This recent post at The n-category Cafe reminded me of an idea for a post I had a while ago. It was partly inspired by these three posts at Adventures in Ethics and Science, but more by a comment a friend of mine made at the math department tea. He said something like the following:
Mathematicians should have some sort of publication that’s halfway between a talk and a paper. The paper is the authoritative source to go to for reference, but the talk is much more effective for actually learning what’s going on. In particular, rather than giving the standard sorts of unreadable definitions that mathematicians normally give, they should give the general motivation for a concept, and the two or three counterexamples from which any competent worker in the field should be able to construct the same unreadable definition for themself.
Since the focus of my research isn’t in mathematics itself, or in the history of mathematics, I haven’t often tried to read journal articles in math, but the couple times I have it’s been really tough going. I don’t know if math is worse off than other scientific disciplines in this way, since I’ve never tried to read anything in any other field (except a couple papers in theoretical physics that are really more about philosophy). But it seems to be connected to the same issues that John Baez brings up in the first link above.
Part of what seems to make it so hard is that, not having worked out the proof oneself, it’s hard to follow what the author is doing, because you have no idea where each step is leading. The individual steps don’t add up to a coherent story. Something about this style is useful – when you do understand what’s going on it’s often easier to refresh your memory by reading the proof in the forwards order, as produced in the “context of justification” rather than the “context of discovery”. It’s tough to train students to write things in this backwards way, and there is some value to it. But it sounds like it’s a habit that one has to moderate to some extent as well, so that others can follow what’s going on.