Does anyone find math proofs easy to comprehend?

Question

Ever since my undergrad, I've been taking challenging math classes. Starting with the sophomore year, I faced some difficult proofs I had to learn (and be able to recite at an exam with slight modifications) and now I'm doing a PhD and still dealing with proofs that take so much of my mental energy.

Now I am in no way the brightest guy in the classroom, but I managed to get through these courses with mostly A's. The problem is when I look back, I remember nothing. Moreover, comparing the time it took for me to be able to do pages of proofs on assignments to how I'm able to absorb information so quickly in my area (AI in vision), I feel like I might be lacking some talents in math. When I read the most recent article DeepMind has published, it is a breeze. But if there is a single proof there, I'll have to be a lot slower, have my pen and paper ready and do lots of iterations.

I had a lot of friends who were extremely smart, but I remember that even they didn't feel too comfortable with proofs (way more comfortable than me though).

So I'd like to ask, is there a thing such as being naturally wired to understand and do proofs? I understand if you do it every day, you'll get familiarized and be a lot more adept in it, but are you or have you ever witnessed someone that picks it up right away?

I always felt it's fun to do this, but I am not going to be the best at it since I spend so much effort (hence I didn't pursue a maths career).

Thanks in advance!

Answer 1

You should read some of Dr Laura Alcock's books such as "How to think about Analysis" OUP ISBN 978-0-19-872353-0. She has an amazing way of talking about how to think about and study proofs. She has written several books but it may just be what you need to help.

Answer 2

I don't have any inherent problem reading math proofs (not that I read them as quickly or effortlessly as novels) as you describe, but I can't make much progress in reading, say, legal documents. My own thought processes are those of a mathematician and naturally line up with math proofs; that's what mathematicians do, and what mathematicians are for. Other people think like lawyers and gravitate toward legal arguments, or computer scientists and gravitate toward algorithms, or novelists and gravitate toward prose, etc. (Though it sounds like you aren't one of them, there are people who breeze through high-school-level math but hit a brick wall when they run into math that's more than just computation. The mindset to churn through integrals is not necessarily one that handles constructing and reading proofs fluently.)

Part of it is also, as you said, practice. Legal documents are written in what seems from the outside like an opaque style because of the need for precise jargon and formularies, and because it makes it clear to lawyers what the relevant parts are and what's just boilerplate. Math proofs are the same way, and after a while you learn to recognize what's just an uncreative computation and which parts are the clever ones, which parts can be extended or tweaked to give a further result, and so on.

I'd also add that there's an art to writing proofs that's rigorous without being pedantic. The style and expectations are different from other fields', even ones that involve a lot of mathematics. I've coincidentally done a little bit of work in AI and vision as well (albeit probably on a different side of things), and there's a different feel to research there than in mathematics.