In mathematics, it is a virtue when a proof is as minimal as possible: one wants to find the fewest assumptions that are necessary to prove statement X, because using unnecessary assumptions may mask the fact that the statement holds in some domain where one of the unnecessary assumptions does

In my experience, mathematicians, having been (correctly) trained in this minimalist approach to

*not*hold.In my experience, mathematicians, having been (correctly) trained in this minimalist approach to

*proofs*, unfortunately carry it over into their*explanations*of mathematical concepts, so that they try to explain a concept employing the absolutely minimum amount of explanatory apparatus possible. But what is a virtue in a proof is not necessarily a virtue in an explanation: students can often benefit from*logically*superfluous stories, examples, counter-examples, metaphors, and so on. Few mathematicians seem to realize that such ornamentation can make an explanation better.
Pretty sure you're just hanging out with the wrong mathematicians. There are many mathematicians who are also excellent teachers, and all of the ones that I've encountered are well aware of this particular distinction between what's appropriate when explaining mathematics versus when doing mathematics.

ReplyDeleteThat has been my experience in a classroom setting, shonk. But it has also been my experience that when teaching oneself advanced mathematics, the only resources available are books of terse explanations and proofs.

DeleteA mathematician (disclaimer: I'm not much of one) might argue that one better learns how to do math themselves by carefully thinking through the proofs and working through the exercises on their own rather than being spoon-fed.

shonk, perhaps the issue is what explanations work for what people. I am not bad at math -- I scored 800 on the math GRE, and that after heading straight to the 8 AM test after spending all night at a bar -- but I find the explanations of many mathematicians frustratingly sparse. I get math when I see the *structure* involved in the mathematics I am trying to learn. But many explanations seem to focus entirely on the formal notation: that is almost meaningless to me until I see the underlying picture of what is going on.

DeleteAnd as Matt notes, I am talking mostly about books.

This is certainly an issue with advanced books in particular. I don't really understand how publishers decide what textbooks to publish, but quality of exposition doesn't seem to be a major factor.

DeleteI suspect a big part of the problem is that many textbooks are just polished-up lecture notes, and a lot of people write minimal lecture notes that contain all the technical details but leave the intuition and analogies out, either because it's so obvious to them that they don't need any notes to give the necessary explanations in class, or (almost the same) because they prefer to do all that stuff extemporaneously.

In fact, some of the better textbooks I've read have been compiled from *students'* notes of what was actually said in class (rather than professors' notes of what they intend to say in class), which seems like it might be a better model.

One trick that I've found helpful on occasion (especially when learning a subject about which there are no textbooks or at least no good textbooks) is to look up masters' or PhD theses on a topic and read the first chapter. The writers aren't necessarily any better, but because they're less comfortable with the material they tend to at least include more of it.

In any case, I don't really disagree; I was mostly just objecting to the generality of your statement.

Gene is right. Concision and abstraction are seen as virtues even in mathematical explanations. My advisor contrasted computer scientists, who he said *want* you to understand.

ReplyDeleteBut you should read Visual Complex Analysis for a brilliant text book that avoids all that.