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Suppose I am interested in really learning / thoroughly reviewing some subject (e.g. the basic theorems of infinite Galois theory, or the classification of compact Lie groups). One approach I might consider would be to write up a set of notes for myself going through all the main definitions, notation, and proofs of the main theorems in the subject, with appropriate references. I can think of several good reasons to do this:

  • I can't find a single textbook that does everything in quite the order I want it done in, or proves things quite the way I want them proven, and I want to synthesize material from multiple sources.
  • Just the act of writing down all the proofs in my own words will force me to understand the flow of ideas better.
  • I can revise the notes as my understanding of the subject improves.
  • I can put the notes online for others to benefit from.

But suppose one day I find a textbook that does things nearly perfectly (e.g. Neukirch for the basic theorems of algebraic number theory), and I am having trouble seeing the point of "competing," as it were. Is it still worthwhile to write up the notes that I was planning? Should I just do exercises instead?

(Feel free to interpret this question more generally, e.g. to ignore the body and concentrate on the title. CW because this question is mildly subjective and I'd like to encourage a broad range of answers.)

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@Qiaochu: as someone who writes a lot of notes, I feel that the question is somewhat directed at me. (I would be quite interested to hear from other prolific note-writers, e.g. Keith Conrad, James Milne.) But for some reason I feel the need to reflect on the question for a while before giving an answer. So let me get back to you... – Pete L. Clark Oct 11 '10 at 7:01
Whenever I try to take notes on just one book, I find myself pretty much rewriting a lot of what's in the book. This is alright for rote-learning, but I learn the best when I take notes and look at examples and do problems from at least 3 or 4 different books at a time... (I'm not a very prolific notetaker though so take what I say with some grains of salt and maybe even some fresh ground pepper.) – Dylan Wilson Oct 11 '10 at 7:22

I have to agree with Martin on the virtue of drawing up personal sets of notes. But sometimes it's hard to keep that going on your own, because there are always so many distractions, so many things to learn and do.

Actually, one of the most effective ways of thoroughly learning a subject is by teaching it. :-) Even if you are an undergraduate, this can be done in the context of a student seminar: find some fellow students at about the same level as you are and organize a seminar on a topic of mutual interest. This will force you to master many details of a subject, and the joint effort helps to keep each individual's energy from flagging.

Or, if you can't teach a class or schedule a student seminar, and your topic doesn't feel quite right to blog about, consider contributing your growing knowledge to a wiki like the nLab. There is plenty of scope to offer your own personal perspective in such a venture, and you will likely get plenty of feedback.

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At the risk of informing (more) people that I'm crazy, I confess to occasionally untangling a proof by lecturing to myself, not always aloud, but standing up, pacing, and gesturing as in a real lecture. – Andreas Blass Oct 11 '10 at 0:43
@Andreas: Talking to oneself is really not so crazy. Talking activates different parts of the brain compared to thinking through logic, and forces you to complete a train of though as opposed to thinking in smaller fragments. Fantasizing that others are listening adds some fake social pressure. Besides, who can argue with the results, if it helps? – Jason Polak Oct 11 '10 at 3:06
I think we need to mention teddy bears somewhere, if only for searchability. – Zsbán Ambrus Oct 11 '10 at 9:50

For me taking notes instantly on a subject for self-study is very, very important. Here I can check proof details, ponder about some relationships with already known topics and get intuition for the theorems etc. and how they interplay. The latter is rather difficult to communicate - therefore the author often does not put an effort into this - whereas the former are just personal issues. This makes these notes rather individual. Everybody has to make his own notes to get new insights from them, right? After having finished the notes, I have internalized them and just don't chuck them because usually some day I realize that they are useful without remembering the details.

Now I must admit that your question goes into another direction, namely you want to make a well-written summary of a theory and put it online. I used to do that very often some years ago (on, but nowadays I think this is only really useful when you know that you are not reinventing the wheel again or when you enrich the theory with some very interesting interpretations which you have never heard of and about which you want to discuss with others. Otherwise it will be "just" an enrichment for the readers who do not have to go through a whole book on the subject or have found your summary instead of one of the thousand other ones online. I think when you put a focus on your own learning success, you should probably do not waste your time with such a summary.

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Thanks for your answer, Martin. I should mention that I do not consider the online aspect a primary part of the question; these notes would be almost entirely for my own benefit. – Qiaochu Yuan Oct 10 '10 at 22:58

Just an idea for the "How" part.

You can try with a local wiki. For example, Instiki is pretty easy to set up and it's the one used in nLab where the purpose is "to provide a public place where people can make notes about [category-theory related] stuff".

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