Comments by "MC116" (@angelmendez-rivera351) on "Insights into Mathematics"
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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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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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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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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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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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@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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@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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@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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@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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@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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