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.


Blogger Jay Bhattacharya said...

"Godel, Escher, Bach" drove me mad. There's a logical puzzle, introduced in the beginning of the book, that looks beguilingly simple. I spent many hours trying to get the thing to work, to no avail. Finally, I gave up and read the rest of the book. At the very end, Hofstader (sp?) reveals that the puzzle has no solution. My whole dorm heard my scream of anguish when I read that. Like you, I didn't feel like I understood Godel's theorem after I read Hofstader's book.

A better book for that purpose is "What is the Name of this Book?" by the logician Raymond Smullyan. It's a book of logical puzzles, which culminates with a simple version of Godel's paradox in puzzle form. I know I don't have all the nuances (I have never read Godel and I never mean to), but Smullyan's book is worth it nonetheless.

I liked Ellenberg's article. It reminds me of the following dialogue:

A relativist says: The world is full of greys. It's impossible to discern whether any statement is true.

His critic responds: Well if that's so, then how do you know whether your statement is true in the first place?

The relativist says: Well OK, the world is almost entirely full of greys. The only statement that is true is "It's impossible to discern whether any statement (save this one) is true."

The critic is left with one paltry truth, while the relativist is awash in grey. Ellenberg's position is sort of the converse (inverse?) of this. While Godel is left with one statement that cannot be decided true or false, Ellenberg plays in a world where many truths can be discerned.

9:46 PM  

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