Red Banana

There is a famous meme: "We do this not because it's easy but because we thought it would be easy". I guess this pretty much explains everything. In our naivety, we don't go in there thinking things are unexplainable, we think we can make a good work understanding it and then... It's extremely hard and nothing can be done! :D

@DanielSchepler I did it in my program looking at the proof you showed me. Thanks!

@Bram28 What program? I made a silly code to make substitutions in Mathematica (which is shown in the question), Schepler uses this Incredible Proof Machine which sounds pretty interesting but I'm not sure what it actually is, I just know that it's possible to handle some kinds of proofs in it.

@DanielSchepler What substitution is happening here?

@DanielSchepler Oh, I once saw this book in a library but never had the time to look at it. I thought you knew something in that direction.

@DanielSchepler I'm curious: How do we deal with logical expressions in terms of topology? What are the open sets and the closed sets?

@PW_246 Until now, I guess the deduction theorem wasn't mentioned in the book.

@OrangeDog He gave you the citation, did you verify it on the book he mentioned? XD

There are entire books devoted to this issue, such as this one.

@Him BTW, Gödel's theorems say something VERY precise and people tend to think they can understand it via vague phrases such as: "Godel's result is a statement about "what mathematicians can possibly ever know" as opposed to "what mathematicians currently do know""

@Him Why are you trying to lecture mathematicians about Gödel's theorem? Anyone with a math major has at least heard about this theorem. It's so funny that you just went there and said: "there is a whole subject called "metamathematics" that rigorously studies the limits and properties of mathematical systems" when this is - basically - common sense for mathematicians.

@OrangeDog We can prove some things about them. Just as AwkwardWhale pointed out. "The experience of the author is not relevant" - OF COURSE it is relevant, you yourself don't have any idea about what are the mathematical issues with Feynman integrals, you probaby just watched some documentaries on YT and think you are well informed about it.

@OrangeDog Also, why do you think that this guy doesn't know that "non classical mathematics can be mathematics"? He has an account on Math Overflow with some activity in there, MO is basically about recent (non classical) mathematics. I guess you're trying to (condescendingly) lecture very basic stuff to people who know way more than you do. Also, "classical calculus" was eventually found to be lacking by Cauchy, Weierstrass, Cantor, Dedekind and others. So it's kinda hilarious that you think that he is saying that "classical mathematics is better". Anyone in a math major learns this.

@OrangeDog "Mathematics" is something specific: It means that we can prove things about the things we talk about. People in physics usually create ideas without too much concern for this. "Mathematics" is not "anything that has numbers in it" as you seem to be thinking.

@bmf No, I need the f's.

@bmf Done. $$$$

@bmf I was going to post the code, but it gets a kinda unreadable. With lots of [Prime], etc.

[1] : youtu.be/PyI_0mJrLlQ

@user21820 Another theory from which a theory happens to me a consequence. For example, it seems we can write Peano axioms in set theory, right? In this case, set theory would be a super theory.

@user21820 I'll ask a - perhaps - vague question: If we have a theory, say PA for example. Can we always find a "supertheory" such that PA is a consequence of this supertheory?

@user21820 Do you teach at university or something?

@user21820 Could you take a look at this? math.stackexchange.com/q/3956382/25805

If in physics, we had a function that when integrated repeatedly, we could obtain other 175 functions which are physically relevant, I guess they'd probably make the students memorize 175 functions.

In the book, they give 3 formulas.

And it's kinda easy to integrate simple examples of polynomial functions, which is all that is seen on HS here.

I am looking at my old physics textbook from high-school: Formulas in mechanics can always be derived from a small set of formulas, right? For example: We have $F=ma(t)$, by integrating we get velocity and position, right?

I don't know if things have changed, but I think this didn't change. I assisted some HS students in the past years and never seen they mentioning integrals or derivatives.

Yeah.

I mean, we see a bit of that on physics, but it's never mentioned they are derivatives or integrals.

@user21820 We don't see it at all.

Correction: In Brazil, seeing integrals and derivatives on HS is "science fiction".

High-school? Where are you from?

@user21820 I guess I mixed things up and thought that "antiderivative" already means "antiderivative in terms of elementary functions".

@user21820 Perhaps, what I am talking about is "an antiderivative in terms of elementary functions", this doesn't always exist, right?

@user21820 It says that: "there is an algorithm that will numerically evaluate the integral of any computable function." But isn't that different from something like finding an antiderivative? Or they are the same thing? When I did numerical analysis, there are functions where you can't find antiderivatives but you can use numerical methods to approximate it, If I remember well.

Do you know anything about it?

The comment from Jacques seems interesting.

@user21820 I just read this:

@user21820 I often hear people saying that a "function" depends on the domain and codomain. These days I was doing some problems, isn't this "function that depends on domain and codomain" the graph of the function?

@user21820 Yep.

@user21820 Yeah, it seems in the book, they use the same definition. They point out that there is a difference between "valid" and "correct". "Valid" depends only on the argument.

A valid argument is only if we have true premises and true conclusion, right?

@user21820 I found this question:

@user131585 No.

@user131585 If you are woke enough, it's 5.

It is a function, no?