What to look for in a proof?

Question

I am a physics undergrad, wishing to pursue a PhD in Math. I am mostly self taught in the typical math undergrad curriculum.

I am looking for more input, in ways I can improve my mathematical thinking. So, my question is once you read a proof of a particular theorem, what should be the important things that you are looking for in the proof. How should you approach it?

So, one of the things I have learnt to do is to look at every hypothesis of the theorem, and see its effect on the proof, or how do the subsections of the proof correspond to parts of the hypothesis, and how they fit together.

However, what I find difficult is to recognize how the proof fits together in the general scheme of things. What should I ask or explore after the proof? How can I learn to solve problems more quickly and effectively through proof - reading? I am very slow are problem solving and wish to improve that.

Lastly, I am terrible at coming up with new examples, and I don't even seem to remember examples beyond very typical ones. How can I improve this ability through reading of proofs?

I hope this question is welcome here. Otherwise, please feel free to close it. I understand this is a vague and difficult question, but any kind of input (however small) will be highly appreciated.

Answer 1

One thing that I find helpful when reading a proof (especially those in papers, where the authors often omit more details than in a textbook) is to take notes and try to restate in my own words the proof as I read it; imagine you are preparing a short lecture on the theorem to present to your classmates, and you expect them to ask "why does that follow?" after every step.

It is also possible that a proof you find in a paper or book simply isn't very well-written, or is simply written in a style that you have trouble following. In this case, see if you can find a different reference and follow the alternate proof (certainly as you're beginning graduate studies, you should be able to find a multitude of references for any well-known result). If nothing else, reading multiple proofs of the same theorem will give you good insights that you might not otherwise get.

Another question you might ask yourself after absorbing a result/proof is "are the methods of proof interesting in and of themselves?"

Answer 2

This may sound trite, but a guiding principle should be "argue so as to convince your intended reader". As you get more experience writing proofs, you learn what will/won't convince people.

Answer 3

Of course it depends on why I read the proof and how familiar I am with the material, but the main things I look for in a proof are those that I would not know how to do myself.

Before starting to read a proof I will think about how to prove the result myself. If it seems standard then I will usually only glance through the proof to see if something unexpected happens. If I would know how to prove some parts of the statement but not others, then I will only look at the proof of those, if the parts are not too intertwined in the proof. Or I might see a specific obstacle in proving the result, then I will want to find out how the author overcomes this obstacle.

Now this was regarding short or medium length proofs. When I want to understand a very long proof, that is more some kind of project, and like with most math reading I will not do it strictly linearly, but first try to get an overview, then look at some parts but skip others and later go back to fill in the gaps.