Hey /sci/, I have a problem with foundations of mathematics.

pure math is autistic, don't worry too much about it

>>8677425
The set is a primitive. You can't define it rigorously.

File: russell2.jpg (36KB, 284x415px) Image search: [Google]

36KB, 284x415px

in fact, I do worry a bit

how far is math from "muh feelings"?

isn't it all some kind of convention, in the end?

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

Go for it from an applied/physicist viewpoint. A theory is only as good as far as it can be applied to the real world with satisfactory results. That, and maybe Ockham's razor is enough.

I think your problem is that you haven't read anything about the philosophy of science, and how any theory we might come up with will only be an approximation of the real world, due to our inability to fully comprehend it. Look into some of that stuff, math history, etc. Some good entry level books are the ones by Ivar Ekeland, e.g. The best of possible worlds.

Unless you're an actual autistic math prodigy. In which case, by all mean, go ahead and do pure math, like I give a shit, last guy that did that came up with category theory.

>>8677484
> A theory is only as good as far as it can be applied to the real world with satisfactory results. That, and maybe Ockham's razor is enough.

So up until the 20th century number theory wasn't "good"?

What about theory of conic sections? Was that also not 'good' until Kepler?

>>8677494

Wow, you're being shallow. Those theories have many other applications than the ones you're referring to. Otherwise they would have never been developed. See, 2-300 years ago and before every mathematician was an applied mathematician, they were trying to solve real life physical problems, that's why they've developed mathematical theories.

>>8677425
> And alphabet is a "set" of symbols etc.?

Except the meaning of set here is very specific : it's just a bag, in which you can pick specific elements. It's much simpler than a set of set theory, in which you can have bags in bags in bags.

What is more, you can write down that alphabet. You don't need any formalism to do so. When it's written down, you can pick elements in it. Here's your simple set. You built it, you didn't describe it.


Or maybe the thing here, is, that we need basic formalism to *describe* the more advanced formalism, but that more advanced formalism doesn't need to be described to exist on its own.

>>8677513
Ok, so what about Hilbert's spectral theory? He didn't do it for any applications. He was an autist who wanted diagonalize an infinite matrix.

You can't build set theory with first order logic because you can't add quantifiers to predicates in 1st order logic.

>>8677534

Sauce? Did he leave behind any manuscripts saying he was just doing it for the lolz? Honestly, I don't know, I'm not a professional math historian, but to me it seems clear that he knew Hilbert spaces were relevant to possible applications in PDEs.

Also, I hope you understand the difference between pure and applied math. Hilbert spaces are very much applied math.

Anyhow, your question was about how to approach set theory without being an autistic mathtard, and that is done via picking ZFC as your set of axioms. Everything else builds naturally on that, starting with analysis.

File: file.png (274B, 10x17px) Image search: [Google]

274B, 10x17px

>>8677425

Sets do not have a formal definition, however you can think about what a set is by what kind of properties it has using axioms that are taken as unconditional true.

In informal language, sets are collections of elements. There is the set of people in this thread, the set of photons in your room, etc.

Axiom : there exists an empty set with no elements.

We can deduce that there is one and only one empty set using first-order logic. It is a proof by assumption that there exists more than one empty set, then proving that any two empty sets, x and x', are proper subsets of each other or equivalent.

Formal proof:

a1 <=> ∀y : y∉x
a2 <=> ∀y : y∉x'

(tautology) q1 <=> ∀y : y∉x => ∀y : (y∉x => y∉x')
(axiom) q2 <=> ∀y : y∉x
(modus ponens, q1 and q2) q3 <=> ∀y : (y∉x => y∉x')
(axiom) q4 <=> ∀y : y∉x'
(tautology) q5 <=> ∀y : y∉x' => ∀y : (y∉x' => y∉x)
(modus ponens, q4 and q5) q6 <=> ∀y : (y∉x' => y∉x)
(modus ponens, q3 and q6) q7 <=> x = x'


So we still don't have a formal definition of sets, but we know that there is a set with no elements and that there is only one of them. Proving other propositions will help you get an intuition for what a set by what is provably true about them.

Any modern rigorous book will be pretty autistic, it just comes with the territory.

>>8677580
>Sauce?
https://en.wikipedia.org/wiki/Spectral_theory

"I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis' without any presentiment that it would later find application to the actual spectrum of physics."[4]
>Honestly, I don't know, I'm not a professional math historian, but to me it seems clear that he knew Hilbert spaces were relevant to possible applications in PDEs.

I doesn't seem that this the case. Notice that the theory of Sobolev spaces was developed way later, notice that even on the space of compactly supported smooth functions, endowed with the L2 norm, the derivative operator is still unbounded.

>Also, I hope you understand the difference between pure and applied math. Hilbert spaces are very much applied math.

Yes, that's true, bu the applications was not the reason for introducing them, Hilbert wanted to extend the euclidean geometry to infinitely dimensional linear spaces.

>Anyhow, your question was about how to approach set theory without being an autistic mathtard, and that is done via picking ZFC as your set of axioms. Everything else builds naturally on that, starting with analysis.

I'm a different guy. I'm sorry if I'm coming off as a douche, the point I'm trying to make is that, occasionally the applicable math comes from autistic, seemingly non applicable shit.

>>8677658

Hah, pretty interesting, thanks, I've learned something.

Good on them, I guess. I still think that the way math has evolved these days there are really no easily accessible questions left in pure maths (that's why you need to be an autistic genius to be good at it), even if it sounds biased, I think developing functional analysis back in the days for instance was a far more easier task than developing stuff like category theory (not to undermine their genius to invent these things at the time of course, new vision always takes some form of genius or another).

Anyhow, applied math masterrace out.