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Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's main source of mathematical interaction. Even a slow course, or a course which covers material which one already knows to some level, can be highly stimulating. However, there are usually a few months in the year when mathematics slows down socially; in the summer, one might not be taking any courses, for example. In this case, one might find themselves reduced to learning alone, with books.

It is generally acknowledged that learning from people is much easier than learning from books. It has been said that Grothendieck never really read a math book, and that instead he just soaked it up from others (though this is certainly an exaggeration). But when the opportunity does not arise to do/learn math with/from others, what can be done to maximize one's efficiency? Which process of learning does social interaction facilitate?

Please share your personal self-teaching techniques!

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I disagree with the premise of this question, particularly the end of the first paragraph: as long as you have internet access, you are not alone! – Qiaochu Yuan Jun 16 '10 at 22:18
Qiaochu: Perhaps not alone, but there are some types of questions that are much easier to ask in person than via electronic media--e.g., "Could you help me understand this proof." – Charles Staats Jun 17 '10 at 0:02
Quiaochu : I agree with Charles, and I'd also like to add that personal interaction is usually richer than virtual interaction. It's great to have a friend introduce you to a new idea, and this doesn't really happen on the internet (at least not at the same level). – Bruno Joyal Jun 17 '10 at 0:35
Dear Qiaochu, The internet is amazing, but it's still much less efficient at transmitting information than face-to-face interaction is. The premise of the question seems perfectly valid to me, but is perhaps under-appreciated by those of us lucky enough to spend >= 9 months of the year in Cambridge, MA? Best, Stephen – GS Jun 17 '10 at 10:47
You also need a social group to interact with online. Sites like this aren't much use for musing aloud or asking stupid questions (its not meant to be for that - I'm not criticising mathoverflow). I'd love to have more individual interaction, but I have to make do with reading books on my own. – Francis Davey Aug 19 '10 at 14:13
up vote 17 down vote accepted

I think the crucial distinction here is not between books and people (after all, books are written by people!) or between physical contact and electronic contact, but between non-interaction and interaction. In my view, the crucial benefit that interaction provides is the ability to have your questions answered and your ideas critiqued. MathOverflow, of course, demonstrates the value of being able to ask a focused question and have an expert reply to it. In a non-interactive setting, you are limited to whatever answers have already been written down somewhere (plus your own ingenuity in answering your own questions).

Having your own ideas critiqued is also important. You may have a good idea (or a bad idea!) but not recognize it as such. An expert can often give you a quick assessment of your idea; you cannot get this without interaction. (Of course, expert assessment is a double-edged sword because sometimes the expert is wrong and you might have been better off without the incorrect feedback!) Even if you have a good idea that you recognize as good, you may have difficulty articulating it properly until you are forced to communicate it to someone else and get them to understand it. Some people are naturally gifted at expressing their ideas clearly and logically, but most people need to be taught communication skills interactively. And even the best mathematicians benefit from the exercise of teaching others what they know (or think they know!). The process of explaining something to someone else is of great value in clarifying your own understanding.

Finally, as for improving your efficiency if you have limited access to interaction, I would try to write down, as succinctly as possible, the questions and ideas that you want to get feedback on. Then you can efficiently send off a batch of questions and ideas and get feedback in a batch.

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Mathematics is not typically considered (by mathematicians) to be a solo sport

I agree and disagree with this statement. I feel that when I'm doing mathematics, then I tend to work best by myself. But if you'll pardon the slightly vague sentence, actually doing mathematics is only a part of doing mathematics. To be a little clearer, when I'm actually trying to prove or devise or learn something, I prefer to do it by myself and work at my own pace. But when trying to figure out what to prove, or what conditions the new thing should satisfy, or how what I've just learnt fits in with the rest of mathematics, or what to learn next, then I need to interact with others.

The difficulties of doing this is pretty much what the nLab is set-up to solve. When working on the nLab, I'm both working alone (and thus at my own pace) and in a group. I like to think of it as like working in a library - most of the time, one can work at ones own pace, but there are lots of other people nearby working as well and that helps. Being surrounded by other people clearly working hard encourages me to work hard as well, and there are experts nearby in case I get stuck or want an opinion on something. As it is a place for current research rather than just-finished-research (aka the arXiv) or out-of-date-research (aka MathSciNet), then if I see something interesting, chances are that the person who wrote it is still working on it and interested in discussing it. And others know that what I write is what I'm currently thinking about so they're much more likely to say, "I see you're thinking about X, have you thought about how that links to Y?".

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I have absolutely no idea of what learning process social interaction facilitates, but I can give you at least a really silly example of what sort of thing you do get when learning something from a person with more experience on the given subject. Everybody has seen at some point the basic expressions for the maximum and the minimum of two given real numbers $a, b \in \mathbb{R}$, say $max(a, b) = \dfrac{a + b + |a - b|}{2}$ and $min(a, b) = \dfrac{a + b - |a - b|}{2}$. I had always found it hard to remember the formulas. Well at one time I just realized that both formulas are completely obvious if you think of them as saying that for instance, if you want to get $max(a, b)$ then you just need to first step on the mid-point between $a, b$, which is $\dfrac{a + b}{2}$ and then you just have to "walk" from there half the distance between $a, b$ which is $\dfrac{|a - b|}{2}$ and similarly for the $min(a, b)$.

So at some point the chance came that a friend of mine needed to use one of this formulas and he said, "Oh but I don't remember exactly how they were". So I explained to him what I just said and he told me that he had never seemed them that way and that now he was sure he will not forget them.

My point is that this is exactly what you can get from an expert or from someone who has already thought about what you are learning at the moment, the hands on experience and the insight they have is what is most precious about this social interaction you refer to, it can give you the necessary ideas that you may not get from reading a book. You'll usually get more from this than what you get from a book or a paper. Obviously everything complements the other part so of course you can't expect to "be like Grothendieck" and learn everything directly from other people and you'll have to start taking the math from books, papers, wikipedia, etc. In a way I think of this as an aid in connecting the ideas I get from reading a book.

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This is the kind of answer I'm hoping for! Your example is good. Books are often very formal (which is good - it's mathematics, after all!), but understanding the formal side of an idea usually requires having an informal and intuitive understanding beforehand. It's usually much easier to obtain this from somebody else who has it already than to work it all out on your own. – Bruno Joyal Jun 17 '10 at 0:31
I disagree that this kind of insight is the sole province of social interaction. Such insights are common on high-quality blogs, e.g. Terence Tao's or Tim Gowers's, and they can also be found in several high-quality MO answers. – Qiaochu Yuan Jun 17 '10 at 0:46
Exactly. I have another example of this in mind. I don't know if you're familiar with Don Zagier's famous article "A One-Sentence Proof That Every Prime p = 1 (mod 4) Is a Sum of Two Squares". Well when I was taking a number theory course last year we saw this proof, and I have to tell you that the proof is amazing, although you do get the impression that everything falls down from the sky because you need to make some really unmotivated choices for some subsets of triples of natural numbers and some functions between them. – Adrián Barquero Jun 17 '10 at 0:52
So after we had completed the proof, the professor came in the next week and said that he had been unsatisfied with the presentation and that he had been looking up Heath-Brown, and Liouville and who knows which other old source and that he had found where the functions and the weird choices for the particular sets in question came from, and then he began explaining everything. It was rather nice to see this since otherwise I would have never done that myself. – Adrián Barquero Jun 17 '10 at 0:55
@Adrian: the motivation behind Zagier's proof is much clearer when expressed in the language of binary quadratic forms so to highlight the connection with Gaussian reduction, see… – Bill Dubuque Jul 12 '10 at 20:49

"learning from people is much easier than learning from books.": I doubt that, my experiences were mixed. Only from one lecture I learned much, because concepts whose reputation of extreme difficulty had inhibited me from reading on them were discussed in a very nice and relaxed 'matter of fact' way. But usually, lecture's contents are covered by books or articles which can more quickly be read instead. Personal communication depends on the participants personalities, which can be distorted by professional envy and selfishness in some cases, some people just can't admit that they don't know everything. Some thoughts on learning etc.: 1 ,2.

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I find myself attracted to Springer's Graduate Texts in Mathematics and buy too many of them. Then they sit somewhere at home so that I can pick them up and read them whenever I get a chance. These are essentially filling in background information that I don't have or don't remember (I'm as CS prof with a rusty major in mathematics). If I find a topic that particularly interests me, I chase it down using and plain old google. This is more or less my forward approach to learning new things. I also have a backwards approach, consisting of finding a paper that appeals to me and then trying to fill in the backwards knowledge required to understand it. Of course, without doing the exercises or applying the stuff I read, my level of understanding is not as deep as it could be.

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I really don't agree with the word "learning from people is much easier than learning from books." Everyone has his own opinion on things he is learning or has learnt. I think at least for me, the best way of learning is to learn through the books first, to ask questions and to form my opinion, and then to discuss with people, either in person or through MSN, blogs or some other ways. sure, one can always get an impression about something he knows nothing by talking with experts, but I don't think one can really expect others to explain in detail and learn a lot from that if his basic knowledge is empty.

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I think there is an important complex relating to research mathematics you could call autodidacticism-contrarianism (A-C), but it is not so clear quite where the dividing line between the two parts lies.

We know (roughly speaking) that some major advances come when a big and rather public research programme reaches maturity and there is a payoff. Others, though, come from largely independent thought developed to some extent in isolation. It is "contrarian", usually, to think that some hard problem can be solved with existing tools when the general view is that it can't. It would be "autodidactic" to base such a view on knowledge of some more obscure parts of the literature. In either case there is an obstinacy that defies conventional expectation.

That said, my own experience and what I know of that of others would suggest a mixture of these independent-minded elements with what you could call the mainstream and socially-supported elements is what is most effective.

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I don't agree that it is always easier to learn from people than from books. I taught myself a lot of mathematics without instruction before college. I think in some cases interacting with people is useful because it is easier for them to see mistakes you are making. But that could be done via the internet. I think for some people social interaction might be needed to learn. The ideal learning situation might vary from person to person.

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I don't know if this counts as a correction, but in Pursuing Stacks Grothendieck mentions reading Quillen's book on model categories and was able to correct a mistake he made by reading a paper about model structures on the category of categories.

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