Sunday, December 12, 2004

Mathematical Errors

This afternoon, I attended a meeting of the local chapter of the WTA where I had the pleasure of meeting Eliezer Yudkowsky. Eliezer is a Director of the Singularity Institute for Artificial Intelligence, and works on ways to build human-friendly, superintelligent artificial intelligence. He's also a devotee of Bayesian philosophy.

We spoke briefly about philosophical certainty (be careful doing this with a Bayesian!), and Eliezer pointed out that there are uncertainties in mathematics similar to those in the sciences. I'm sure I raised an eyebrow in response, but my poker face hide a fairly deep gut reaction to his claim. Mathematics is not science!After a bit of computation, I'm convinced that we're both right.

Less than a decade ago, Fermat's last theorem was proven to be true. In 1637, Pierre de Fermat scribbled in the margin of his copy of Arithmetica that he had proof that there are no natural numbers a, b, and c such that

an + bn = cn


where n is a natural number greater than two. For 357 years, the theorem remained no more than a conjecture. Over the centuries, numerous attempts by mathematicians to prove the theorem were claimed, but later shown to be false. Finally, in 1995, in a 200-page paper, Andrew Wiles and Richard Taylor successfully proved Fermat's last theorem. Of course, their proof was not accepted until it could be thoroughly verified!

This story shows us that mathematical calculations are uncertain because mathematicians can make mistakes. You might say that the conjecture that a particular theorem is true is analogous to a scientific theory. But is it exactly a scientific theory?

Scientific theories are mathematical models that explain experimental data and make new predictions. The physical world is so complex that it is extremely difficult to isolate particular properties of nature in controlled laboratory experiments. All of our theories make the assumption that the effects of all phenomena apart from the one we are studying are small. For example, Galileo's famous experiment at the Tower of Pisa, which showed that gravitational acceleration is independent of mass, neglected the effects of air resistance. Such uncontrolled influences in research create systematic uncertainties in our experiments.

As far as I can tell, mathematics has no such uncertainties. I think that there are several reasons for this. The first is that mathematics takes a "constructionist" approach instead of a reductionist one.

Nature is very complex, and natural scientists spend their time trying to break down and isolate the fundamental (and, we hope, simple) mechanisms underlying the complexity. Statistical errors creep into their experiments (and systematic errors due to imperfect experimental design may appear), but scientists must also deal with systematic errors that result from an inability to perfectly isolate the phenomenon being studied. In contrast, mathematicians are already familiar with all of the fundamental building blocks of mathematical theory. Their goal is to identify the relations and categories of all of the complex objects that can be constructed from those building blocks. Mathematicians are subject to statistical errors because they might make mistakes in computation, but they don't have the systematic error inherent in reductionism.

My conclusion is that there are statistical uncertainties in mathematical results, but mathematics does not suffer from the same kinds of systematic errors that plague the natural sciences. Mathematical experiments are perfectly controlled.

3 comments:

Richard said...

Hi, doc: Challenge to the modifier "all" in ". . . all of the fundamental building blocks . . . ." and to the modifier "perfectly" in ". . . perfertly controlled."

References: MATHEMATICS: THE LOSS OF CERTAINTY, Morris Kline. Even more relevant and much shorter:
"Devlin's Angle": 'When is a proof'? June 2003 (GOOGLE)

Challenge to logical positivism/empiricism: THE COLLAPSE OF THE FACT/VALUE DICHOTOMY, Hilary Putnam.

Richard said...

Devlin has an excellent paper relating to "The Conscious Clock": 'The Amazing Ahmed'; "Devlin's Angle", December 2004 (GOOGLE)

Doctor Logic said...

Richard,

Thanks for the references!

Devlin's "When is a proof?" is a nice article, especially in the way it describes brute force proof by computer being susceptible to semantic doubt.

http://www.maa.org/devlin/devlin_06_03.html

This month's essay is an excellent example of nonconscious mathematical modeling in nature. As Ahmed demonstrates, there can be no perception of structure (conscious or otherwise) without mathematics. :)

http://www.maa.org/devlin/devlin_12_04.html

doctor(logic)