Thursday, March 10, 2005

Spreading the Gödel news

Mathematics doesn't get too much attention in the popular press, but Slate magazine today has a column about Gödel's incompleteness theorem, written by Jordan S. Ellenberg, and honest-to-goodness mathematician at Princeton. I see it's part of a series of columns called "Do the Math," although most of the columns are more about statistics and common sense than they are about serious theorems. Much like relativity and the Heisenberg uncertainty principle, Gödel's theorem tends to capture the imagination of literary types and other armchair philosophers. Also like those other landmarks of 20th century science, it is easy to misconstrue. Ellenberg writes:
But what's most startling about Gödel's theorem, given its conceptual importance, is not how much it's changed mathematics, but how little. No theoretical physicist could start a career today without a thorough understanding of Einstein's and Heisenberg's contributions. But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel's work. So far, I've done it myself.
This sounds right to me, although I'm not sure I quite understand Gödel's theorem myself, despite actually finishing Gödel, Escher, Bach. I think mathematical dilettantes should pay more attention to another eccentric genius who's work revolutionized mathematics in the last century, Georg Cantor. Cantor's set theory really is fundamental to all of modern mathematics, and his simple ideas about the differences between countably and uncountably infinite sets are both simple to understand and profound, yet little known by non mathematicians.