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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?

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    $\begingroup$ Don't memorise them, learn them. $\endgroup$ Sep 7, 2020 at 11:00
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    $\begingroup$ @AnginaSeng i learned perfectly what I did on holiday last year but I can't remember the detail anymore anyway.. $\endgroup$
    – user820091
    Sep 7, 2020 at 11:13
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    $\begingroup$ What you need to do is to start recognising the ideas which motivate the theorems - then you will start understanding why the theorems are there and what they are trying to achieve. You will find a sense of direction and purpose which powerfully structures your thinking - and you will not just be remembering things, they will come naturally. After all, did you learn to speak your native language by reading a dictionary? Mathematics is very like a language in some important respects. $\endgroup$ Sep 7, 2020 at 11:18
  • $\begingroup$ See also math.stackexchange.com/a/63735/589 $\endgroup$
    – lhf
    Sep 7, 2020 at 11:20
  • $\begingroup$ Thanks for the question !! I am myself a student and I do agree with what the experienced members have said there. But honestly, things have turned so competitive that students are expected to learn a lot in a very short time and so we don't get enough time to have a clear understanding of everything . I would like to know if someone would say anything on this .. $\endgroup$ Sep 7, 2020 at 11:53

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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.

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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.

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  • $\begingroup$ Thanks for the answer! That sounds like a good way of studying, but for me personally, I don't like to take notes, I think it's a waste of time.. if i own a textbook and something that i understand is written down in it already, why would I write it down again risking of making some mistakes too? I used to take notes too but then I realized that I was too slow and now i study without doing it 🤷‍♂️ $\endgroup$
    – user820091
    Sep 7, 2020 at 11:19
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    $\begingroup$ @Vikise I would highly recommend getting back into the habit of writing notes. I'm slow with notes as well but writing things down in your own words does actually help solidify your understanding. And making mistakes in your notes is not a problem at all. You will never be free from making mistakes, and they can be extremely valuable in not only testing you but also in steering you towards strengthening the areas that you are less proficient in. $\endgroup$
    – SeraPhim
    Sep 7, 2020 at 11:29
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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.

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