Youtube comments of Anonymous (@Anonymous-df8it).
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@ChibiNyan Technically, that's a bad idea as a binary search is probably better as the expected value of information (entropy) is higher. Also, Guess Who? has (had?) an actual problem; the completely antiscientific 'male as default' for humans, which is like stating that water flows uphill
Before I clarify the expected value of information, let log(b,x) be log base b of x.
Also, let's work out the amount of information in bits of finding out a trait:
If the trait occurs in half the sample, knowing whether or not it's present gives one bit of information by definition.
If you found out that a sample member had two independent traits with (not necessarily) different probabilities, the probability of having both traits is the product of the probabilities for each trait. Since we want the amount of information to be additive, not multiplicative, it's going to involve some base of logarithm for the probability of the trait.
Therefore, the amount of information in bits of finding out a trait is -log(2,[probability of trait]).
Finally, the expected value of something is defined by taking the probability of getting a certain value multiplied by the value itself, repeating this for all possible values, then adding them up. In other words, the expected value is the sum of the values weighted by the likelihood of each value occurring.
In the case of information, given two possibilities 'yes' and 'no' with probabilities p and 1-p respectively, the expected value of information is p*[amount of information gained if 'yes']+(1-p)*[amount of information gained if 'no'], or p*(-log(2,p))+(1-p)(-log(2,1-p)).
Therefore, in your example, if you had a binary search, then the expected value would be (1/2)(-log(2,1/2))+(1/2)(-log(2,1/2)), (1/2)+(1/2), or 1 bit of information. If you ask a question that takes out either 2 or 435 possibilities, the expected value would be (2/437)(-log(2,2/437))+(435/437)(-log(2,435/437)) or (approximately) a whopping 0.042155 bits, which shows that it's a terrible, terrible move.
Also, Guess Who? did run into a problem, though it wasn't the lack of non-binary genders; the problem was one would expect that 'is it [sex]?' would form a binary search (i.e., either sex would show up in 12 cards out of 24), but it didn't as femaleness (as opposed to maleness) was treated as a 'characteristic' and thus only showed up in 5 cards out of 24, which is completely antiscientific (not to mention sexist) as humans are mammals, not birds, and this is how mammalian sex determination works:
Female mammals have XX chromosomes and male ones have XY chromosomes. When mammals procreate, one of the female's X chromosomes unite with the male's X or Y chromosome with equal probability, giving rise to (approximately) equal proportions of male and female offspring. The X chromosome, like the autosomes, are essential for life, whereas the Y chromosome simply signals to produce testes that release androgens, triggering masculine development, as opposed to feminine development, which is the default mammalian pathway. In birds, this is reversed, as I alluded to in the previous paragraph. If you absolutely must treat a human sex as the default (which you shouldn't), it has to be femaleness
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a) The male to female ratio is 19:5, as they had the completely antiscientific 'male as default' assumption for humans, which is like stating that water flows uphill. and b) You'd only need to make eight extra guesses in the modified game
a) Femaleness (as opposed to maleness) was treated as a 'characteristic' and thus only showed up in 5 cards out of 24, which is completely antiscientific (not to mention sexist) as humans are mammals, not birds, and this is how mammalian sex determination works:
Female mammals have XX chromosomes and male ones have XY chromosomes. When mammals procreate, one of the female's X chromosomes unite with the male's X or Y chromosome with equal probability, giving rise to (approximately) equal proportions of male and female offspring. The X chromosome, like the autosomes, are essential for life, whereas the Y chromosome simply signals to produce testes that release androgens, triggering masculine development, as opposed to feminine development, which is the default mammalian pathway. In birds, this is reversed, as I alluded to in the previous paragraph. If you absolutely must treat a human sex as the default (which you shouldn't), it has to be femaleness
b) A yes/no question's response provides one bit of information, the standard two-gender game requires one bit to state the gender (since 2^0<2<=2^1; in fact 2=2^1), the modified version needs nine (since 2^8<437<=2^9, and more specifically, 437<2^9). Therefore, only 9-1, or eight additional bits of information are required
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@loganskiwyse7823 I'm fairly certain that the ability to speak is a prerequisite to singing. If we assume that someone who can't speak can sing, then this person could sing the words to communicate, whilst holding their pitch steady (if one can't control the pitch of one's voice, then one can't sing; by taking the contrapositive, we can show that if one can sing, our original assumption, then one can control the pitch of one's voice), showing that said person can speak, a contradiction
Also, why couldn't wikiHow or a similar website teach you how to do the things that your carers are currently doing for you? I don't mean to be rude, but I don't see how it can be anything other than learned helplessness, assuming that your belief that you're highly gifted isn't attributable to the Dunning-Kreuger effect
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@AmericanBrain I think I've understood everything.
I never said that the workplace is democratic. I said that it should be democratic. There is force in the contract. Sign it or there may be weeks or months before you find a better one, during which, you could starve, be rendered homeless etc. Also, I don't disregard science. Economics is the study of economic systems, most often capitalism. Whether it's the best is subjective and up to debate. Also "Just take them to court". Yeah, when all the judges are appointed by a government in which both viable options (Democrat and Republican) a) have a lot of corporate lobbying and b) have a lot of CEO politicians themselves. They need to be held accountable directly. And if "You can just work harder to get to places", you conveniently ignore the fact that inheritance money is a thing, meaning that yes, lower classes staying in lower classes is a property of capitalism. It doesn't take much research to find out that Sri Lanka is capitalist, proving MY point. Do you have any sources suggesting otherwise? Also, calling me delusional because I disagree with you? I don't call you delusional just because you have a different opinion to me. That's just rude.
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@AmericanBrain I never said that China, Russia etc. is good. Also, non-US countries are not democratic socialist places. It's not force if it's a contract doesn't refute my point. According to your definition, it was never a contract in the first place. Also, actively seek out worker cooperatives? Ok then.
Let's look at the equivalent of capitalism, but in government:
People get tokens and get into high-ranking jobs if they obey laws (like CEOs do to who listen to them)
Most governments are dictatorships (like how few businesses are worker co-ops)
If you don't like your country (like businesses in this analogy), you can:
a) move to a different one but your tokens are worth less the longer you're outside a country, not to mention the lack of tokens will make them reject as you're not a 'good citizen' (like how your money slips away when you're out of a job and poor people have less opportunities)
b) start a new country but you need to use up a lot of tokens in creating said country, or it will fail (like how most businesses fail due to insufficient capital).
It should be a utopia as it's inspired by capitalism, but it isn't and the flaws are glaringly obvious, hinting at the fact that maybe, just maybe, there's something better than capitalism.
Please correct me if I made mistakes with my analogy.
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@AmericanBrain Why should it (social democracy, NOT democratic socialism) not be brought into the US? I really don't see how it's worse then what America has. I really don't.
"It's not force if it's a contract doesn't refute my point. According to your definition, it was never a contract in the first place.
Me: I am lost . I am sure you are not doing it on purpose but it feels like your second strawman argument. But there is nothing to indicate you are doing it on purpose."
a) By 'the point' I was referring to the fact that you have to take any job opportunity, no matter how bad (especially if you're in poverty and you're trying to work your way out of it) because a lot of things could happen before you get another one. If you define a contract as something that isn't coercion, it's only a contract by name.
b) It implies that's this is a strawman and I've strawmanned your arguments before. Dare I ask for evidence?
"Let's look at the equivalent of capitalism, but in government.
ME:
1. government jobs are not part of capitalism per se. I fear the analogy will not work.
2. There should be NO pay whatsoever for government jobs. The ONLY way to pay people for government jobs is to STEAL money USING FORCE called taxation from "fair earners' like CEO/FOUNDER of Amazon and all other people in the U.S.
You can Not take 1 penny - not even 1 penny from a billionaire (nor any other American)>"
1. You haven't even looked at the analogy. You just said it won't work then took a tangent, which I'll address in points 2 and 3.
2. There is also theft in private companies. The workers provide goods and services to the consumer. Those who don't (CEOs) somehow manages to get most of the money earned and call it 'profit', and you don't seem to want to convert all conventional firms to co-operatives.
3. Economics Explained has a video on taxation. Under a capitalist system, there is no way you can remove taxes, period. If you support capitalism, you support taxes. So pick one.
"except if it involves any "unionization" that's supported by law for that is socialism." No. Having a government that is sensible enough not to bust unions (workers that are unified) is not necessarily socialist, just social democratic.
This was a long one.
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@AmericanBrain "You are sovereign and can do whatever you like. As CEO - hire people but pay them freaking low and work them incredibly hard. If they leave, have a system to get others in place. You will out-earn, out-do, "out-mate" , out-compete all your peers by magnitudes whilst working a fraction of the time. Your job in life is to be the best and form a monopoly."
Imagine if you lived, as a worker in this true free-market world. You work very tirelessly, 24/7, but you hardly get a few pennies, as it's not profitable to pay workers much and there are no minimum wage laws. You have an amazing business idea, but despite toiling away your entire life, never stopping, even once to see your family, you'll never be able to pursue your dreams, and you can barely afford basic needs. You ask others for help, but they're in the same situation. You can't afford any technology or internet, so you can't contact anyone with much money.
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@ChibiNyan log(b,x) is defined such that b^log(b,x)=x. Now, let's define b^k. For positive integer values of k, b^k is defined as b*b*b*b... k times, or more formally: b^1=b, and b^(k+1)=b*(b^k). Here, I'll outline how it is typically defined for arbitrary real k
Before we start, let's make it clear that b^(m-n)=b^m/b^n. This is because the right-hand side looks like b*b*b*b... (m times)/(b*b*b*b... (n times), that is, n copies of b in the denominator cancel with m copies of b in the numerator, leaving (m-n) copies of b in the numerator. Similarly, b^(m+n)=(b*b*b*b... m+n times)=(b*b*b*b... m times)*(b*b*b*b... n times)=(b^m)(b^n).
Also, let's make it clear that, b^(mn)=(b^m)^n. This is because the left-hand side looks like:
b*b*b*b*b...*
b*b*b*b*b...*
b*b*b*b*b...*
...
where each row has m b's and each column has n b's.
We can add parentheses as shown:
(b*b*b*b*b...)*
(b*b*b*b*b...)*
(b*b*b*b*b...)*
...
=(b^m)*(b^m)*(b^m)... n times, or (b^m)^n, the right-hand side.
Now, we can use these two identities to define b^k for all rational k. Let k be equal to m/n, where m and n are integers and n is positive. We want to define b^(m/n) such that the above properties still hold, allowing us to conclude that (b^(m/n))^n=b^(mn/n)=b^m.
If m is positive, we can define b^(m/n) such that when you multiply it by itself n times, you get the same result as multiplying b by itself m times.
If m is zero, and since we want the above properties to still hold, we can define b^0 as b^(1-1) or (b^1)/(b^1)=1. Also, as a bonus, if m=0, m/n also equals zero, so we're done here.
If m is negative, and again, since we want the above properties to still hold, we can define b^m as b^(0-(-m)), or (b^0)/(b^(-m))+1/(b^(-m)). In other words, we can define b^(m/n) such that when you multiply it by itself n times, you get the same result as multiplying b by itself -m times, then taking the reciprocal.
Now, assuming that we want b^k to be continuous, we can define b^k for all real k. Let k be equal to the limit of k_1, k_2, k_3... where k_n for all n are rational. b^k can then be the limit of b^(k_1), b^(k_2), b^(k_3)...
If we set k such that b^k=x, then k would be log(b,x).
Now finally, to see how logarithms turn multiplication into addition, consider that b^(log(b,x)+log(b,y))=(b^(log(b,x)))(b^(log(b,y)))=xy. Taking the base b logarithm of both sides gives the following: log(b,x)+log(b,y)=log(b,xy)
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@egilsandnes9637 Real numbers define sliding (addition; if 0 is dragged to a, b is dragged to a+b) and stretching (multiplication; if 1 goes to a, keeping 0 fixed, every b goes to ab) of a line. Complex numbers do the same with a plane (0=(0,0), 1=(0,1)). You may ask, why can't an equivalent thing be found in 3d? Well, sliding is fine, but stretching/rotating fails. If you keep (0,0,0) fixed, and move (1,0,0) to (0,0,1), then another, different action can be found defining (0,0,1)(a,b,c); rotate your solution, then you have another one. This is due to the fact that rotation happens on a plane, and 3d gives us a 'spare direction'. Quaternions do get around this problem, though
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@AmericanBrain "Global weather changes have ALWAYS happened, sometimes changing far stronger year to year, other times years of a "range"." Yeah, but this time, it's caused by man. Just because last time, the extinction event was caused by nature, doesn't mean it is this time round. We have a choice as it is us who invaded the environment this time. We are responsible for a sixth mass extinction. We could, and should save those animals.
"using capitalism such as more air conditioners, more heaters ; fortifying sea shores [which may be a waste as the sea will come on in] OR building further inland - building HIGHER and if need be, lower into the ground too. There are many other ways."
Two things:
a) Capitalism doesn't make those products, labor does.
b) What about desertification, turning the earth into a desert square mile by square mile? We can't just technology-ize our way out of this one. There is less and less arable land, and also less fresh water. If you think that hydroponics will solve everything, you're wrong. There is less fresh water and desalination is expensive. Increasing the costs of food production just means that poorer countries are left in the dust.
"I have invested so much into earth destruction, and a fortune burning the Amazon [but via investment in Brazilian firms that do it UNDER LICENSE]. I made FAR FAR FAR more investing in petroleum. This is a GREAT opportunity to now." You d1$gu$t1ng sw1ne.
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@allylilith5605 What about climate change? I wouldn't say that long- or short-term thinking have a right- or left-leaning bias respectively. Whilst the long-term view may be more right leaning for some issues, it's more left leaning for others
Also, regarding your example, according to Sabine's video on gender-transitions, the conservative view seems to be more about weariness to change, which seems consistent with conservative positions on most topics, as opposed to maybe a few. Though, as you state, the leftist desire to change does stem from the idea of not hurting others. Examples include, but aren't limited to the pandemic response, refugee crisis management, climate mitigation, and, of course, the original example of transgender issues.
All of these examples demonstrate the leftist view of not harming others. as you say, (at least in the immediate), since the leftist view on pandemic responses is centred on preventing illness, the leftist view on refugee crisis management is on helping the refugees, leftists advocate action on climate mitigation to prevent crises, and leftists, as you state, advocate gender transitions for gender dysphoria, hoping that it's effective in treating the condition, or at least, more effective than for instance, counselling alone.
All of these examples also provide evidence of the conservative view being about weariness of change, since the conservative view on pandemic responses is maintaining a sense of normalcy, for fear that action may set precedent for the systematic stripping of freedoms, the conservative view on refugee crisis management is to minimise the change in the makeup of the native population, conservatives advocate for climate inaction for fears of the effects of other electrical sources, as well as (as you state), not providing gender transitions, due to long-term side effects, some known, and some, potential.
Furthermore, whilst your example of gender transition seems to provide evidence that the conservative view is "about giving people the best possible outcome, even if the first steps may be unpleasant", since the conservative position on gender transitions is about avoiding long-term side effects for relief that may potentially be short-term, two of the other examples outlined here (pandemic responses and climate mitigation) provide evidence against this notion, since your description of the general conservative perspective better fits with the leftist viewpoint; sacrificing short-term comfort to prevent mass illness (and the deaths that go along with it), and climate disasters, respectively, are examples of long-term thinking.
The left leaning viewpoint, is therefore, as you've stated about minimising harm; however, the conservative view is one about general weariness of the potential consequences of change, and these viewpoints are probably independent of short- and long-term thinking, since the left and the right both exhibit short- and long-term thinking depending on the situation, thereby indicating a lack of correlation, either as you suggest, or opposite to what you suggest.
An analogy that I've read somewhere is that the political left and right are like the accelerator and the brakes of a car respectively (so reversed compared to the actual positions of the brake and accelerator pedals on a car); you need the accelerator for the car to start moving (or to increase the velocity, and thus, momentum, of the car in general), including to get out of undesirable places (faster, in the case of the car already moving), and the brakes to stop the car from crashing
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@rianczer Except I'd argue that it isn't perfectly argument, as per the following response:
@allylilith5605 What about climate change? I wouldn't say that long- or short-term thinking have a right- or left-leaning bias respectively. Whilst the long-term view may be more right leaning for some issues, it's more left leaning for others
Also, regarding your example, according to Sabine's video on gender-transitions, the conservative view seems to be more about weariness to change, which seems consistent with conservative positions on most topics, as opposed to maybe a few. Though, as you state, the leftist desire to change does stem from the idea of not hurting others. Examples include, but aren't limited to the pandemic response, refugee crisis management, climate mitigation, and, of course, the original example of transgender issues.
All of these examples demonstrate the leftist view of not harming others. as you say, (at least in the immediate), since the leftist view on pandemic responses is centred on preventing illness, the leftist view on refugee crisis management is on helping the refugees, leftists advocate action on climate mitigation to prevent crises, and leftists, as you state, advocate gender transitions for gender dysphoria, hoping that it's effective in treating the condition, or at least, more effective than for instance, counselling alone.
All of these examples also provide evidence of the conservative view being about weariness of change, since the conservative view on pandemic responses is maintaining a sense of normalcy, for fear that action may set precedent for the systematic stripping of freedoms, the conservative view on refugee crisis management is to minimise the change in the makeup of the native population, conservatives advocate for climate inaction for fears of the effects of other electrical sources, as well as (as you state), not providing gender transitions, due to long-term side effects, some known, and some, potential.
Furthermore, whilst your example of gender transition seems to provide evidence that the conservative view is "about giving people the best possible outcome, even if the first steps may be unpleasant", since the conservative position on gender transitions is about avoiding long-term side effects for relief that may potentially be short-term, two of the other examples outlined here (pandemic responses and climate mitigation) provide evidence against this notion, since your description of the general conservative perspective better fits with the leftist viewpoint; sacrificing short-term comfort to prevent mass illness (and the deaths that go along with it), and climate disasters, respectively, are examples of long-term thinking.
The left leaning viewpoint, is therefore, as you've stated about minimising harm; however, the conservative view is one about general weariness of the potential consequences of change, and these viewpoints are probably independent of short- and long-term thinking, since the left and the right both exhibit short- and long-term thinking depending on the situation, thereby indicating a lack of correlation, either as you suggest, or opposite to what you suggest.
An analogy that I've read somewhere is that the political left and right are like the accelerator and the brakes of a car respectively (so reversed compared to the actual positions of the brake and accelerator pedals on a car); you need the accelerator for the car to start moving (or to increase the velocity, and thus, momentum, of the car in general), including to get out of undesirable places (faster, in the case of the car already moving), and the brakes to stop the car from crashing
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@ratpatooti5080 Then they'll make two more fake caves so people can be at awe as to how similar they are. Then several more thousand years pass, they find 4 nearly identical caves, and recreate the 4, again to impress people about how similar they are. Then they find 8, then 16 etc. All of this relates to powers of 2. It's like how you have 2 parents, 4 grandparents, 8 great-grandparents etc. Maybe people millions of years from now will look back and perceive the entire history as a representation of our family tree...
Maybe, they think, the first cave was made, then the replica, both being metaphorical siblings. Then there are two more replicas, with them being metaphorical parents, but time reversed. Then there are four more replicas, being metaphorical grandparents, if not time reversed. Then 8, being metaphorical great-grandparents etc.
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@Homer-pc6oj "If you visualize infinity, [infinity] goes to a singularity." Does it? It seems to be well-behaved as infinite cardinals and ordinals are used in set theory. "We call [the singularity] the tenth dimension in quantum physics." Even assuming that there is a 'singularity' involved, there are still so many problems with that statement, I don't know where to start.
Firstly, mathematical pathologies aren't physical dimensions. The number of dimensions is simply the smallest number of vectors in a space (or space-time in physics) required to represent every point in said space via linear combination, regardless of how tightly curved some directions may be (I'm assuming you're referring to string theory here).
Also, even in an at-least-ten-dimensional space, no dimension would be 'tenth'. For simplicity, let's consider the three-dimensional case (and trying to pin down the first, second and third dimension) before moving on to the notion of a tenth dimension in a space that's at least ten dimensional. What is the first dimension in 3d space? How about the second? How about the third? Notice how you can't answer a specific direction for each.
If you, for instance define the 'third dimension' as the vertical dimension, you'll disagree with anyone who is neither near you, nor at the antipode of your location, not to mention that there's no real reason for that dimension to be 'third' as opposed to being 'first', 'second' or a linear combination of the 'first', 'second' and or 'third' dimensions. Similarly, if you, for instance, define the 'second dimension' as north, you'll disagree with anyone not on your latitude (and also anyone not on your longitude, provided you're not on the equator), as well as running into the arbitrariness of calling it 'second'). If it's not possible to pin down a 'first' 'second' and 'third' dimensional space, why on earth would you think that it's possible to pick out a 'tenth' dimension in a space with at least ten dimensions?
To summarize, there are no singularities involved here, and even if there were, they a) aren't physical dimensions, and b) there is no 'tenth dimension' even in spaces with at least ten dimensions
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May you please calculate and compare the proportion of handicapped minority female rural people in China as a whole, and the proportion of handicapped minority female rural students in elite universities, so that I know how to interpret your comment?
I'm well aware of discrimination existing; however, it's just that "handicapped minority female rural students" reeks of overspecification to me, since, even in isolation, a small proportion of people are handicapped (most are able-bodied), a small proportion are minorities (if that wasn't the case, they'd be the majority), a small proportion are female (due to the one-child policy's unintended effects), and a small proportion are rural (due to increasing urbanisation). Assuming that these probabilities are independent, the combined proportion would be the product, which would be quite small
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@AwesomePlaylists888 Do you have any evidence for the notion that your sky daddy's son, who is also sky daddy, somehow, spoke celestial objects into existence and set them in motion?
Oh, wait. I forgot that blind faith is a virtue to you. You're going to cite the Bible, to which I ask for proof that it's a credible source, and you'll just state that I just have to have faith in it (i.e., believe it without questioning). Sure!
Besides, your god isn't very moral, as evidenced by the following scenario posed in your holy book:
An omniscient god sets up a paradise, including Adam, Eve, a tree of knowledge with forbidden fruit, and a talking snake. God tells Adam and Eve to not eat the fruit, and the talking snake he set up tells them to eat the fruit. They oblige with the snake, eat the fruit, and now, all their descendants are destined to die, the females must suffer a really painful childbirth, with many experiencing premenstrual syndrome every month outside of that, as punishment for eating the fruit.
When Adam and Eve ate the fruit, they had no sense of right and wrong (as that came from the forbidden fruit), and so had no idea that they were supposed to follow your god and not the snake he set up. Also, your god would have known that this would be the result due to his omniscience. He deliberately set up the scenario, which is a trap, for they didn't know any better, and he, being omniscient, knew that they didn't know any better. Then, if that wasn't bad enough, he then punished every future innocent person, who did not, and will not eat the fruit, to certain death and half of them to some of the most painful childbirths in the animal kingdom.
Even if your holy book is right, your god is a piece of $h!t, and isn't worth worshipping and giving up a seventh of your life for. And if it's wrong (which is the stance I take), then it's not worth worshipping and giving up a seventh of your life for a fictional character. So, either way, your god is not worth worshipping, at all, and you can stop trying to convert me. Bye!
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There is no objective morality- see Hume's guillotine. However, we all generally share basic moral axioms (e.g., non-hypocrisy (if x claims y should be true, x ought not to be y's counterexample), empathy (if x claims for all y, y shouldn't do z to x, for all w, x ought not to do z to w), and self-preservation (if x being alive doesn't kill others, x ought to remain alive)). Without these, society would collapse. However, they're not objective like some universal law; that's why you can't convince a psychopath to be empathetic. The hope of holding criminals accountable is to influence their morals and whilst isolating them, such that they don't cause harm. Your morals, reality, and logic all shape what we do. You're not going to kill your child, regardless of free will's existence
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@minimo3631 "nope, talk therapy can of course make it easier to handle a lot of different stressors in life, but it can't cure dysphoria." Dysphoria, "A state of feeling unhappy, dissatisfaction or frustration." I take it to mean that you're referring to gender dysphoria specifically, as body integrity dysphoria has a treatment that isn't parallel to gender transitioning (no offense to sufferers of the former; I know that the comparison used by some to hate against them, but I haven't a better example). Gender dysphoria, iirc, is the desire to become the opposite sex, so why would women with beards desire to become the opposite sex, i.e., male? Also, I read the last paragraph and there's actually a phrase for it, 'social model of disability'
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@MuffinsAPlenty These problems can be resolved:
Motivation: Take the set of natural numbers and put it on the number line. There are gaps everywhere, leaving practically the entire number line unaccounted for. It seems natural to repeatedly halve intervals to fill in the gaps. The gap issue seems solved. However, take the sequence: 1/4, 1/4+1/16, 1/4+1/16+1/64... This sequence clearly has an upper bound: 1/2 (all successive halvings are done to the left of the first halving of the unit interval), yet it's not a number accounted for. Thus, we could 'fill in' these gaps: {-2,-4,-6,-8...} (the number is to the right of the second, fourth, sixth, etc. halving, but not the first, fourth, fifth, etc.)
Set restrictions: Yes, it's clear that the sets must have an upper bound. {1,2,3...} for instance makes no sense: the numbers seem to be unbounded. However, if you replace the upper bound rule with a lower bound rule, you get the 2-adic numbers.
Equivalence classes: All sets where there's an absent integer with all lower integers present mean the same thing as another set where all the lower integers are replaced with the absent one. This doesn't seem to be an issue either. The motivation for this equivalence relation is that if you took the unit interval and subdivide the right-hand side every time, you'll just get one.
Negative numbers: You didn't point this out, but the current system is not a field; we need more numbers. One solution is to switch to negabinary. However, that makes equivalence classes more complicated. Another solution is to re-introduce some of those pathological sets. The complement of the set of all integers and a non-pathological set behaves like its negative and the set of all integers behaves like 0, the empty set. To see this, adding 1 to '...111111.111111...' gives '0.111111...', or 1. However, this solution requires another condition, that the integers and the empty set mean the same and an exception to the set restrictions: that it doesn't apply if there's an integer such that all integers above that are represented. A third solution is to denote the negative of a set as {[empty set], A}, which is I think is the most intuitive solution. However, a new set of operators is required to define interaction with negative sets, like subtracting a positive set
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You'll find that it's not "1, 2, 3, 6, 12, 24, 48, 96...."; in fact, it's "1, 2, 4, 8, 16, 32, 64, 128...."
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