18 September 2007

how to categorize mathematicians, part N

Andrew at Statistical Modeling, etc. talks about the hedgehog-fox distinction (also here). The idea is that scientists of whatever type can be divided into "hedgehogs" and "foxes", with the "hedgehogs" being those who work on one problem and the "foxes", those who work on many. Andrew claims that statisicians, for the most part, are "foxes". (I can't intelligibly comment on this because in order to do so one has to have a sense of which results come from which people, which I don't in that field.)

There's a similar distinction that some people make in mathematics between people who prove lots of facts and people who produce an overarching theory that unites those facts. Any discipline needs both, I think. In physics one hears the metaphor of experiment and theory "leapfrogging" each other; the theory suggests new experimental approaches, which eventually push the boundaries of the existing theory, which lead theorists to come up with new theory which explains the new experimental results, and so on. The distinction exists in mathematics, too, although it's a little harder to see because what the "theoretical" mathematicians do and what the "experimental" mathematicians do look superficially similar. They both involve lots of staring off into space and occasionally scribbling illegibly on a piece of paper.

But this isn't exactly the same distinction. A mathematical "hedgehog" could, for example, work on solving a few "problems" which happen to be very hard; a "fox" could theorize and systematize in a bunch of different areas. This is perhaps made difficult by the fact that although it's easy to do a problem in an area not one's own, it's harder to pull together many problems in an area not one's own to make a theory without taking a long time to understand the large set of disconnected problems that have been done well enough to see the underlying structure.

Timothy Gowers has talked about the distinction between problem solvers and theorizers in his essay The Two Cultures of Mathematics (via the n-Category Cafe). Gowers also gives some examples of "organizing principles" of combinatorics; a lot of his examples would be difficult to write as "theorems", but are more in the nature of "tricks that work an awful lot of the time":
I have been trying to counter the suggestion that the subject of combinatorics has very little structure and consists of nothing more than a very large number of problems. While the structure is less obvious than it is in many other subjects, it is there in the form of somewhat vague general statements that allow proofs to be condensed in the mind, and therefore more easily memorized and more easily transmitted to others.

Were other areas that now appear to have some sort of big overarching theory like this in the past? Is combinatorics only like this because it's relatively young, or is this a fundamental trait of this part of the mathematical universe?

No comments: