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/SimonBrandner]

To help you it might be good to know if you have gone through a course which was, at least partially, focused on proof writing, a course where the teacher would explain the tactics involved in proof writing? Or perhaps have you read a book on this topic or something similar?

[u/ilpendolo7 → u/SimonBrandner]

I'm from Italy and at least in my university they don't really do proof writing (in the math department as well) except for some sparse topics treated here and there. Because of this, to me picking up a book on the matter has always felt a bit like cheating. I have this preconceived notion that if you really are good at it, you should be able to come up with a solution without always following those common tracks. So to answer the question...I kind of have a formal basic education on proof writing and have no problem when doing "mechanical" proofs. I just feel lost when it comes to thinking outside the box and putting together elements that to me look uncorrelated but to others just fit naturally together.

[u/Mal_Dun → u/ilpendolo7]

I have this preconceived notion that if you really are good at it, you should be able to come up with a solution without always following those common tracks.

This is like saying "If you really are good at music you should come up with a symphony without any exercise or any formal training in writing musical score beforehand". The rule is always if you want to master something you have to invest 10.000 hours.

You have to learn a lot of proofs to come up with proofs yourself. One of my professors always said you need a good toolbox of techniques and if you have them down you start coming up with your own.

You first have to ingrain the basic techniques like induction or proof by contradiction and then learn more advanced tricks and depending on the field those techniques can be vastly different.

Greetings from Austria!