What is math good for?

This is a response to Aaron's post, as well as a chance to link to some of my stuff, buried deeply in the comments on other blogs. Every mathematician, at some point or other, is asked to justify his occupation. A common (and, I believe, sufficient) answer is that mathematics is ars artis gratia and requires no more justification than poetry or painting. Some mathematicians even take a (perverse) pride in claiming that their work has no practical applications whatsoever.

I consider myself an applied mathematician, and I take great pride and satisfaction in my work. My motivation to work on a problem is directly proportional to how useful people will find the result. Recently, Anthony Brockwell has been applying some of my rather theoretical results to the very practical and obviously important problem of decoding neural signals. Now, I never had this particular application in mind when proving the inequalities, but it was obvious to me that many people will find this stuff quite useful, and that certainly added to my motivation; it was very satisfying to see my work finding a use.

It's a common false dichotomy to distinguish "useless" math from "useful" engineering. I hate to repeat myself, so I refer the readers to the relevant discussion on Olivier's blog. If I may be forgiven the cliche: "There is nothing so practical as a good theory" -- so trite yet so true. I am opposed to snobbery between the different fields of human endeavor. I readily recognize the importance of bread-bakers, bridge-builders and code-crankers. Can we recognize, once and for all, that theoretical math is an important thing to study in and of itself? Need I remind you that the most basic tools in every engineer's chest (calculus, Fourier analysis) were once nothing but mathematical esoterica? Even that genteel queen of mathematics, number theory, has recently found herself rolling up her sleeves and taking her turn in the sordidly applied field of cryptography. Did Higman have any idea in 1952 that his result on partial orders would be used in 2005 to prove that a certain class of languages is linearly separable (and therefore learnable)?

Any way you approach math -- an intricate art, an indispensible tool, some combination of the two -- it unquestionably merits (nay, demands!) intense study.