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Thursday, January 17, 2008

Mathematicians are Platonists

In this past Sunday’s New York Times Book Review, Jim Holt wrote a mildly interesting review of the new book by John Allen Paulos, Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up. Since I haven’t yet read the book, I can’t comment on Holt’s negative evaluations of Paulos’s logic, but I want to discuss his observations on Platonism.

From time to time, some of my students object when I say that the Pythagorean Theorem or prime numbers or the quadratic formula or whatever was discovered. They want me to say that it was invented. And there’s a profound philosophical disagreement buried in that distinction. Here’s an excerpt from Holt’s review:
Mathematicians believe in God at a rate two and a half times that of biologists, a survey of members of the National Academy of Sciences a decade ago revealed. Admittedly, this rate is not very high in absolute terms. Only 14.6 percent of the mathematicians embraced the God hypothesis (versus 5.5 percent of the biologists).

But here is something you probably didn’t know. Most mathematicians believe in heaven. Not a heaven with angels, but one populated by the abstract objects they devote themselves to studying: perfect spheres, infinite numbers, the square root of minus one and the like. Moreover, they believe they commune with this realm of timeless entities through a sort of extrasensory perception. Mathematicians who buy into this fantasy are called “Platonists,” since their mathematical heaven resembles the realm of the Good and the True described in Plato’s Republic. Some years ago, while giving a lecture to an international audience of elite mathematicians in Berkeley, I asked how many of them were Platonists. About three-quarters raised their hands. So you might say that mathematicians are no strangers to belief in the unseen.
Perhaps the reason that a majority of students view mathematics as a meaningless game is that they don’t believe in its reality: if you’re inventing a formula or a theorem, you could just as well invent a different formula or theorem. But those of us who are Platonists believe that mathematical objects are truly out there, waiting to be discovered. If and when another intelligent species is discovered on another planet, they too will have the same prime numbers, an equivalent quadratic formula, and so forth. (Of course it’s likely that they will have discovered things that we haven’t, and vice versa, but they won’t have contradictory findings; 42 won’t turn out to be prime on Gliese 581c.)

There’s a slight twist to this issue, apparent to anyone who has explored non-Euclidean geometry. Isn’t there a contradiction between Platonism and the knowledge that other valid geometries can be formed by varying the Euclidean postulates? Anyone who has studied enough math knows that the conclusions of math are contingent: the Pythagorean Theorem and other familiar theorems of Euclidean geometry will change if the postulates change. More on this apparent contradiction later on...

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