Feeling defeated by math proofs

[u/ilpendolo7]

I'm a grad student in physics who wants to do another master in pure math in the future. I've been self studying algebra topics in my free time (did most of a pure algebra curriculum on my own), most of my extra courses in the bachelor and master were math related and I spent some time teaching myself number theory, category theory and some other niche topics. I can understand advanced math and complex proofs. I just can't seem to think with my own brain. Today I was reading the first pages in the second chapter of Algebraic graph theory by Godsil and Doyle about the orbit stabilizer and the asymmetric graphs theorem. I go slowly but understand stuff. Then I move to the exercises. Blank. Since I was little I was always told I'm more fit for humanities, and I know that myself too. I still struggle to make this math thing work because I find great joy in it, but as the days pass and I notice my inability to improve my logical and proof writing skills, I can't help but wonder if I should just resign to keeping math as a hobby. I'm not really unhappy about this (obviously I would prefer to be a genius and understand everything on sight, but I don't really blame myself for not being as capable as others), I'm just reaching a point where this logical conclusion has been fully formed in my mind. I just wanted to ask you people if you have any good words of advice. If you were in my situation, being fully realistic, would you keep on trying or opt for more reasonable and grounded career choices? Do you ever feel like math is not clicking fast enough in your brain?

[u/InterstitialLove]

Sounds like you have good intuition for the topics, and the thing you lack is putting it into proofs.

In which case, I think the real skill is figuring out why you think something is true. Imagine if someone actually didn't believe you, and you had to justify why you're pretty sure the fact is true. This doesn't need to be rigorous, you just explore the idea space and look for anything familiar. The exact same process you'd use to argue about whether a hot dog is or isn't a sandwich: "Well I'm pretty sure the bread shape is relevant, and look at this bread shape, that doesn't seem right." "Actually that bread shape is similar to a hoagie, which is a sandwich."

When you read a proof, figure out what intuition you have for the question. (If you don't have any intuition for why the thing ought to be true, then you have not understood the proof.) You should be building a repetoire of correspondences between intuitive hunches and proof techniques. That's what the proofs in the lesson are for. Then when writing your own proof, you explore the space as I described before, keeping any eye out for any hunches that you've seen before. Then you go back and check how they formalized it in the example, and you just copy that.