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I'm beginning to run into work where I have to do a significant amount of learning of math by myself, with a book rather than with a teacher. Now, I do know that doing problems tends to be the best way to learn these things, but my question is a bit different.

How do you pace yourselves when you're learning new mathematics? Or reading a paper, let's say. Any pointers for reading large swathes of mathematics? (I'm running into this problem with my category theory references).

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Could you specify a little bit what or how much exactly you have to learn? It's hard for me to compare the notion of "much to learn" to others. And for a short answer: Since I normally don't work much, I just work harder if I have to learn much in a short time. That works pretty good. –  Konrad Voelkel Nov 8 '09 at 10:15
    
Sure, I'm trying to learn enough category theory to understand the connections between it and programming language theory in a strong way. I've been reading through books on category theory and programming languages, and topos theory recently. –  Michael Hoffman Nov 8 '09 at 15:41
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Here are my advise that are mostly based on experience:

If I start a totally new math, especially in the graduate level. I'd give my self at least 1.5-2 years (especially if it's an area in which a lot of lot of reading is involved.. say algebraic geometry). One of the things I find important, is not getting frustrated that you haven't learned to the level you need to learn. It is indeed frustrating, but when I look back in my PhD years.. I indeed took 2 years of just reading before even being able to start any new ideas of my own. You just don't have enough knowledge to make a ground-breaking mathematics and until you do be patient and learn it and try new ideas and create new examples (out of the book). I personally hardly answer excercises in the book, but created my own questions and tried to answer them first. If you were able to make new ideas and even publish a paper or two during these 1.5-2years then thats a bonus but you shouldn't feel incapable during that time.

The other advise I'd give is the references. Never stick to one or even just two single reference. Especially if they are the references that are difficult to digest. You should collect as many of the references in that area of mathematics as possible. If this is math that people have done already, chances are there are many many references about it that you don't probably know yet. And never read linearly through the references (esp. textbooks). I don't know of any professional mathematicians that has actually finished reading an entire book that he has not written himself. Switch from the different references as much as possible and try to get as much goodies from each as possible. There is no ONE book in homological algebra and different mathematician find different book suitable, you should find one that writes in a style your prefer and every now and then look at the other books as well. There is NO one book in commutative algebra, you can read certain characterization of testing for flatness of modules/algebras in commutative algebra books but you can hardly find ALL of them in ONE single book and some of them don't even have all of the proof.

Third advise, is to collaborate or speak as often as possible with people very knowledgeable in the topic. Attend seminar and conferences in that area of mathematics, even if you don't understand a pea. Chances are you learn something new or you learn about a question you think you find interesting in that area that is unanswered. There are some people who are knowledgeable in some area and make me feel like sh*t when I speak to them, I tend to avoid them.. but sometimes I mingle nevertheless. For me, true authentic mathematicians must good educators as well, so that if they find someone not knowledgeable in one thing they actually help him become knowledgeable instead of making him feel bad about it.

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"Never stick to ... just two single reference.": Yes and never forget that some standart texts are really bad written. E.g. I once spent weeks in vain to understand an important standard article of a very well known specialist, then again with a seemingly explaining article by other famous specialists - only then I found out that both are regarded by all as completely unreadable. Maybe someone should invent for such texts a math version of the former "rusty electron"-award in computer science... –  Thomas Riepe Nov 8 '09 at 18:00
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And don't be afraid to take a look at the original papers on a topic. Sometimes they are unreadable, of course, but sometimes they are very readable, with a wealth of motivation that tends to get left out more efficient textbook accounts. –  Charles Rezk Nov 8 '09 at 18:22
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"Read the masters." Indeed good advice,but not for the beginners.That's the key.Read the STUDENTS until you've learned enough to be able to read the masters.Remember,the reason the masters are unreadable isn't because they're inhumanly brilliant. It's because the concepts are only in the formative stages and the masters themselves aren't clear yet what the best way to lay it out is,they're groping for the ideas.But that's why reading the masters is so enlightening-they are sources of IDEAS. And they're great models of how math is actually done in the real world. –  Andrew L Apr 23 '10 at 16:40
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@AndrewL sometimes the masters really are a lot more readable than the students. For example, I think Hormander is really a master of exposition, and stays close to the essential. Later reformulations layer on more abstraction than needed, and can obscure the ideas. –  Steven Gubkin Feb 19 at 22:53
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Having learned everything through reading by myself, my impression is that it helps to read in a way which enables the subconsciousness to cooperate. E.g. reading not only one text, but browsing related surveys etc. too, seemingly 'just very superficial' browsing gives the subconsciousness de facto a lot of usefull inputs. Not ignoring and counteracting signs of exhaustion or tiredness, e.g. during sleep one processes the read texts. Continuing reading even if something puzzles one or one can't solve an exercise on spot (I found that most difficult to do)- in most such cases one just forgot some tiny info or one's mental image of the issue in question is a bit confused, what corrects by itself anyway. But stopping reading would just waste time. And, of course, puzzlement is a good way to activate the subconsciousness. When I tutor students, I express it by an analogy: "If you climb a mountain, you stumble only over small stones, not over mountains. So, one should notice such stones, but not take them too serious".

Edit: A study on the usefullness of reading before sleeping.

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I couldn't have given a better advice! This strategy has revealed itself as the "best" strategy for myself many times. –  Konrad Voelkel Nov 8 '09 at 16:42
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Of course things may vary considerably from one person to another. Yet I can give you my experience of the last two years with category theory. For applied category theory anything goes as standard maths : rest from time to time etc... But for categories theory itself (or its first applications) I believe the pace is a bit special because it "recables" your brain in a way different from that other of fields (in which re-cabling is due to focusing on one type of object).

Typically I tend to describe categorical definitions as rather short or almost trivial yet THICK! You have to use them several time before being cabled.

I believe that the reason lies in the 'abstract nonsense': a categorical definition make sense only when applied to specific examples which are yet abstract. Hope it helps.

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Something could be useful when Special Force Soldiers are trained, named Devil Training.

It seems the good reasons are related to transcend extreme limit of capacity to quickly extend the comfort-zone and equipped with massively difficult practices in very complicated environments NOT easy ones.

Some parts the same as good mathematical training, one could absorb both broad and important knowledge quickly, and move on quickly, intensively practices to be done in advanced and complicated 'environment' to train both elementary and advanced skills at the same time. Never stay in easy level but just move on even if some skills/knowledge are not absorbed well, since it'll be better as we go further. There's only a fine line between Wisdom and Foolish, wise persons absorb and move on fast with playing around with knowledge like a swimming fish and keep on deepening the understanding; while fool persons only suffer from the pain passively and repeat everything mechanically without thinking.

While some parts are apparently opposite, in Military Training it's obey the orders, no question asked, but in mathematics(natural science) it's break the orders, endlessly ask questions. Plus some passion, curiosity, self-automatic willingness to explore and some luck, it can generate a very beautiful road in mathematical world.

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I am just a leisure-mathematician but I almost never read any texts which do not contain exercises. Basically what I will do is to select some interesting unit (which could be a book chapter or lecture slides). First read through the exercises (respectively lecture exercises), then quickly read through the content accompanying the exercises to get an overview (no notes taken as this will just slow you down at this stage). Then I will try to do the exercises on paper. When I am happy with the solution I will type it up on LaTeX. Obviously while solving the exercises I will browse the chapter, but in a selective way. Sometimes I'll also browse other sources like Wikipedia or survey papers at this stage.

The advantage of this technique is that you will not be wasting your time for writing extensive notes of things which have already compiled in the textbook but rather stimulate your mind and put you in pseudo-research situation. In addition you create your own content, which helps to build your mathematical taste / personality.

Also a huge mistake you can make is to force yourself linearly through one or two textbooks only and not consider alternative sources. In the end the goal is to solve the exercises, get your own hands dirty and ultimately formulate your own research conjectures etc. Which resources you have used on the way is secondary.

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