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This question was asked earlier on Mathstackexchange but was closed very very soon without any answer and then deleted by the system!

I am a PhD student (1st year) in a poor country with a corrupt academic system (very nepotistic, racist,open bullying and other serious issues). So, a lot of mathematics I used to study was through textbooks of Western authors (Springer, Wiley and others). Local textbooks are of poor quality. I had this question in mind for 4-5 years but I don't have any help of colleagues as everything is a competition in the country and I think this question should be asked to many people and also because I don't have much guidance in real life. Journey to even get admission in a PhD program was very hard due to racism, lack of guidance and personal issues. My specialization is Commutative Algebra. Most things I learnt was through self study only.

Question: What approach should I have while studying advanced mathematical textbooks (PhD level and beyond)?

My Approach: I start self studying from page 1 of chapter 1. Remember at the definitions and also the intuition. I go through statement of theorems and the proofs and see if I can understand the logic in the proofs. If I can then I go towards corollaries and next theorems and in the end I try to do 50% of the exercises at the chapter end. Then same happens in chapter 2 till the end of the book.

After some time I don't remember the core ideas of the proof of theorems as I don't try to memorize the crux ideas of the proof but always make sure that I can understand how the proof works in the book but I can remember definitions, statements of theorems and some corollaries.

What are the pitfalls in this approach? Do I need to remember the core ideas of the proof of theorems?

I shall be very grateful for any answer!

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    $\begingroup$ Very carefully. And let me reiterate: you're a phd student. $\endgroup$
    – no upstairs
    Commented Oct 27 at 7:36
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    $\begingroup$ Doing a PhD is not passively reading books. There are way more mathematics than one can assimilate during a lifespan, let alone PhD. It is more about guided investigation: you basically learn what your advisor suggests, and see that, with this limited learning, you are still capable of proving something nontrivial. This is highly personal: different advisors have different opinions. $\endgroup$
    – Z. M
    Commented Oct 27 at 8:37
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    $\begingroup$ @non-euclideangeometry I don’t think your indirect, slightly cryptic comments are very helpful to OP at all. Could you rephrase them in a more direct way? $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Oct 27 at 12:50
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    $\begingroup$ I think that sometimes even more important than remembering statements by heart is remembering some key ideas in the proofs, because they become new tools in your toolbox. Another thing that helps is to try to make variations of the theorems in the book and try to prove or disprove them. Self study is hard but it is essentially the only way from the PhD and on wards. Also try to discuss with other mathematicians as much as you can. $\endgroup$
    – an_ordinary_mathematician
    Commented Oct 27 at 13:05
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    $\begingroup$ Commenting a bit on the approach part of your post, it is of course very important to know the relevant definitions, so that's good, but I personally do not think that memorizing entire proofs is very helpful. However, understanding the basic ideas used in proofs is very important, as those can often be reused elsewhere. I don't think math books are to be read "linearly", so to speak. And definitely do talk to some more experienced mathematicians, especially those who could potentially be thesis advisors. $\endgroup$
    – Malkoun
    Commented Oct 27 at 17:30

2 Answers 2

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If the books have problems sections, the most important thing is to do the problems to the best of your ability -- "tell me, and I forget; show me, and I remember; let me do, and I understand" [not actually said by either Confucius nor by Maria Montessori, to either of whom this is often misattributed, but true nonetheless] -- this will enable you to understand the concepts much better than a passive reading ever could.

Otherwise, yes, indeed it is very important to understand the core ideas of the proofs. Memorizing entire proofs is mostly quite pointless and just burdens your brain with too much detail, but understanding what the core idea of the proof is enables you do use the same idea in your own work.

Lastly, there are too many books out there to read them all -- "of making many books there is no end; and much study is a weariness of the flesh" [Ecclesiastes 12:12, showing that this was held to be true already even before the advent of the printing press] -- and if trying to read too many books keeps you from doing your own research, then you will have to change your approach beyond just the way in which you read books.

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Remembering the ‘core ideas of proofs’ essentially amounts to remembering all the relevant definitions and understanding how they relate to one another; this is essentially the content of any given theorem, “here are some definitions and here is how they relate to one another”.

Understanding relationships between definitions is generally much trickier than remembering definitions alone; it helps to really understand the intuition behind each definition, in addition to the formal symbolic representation you’ve chosen to work with. Often reasoning intuitively about definitions reveals subtle relationships that can then be formalized symbolically, and this amounts to a new theorem.

Long story short: hang in there and keep up the good work! You’re already most of the way there; if you presist in good faith, you will find treasures waiting for you.

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