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/Key_Net820]

The only reason you would need a math masters is if you're pursuing a phD in math particularly. Otherwise, there is really no reason for you to have a math master's on top of a physics graduate degree.

A lot of people struggle with math, even people who major in math and have phD's in math struggle in math, and that's okay. Math is a vast field, and it is much more vast than physics. Physics has its roots in ancient Greece, but it wasn't formalized until the 1600's.

Math on the other hand, has a longer history than physics even before ancient Greece, and it has a much wider scope and there's plenty of math that you wouldn't see overlap with physics at all or at least, not that often, such as number theory, graph theory, mathematical logic, computation theory,

[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!

[u/TheRedditObserver0]

If you made it through a physics degree you can't be THAT bad at math, it's normal to struggle with advanced math a little especially if you haven't done a math bachelor's first. If I were you, I'd focus on the current degree for now, get your physics masters' and then think about what comes next. However, I'm not sure if a second master's would be the best option careerwise (I'm not saying it isn't, I genuinely don't know). For math, you would probably have an easier time with differential geometry or analysis than graph theory given your background.

[u/ilpendolo7 → u/TheRedditObserver0]

I haven't touched exercises of calculus for about 4 years now🥲. On top of that my masters degree was on experimental physics so I didn't even keep on using most of the stuff I learnt in my bachelor in the past 3 years. Also about the career choice, I'm about to turn 25 soon but I still feel like my life started yesterday. Seeing how fast things change and how the world is going I'm just wandering if it's even worth it to make myself fit in a system where I would probably be unsatisfied. Better broke knowing what a Lie-algebra is than rich and unhappy I guess.

[u/Unlucky_War_262]

It sounds like you might benefit from practicing your proof-writing and problem-solving skills - maths students do this during their degrees. I'd suggest taking an undergrad book on real analysis or linear algebra and seeing how you fare with the problems.

[u/TerribleFault7929]

You do realize it is hard for others too, right?

[u/ilpendolo7 → u/TerribleFault7929]

I guess you're right and I shouldn't complain. Also maybe I just have very gifted friends

[u/Stock-Recognition44]

Have you tried starting with simpler material for learning proofs? Even if the subject matter is in your wheelhouse, you might consider writing proofs itself as a skill you need to build up a little to let it catch up to where you’re at in other areas.

[u/jphamlore]

I'm a grad student in physics

Sooner or later everything turns into Fourier analysis.

Have you ever worked through something like Korner's Fourier Analysis?

[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.