How do you go from proof to intuition in math? I can quite easily

>>8936622
You can't, really. All you can do is try to prove it yourself, see what walls you find blocking your proof, and then read the proof you have to see how the author resolved those issues. Maybe that will illuminate the motives and inspiration of the author.

>>8936622
If there was, learning math wouldn't be as difficult as it is.

>>8936647
Its really this. You have to not only see the proof itself and its individual steps but why the particular mathematician used that step in the first place. You have to really get, initally, a broad overview of the game plan- so to speak. For example, this game plan for the proof of Stokes Theorem (not the generalized version) in certain contexts involves paramaterizations that make an application of greens theorem possible (assuming you have already proved greens theorem in the plane). Then once you see this motivation, you can go back to the nitty gritty of the steps of the proof to see how it all ties in. Drawing pictures or really just not writing math and instead imagining math helps.

>>8936622
By thinking about it really hard. Also do exercises involving the concept.

>>8938132
Yeah, I can often understand the "game plan" of proofs, why certain approaches have been taken, but that still doesn't help me understand the concept being proved intuitively.

Are there any good books on this subject? Like a reverse "how to prove it"?

>>8938203
Like you can believe the proof and everything but you don't know exactly what has been proven (from an informal standpoint) or why?

>>8936647
I disagree with this anon but I'm not going to elaborate on it as I don't think it's what you're asking for.

>>8938208
No, I can understand what has been proven, and why, most of the time.

What I'm talking about is an intuitive grasp and understanding of what has been shown. I guess informal explanations of concepts would be another name for it. Let's take irrationality of the square root of 2 as a super simple example. The proof of it is short, and easy to understand, but I find (at least personally) that intuitively grasping why it's true is rather difficult. I can easily see why the steps taken are correct, why they were chosen, and how they lead to our conclusion, but as for any informal understanding of it I'm at a loss.

>>8938241
May it be perhaps because the proof of irrationality of the square root of 2 is traditionally a proof by contradiction?

Maybe you're getting caught up on the hidden jump at the end where one goes from claiming that the number is not rational to claiming that it is therefore irrational (i.e. is it possible that it's maybe not a number at all, or somehow ill-defined)?

Do you have trouble with proofs that don't use contradiction?

>>8936622
I think I understand what you're saying OP: you only feel like you really understand things when they seem "intuitively obvious", right? You might see a proof and understand it, but you have to "feel" it to really accept it.

I have that problem in particular with the halting problem. I've read both detailed and "intuitionistic" proofs, but I still refuse to accept that it's impossible to design an algorithm to tell whether an arbitrary program (ignoring I/O operations and multithreading, I'm talking about pure and "simple" computation) will eventually halt or not. "Surely" you can reason about a program's "flow of execution" and see whether it terminates or not for any arbitrary well-structured program.

>>8938241
(1/2)

What would you consider to be "intuition" for a proof? In lieu of a formal definition, I suggest thinking of intuition as a path in a directed graph (well, really a multicategory) where vertices are mathematical propositions and edges are proofs.

A formal proof of "If A, then B" would consist of a set of instructions for getting from A to B, in a purely mechanical fashion. Extending the navigation analogy, these instructions would look something like "Move 100m forward, then rotate counterclockwise 90 degrees, then move 200m forward..." etc.
Informal intuition would resemble directions for humans rather than machines, i.e. pointing out key checkpoints (lemmas) along the way, and describing the path to follow in "high-level" terms (e.g. "use the Euclidean algorithm to obtain a pair of natural numbers [q,r] such that XXX, then...").

"Developing intuition for a proof" would then manifest in being able to describe to another human how to get from antecedent A to conclusion B, in contrast to a machine-readable formal proof.
For example, developing intuition for the proof of the irrationality of root(2) would consist of developing intuition for each of the proofs of the lemmas involved:

(cont.)

>>8938355
(2/2)

(A) Assume there is a pair of natural numbers a,b such that [math]a/b = \sqrt{2}[/math]
(A->i) If [math]a/b = \sqrt{2}[/math] then [math]a^2=2b^2[/math] (intuition: multiplication distributes over equality)
(i->ii) a^2 is even (intuition: if x=y then x and y have the same parity)
(ii->iii) a is even (intuition: a^2 has the same parity as a)
(iii->iv) b is even (intuition: p|a --> p^2|a^2, fundamental theorem of arithmetic to conclude that 2|b^2 -- I don't know if there's a simpler way)
(iv->v) Existence of a pair of naturals a',b' such that [math]a'/b' = \sqrt{2}[/math], a>a' and b>b' (intuition: cancellative property of multiplication)
(v->vi) Existence of an infinite sequence of pairs a>a'>a''>... and b>b'>b''>... with the same ratio [math]\sqrt{2}[/math] (intuition: definition of [math]\mathbb{Q}[/math] as equivalence classes over [math]\mathbb{N}^2[/math], ability to apply the algorithm [i-iii] repeatedly on the equivalence class [math][\sqrt{2}][/math])
(vi->B) Proof that the natural numbers are not well-founded, hence contradiction (intuition: the natural numbers are well-founded)

I've tried to make each step as trivial as possible by reducing it into a single application of a definition, an algebraic rule, or a logical rule of inference. I don't think it'd be controversial to claim that "if a^2=2b^2 then a^2 is even" is intuitive. But if the number of definitions and mathematical concepts required is any indication, the irrationality proof is not as simple as it is often claimed to be.

>>8938324
I have more trouble with proofs by contradiction, yes. However, sometimes I can have the same trouble with regular proofs as well.

>>8938351
Yes, exactly. I'm wondering if there is any sort of method to developing this "feel".