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April 06, 2004
Like Comparing Apples and 01001100s
A paper by University of Pittsburgh mathematician, Thomas Hales, is finally going to be published after having spent six years in review. Why did it take so long for the paper to make it through the review process? Well, because it hits on a controversy which sits at the heart of post-millenial math, namely the debate over the proper role of computers in theorem proving.
Hales has used some machines to prove a theorem known as "Kepler's Conjecture" which states that the best way to pack oranges in a crate is in the form of a pyramid. The result seems intuitively obvious, but self-evidence rarely squares well with mathematical rigor.
Apart from being an interesting (right? RIGHT?) case study in how technology is affecting the intellectual development of our species, the issue is one of the few points at which abstract issues in the philosophy of math overlap with practical considerations of the methodology of working mathematicians. At the heart of the matter is the question of whether mathematicians should be allowed to advert to alternative methods of proof beyond the classical deductive and inductive methods. Powerful computers allow theorists to, in essence, do all (or an extrememly large number) of the calculations "by hand," as it were, to check whether a theorem holds in all of those cases. Obviously, this method was unavailable to mathematicians of bygone eras. But the question remains of whether it meets the standards of rigor that we require of our more traditional kinds of proof, especially when, by their very nature, these new proofs can't be checked by humans.
We here at greenideas are very far from having any strong intuitions one way or the other on this one, but this stuff is at the cutting edge of math and philosophy, and whichever way it goes will undoubtedly have massive implications for the future of scientific inquiry. Still, if we had to pick a horse to bet on, we'd just point out that people who stand in the way of using computers for a novel purpose almost inevitably end up as footnotes in the books by the advocates of new technology.
If you made it all the way to the end of this post, give yourself three gold stars.
"In Math, Computers Don't Lie. Or Do They?" (NYT)
Posted by matt at April 6, 2004 02:01 PM
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