you actually knew about apologetics.
The fact that you do not know biblical scholars use modern translations is a demonstration of your ignorance, not mine.
Alexandrian manuscript refers to the new testament.
Not necessarily, no.
The old testament was written in Hebrew, except for the Septuagint, apocrypha.
Again, this demonstrates your ignorance. The Septuagint is not part of the Tanakh. The Septuagint is a translation of the Tanakh into Greek.
Psalms is written in Hebrew because its part of the old testament, the Tanahk.
It's "the Tanakh." It is sad that you cannot transliterate the name of your sacred books correctly.
Little things like miracles and wonder makes a big deal. For our reason exactly. You're saying miracles and I say wonder. Apparently you attribute miracles to just that.
No. I am saying the word used translates to "miracle." You using the King James Version does not disprove my point. If you want to have this discussion, then you better start demonstrating at least a basic understanding of Hebrew, which you clearly lack, because if you had a basic understanding of Hebrew, you would not be using the King James Version of the Bible.
I'm saying it's not a miracle because God doesn't do miracles,...
And I am saying the Bible disagrees with you. You have not actually addressed the problem here. You just chose to ignore it.
...he created the universe and it's laws so he knows how to use them.
If God created the universe, and intervenes in it freely, then there is no such a thing as "the laws of the universe." All this tells me is that you have no clue as to what word "law" means in the context of science. A law of nature is simply a description of a behavior of the universe that is consistent and the universe never deviates from. There are no such behaviors if God intervenes in the universe at will.
So to the ignorant, it seems like a miracle but it's actually not.
It is a miracle, by definition. The word "miracle" is defined as an extraordinary event manifesting divine intervention. What makes the intervention divine is precisely the fact that it is an act of God.
God is omnipotent, he doesn't need to perform a miracle when he knows how everything works.
God does not obey the laws of nature. If he did, then he would not be omnipotent.
Doing a wonder, is acknowledgment of something that we don't know how it's done, so it's a wonder. One can say it's a miracle but that's not the correct interpretation or meaning.
Yes, it is correct. That is literally the definition given in the dictionary, and it is the definition that most Christians use too.
In the bible, Jesus talks about the things he does, these "miracles," but he tells his disciples how to do them.
Even if he tells everyone how to do them, they are still miracles, by definition, since they are a manifestation of God intervening in the world.
How hard is it to realize that if God is omnipotent, knows everything and has authority of everything; if he made a limb grow back, you would say it's a miracle. Because you don't know how he did it, he just did it.
No. Me knowing how he did it has nothing to do with it. If I knew how he did it, it would still be a miracle, according to the dictionary definition, as is still counts as an act of divine intervention. Knowing how God did it does not make it not divine.
But if you become a disciple of God, meaning to give your life and follow every word he spoke, you would understand how he did it.
This is false. I know this is false, because I used to be a very devote Christians many years ago, but I never learned how to regrow limbs.
It's not magic.
Okay, sure, it is not magic, but as it still divine intervention, it is, by definition, a miracle.
This the teaching of the bible. Any practicing biblical scholar of christianity would tell you, an educated Muslim and educated Jew, would also tell you the same thing about God.
No, they would not, and I know this, because despite being nonreligious, I have many highly educated friends who are Christian, Muslim, and Jewish. I even have highly educated friends from other religions, such as Buddhism and Hinduism.
I bet you didn't know that Jews, Muslims and christians, have the same God.
I have literally known this since I was in elementary school. What, you think you are smart because you know this? 🤣 That is pathetic. Virtually every self-respecting individual who knows about Islam actually knows this. This is not some "fun fact" or secret knowledge. It is common knowledge.
I pretty sure I'm going to have to get technical with you.
Even though you do not know Hebrew? Yeah, right.
Psalms 77 is someone saying God doest wonders, in your case, miracles. I'm should say that God never says that about himself.
And? This is irrelevant.
It's only someone else who says it about God or Jesus.
Again, this is irrelevant.
Neither does Jesus claim he does miracles. People today claim God does miracles. Because they can't explain how he did it.
No. People call them miracles, because by definition, that is what divine acts of intervention are.
But I assure you it's not by magic.
Again, you are conflating the words "magic" and "miracles." They are not the same thing.
How do I know.............. Because sorcery is forbidden.
You clearly have very little knowledge of paganism. Magic and sorcery are actually not the same thing.
You would know this if you actually read the bible.
You are the one who has not read the Bible, clearly.
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@ruaraidh74 No, I get it perfectly fine. You are the one who does not get it. The reason I have kept repeating myself, is because you continue to understand the argument. In fact, you have completely failed to address my two objections, and are now just ignoring them, on the alleged basis that I am missing the point, despite failing to explain how this is the case.
"Fine tuning" is another way of saying "the universe needs to be a really specific way for this to happen, GIVEN what we know about the natural world."
No. If there is no omnipotent, omniscient creator, then what we know about the natural world does not imply anything about how the universe needs to be in order for life to exist. The claim that the universe needs to be a specific way for life to exist is an unfalsifiable claim. This is what you are failing to understand. However, if there is an omnipotent, omniscient creator, then there demonstrably does not exist any specific way the universe needs to be for life to exist. This is what you are failing to understand.
In summary: if classical monotheism is true, then the assertion you make is demonstrably false. If we do not assume classical monotheism is true, then the assertion is unfalsifiable, and entirely speculative. Either way, your argument is unsound.
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@user-fb2jb3gz1d What is wrong with fine tuning something?
No one has claimed there is anything wrong with a non-omnipotent being fine-tuning a parameter. The claim is that it is impossible for a parameter to be fine-tuned if the entity doing it is omnipotent. See? Literacy goes a long way in a discussion.
So, it has limits, but that does not mean the creator does.
Nope, you are wrong, because that is not how that works. If a creator has no limits, then neither does its creation.
I've been saying that there are laws, boundaries that we, the universe, have. And that God knows these limits, and si he abides by them.
Then he is not omnipotent, by definition. If God abides by the laws of nature, then God does not unlimited authority or influence over the universe. In order for the universe to actually have laws at all, God cannot be omnipotent. The universe has no laws at all if God is omnipotent.
But being omnipotent,...
No. You just said God is not omnipotent. You just said God abides by the laws of nature. Since God abides by the laws of nature, God is omnipotent. You are the one who said God abides by the laws of nature, not me.
If I were God, I wouldn't do anything to amuse you.
No, but unlike God, you are not interested in having a personal relationship with me. In any case, no one has asked God to do anything in this conversation, except for you.
If we as humans can create fine-tuned things and then improve on them... then an omnipotent entity should be able to do the same.
Nope, that is not how that works. The reason humans can fine-tune parameters is because we are not omnipotent, and we are constrained by the laws of physics. If we were omnipotent, then there would not be such a thing as fine-tuning, since the values of a parameter would be irrelevant, and in fact, meaningless.
When he creates, he automatically creates something with detail, fine tuning.
Once again, you are demonstrating your ignorance. You are demonstrating that you do not know what the word "fine-tuning" means. Creating a detailed universe is not "fine-tuning."
What's interesting is that whenever something gets created, it automatically has limits.
Nope, that is not how that works. Now you are just pulling nonsense out of your butthole, and pretending it is a fact.
the Septuagint is not part of the Tanahk.
Yes, I know. I am the one who taught you this. You are the one who implied it was part of the Tanakh earlier, not me.
The Tanahk is all in Hebrew. The Septuagint is all in Greek.
Yes. Again, I am literally the one who told you this.
You would know this if you read the bible.
I do know it. I am the one who taught you these things. I guess you fail to realize this because you are illiterate, but there is nothing I can do about that, since I am not omnipotent. Why did God send an illiterate messenger to talk to me? Who knows?
Protestant bibles don't use the Septuagint. Catholic Bibles do.
No, this is false. Bible translations are not divided into Catholic or Protestant translations, and there are no Bibles today that use the Septuagint. All Bibles use the Alexandrian manuscripts, or in case of the older translations, the Textus Receptus, which is not in Greek, but in Latin.
I say nobody uses modern translations in apologetics... yes they do. I mean to say that when it comes to certain words or meanings, the practice is to go back to the Hebrew and Greek manuscripts.
Ah, okay then. Well, that is the same thing I did. I did not use a particular translation. You did.
They don't use the modern translations for that.
Neither did I.
I do associate miracles with magic. Because the miracles in the bible are wonders not magic.
Why do you associate miracles with magic when the Bible contradicts? How are you typing this, and not realize how incoherent your argument is?
So why are you still using miracles for Psalms 77:14 when the majority of bibles use wonders?
What the majority of bibles use is irrelevant. You literally just said that apologists "don't use modern translations for that," and that "miracles in the Bible are wonders, not magic." Therefore, you literally proved my point, while failing to realize that this is exactly what you did. Geez. Put some thought into what you write.
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@user-fb2jb3gz1d How can a loving merciful god allow such evil?
Well, there is an easy solution: said god is not omnipotent, or not omniscient.
Science also knows that once we know these laws, we can manipulate them, use them in different ways.
No, science does not say that at all. Citation needed. Certainly, you can use knowledge of these laws to build technology, but this is not "manipulation" of those laws.
Like a child doesn't know good or bad, until experiences shows them or someone teaches them.
This is false. Many aspects of morality need to be taught to children, yes, but some aspects are just innate to human biology.
I look at creation with balance, everything created has balance because that is what is required for it work as it was intended to work.
No, not if God is omnipotent, but I suppose you have already made it more than clear that your God is not omnipotent.
I get my views because I am a mechanic.
Too bad God is not a mechanic. Mechanics are human. They are fallible, they are not omnipotent nor omniscient, and cannot behave in a fashion contrary to what the laws of physics have predicted.
I have to fix vehicles, troubleshoot issues.
God does not have to fix vehicles, or troubleshoot issues. Nothing God creates can have issues at all, since God is omnipotent and omniscient.
So, that's how I see things, because it has to be that way...
It only has to be that way for humans, who are not omnipotent, nor omniscient. None of your analogies work, because again, God is omnipotent and omniscient. There is no "balance," because God does need to achieve a particular thing in order for the universe to work. In fact, to the contrary: no matter what God does, it is impossible for the universe to "not work," since God is willing the universe to exist.
So, God says he gave us free will.
Nowhere in the Bible is this stated. Furthermore, the scientific evidence most definitely does not support the thesis that we have free will.
I see God as the parent.
You cannot do this. You already compared God to a mechanic. Now you are comparing God to a parent. Make up your mind. You cannot have it both ways.
Yes, he would love to keep us from harm but then we would eventually hate him for not letting us experience life.
Well, maybe you would, but that just speaks to your emotional maturity. I know I would not hate him at all! In fact, I would very much prefer it. Besides, a competent parent actually has a conversation with their child, explaining to them why they are not allowing a particular behavior. Most children are fairly understanding when you take this approach. No, the success rate is not 100%, but again, humans are not omnipotent, nor omniscient, so this makes sense, and children often have disabilities that also prevent them from understanding. However, mental disorders are nothing to God, and God is omnipotent and omniscient. Having his children understand is not merely very easy, but actually trivial. In fact, if God actually bothers explaining, and if God is truly omnipotent, then it is simply impossible that we will not understand his reasons at all and fail to abide by them. The truth would be so compelling, even us mortals would not be able to resist it. Your suggestion that we would hate God only aligns with the idea that God is incapable of making us understand, and therefore, not omnipotent.
Just like having your own child. You can't protect them from everything, because then you are controlling them, taking away their free will.
Yes, but human parents are neither omnipotent nor omniscient, and are bound by the laws of biology. We have irrational brains, and can suffer from problems of mental health that can be obstacles in us making safe or sound decisions. None of this is true for God, so your analogy fails: if God truly is omnipotent and omniscient, then it is literally impossible for us to not be compelled to by God's truth. Since we do observe that there are people not convinced by God's truth, it proves at least one of two things: (a) God is neither omnipotent nor omniscient; (b) God has not actually revealed the truth to all of us. Also, God actually explaining his reasons to compel us does not violate our free will, not any more than him having omniscience and creating us already does (because it is already not possible to have free will if God created us and is omniscient).
Just because there is a God, that doesn't mean he has to intervene in every bad thing.
No, but it does mean that, if God is omnipotent, omniscient, and loving/benevolent, then it is simply impossible that bad things could exist at all.
Imagine if science mocked or hated things they didn't understand?
This is a strawman. No one here is mocking God. We are presenting you with a logical contradiction, and we are asking you to either accept that it is contradiction, or demonstrate that it is not a contradiction. You have failed to demonstrate it is not a contradiction. Saying that "we do not understand God's motivation" is not an argument, and does not address the contradiction. After all, our capabilities to understand God are completely irrelevant to the contradiction.
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@user-fb2jb3gz1d But sometimes, God does intervene.
You say that, but there has never been any evidence that divine intervention has actually occurred.
I say that 95% of people who call themselves Christians are not Christian, because they pick and choose where, when, and to whom they practice Christianity.
This is a No True Scotsman fallacy.
Jesus specifically said that we will know his disciples by their fruit, and their fruit will bear things like healing the sick... you know, things that sound like miracles. Yet we don't see any of that today.
The Bible claims Jesus said this, although no one actually knows if Jesus really did say it. But yes, the Bible does explicitly say that. And as you say: no one in recorded history has actually been confirmed with having these capabilities.
It's because of what I see that no one is a Christian...
Well, that means you are not a Christian either, then. In fact, this just means Christianity is dead, it does not exist. You do not realize it, but you have literally just accidentally admitted that Christianity is effectively false.
Because God listens to those who obey him.
And when you say this, why should anyone believe you?
I have seen some people pray for hours, pray devoutly, and these people have incredible stories of God answering their prayer.
Just because a person believes God answered their prayers does not mean God actually did answer the prayers. These things can always be explained away by things like the Barnum effect, the placebo effect, and other such things. Besides, I have known many devoted Christian people who have prayed intensely for hours, and yet their prayers went unanswered. As such, this is a counterexample.
My only true issue is, condemn me on what I actually believe, not on what you think I believe.
The problem is that you actively refuse to disclose in full formality what your beliefs are, you actively refuse to define your terminology, and when we define our terminology, you refuse to engage with those definitions. You continue shifting the goalpost, and you continue to contradict yourself in every sentence. In one sentence, you say "God is omnipotent," but in the next sentence, you directly contradict it. When I make an assertion, you pretend that I made an entirely different assertion, or that I made none, or you simply throw the assertion back at me, as if you came up with it, when I was the one who initially made the assertion. When I ask you to argue against my particular thesis, you throw all these red herrings about biblical translations, the definition of "miracles," and other such things, which are completely irrelevant to the discussion, since this discussion is about fine-tuning and God's omnipotence, not any of those other things you decided to bring up. In summary: you are engaging in massive mental gymnastics and employing intellectually dishonest rhetorical tactics to avoid facing the criticisms to your beliefs. It makes it obvious that you are experiencing some very heavy cognitive dissonance, and I am willing to bet that many of the things you have said are just you parroting what other pastors or apologists on the Internet have said, as a desperate resort, despite not being totally convinced about those things being parroted. Last, but not least, saying we should condemn you for what you actually believe, and not for what we think you believe, is hypocritical, since this is a principle you yourself are not willing to follow when it comes non-Christians.
I believe in the God of the Bible.
No, you definitely do not, since so many of the claims you have made are in direct contradiction with the Bible. The Bible does not present God as being omnipotent or omniscient, and in fact, prior to the destruction of the Second Temple, there were no Jews or Christians that ever spoused the idea of an omnipotent or omniscient God. Classical monotheism in Christianity did not come to exist until much later, and by that time, all the texts in the Bible had been written. Christians who subscribe to classical monotheism will take the Bible and reinterpret it to suit their needs, cherry-picking the verses they can easily contort as if they supported classical monotheism, while completely ignoring the verses that directly contradict it. In the process of doing this, they have eliminated alternative interpretations, bashing them as "heretical," which is ironic, since classical monotheism did not originate from the Abrahamic religions: it originated with Hellenistic tradition. This reinterpretation and contorsion has erased the cultural context and ancient theological connotations of the text, while also taking a naive approach of taking the Bible literally, rather than treating it as a work of literature, the latter being exactly how the writers of the text treated these texts. This is how you get to unscientific nonsense like the "biblical flat Earth movement" and "young Earth creationism."
Additionally, while the Bible actively describes God as a god of miracles, you actively refuse to accept this. We had an entire discussion around this, which proves my point. Furthermore, you insist that free will is biblical, even though it is very much not biblical. The entirety of Christian theology is based around misinterpreting the Bible and misunderstanding the text; so having the audacity to then say that you believe in the God of the Bible is inaccurate, disrespectful, and could even be taken as antisemitic, as, in the process of contorting the Bible to fit your unbiblical beliefs, you are actually demeaning a work of Jewish literature, and erasing the Jewish social culture and history that gives connotation to the text as written.
He is the same yesterday and tomorrow.
This is not how the Bible portrays God. The fact that there is an Old Covenant, and then a New Covenant, is enough of a rebuttal.
A lot of Mosaic laws, like stoning a chick in her menstrual cycle...
...and you are also misogynystic, to top things off. Great. Look, you cannot be unironically talk about women as "chicks" in a discussion about Christianity. This is pathetic.
...it's because the people wanted it, so God allowed it. Not because he wanted it so.
No, this is false. God explicitly commanded these things to be upheld. These were not laws introduced by humans in the biblical narrative.
We can argue that because he allows it, he condones it.
God condones something he does not like? Now you are really contradicting yourself.
To me, the 10 Commandments are his laws...
Which 10 commandments? The ones from Exodus 20? Or the ones from Exodus 34, which are different, and which the biblical text actually calls "the Ten Commandments" in verse 28?
The Mosaic laws are the laws his people wanted.
No. The Bible is explicit: God commanded these things. God wanted them.
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@user-fb2jb3gz1d I don't see anything wrong with God's laws.
You mean you seen nothing wrong with Numbers 5:11-29, the torturing of a woman solely to find out if she cheated or not, without even requiring any evidence to justify being suspicious of her, and forcing her to have a miscarriage if she did cheat? You see nothing wrong with Exodus 21:2-6, explicitly delineating a God-given loophole for masters of Hebrew slaves to permanently enslave the entirely family, including women and children? You see nothing wrong with Exodus 21:17, where God commands that someone being having an outburst at their parents be punished by the death penalty? You see nothing wrong with Exodus 21:20-21, where God explicitly commands the exoneration of masters who beat their slaves? You see nothing wrong with Exodus 22:16-17, where a woman who is sexually violated can be forced to be married to the offender if the offender simply pays her father? You see nothing wrong with Exodus 22:18, one of the verses that justified the European witch hunts of the second millenium, where millions of people in Europe were killed because of it? You see nothing wrong with Exodus 22:20, in which God literally commands ethnic cleansing? You see nothing wrong with Exodus 23:23-24, where God explicitly commands ethnic cleasing again?
Because, if so, then I want you nowhere near my family ever. Forgive me, but I do not want to be in contact with anyone who sees nothing wrong with torturing women, permanent enslavement of entire families, beating up slaves with no repercussions, executing to death people who just had a bad day and ended up taking out on their parents, forcing a raped woman to marry her rapist via payment, executing people who practice sorcery or who worship other gods, and committing ethnic cleansing.
Especially when Jesus summed them up with 2 commandments... love God with all your being, and love your neighbors as yourself. All of the other commandments of God will follow if you follow the 2.
No, they most definitely will not. Loving your neighbors necessarily entails things such as not committing ethnic cleansing, not forcing raped women to get married to their rapists via payment to their fathers, not torturing women, not owning people as slaves, not giving the death penalty to people committing petty actions, not torturing women, etc. Also, I should mention, commanding someone to love you, regardless of whether you are responsible for their existence or not, is inherently manipulative, and instutiting a punishment for if they do not love you is called domestic/child abuse. Also, commanding people to love everyone is also toxic. For example, no one should be obligated to love, or even forgive, their abuser. This leads to further trauma in the victim, and in the worst case scenarios, this can make the victim completely unable to feel emotions, or overwhelm them so much that it makes them suicidal. If God is omniscient, then God knows this. If, despite knowing this, God is still commanding you to love your abuser, as he does, then God does not care about you well-being, and is not benevolent.
People put things above God, and limit who they love.
Well, no. I do not put things above God, because I do not believe God exists.
I'm omitting the things like what happened to Jericho, the killings God commanded.
Yes, you are, because they very clearly pose a problem for your worldview, so they only way you can continue to pretend your worldview is tenable is simply for you to ignore these things. You are doing that thing again where you cherrypick the Bible. Almost all Christians do this, even the reasonable ones.
I acknowledge the slavery and killings he commanded.
Not only do you acknowledge them, but earlier, you admitted you see absolutely nothing wrong with these things, which to me, suggests you are perhaps a psychopath.
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@user-fb2jb3gz1d There are reasons for them.
Then, present those reasons. It is that simple... No, no such reasons exist. We can prove this logically, and omniscience is not at all required to do so. Why? Because one can understand how omniscience works without being omniscient. Obviously, this must be true, since otherwise, it would simply be impossible for Christians to "know" that God is omniscient. Fact 0: there do not exist any circumstances under which committing genocide is an act of love. Fact 1: an all loving, benevolent entity, by the very definition of these words, will always, to the best of their ability, seek to commit only those acts which are loving, and prevent those acts which are unloving. Deduction 0: it follows that, an all loving, benevolent entity, by the very definition of these words, will always, to the best of their ability, prevent acts of genocide. Fact 2: an omnipotent and omniscient entity can always successfully prevent any act from hapening. Deduction 1: it follows that an omnipotent and omniscient entity can always successfully prevent acts of genocide from happening. Fact 3: an entity which is omnipotent, omniscient, and all loving, benevolent, by the very definition of these words, will always, to the best of their ability, seek to commit only those acts which are loving, and prevent those acts which are unloving, and will always succeed in doing so. Deduction 2: it follows that an entity which is omnipotent, omniscient, and all loving, benevolent, by the very definition of these words, will always, to the best of their ability, seek to prevent acts of genocide, and will always succeed in doing so. The God of the Bible does not satisfy the statement laid out in deduction 2. Therefore, the God of the Bible fails to omnipotent, or fails to be omniscient, or fails to be all loving and benevolent.
You may respond to this by saying "You are ignoring the fact that God has a plan for us, and than plan, in the longterm, involves enduring acts of genocides." However, this does not help: if such is the nature of God's plan, then God's plan is not all loving, and thus, God is not all loving, for if God were all loving, God would not choose to have a plan that requires us to endure acts of genocide. Also, there is another reason why such a defense fails: God is omnipotent and omniscient. Therefore, a plan by God cannot "require" anything at all: there is nothing that needs to be fulfilled for the plan to be achieved, since the plan can simply happen, just because God says so. Requirements are an emergent property of entities with limitations, which God allegedly lacks (although, I should reiterate, the Bible never actually portrays God as limitless, at least not in a non-hyperbolic context).
Though you may not agree with the reasons.
Since the reasons cannot exist, as demonstrated above, it is impossible to agree with them. Saying that one could agree with them is like saying that you can paint an existing wall with a nonexisting paintbrush.
I may not agree either, but if God is omnipotent, then it is impossible to comprehend why he commanded such an evil thing.
It is impossible to comprehend, because it is logically impossible for that to happen. All this tells me is that you are continuing to fail to understand how omnipotence and omniscience work as properties of an entity.
To simply say God is evil, therefore not omnipotent, then I say one is changing the definition of omnipotent.
Nope. No one is saying that, if God is evil, then he is not omnipotent. We ars saying that God is evil, or God is not omnipotent, or God is not omniscient.
We don't know what's beyond 100. God does.
We do not need to know what is beyond 100, because what little we do know is more than sufficient to prove that a logical contradiction occurs.
So how can we say for certain that what God does is evil or wrong, or that he is no omnipotent, based on our limited understanding?
We can do it in exactly the same way you managed to "know" that God is omniscient to begin with. Meaning, if our knowledge is not sufficient to prove that a logical contradiction exists, then it is actually impossible for us to know that God is omniscient to begin with.
That's just stupidity on our part.
Yes, I agree that claiming that we can know God is omniscient is stupidity on humanity's part. Massively stupid, in fact. Which only makes me wonder why you do it anyway.
I don't see knowing the future as not having free will.
That is because you do not actually understand what it means to know the future, or what it means to have free will. Let me explain this for you. God is omniscient, and created the universe. Therefore, God knows exactly what each created entity in the universe will do, and furthermore, what each created entity in the universe will do will be done precisely because God ordained it so. As such, no entity in the universe can do anything differently than how God ordained it will do things at the moment of creation, since God is omniscient. As such, all created entities in the universe are just automata: deterministic systems whose future is completely set in stone. In other words, we cannot choose our future actions: whatever we do at any given point in time, it happened because God ordained it would happen so at the moment of creation, and because God knew it would happen, and as God's knowledge is absolute, since God is omniscient, nothing could ever deviate from it. In other words, it was fixed in stone, since the moment of creation, that we would behave in the way that we do behave and have behaved. Since it was fixed in stone, we had no "choice" on the matter, even if we had the illusion of choice. Therefore, if God is omniscient and created the universe, then we have no free will.
To me, it's like me creating a game with all the possibilities that can occur, and you choosing one,...
No, this is not at all analogous to God creating the universe. In this case, yes, you created the game, but you did not create me, and you are not omniscient, because you do not know the future: since you do not know what choice I will make, you do not know my brain chemistry. It seems like you are confused, so let me make this clear: knowing all of the possible outcomes is not the same as knowing the future. Knowing the future also includes knowing which outcome actually will happen.
I know all the possibilities of your choice, but it's up to you to choose, not me.
Nope, this is not omniscience. Knowing what are the possible choices I can make is not omniscience, because that is not the same as knowing the future. Knowing the future means knowing exactly which choice I will make. Also, in this case, you did not create me, so again, the analogy fails.
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@user-fb2jb3gz1d Yes, religion is written by man, so it's correct to say religion is manmade.
I am glad you understand. So, now, why should I accept any religion is true, if they are just inventions by humans? For the religion to be true, it would actually have to be initiated by the gods the religion proclaims to worship, unless the religion is deistic or atheistic.
So, that's how see religion. I base it off their actual doctrine, not the people who pervert it.
Nope, you completely missed the point of the argument. The point is, the doctrines are manmade, and so, are themselves just results of human's perversions of reality. And by the way, this also applies to the Constitution. This is why the Amendments exist. Even with the Amendments, there are many, many problems with the U.S. Constitution.
We have the written manuscripts or how the teachings and meanings are meant to be.
No, we most certainly do not.
* We do not have any of the original manuscripts used in the text of the Bible. No manuscript in our possession dates to anywhere near as far back as the text themselves do.
* We also do not have the original teachings. The text itself was written decades, sometimes centuries, after the theological idea it was meant to convey originated, only having been communicated by oral tradition otherwise. Most people were illiterate during those times, so traditions could not be written down until long after they emerged.
* We also do not have the original meanings. Ancient Hebrew, Aramaic, and Ancient Greek are all dead languages, and on top of that, the texts contain many words that exist nowhere else in the literature of these languages, words that were literally invented by their authors. We do have some of the meanings, yes, but definitely not all of them, like you say.
Also, I should mention, these traditions evolved from previous religions. Christianity evolved from a new radical Messianic sect of Judaism, combined with theology and traditions of the Essenes. These, in turn, where traditions that had evolved from Second Temple Judaism, having been influenced by the Greco-Roman religions that surrounded them, and also influenced heavily by Zoroastrianism. Second Temple Judaism was itself evolved from early Judaism, having been influenced by Hellenistic philosophy, local post-Babylonian religions, and the religions of the empires that seized control of the Jews, which included Zoroastrianism, again. In turn, early Judaism originated from the ancient Canaanite religion being heavily influenced by the ancient Sumerian religion that pre-existed it, and the Egyptian religion, and then later became syncretized with Yahwism. Would you not say that Judaism is just a perversion of the Canaanite religion? Are they not the heretics who deviated from the real, original doctrines?
We have writings of disciples of disciples of disciples of disciples, and so on,...
No, what we have are writings of Church fathers whose relationships to the disciples of Jesus is highly questionable, according to the scholarly consensus, and then also, lots and lots of anonymous writings and forgeries.
Doing this is why I'm Catholic. Only Catholicism goes right back to Jesus himself...
This is false. Even by the time of the destruction of the Second Temple in 70 CE, at least 8 different major Christian sects had emerged, with their origins being unclear. Among these were the Pauline sect, the Marcionist sect, and the Ebionite sect, but there were several others too. All of these have equal right of claim to being traceable to as far back as history has allowed us to do. Pauline Christianity went on to split into Nicene Christianity, which is the most recent common ancestor of all Christian sects today, and Arian Christianity, which later went extinct due to being persecuted to extinction. There are many non-Protestant sects of Christianity today traceable all the way back to the Council of Nicaea: all of the Chalcedonian sects, the Oriential Orthodox sects, and the Church of the East, which follows the Nestorian theology. These ramifications of all happened in the 1st millenium, long before the Great Schism of the 1000s, where Chalcedonian Christianity eventually ended splitting into many churches, 24 in the Roman Catholic Church and at least 15 in the Eastern Orthodox Church. And this was all before the Protestant Reformation.
...and anyone can read the early church writings...
You mean the ones that the Nicene Church did not burn down or bury in deserts?
...to see Catholicism aligns with what Jesus taught his disciples...
You mean with what the Gospels and Epistles claim Jesus taught his disciples. We have no way of knowing what Jesus actually thought, as he wrote no texts of his own, and neither did the 12 apostles, except for maybe (and this is a big fat "maybe") the epistles that maybe Peter wrote, although most scholars still think those were actually cases of pseudepigrapha. Neither are there writings of anyone claiming to have been a disciple of any of the apostles. We do have writings of someone claiming to be a disciple of a disciple of an apostle, but the person they listed as their master never claimed to be a disciple of an apostle, despite being a fairly accomplished Church father.
Also, no, your claim is false. Even the work of theologians from the second century do not align all that well with the texts of the Bible. The Church does mental gymnastics to try to pretend that it does, but this is just typical of Christians, reinterpreting works to make it seem like they say things they do not say.
All the other Christians come 1500 years later.
Nope, this is false. Gnostic Christianity, Pauline Christianity, and Ebionite Christianity, all existed as mutually exclusive sects of Christianity by 70 CE. Yes, Gnostic Christianity and Ebionite Christianity went extinct, and so did Arian Christianity, but the split between non-Nestorianism and Nestorianism occurred as early as 431 CE, and Nestorianism still exists to this day, in the form of the Ancient Church of the East.
Anyone who picks up a Bible will have their own interpretation on things.
Yes, which is a problem. If the Bible is the Word of God, then it should not be subject to interpretation: it should be objective, and unambiguous. Since this is not the case, I conclude the Bible is not the Word of God.
But how can you simply read a religious book without proper instruction?
It's called "being literate and having reading comprehension skills." Also, I thought the Bible was the Word of God, according to you? Is it not? Because if it is the Word of God, then there is absolutely no reason you would need anyone to give you proper instruction. After all, the Word of God IS the proper instruction.
Well, I say that, but this is only me hypothetically granting the absurd assertion that an omniscient, omnipotent creator who wants to have a relationship with us would actually do something as mind-bogglingly dumb as communicating hyper-indirectly by ordaining other people to write an arbitrary collection of books. In actuality, an omniscient god would know that books are not the best way to communicate knowledge, and are horribly ineffective way of initiating relationships, even by omnipotent standards. So, anyway, this already disproves the idea that any manmade book could ever actually be the Word of God.
And without proper instruction, people will interpret things in ways that were not meant to be.
If the Bible is the Word of God, then it cannot be misinterpreted by even illiterate people. It is called the Word of God for a reason, do you not think? The fact that so many misinterpretations of the Bible exist is sufficient evidence that the Bible is not the Word of God.
Also, this thing where people misinterpret books happens with all books. It is not exclusive to religious books. This happens because most people know how to read at only a moderate level, not a proficient level. And this is why I said that using books to communicate information is ineffective for a God that is allegedly omniscient. An omniscient God would know better than to use books.
...just shows that people do not do their research.
Yes, your replies are a great example.
You need proper instruction.
Not really, no.
It's stupidity to say we don't.
No, it is not. What is stupid, however, is to assert that somehow, it is metaphysically possible for an inerrant book made by God himself, to be misinterpreted by a measly human being. I thought God is all powerful. Is God not capable of making a book that no one can misinterpret? Better yet, if God is omnipotent, why is he relying on books? There is this thing called telepathy. God could just use this. Then your ability to understand books becomes completely irrelevant. Now, does God have to use telepathy? No, God is not obligated to do anything at all. That is not my point. My point is, since he is not using a method of communication that is impossible to misinterpret, it means he clearly has no interest in us actually knowing the truth about him and about the world. Sure, God has a plan, but that plan certainly does not involve us knowing the truth about him and the universe, that much I can be confident of. Maybe God wrote the Bible to confuse us because he thought it would be amusing (whatever that means for an omniscient entity, anyway). You cannot disprove this possibility.
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And in the end, you ignored my previous comments to you. How oddly convenient for you. I think I am going to end my conversation with you here as well.
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@user-fb2jb3gz1d Oh, so now that I have told you that I am going to end the conversation, you finally decide to engage with my argument to try to make me look like a fool? Nice try. Okay, fine, I take back my decision. I will give you one last chance to be intellectually honest here.
So, would you say that because he is omnipotent, that he cannot make things that are fine tuned?
This phrasing is all wrong. You are trying to make it seem like I am imposing a limitation on God's omnipotence, when that is not at all what my argument is doing.
Here is the point: the point is that there cannot exist such a thing as a "finely-tuned universe which is created by an all-powerful and all-knowing entity." Such a "thing" is, in fact, not a thing at all, it is just meaningless word salad, an incoherent notion. Talking about a "finely-tuned universe which is created by an all-powerful and all-knowing entity" is exactly equivalent to talking about a "square circle" or a "married bachelor" or a "blonde redhead" or a "feline insect."
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@lewkor1529 The idea of a potentially infinite collection is nonsensical, so naturally, any definition that any theologian tries to give will be nonsensical, or circular. The nuance that is being missed here is that the idea of actual infinity versus potential infinity originated in the literature millennia before axiomatic set theory was developed, and so, the idea originated during a time where the concept of the infinite was poorly understood, if not rendered invalid. This is the reason why real analysis was invented: there was a requirement to set calculus of real-valued functions of real numbers on a rigorous foundation, one which did not include a concept of infinitesimal or infinite quantities, but which solely relied on the properties of the real numbers. It was only after the end of the 19th century that the study of infinite objects in mathematics was put on a rigorous foundation, thanks to Georg Cantor and other pioneers. This also allowed the development of the hyperreal numbers and nonstandard analysis, and then later, the development of combinatorial game theory, giving rise to the theory of surreal numbers. Today, infinite quantities are understood in terms of these theories, all of which rest on set theory as the ultimate axiomatic foundation.
With this modern understanding, we know one thing: some sets are legitimately infinite. For example, the set of natural numbers is an infinite set, by definition, and it is an axiom of set theory that the set of natural numbers exists. Some sets are infinite, some sets are not infinite. That is all there is to it. The distinction between actual infinity and potential infinity simply does not exist in mathematics, and the idea of a set with indefinite cardinality is also impossible, and this can be proven from the axioms. Note: indefinite cardinality must not be confused with infinite cardinality. Indefinite cardinality refers to a cardinality that is not fixed. This ancient theological notion should therefore be finally put to rest, but old-school thinkers like WLC and Islam theologians just refuse to let this already obsolete, disproven idea, die out for good.
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In axiomatic set theory, there exists an axiom called the axiom of infinity. The axiom of infinity states that there exists a set N such that, (0) {} is an element of N, (1) if x is an element of N, then the union of x and {x} is an element of N. This set N is called the set of natural numbers, also known as the simplest inductive set. Assuming the Zermelo-Fraenkel axioms, or any strengthening thereof, a set S is called an infinite set iff there exists an injective function f : N —> S. A set that is not an infinite set is called a finite set.
Given how finiteness and infiniteness are defined, a set is either infinite, or finite, but as one is the negation of the other, a set cannot be not-finite and not-infinite. A set also cannot be both. This renders the ancient theological distinction between actual infinity and potential infinity as being obsolete, inaccurate, and useless.
Aleph(0) is defined as being equal to the set N. Additionally, it is said that, for a set S, card(S) = Aleph(0) iff there exists a bijective function g : N —> S. If α, β, γ are cardinalities, then γ = α·β iff γ = card(A*B), with α = card(A) and β = card(B). From this, it can be proven that if α = 30, β = Aleph(0), then γ = Aleph(0) as well, by constructing a bijection. So it is a theorem of set theory that Aleph(0) = 30·Aleph(0). There is no contradiction here.
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@lewkor1529 is the Kalam question begging when it states "everything that it begins to exist has "A" cause"? It seems to me that this premise smuggles in the idea of "A" (=single) cause in order to set the stage for "A" god further down the road. Without the usage of the "A" in, is it conceivable or possible that multiple causes could concur or act concomitantly in bringing about an effect, in this case the universe?
This is a very good question, and I think the issue here is that the Kalam argument is just worded poorly, and its premises are not formulated rigorously. In particular, the Kalam argument does not provide a definition for what it calls a "cause". Depending on which of the various definitions you adopt, what the Kalam does could be considered begging the question, although this could be fixed by presenting the argument with a more careful wording, or by simply providing an adequate definition of "cause" beforehand. This is in general a problem with every argument for the existence of deities, not just with the Kalam specifically: these arguments fail to carefully define their terms, and it is by way of this semantic obfuscation that they often get away with making the premises seem known-true when they are not. The classical ontological argument used to suffer from this problem, which is why theologians now have often for Alvin Platinga's modal ontological argument, which has its terms all rigorously defined, though that argument has a different set of problems.
The reason theologians avoid defining "cause" in the argument is because they want to appeal to this vague, intuitive notion that the universe "at some point, came to exist, and it happened in some way", to get us to agree with their premises, because they know that if they actually provide a rigorous definition of "cause" that we can agree to, then the premise of the argument will be easily exposed as unsubstantiated. The truth is that too little is known about the universe, and even about causation as a whole, to actually understand how the concept of cause should be applied here, but theologians do not want to simply present a God of the Gaps argument, because they already know this is not convincing. Every cosmological argument is just an attempt to turn a God of the Gaps argument into not God of the Gaps.
I was told that because of the principle of parsimony, multiple causes can always be boiled down to one, but I disagree.
Well, as I said, this does depend largely on how you are defining cause, in the first place. If a cause is a set of factors, then a pair of sets can always be consolidated into a single set, but this also depends on how causes behave with respect to propositions. However, if the way you are defining a cause is by the factor themselves, then the argument definitely needs to be worded so as to not beg the question. Though, even if it did get reworded, it would still be flawed.
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@donnievance1942 I completely agree. Apologists are a joke in general. To be an apologist, you must misrepresent and reject well-established scientific and mathematical concepts. To mind comes the example of the second law of thermodynamics. Every apologist ever uses the second law of thermodynamics in their arguments at some point, but leaving aside the fact that they always misunderstand the law itself, apologists also refuse to understand that the second law of thermodynamics is not some type of unviolable principle. The second law of thermodynamics is a statistically-attested observation of thermodynamics within the specific context of classical physics, and the moment you abandon classical physics and you introduce quantum phenomena and the theory of general relativity, the law becomes incoherent, or inaccurate, for a multitude of known situations. This is also true of the law of conversation of energy. Really, this is true of any law of physics studied in classical physics. Nature does not operate by these laws: these laws are merely approximations of reality that were once considered valid, back when classical physics was the dominant paradigm, and physicists did not know any better.
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Technically, a limit to infinity is a limit to 0 of the limiting function composed from the right with the reciprocal function. Symbolically, we define h : R\{0} —> R\{0}, h(x) == 1/x, and we define lim f (x —> +♾) := lim f°h (x > 0, x —> 0), and lim f (x —> –♾) := lim f°h (x < 0, x —> 0). This actually helps clarify what is going on. If you have polynomials P and Q, then lim P(x)/Q(x) (x —> +♾) := lim P(1/x)/Q(1/x) (x > 0, x —> 0). Now, let P' denote the polynomial that has coefficients of P in the reversed order. So the nth coefficient of P' is the [deg(P) – n]-th coefficient of P. Let Q' be defined similarly. Then lim P(1/x)/Q(1/x) (x > 0, x —> 0) = lim [P'(x)/x^deg(P)]/[Q'(x)/x^deg(Q)] (x > 0, x —> 0) = lim P'(x)/Q'(x)·x^[deg(Q) – deg(P)] (x > 0, x —> 0). With this, whether the limit exists, and what its value is, depends on the value of deg(Q) – deg(P) and on the value of the lowest order nonzero coefficients of P' and Q'.
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@x-popone6817 That makes it seem like you think that you need to have a degree in physics to prove you wrong, which is absolutely false.
If you think that is what my statement means, then that is an issue with your reading of it, rather than with my statement itself.. In particular, I never said anything about proving me wrong, and even more in particular, there is surrounding context that dictates the implications of my comment that would be obvious to any reasonable person. You claimed you know more about cosmology than another people here, and I responded, by indicating that as someone who actually is professionally trained is cosmology, your statement about understanding cosmology is completely laughable, based on your cosmological claims, which provide strong evidence of your lack of education on the subject. Thus, the only way you could know more about cosmology than me is if you could prove that the education system as a whole is wrong about cosmology, precisely because you do not have a physics degree. This is no way implies that I cannot be proven wrong by you or anyone else about a particular claim in cosmology, but as we were talking specifically about education and knowledge, and not individual claims themselves, such is irrelevant.
You say that if we have free will, that doesn't mean we can create a universe. That's a total misunderstanding of what I said. I said that if God (an omnipotent being) has free will, then He can create the universe as He pleases,...
No, that is not a misunderstanding of what you said. In particular, you never talked about an omnipotent being in your claim. You only talked about a god with free will, and you never provided any definition or descriptions of god being omnipotent. I have no valid reason for assuming that you are defining god as being omnipotent, because plenty of theologians do not do so, and in fact, it is not necessarily true that omnipotence is necessary for a being with free will to create a universe. Such a being would only need sufficient "power", in whatever sense this is even meaningful. How much power would that being need is unknown, and currently, unknowable. So, I am perfectly justified in not assuming you were talking about an omnipotent being, or a being with any particular amount of power at all. I did not misunderstand what you said. What you said simply does not communicate the point you thought it communicates.
...opposed to an eternal impersonal force, which wouldn't be able to create anything, because if so, the effect should be eternal, which clearly isn't the case.
No. I already disproved the validity of your claim that "if a cause is eternal, then its effect must be eternal". You also have not proven our universe is not eternal, to begin with.
As for my claim that the effect should be eternal as well, I think you totally misunderstood me. I never said that a cause's effect needs to be he same as the cause.
Yes, you did say it. I even quoted the exact words in which you said it, more than once too. You may not think you said it, because perhaps you intended to communicate something different. But I am quoting your words, not your intent. It is impossible for me to know your intent through words if your communication is ineffective.
I said that if there was an impersonal force (...), then that wouldn't be able to freely create.
Yes, you said this, but I questioned the validity of this statement, and you replied by saying, that this statement is true because, "if the impersonal cause is eternal, then so should its effect be eternal", implying that you think that the properties of an impersonal cause must also be held by its effects, a claim which is demonstrably false, and for which I provided counterexamples. So no, it is not true that, if a cause is impersonal and eternal, then its effects should also be eternal. The latter does not follow from the former. This is a non sequitur.
If you claim this is not the case, then it is the equivalent of saying that there was water [at] 1°C since all eternity, but it became ice at a certain point.
No, it is not equivalent to this whatsoever, because this analogy has no causality involved, and no transferring of property involved. You are saying that your particular example of a non-contradiction is equivalent to this example of a contradiction, which is obviously impossible. It is not a logical contradiction that a cause be eternal and its effect not be eternal.
Only a being with free will could stop this problem by DECIDING to create the universe.
You are begging the question, by assuming that the only process by which causation can occur is via decision making. No, deciding to create the universe is not the only way an eternal cause can cause a non-eternal effect. This is just an assumption you are making, and it is completely baseless.
Apologetics does not require scientific data and mathematical concepts to be misrepresented to prove a point.
It absolutely does, and this very discussion we are having is an example, where you are grossly misrepresenting the Big Bang theory, and claiming that it stipulates that the universe had a beginning. It does not. You will not find a single peer-reviewed scholarly scientific work that makes this claim. Yet you do this misrepresentation, because it is the only hope of being convincing apologetically.
You seem to presuppose that everything theists say is false, dishonest, misrepresentation, etc.
You seem to make a crap ton of assumptions about people based on things they did not say because you decided to twist and misrepresent their words. For someone who claims I am presuppositionalist, you are very much one yourself. This is an example of it: you claim that I think every theist is dishonest or mistaken, which is a claim I never made. My claim was specifically about apologists. If you think "applogist" and "theist" refer to the same thing, then you are just an idiot.
I never said the scientific method is biased, I said scientists are biased, which you even admit.
Yes, and the fact that you said precisely that proves your ignorance about the scientific method: for if you actually understood the scientific method, you would understand that the bias of individual scientists is actually completely irrelevant to the discussion and has no bearing of the results of the scientific method, and so you would have never even attempted to mention it in the first place.
Bias in scientists can influence their findings, as it can in everything.
No, not really, because the scientific method exists. The scientific method is not perfect, but it certainly cannot be meaningfully influenced by the biases of individual scientists.
To claim otherwise is just ridiculous.
No, to claim otherwise is to understand how the scientific method works: it works precisely by filtering out the biases of the scientists doing the work.
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@x-popone6817 You assume that physical measurements of time are actually time itself.
No, I never said such a thing, but yet you continue to twist my words and be dishonest, as you have been doing this entire discussion. It is not true that the measurement of time is time itself. However, the fact that time can be measured at all, proves that time is physical, because that which cannot be measured, directly or indirectly, is by definition, unphysical. Hence, why I said, that time is physical. But it is not just the measurements of time themselves that indicate that time is physical. In fact, I never appealed to measurements of time in my comment, so the fact that you are pretending I did is another example of dishonesty on your part. What I actually appealed to in my comment was Einstein's theory of general relativity, hence the fact that we have a precise understanding of how time works beyond just measurements by clock.
People like William Lane Craig argue that physical time is like a clock, roughly measuring metaphysical time.
There is absolutely no evidence that there is such a thing as metaphysical time. His interpretation is irrelevant if it is unscientific.
In this same way, it could be argued that physical time is just a "clock" that roughly measures physical time, but it can be faulty at times.
Sure, it can be argued, but that does not imply any such argument is valid, let alone sound. Besides, to say that metaphysical time is the "true time" and that physical time is a "faulty" measuring of it begs the question: it presupposes that physical time is not itself correct to begin with. This is a ridiculous presupposition, given the lack of evidence for the existence for metaphysical time.
I know there are verses used to support this, such as Thomas, and "faith is what is hoped for, but not seen".
See? The Bible does say it. Why are you being a contrarian, then?
However, it has been argued by the likes of InspiringPhilosophy and Whaddo You Meme?? that this can be interpreted differently, especially in the context of the Biblical word for faith, "pistis", which generally means trust.
I have not watched the source material you are referencing, so I have no idea if this is really what they have argued, or if you are misrepresenting their argument, but regardless, it should be noted that simply declaring that the word "pistis" is generally synonymous with trust, and so that whenever the word is translated to "faith", it implies that faith is synonymous with trust, is more than just a bit problematic. Firstly, as it is well-known that much of biblical scripture was at least partially allegorical, especially with these kinds of verses, structurally, claiming that the word in the verse means trust, as it would normally mean, is unwarranted, and analogous assumptions with other words in the scriptures would produce false results in this regard. Secondly, while we have no way to know for certain what the word actually meant to the original authors and their culture, we know that the word, while it can overlap with the meaning of trust, is not actually the same thing as trust. Thirdly, "pistis" having referred to trust back then does not imply it refers to our modern concept of trust today, as the understanding of trust has itself evolved througout history, as has the understanding of most other concepts. This relates to my second point. If the sources you cite did indeed say what you say they did, then their argument was likely a lot more nuanced than you are leading on, and while it would be unreasonable to expect you to reproduce their exact arguments in your comment, it certainly would be a stretch that there is sufficient evidence indicating that faith, as defined in the Bible, is actually equivalent to trust in any way, especially when it is taken in the context of the verses the word was used in.
You don't need a Bible verse that defines faith as trust.
Yes, you do, that is part of what is required to have it be biblical.
All you need is to look at the original language and what the word meant.
The word in the original language likely was not synonymous with trust as it is understood today.
Do you really think they suddenly changed the meaning of the word when used in the Bible?
It is more than plausible that this could have happened, considering that we do have confirmed examples of it happening. Not only that, we even have examples of words that were coined specifically in biblical texts. Besides, this is just a natural occurrence with languages as a whole. Books coining new terminology, especially when they are non-mundane, is the norm, not an exception, and we know this has been happening for as long as writing itself has existed.
Furthermore, I do think there are verses using faith as trust. If I remember correctly, this was brought up in one of InspiringPhilosophy's videos regarding this.
For someone who insists that such verses exist and have been used by a source that you watched, you are struggling an awful lot at providing such a verse.
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1. Crystal are actually defined in terms of their structure, which includes their spin, varying periodically in space. If some non-structural feature varies periodically, then it is not actually a crystal. Similarly, a time crystal is defined analogously, but with variation in time. It would be a huge misnomer to use the name "time crystal" to refer to what you are talking about, since there is nothing crystal-esque about non-structural features varying periodically (not that it is even clear that this actually can be achieved without periodic structural variations anyway). Also, the name "impossible" is being used to describe time crystals in dynamic equilibrium, which, as stated in the video, are genuinely truly impossible (by the No-Go Theorem). The time crystals recreated in the lab do not fall in the same category, because they are not in dynamic equilibrium.
2. You said "splitting and/or combining polarized light into orthogonal polarized beams in which the total input power is not oscillating does not eliminate the oscillating nature of the input energy." This is incorrect: it actually does eliminate the oscillations in the energy. The only thing that is actually oscillating is the electromagnetic field itself, not the energy density it carries. You can prove this just by taking the electromagnetic field of monochromatic plane waves and computing the energy density from it. The energy density depends on the amplitude of the wave, but as long as the amplitude itself is a constant (which it is for a laser), the energy density is constant too. This is a characterizing feature of lasers, and it differs from other forms of radiation, in which the amplitude itself is a function of time, and therefore, so is the energy density. The distinction is analogous to the difference between direct current and alternating currents. All currents are caused by periodic oscillations in the electric charge density across the wire, but direct currents are non-oscillating, whereas alternating currents are oscillating. The fact that the electric charge density itself is oscillating is irrelevant: we are not concerning ourselves with the electric charge density variations, especially as measuring them directly is nearly impossible anyway.
3. He explained in the video why a ringing bell is not an example of a time crystal, and not of any interest to the research in question. Any material can experience vibrations, but those vibrations are not intrinsic properties of the material, unless we are discussing time-crystal, in which case, the characteristics properties of the material are sufficient to predict all vibrations it will undergo given a constant energy input. This is different than a resonance phenomenon, which any material can experience, provided the energy input is non-constant and oscillates.
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@joda7697 Well no, if you look at it as a function to and from R, then use the formal definition of the derivative on an exponential function, you can derive the limit definition for the number e.
You cannot use the definition of the derivative on x |—> e^x : R if e^x is not defined. You need to first define what the symbol e^x means.
And then it has seemingly nothing to do with a power series anymore at all.
No, it still has everything to do with a power series, especially since the Maclaurin series expansion converges to e^x everywhere, which means e^x and its Maclaurin series expansion are actually one and the same thing. This is why no mathematician bothers with the limit definition, and instead define e^x as a shorthand symbol to refer to the power series. Hence, again: e^x is equal to the power series by definition. What I said is an indisputable fact.
It then becomes simply an exponentiation with a certain base which is defined by said limit expression.
No. You have been misled. e^x does not denote exponentiation. Historically, it never has. This is also why many mathematicians today, and many computer softwares, denote the output as exp(x) instead of e^x: because using the e^x notation is extremely misleading to students who have not studied real analysis, as those students are not aware that e^x is not what they think it is. The notation e^x makes you think you are working with exponentiation, since in general, the notation a^b, at least when a and b are specifically integers, does denote exponentiation. However, it just refers to something different altogether. There is no meaningful way to even define exponentiation for non-rational exponents, not without actually losing the meaning of exponentiation. So talking about a^x being an exponentiation for real x is nonsensical. This is something that mathematicians, and maths teachers, often take for granted and as being blatantly obvious, so they always forget the fact that laypeople are not actually aware of this. Yes, we do call it "the exponential function", but this is just a case of a really bad naming convention that has stuck for historical reasons, despite being objectively nonsensical. Such cases are common in mathematics. e^x, or well, exp(x), it not defined "as an exponentiation", it is merely a convenient shorthand for denoting a far more common yet naturally ocurring expression: the power series.
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@redx11x How is fine tuning not a problem for atheists?
Because (a) fine-tuning does not exist (b) even if it did exist, it has no logical implications regarding the existence of some ill-defined entity that is meant to be capable of reality-warping powers.
I don't see many atheistic physicists say the laws of physics came about by chance; rather, they posit the possibility that we are one universe in an infinite number of universes.
To be clear: there are no atheistic physicists that believe the laws of physics "came about by chance," because the laws of physics did not do any "coming about" at all. The laws of physics are not entities in the universe. They are abstract concepts existing only in the human mind. A law of physics is nothing more than an attempt at an empirically-supported mathematical formulation of some observed phenomenon. In other words, the laws of physics are descriptive, not prescriptive. The laws of physics do not cause the universe to behave how it does. The laws of the universe are our attempts at describing the way the universe does behave.
As for the various multiverse hypotheses, those would indeed solve the fine-tuning problem, if it were a real problem, but the idea of a multiverse is much older than the fine-tuning argument. The multiverse idea originated from physicists' attempts to unify the general theory of relativity with the quantum paradigm of physics. Research in quantum gravity and in string theory has suggested the existence of a multiverse, and there are many different models for what this means. When the idea of fine-tuning was first proposed, many models of the multiverse concept already existed, which is why physicists have never been particularly concerned about fine-tuning.
If fine-tuning were to exist (and it does not), there are many atheistic solutions to the problem besides the multiverse concept. Many such hypothetical models exist in physics already.
Most religious people do not ascribe to the belief that life came about through a naturalistic process, formation of DNA from complex protein chains.
With the exception of biologists, religious people have yet to demonstrate that they have a basic understanding of biology and chemistry. I have yet to meet a religious person who is not a biologist or chemist who understands that chemical reactions are not posited to be random by the scientific conclusions.
I have also yet to meet a religious person who can provide me with an ontologically-coherent definition of the word "naturalistic," and also provide me with an ontologically-coherent definition of the word "non-naturalistic."
With that being said, whether religious people accept basic scientific conclusions or not is irrelevant to this discussion, although they should be ashamed of denying the science, regardless. The fact to the matter is, proponents of the fine-tuning argument for the existence of an all-competent creator (not spaceless or timeless, since a spaceless timeless creator is a logical contradiction) declare that the universe was designed for life, not merely that the fine-tuning barely allows the existence of life. Also, the fact that life exists in a universe that permits life to exist is unimpressive, and does not costitute evidence for the existence of a creator, much less an all-competent one. If the universe were such that it does not physically permit the existence of life, and yet life were to exist anyway, then this would be evidence that some phenomenological entity with powers capable of ignoring the universe's typical behavior is responsible for life existing. Furthermore, if the creator is, in fact, all-competent, then it is impossible that the universe be fine-tuned, by definition. Fine-tuning, by definition, refers to the process of taking some free parameter, and adjusting the parameter to abide by specific constraints in order to achieve its goals of life existing. However, if the creator is all-competent, then no such specific constraints exist: any values of the free parameter, including no value at all, is consistent with the existence of life, simply by virtue of the fact that the creator is all-competent.
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I'm dissatisfied. When he said "let's get rid of carrying", I took that very literally, so what I envisioned is that when adding 6 + 7 in base 10, the value you would get in this new arithmetic would be 3, because normally you would get 13, but the 1 comes from carrying, so you get rid of it. Similarly, 12 + 9 = 11 because normally you get 21 but the 2 in front is the result of the 1 from 12 plus a carry from the summation 2 + 9, so instead, get rid of the carry and you get 11. That is what I expected when he said it. So when he then introduced addition as actually being fundamentally different altogether, I felt like it was all just false advertisement LOL. Then multiplication would follow similarly.
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I would have included Yasuri Nanami from Katanagatari. She is VERY deadly. Allow me to explain why. WARNING: SPOILERS AHEAD.
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1. She destroyed an entire village by herself. This would not be so impressive if not for the fact that this village was the village of some of the most powerful warriors in history, according to this plot.
2. She is so powerful her own body cannot handle how powerful she is, which is why she is constantly in pain. The fact that she is always in pain meant she always wanted to die amd had no will to live, so the reason she even searched for opponents to fight in the first place was so they could kill her. Despite not putting any effort into the fight because she wanted to lose, she always won every fight because of how powerful she was... every fight except 1. Need I mention that whenever she used her Observation skill, she always used it to try to become WEAKER? And yet she was still deadly.
3. The only fight she ever lost was against her brother Shichika, whom is the main character, obviously. Yet we all know she only lost that fight because she wanted to lose it. She allowed him to kill her. If she had not wanted to die from his hands, then she could have killed him in the bat of an eye with one of her basic moves, the move which she literally described as "In the time it would take you to fall to the floor from a single hit, my move will kill you 272 times". And in fact, in their first fight, Shichika, who is already an extremely OP character, couldn't even touch her, and upon using his finishing move, she parried it completely and told him "Your move is trash" (I'm paraphrasing, those were not her real words). The protagonist, who was so much more OP than their enemies, couldn't even step a stone to his sister.
3. Did I mention she has never needed physical training to fight? That is right.
4. Did I mention that, by virtue of being a member of the Kyōtorū family, she is literally a SWORD? Not
just a sword, but what was known as a Complete Deviant Blade, a perfected version of the other Deviant Blades. As such, despite the fact that her body is too weak and frail to withstand her own power, even though her body is weak compared to that of the rest of her family, she still cannot be killed the way any normal human can.
I honestly think she is so deadly and OP that she is not only one of the top deadliest young girls in anime, but I genuinely believe she is one of the top most OP and powerfully broken characters in the entirety of the fiction omniverse in the world. Any ability you can try to perform on her, she will observe it, and then immediately master the ability without any training or usage whatsoever, understanding all its flaws and learning its improved, perfected version. Goku doesn't even stand a chance against this girl if I'm going to be honest.
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@timothyhicks3643 Your understanding of Mormonism is incorrect.
No, it is not. My understanding of Mormonism is backed by the scholarly consensus, and this is reflected by the sources I have cited, sources which I know you have not consulted in a critical and comprehensive capacity prior to replying to my response.
But they do not view Jesus as simply human but rather as a distinct but still very much divine god of his own. Consider the following quote from Joseph Smith in “History of the Church” Volume 6 Chapter 23: “I will preach on the plurality of Gods... I have always declared God to be a distinct personage, Jesus Christ a separate and distinct personage from God the Father, and that the Holy Ghost was a distinct personage and a Spirit: and these three constitute three distinct personages and three Gods.”
This quote is not the powerful counterargument you think it is. As you will see in Bushman, Richard Lyman (2005). Joseph Smith: Rough Stone Rolling. New York: Knopf. pp. 455-456, 535-537. Smith only declared the existence of a unique Supreme God. However, he also believed in a hierarchy of inferior deities subordinate to this Supreme God. Jesus is among these inferior deities, as are all humans, ultimately. Smith had a tendency to call these entities "Gods" as well. See also Widmer, Kurt (2000). Mormonism and the Nature of God: A Theological Evolution, 1830-1915. Jefferson, N.C.: McFarland. pp. 119. for more details on the matter. Besides, not everything Smith said is representative of Mormonism today. Even among Mormon fundamentalists, a far more stark distinction is made between God the Eternal Father, and any other notions godhood. The only thing this argument proves, at best, is that Mormonism has Christian origins. I do not disagree, but as you have already agreed that having Christian origins is not sufficient for being a sect of Christianity, this is irrelevant.
To be fair, Mormon doctrine has changed a great deal since the lifetime of Joseph Smith and even within his lifetime, but this belief in Jesus as a god distinct from his father Elohim has persisted.
Not in the form stated above, it has not.
If I recall correctly, I believe that current LDS Church leadership has tried to make this polytheism less prominent in official channels in recent years, but I don’t think they have explicitly denied it...
To the contrary, the leadership has explicitly made many statements directly contradicting Smith's beliefs regarding what you just called "polytheism" (which I admittedly agree it is). See, for example, Widmer (2000). pp. 139, as well as Alexander, Thomas G. (1980). "The Reconstruction of Mormon Doctine: From Joseph Smith to Progressive Theology" (PDF). Sunstonse. Vol. 5, no. 4. pp. 24–33 for various examples. It is true that they do still have polytheistic tendencies. I never denied this, and in fact, I explicitly acknowledged this in my previous reply (and by now, I am beginning to question whether you actually read my reply closely at all or not). But, if anything, this only actually supports my point about Mormonism being a different religion from Christianity: Mormonism is henotheistic, while Christianity is undoubtedly monotheistic, and their theologies differ fundamentally, with the similarities only being superficial. The attempts you have made to establish these similarities as more than superficial, though, simply do not work, and I have no idea why you are appealing to Mormonism's henotheism to try to prove that it somehow has "similar" theology to Christianity. It simply does not.
...and the doctrine can still be found stated clearly as in other relatively modern sources such as the LDS believer-run website mormonchurch dot com, which published an article in 2009 explaining the belief to non-Mormons in the following words:...
If you think I should take you seriously for citing a website that is ran by a relatively small number of believers, and take that as having more weight in accuracy than the half-dozen scholarly books I have cited by experts who are deeply acquainted with Mormonism and have been for decades, then I think this conversation is a waste of my time, and I should end it here. But for anyone else reading this, I should mention, that again, this does not actually constitute much of a counterargument, and it makes the same mistake with failing to acknowledge "inferior" notions of godhood from the Supreme God, God the Eternal Father, which the LDS movement-theology has always singled out, in all of their sects (and yes, the LDS movement actually has multiple sects, which you seemed unaware of. The fact that one dominates over all the others in popularity is not relevant to the discussion).
Though he is a different deity from God the father, Jesus still holds a central position in Mormonism (he is named directly in the name of the Church of Jesus Christ of Latter-day Saints) as the being through which humans can be saved.
This is just an inaccurate simplification of their soteriology, as mentioned. And please, really? You are appealing to their name as a piece of evidence? Okay, so I take it you also believe that North Korea, a.k.a the Democratic People's Republic of Korea, is actually a democratic republic, since it says so in their name? And I take it that you think that the Nationalsozialistische Deutsche Arbeiterpartei, a.k.a the National Socialist German Workers' Party, despite their anti-Marxist policies, were actually socialists? This is just a terrible argument. I am not denying that Jesus is an important figure in Mormonism, but this alone does not automatically make a worldview a sect of Christianity. This is an extremely myopic way of thinking about classifying religious worldviews, so there is no surprise here that the scholars disagree with you on this. Different worldviews treated as sects belong to the same religion when there are common fundamental tenets all the sects share, share the same foundational scriptures with limited variability, are viewed as having similar traditions, and do have some tolerance (extremism aside) for cooperation between sects. What you are doing has essentially nothing to do with that.
I am aware of the other differences you listed between Mormonism and most other Christian sects, but I don’t see why the definition of Christianity should be drawn upon those lines as opposed to any others.
It is done for the same reason nearly everyone treats the Druze as a separate religion from Islam. Besides, Christianity is the only religion in the entire planet with this kind of problem: of seeming ill-defined, for the fact that it essentially is comprised of multiple religions being treated as if they are one. Many scholars have started to recognize this, though, and how we think about Christianity has started to change in the past few decades.
Not all Christian denominations are trinitarian; other examples include Oneness Pentecostalism and Jehovah’s Witnesses.
I never said that Mormonism being nontrinitarian makes it not Christian. I explicitly avoided making this point, while still explaining my actual argument. Besides, this comparison is outrageous. Oneness Pentecostalism is much closer to Trinitarianism than Mormon henotheism, and while they are indeed not Trinitarian, they do, in all other aspects, have fundamental similarities tying it together to Christianity, which is not something we observe with Mormonism and the LDS movement. As for Jehovah's Witnesses... there is legitimate discussion to be had as to whether they are their own religion or not, and scholars are having such a discussion as we speak; but even so, again, they are far closer to Christianity in core tenets than anything that Mormonism is doing. They have many more fundamental similarities from other Christian sects which are restorationist.
Not all Christians agree on the same set of scriptures; Catholic Bibles contain more books than Protestant Bibles, and Eastern Orthodox Bibles contain more books than both.
This is not at all analogous to simply having a whole other religious book that, in the very religion, is seen as meant to salvage what was corrupted and lost from the Bible.
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@daneaparker The Church of Jesus Christ of Latter-day Saints (not Mormonism) presents the Father, the Son and the Holy Spirit as three separate Gods (Gen. 3:22 "...has become as one of us").
Yes, I know that. I was literally the first person in this thread to bring this up. Your choice to start this reply with this already proves that you did not read my reply, or that you might be illiterate. Whatever the case is, I do not care, because I already know that talking to you for an extended period of time is probably going to be a waste of my time. You know, if you want to have a conversation with someone, this is not how you do it.
Also, the Church of Jesus Christ of Latter-Day Saints is a sect of Mormonism. Being delusional enough to say "No, it is not" does not change the facts in question.
Jesus is not an "angel", except in the sense that those he appeared to after his resurrection might have perceived him as such.
I never claimed that he is an angel in any other sense. Again, proof that you did not read my comment.
I get the sense that your concerns boil down to the Church of Jesus Christ of Latter-day Saints doesn't accept the concept of a Trinity.
Nope. I explicitly proved otherwise, you know, in the comment that you did not read.
This discrepancy of considering Jesus a resurrected (with a body) God (the New Testament is very clear that Jesus is not a spiritual being - telling Thomas to touch the prints in his hands and eating some fish after his resurrection) separate from his Father in no way precludes this church from being Christian.
I do not care. I am not Christian, so I am not here to have you preach to me about your personal beliefs about Jesus Christ. My comment is concerned with one thing only: categorizing religious systems, and determining whether two systems are related enough to be classified as the same religion, or whether they are simply different religions. Once again, I made this clear in my comment. You would know, had you actually read it sigh I seriously need to stop interacting with people on YouTube. This type of incompetence is going to give me an anneurysm one day.
If it did, I suppose the Church of Jesus Christ of Latter-day Saints could claim other religions are not Christian because they disbelieve the Biblical testimony that Jesus still has a body, albeit a resurrected one.
That is not how classification of religions works, but also, the Church already does that.
If the Church of Jesus Christ of Latter-day Saints is respectful enough to accept Christianity in all forms, perhaps that respect should be reciprocated.
Classification of religion has nothing to do with respect, and the Church of Jesus Christ of Latter-Day Saints is among the ones most strongly employs the No True Scotsman fallacy to exclude other Christians from being Christian.
Anyway, I am not going to waste my time further. If you are not going to read my comments, then I will not read yours either. Bye.
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