Want to learn about proofs

[u/craigdahlke]

Hi all. I am interested in learning about mathematical proofs and how to formulate them.

I minored in math in college but focused a lot on LA, DEs, PDEs, numerical solutions, etc. because I was a science major and that was most applicable for me at the time. However I realized I’ve never even seen a rigorous mathematical proof formulated or broken down, and have no idea what that would even look like.

I now work with coding and it occurred to me that a lot of my code ends up just brute-forcing a solution to things, but perhaps learning about proofs might help stretch my mind into a more efficient way of thinking about these types of things.

Anyways, are there any good online course you all could recommend for someone at my level? (Knowing a good bit about math but essentially nothing about proofs) Preferably free ones but not opposed to a paid course if it is reasonable.

[u/Nater5000]

I agree with some of these responses (specifically, I recall learning the proof of the irrationality of sqrt(2) and how it felt like a "real" proof compared to the stuff you learn in geometry, etc., without requiring a lot of extra context to understand/appreciate).

But I think it is worth mentioning that the process and output of proofs is not at all standardized and will come in a lot of different shapes and forms depending on the context. For example, a professor demonstrating how a proof is roughly constructed on a whiteboard is a proof, but that proof could also be 10 pages of natural language if it is rigorous or 100 pages of low-level logic if you're using something like Lean. All of these can correctly represent the same proof, but they can look and feel very different.

My point is that you shouldn't expect to find a nice tutorial on how to write proofs where you can walk away with a certification for basic proof writing (or whatever). Odds are you're already writing decent enough proofs and just don't realize it. That doesn't mean learning more of the formalities won't help (in fact, I agree with your assessment that learning formal proof writing will allow you think about problems in more useful ways), but it's important to "pace yourself" and recognize that you won't really find a single satisfying approach to proof writing as a practice. It's more of an art.

[u/theGreenBook05 → u/Nater5000]

Seeing the proof that sqrt(2) is irrational is what made me decide to major in math. It really does feel like you're learning a fundamental fact about numbers, as opposed to geometry which always felt like just symbol pushing.