Why do I just not "get" math proofs?

  • #1
The only proof-based math class I've taken so far was on abstract algebra. Concepts were easy for me to understand, but I was constantly having trouble with some of the proofs.

I so frequently get this feeling that the last, tiny trivial step left in my proof is just "right there," and yet I can't find it. Or I might drastically over-complicate things and write a page proving something that my professor's solution states in one line as an obvious conclusion (which it was). It's just ridiculous; there were a few problems assigned for the class that I might spend four hours just messing around with until finally figuring out a convoluted (though valid) proof to when they could have been solved in a few lines.

Technically, grade-wise, I did well in the class, but in the end, I still feel like proof-writing just doesn't come naturally to me. Now I'm working on learning topology and differential geometry, and while, again, I can understand the concepts and proofs and do some of them on my own, none of the proofs ever come naturally to me.

Honestly, is it common for people to take so long to get used to writing proofs? I'm just working on getting the hang of things (self-studying topology and differential geometry now) but it's not really happening.

Answers and Replies

  • #2
Simon Bridge
Science Advisor
Homework Helper
Honestly, is it common for people to take so long to get used to writing proofs?
Yes - that is very common.
It can get easier - but there are other ways to learn maths that do not rely so heavily on learning proofs.
  • #3
you honestly have to stare at it for a while and do it over and over and actually know what everything means in the proof.

i took abstract too and had a hard time with the proofs but i just did my best.

just remember some of those professors have been teaching those same proofs for like 20 years and that's why they actually know them.

also it really really helps if you have a TA or professor to sit down with you and actually EXPLAIN them.

i had an amazing TA in number theory and I went to her 3 days a week. she helped me with almost all my homework, however I got a solid A in the class and really understood the material.

it takes lots of time and a good teacher by your side.
  • #4
Don't proofs always look so beautiful in the book or when your professor is writing them at the board? Well those proofs in the book are all later drafts and the proof your prof writes so cleanly is the same one he's been writing on the board every semester or two for years. A few years ago I went and did every problem from the first 7 chapters of Baby Rudin (lousy textbook but amazing problem sets), and my first drafts of my solutions to every problem were so hideous to read. Just like when you're writing your English paper, you got to do drafts of your proofs too to tidy them up, flesh out hidden assumptions, tie related things together more cleanly, fiddle with the tone to make it read more easily, and so on.

I like what Alan Kay (creator of the GUI computer as we know it today) had to say on human learning. He said almost none of us really do math with the logical parts of our brains. Instead we do it with diagrams, illustrations, or sometime just thinking how things would feel to our bodies. That the logical part of our brain really wasn't the most powerful. So maybe what makes proofs hard is it's not how we're really thinking of the problem; instead we're like lawyers trying to make an airtight case for why this intuitively clear argument actually is true. A lot of times I try to draw a picture, even if it's just drawing say the modular integers Z/nZ in a ring to make my brain realize they loop over.

Abstract algebra definitely isn't the place I would want to see undergrads introduced to proofs. At my school you had to take a theoretical linear algebra course before any of the really interesting upper division courses. It made sense because we all had to take the linear algebra + differential equations course with the engineers before, so we knew a lot of linear algebra when we came in. So at least we were in somewhat familiar territory taking the theoretical course, so that we weren't trying to learn a whole new subject while learning how to prove things too.

My best advice:
1. Draw pictures
2. Read a lot of math proofs and work along on your own sheet of paper as you do it
3. Write a lot of math proofs and edit the hell out of them until they look pretty (well, as pretty as you can get them; some proofs can't be made beautiful!) and are readable

I don't think there is any shortcut. I mean it was a lot of effort for me to get used to it. As you read a lot of proofs you come across a lot of tricks to add to your arsenal: e.g., let me add 0 = x + (-x), or let's take epsilon/2 here, stuff like that.
  • #5
I was a "natural" at doing proofs, one of the people who just got it instantly (although it turned out I was astonishingly bad at actual research, maybe partly because of lack of interest in current research). I don't think this was entirely due to innate talent. Instead, I think it has to do with several things.

First of all, before I took a difficult proofs class, I took an easy proofs class, which was basically naive set theory. That helps to separate the difficulty of learning how to write a proof from the difficulty of learning whatever subject it is.

Secondly, studying subjects which involved a lot of visualization and conceptual understanding were helpful, just as the electromagnetism class I took and reading Visual Complex Analysis. If you read VCA, because it's not completely rigorous, it teaches you how to deal with the conceptual part of it before you write down the final proof. So, that's usually step one. Just get the idea of the argument and then translate it. When you have good pictures of the main proofs in the subject, it's easier to remember all the theorems that you are going to need.

And that's the third thing that made me do well--I had an excellent recollection of all the previous definitions, results, and proofs in each class (wasn't able to keep that up as well in grad school because the pace was faster and the material was more difficult to make memorable). In keeping with the second point, to remember things, you have to make them memorable, often with some sort of visualization or at least tying it into something intuitive, if possible. Another thing that I think gave me an advantage here is that I didn't rely on just doing problems. I spent a lot of time thinking about the concepts and how they relate to each other. You can do that while you're doing problems as well, but I don't think it's wise to restrict yourself to whatever problems the textbook writer has arbitrary chosen to help you, and if you only do the problems, I don't think you're going to see the big picture of the subject and how things fit together.

Another thing is that it can help to think of what the purpose of each theorem is and what it is good for, so when you come across a place where you need to use it, it will naturally come to mind. When all else fails, you can go through your checklist of relevant theorems to see which one might help. That, and proof by contradiction.
  • #6
One trick is to try to prove it yourself before reading the proof. The goal isn't to actually prove it, but to simply see how far you can get, since attempting to solve the problem will auto-generate the context of the problem (assuming you understand the problem that the result is trying to solve).

This works for me with results in physics anyway.

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