Definitional Drift: Math Goes Postmodern

A baker knows when a loaf of bread is done and a builder knows when a house is finished. Yogi Berra told us "it ain't over till it's over," which implies that at some point it is over. But in mathematics things aren't so simple. Increasingly, mathematicians are confronting problems wherein it is not clear whether it will ever be over.

People are now claiming proofs for two of the most famous problems in mathematics -- the Riemann Hypothesis and the Poincare Conjecture -- yet it is far from easy to tell whether either claim is valid. In the first case the purported proof is so long and the mathematics so obscure no one wants to spend the time checking through its hundreds of pages for fear they may be wasting their time. In the second case, a small army of experts has spent the last two years poring over the equations and still doesn't know whether they add up.

In popular conception, mathematics is the ultimate resolvable discipline, immune to the epistemological murkiness that so bedevils other fields of knowledge in this relativistic age. Yet Philip Davis, emeritus professor of mathematics at Brown University, has pointed out recently that mathematics also is "a multi-semiotic enterprise" prone to ambiguity and definitional drift.

Earlier this year, Davis gave a lecture to the mathematics department at USC titled "How Do We Know When a Problem Is Solved?" Often, he told the audience, we cannot tell, for "the formulation and solution of problems change throughout history, throughout our own lifetimes, and even through our rereadings of texts."

Part of the difficulty resides in the notion of what we mean by a solution, or as Davis put it: "What kind of answer will you accept?"

Take, for instance, the task of trying to determine whether a very large number is prime -- that is, it cannot be split evenly into the product of any smaller components, except 1. (Six is the product of 2 by 3, so it is not prime; 7 has no smaller factors, so it is.) Determining primeness has huge practical consequences -- prime numbers are widely used in computer security codes, for instance -- yet when the number is large it can take an astronomical amount of computer time to determine its primeness unequivocally. Mathematicians have invented statistical methods that will give a probabilistic answer that will tell you, for instance, a given number is 99.99% certain to be prime. Is it a solution? Davis asked.