How should someone remember theorems?

Question

I'm reading a little textbook on axiomatic set theory, $296$ pages, and more than $270$ theorems are given (I counted them!), there is no way a person can remember $270+$ theorems (let alone the proofs!) just from one little book. How should someone study a textbook then? I mean, math at the end is just a list of theorems, proofs and definitions, but if it's impossible to remember them all, how should someone learn this subject? Should I only remember the important ones and if needed go back to that textbook? Should I write them down and keep them in a notebook? How do you guys do it?

Answer 1

Memorising theorems and proofs doesn't make a good mathematician. If you focus on just rote learning every proposition, lemma and corollary that crosses your path then you will be severely missing the point. In my experience many results that you find in textbooks are there for two reasons:

1. To set up more important results later in the text, and
2. As an exercise to test your understanding.

Many of these types of theorems can be learnt and understood during your studies and then safely forgotten, and, as you say, returned to later if there is need for them.

Your focus should be on cultivating understanding and mathematical intuition about what you're studying. That's not to say that you shouldn't take time to learn theorems and proofs and make notes on them etc, you definitely should, but if you try to learn, and remember, every single one of them then you'll quickly become overwhelmed.

As you learn more you'll naturally remember the results that are most important for your interests and studies. So don't worry too much about it. If you want to know what I do, I copy from textbooks by hand, attempting the proofs before I read them and attempt the exercises, and then I'll rewrite my notes in my own words in a tex document, adding in or leaving out details depending on how important I think they are. That seems to have worked fine for me.

Answer 2

A good principle to keep in mind: everything in mathematics is obvious from the right perspective. You want to strive to have this perspective as often as you can (without misleading yourself, of course).

-- This depends on your style as a mathematician, too! I am very geometrically minded, so often I have to have some geometric picture for what is going on: Expressing theorems in terms of curvatures, symmetries, rigidity, and so on.

Something that is very much undervalued in the present state of things, at least as I see it in the research literature, is the importance of understanding theorems and their applications to examples. If you really want to understand a theorem, first generate some examples where the theorem provides some insight, generate some examples where the theorem fails. And if you really want to understand the theorem, try to construct counterexamples. In the course of failing, you will understand more and more.

Answer 3

You say

I mean, math at the end is just a list of theorems, proofs and definitions.

I think that there is lot more structure than this suggests.

What I find helpful when working through a chapter of such a book is to sketch a graph of the relationships between the theorems: theorems are vertices, with arrows to indicate which theorems are used to prove each new result. The results you need to develop the big picture stand out; and you can ignore (for a bit) the others.