By the law of identity, two distinct sets cannot be the same. If they actually are the same, then they are necessarily one, the same set. It's contradictory to say that two things are the same. If it is the same, it is only one. — Metaphysician Undercover
The categorical vocabulary itself, however, seems to be spreading like wildfire. — alcontali
It is almost literally what you will find mentioned in the page on the "Brouwer-Hilbert controversy" — alcontali
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Concerning "no bearing on the topic at hand", you undoubtedly say that, because you are not aware of that famous discussion between Hilbert and Weyl in 1927, which was exactly about this. Could that have something to do with "weaker" Wiki skills? ;-) — alcontali
Arbitrary axioms are the hallmark of creativity! — alcontali
I don't think that can really be true though. Math IS useful and meaningful because it takes human effort to determine whether two different representations of a thing are actually the same thing. Don't you agree? 2 + 2 = 4 is formally a tautology. But historically, it was a really big deal for humanity. Agree or no? — fishfry
I seem to recall the old philosophical standby of the morning star and the evening star, which appear to be two different things but (upon astronomical research that took millennia) turn out to be the same thing, namely the planet Venus and not a star at all. — fishfry
If you reduce everything to the law of identity, you are saying those millennia of observation and theory and hard work by humans means nothing. I don't accept that. — fishfry
Well, I really don't agree, and I think you misunderstand creativity. Art is not a product of arbitrariness, there are reasons for what the artist does, purpose, so arbitrariness is not the hallmark of creativity. — Metaphysician Undercover
Functional languages are the big thing now and they have monads — fishfry
I'ts okay. Philosophers use no categorical language; they say, "That's post-modernist regressivism" or something of the like, and they leave it at that. It's us, dilettante, who spell everything out for each other.One problem is that programmers who discuss pipelining rarely use categorical language in their discussions. So, I cannot determine if both things are related (monads versus typical pipelining practices). — alcontali
I'ts okay. Philosophers use no categorical language; they say, "That's post-modernist regressivism" or something of the like, and they leave it at that. It's us, dilettante, who spell everything out for each other. — god must be atheist
That would almost amount to saying that an artist's design choices are exclusively rational, and could therefore even be expressed in formal language. My own take is that I do not believe that. I believe that artists make use of other mental faculties, that are not rationality, when making their design choices. I also do not believe that it is possible to express, even in natural language, the output of these other mental faculties. — alcontali
Well, that would depend on how you define "arbitrary". Use of mental faculties in one's decisions negates randomness. If such decisions are arbitrary, then how do you understand "arbitrary"? — Metaphysician Undercover
It is the same situation as with a sequence generated by a Mersenne Twister. From the outside, it looks random. From the inside, we can see that you will always get the same sequence depending on the seed that you use. Is the sequence random? For outsiders, yes. For insiders, no. — alcontali
It's not arbitrary then, it just looks arbitrary, in appearance, but it really is not. That it is arbitrary is an illusion. Would you see mathematical axioms in the same way? They look arbitrary, but they really are not. What is required to get beyond the illusion of arbitrariness is to get inside of the head of the artist. This does not mean to literally get inside, but to learn how to think in the same way as the artist. Then you will no longer be an outsider who sees mathematical axioms as arbitrary. — Metaphysician Undercover
Furthermore, their assumed input could still truly be random, because there is no method available to distinguish between the output of unknown mental faculties and sheer randomness. — alcontali
Still, the uncanny sensation of recognition suggests that this link is not necessarily, completely out of scope for other, unknown mental faculties. — alcontali
No, this is exactly the opposite to what I am arguing. When we adhere to the law of identity, then everything has an identity proper to itself, therefore its own meaning. This does not rob meaning from mathematics, it only establishes clear limits to the possibilities of mathematics, so that mathematicians will not believe themselves to have accomplished the impossible, like putting the infinite within a set. — Metaphysician Undercover
ZFC theory allows that two distinct things are the same, contrary to the law of identity. — Metaphysician Undercover
You made the statement that ZFC allows two different things to be equal. I said I know of no such example and you have not backed up your claim or put it in any context that I can understand. You must be thinking of something, I'm just curious to know what. — fishfry
S1 and S2 describe the same set. Therefore, S1 = S2. — alcontali
2 + 2 and 4 represent the exact same mathematical set. '2+ 2" and '4' are distinct strings of symbols. I don't know any mathematicians confused about this. And, as you agree, the discovery that these two strings of symbols represent the same set, is a nontrivial accomplishment of humanity and is meaningful. — fishfry
I really don't understand your remark that ZFC allows distinct things to be regarded as the same. Unless you mean colloquially, as in the integer 1 and the real number 1 being identified via a natural injection. — fishfry
The point I made is that 2+2 is not the same as 4. So if set theory treats them as the same, it is in violation of the law of identity. — Metaphysician Undercover
Of course 2 + 2 is the same thing as 4. I cannot imagine the contrary nor what you might mean by that claim. — fishfry
But more importantly, they are the same set in ZFC. So it's not an example of your claim that ZFC allows two distinct things to be regarded as the same. — fishfry
But you hold that 2 + 2 and 4 are not the same? How so? Without quotes around them they are not strings of symbols, they are the abstract concept they represent. And they represent the same abstract concept, namely the number 4. You deny this? I do confess to bafflement. — fishfry
Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same? — Metaphysician Undercover
This is absolutely false. The symbol 2 has a meaning, the symbol 4 has a meaning, and the symbol + has a meaning. Clearly 2+2 is not the same concept as 4. — Metaphysician Undercover
The point I made is that 2+2 is not the same as 4. So if set theory treats them as the same, it is in violation of the law of identity. — Metaphysician Undercover
I walked through this in detail a few posts ago. In the Peano axioms they are both the number SSSS0. In ZF they are both the set {0, 1, 2, 3}. = { ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }. — fishfry
We must be talking past each other in some way. I cannot conceive of anyone claiming 2 + 2 and 4 are not the same thing. — fishfry
Perhaps you have a reference to support your point of view. — fishfry
But you claim that 2 + 2 and 4 are not the same object in ZFC. And THAT is an area where I am not ignorant. You're just wrong. 2 + 2 and 4 represent the same set in ZFC. — fishfry
Well, first there is the understanding that the "=" symbol pretty much never means "identical". — alcontali
You tell me how 2 + 2 is not 4. If it's not, what is it? — fishfry
I fully acknowledge, that in ZFC 2+2 is "the same" as 4. I am not denying this. I am saying that it is wrong, because it violates the law of identity, without any justification. — Metaphysician Undercover
I will depart from this thread, feeling on my side that I can't talk to someone who is claiming that 2 + 2 is something other than 4. And also feeling deep down that I must be missing something really profound, but I don't think it's something I'd want to get even if I could. — fishfry
And in visual psychology, it should not be regarded as an error if a test subject reports that he saw 5+7 as 13. It simply means that visual phenomena are not a good model of ordinary arithmetic and vice versa. — sime
I suppose the feeling is mutual. I really cannot believe that there is a rational human being who truly believes that 2+2 is the same thing as 4. — Metaphysician Undercover
Isn't this what we learn in basic math, first grade? You take two things, add to them another two things, and you have four things. — Metaphysician Undercover
Very good. But we can get four by adding three to one, or by subtracting two from six, and an infinite number of 'different' ways. — Metaphysician Undercover
So it is impossible that 2+2 is the same as 4, because there would be infinitely many different things which are the same as four. — Metaphysician Undercover
Does it make any sense to you, to believe that there is an infinite number of different things which are all the same? — Metaphysician Undercover
Or can you see that 2+2 is not the same as 8-4? — Metaphysician Undercover
I think I understand your point but I have some counterpoints. I believe you are saying that when we say 2 + 2 = 4 we are saying two things: One, that they represent the same natural number; and two, that 2 + 2 is a legal decomposition of 4, which is not necessarily known beforehand. So 2 + 2 = 4 asserts something more than merely saying 2 + 2 or 4 by themselves. And you're right about that. — fishfry
However it's not an ontological fact, it's an epistemological fact. — fishfry
That is, the partition of 4 into 2 + 2 is literally a matter of definition. — fishfry
It was always an identity, even before we learned it. — fishfry
So I would say that 2 + 2 = 4 is an expression of the law of identity; but we did not always KNOW that until someone discovered it and taught it to others. — fishfry
But there aren't. There are infinitely many different representations of the concept of 4, just as schnee and snow are two representations of the white stuff that falls from the sky in the winter. And you are right that it may sometimes take hundreds or thousands of years for us to discover that two representations represent the same thing. But they were always the same even before we knew that. — fishfry
Do you agree that schnee and snow are identical, even though one has to pick up a little German (or English as the case may be) in order to discover that? — fishfry
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