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.
Other problems can also be addressed by brute computational force, but many mathematicians feel intrinsically uncomfortable with this approach. Said Davis: "It is certainly not seen as an aesthetic solution." A case in point is the four-color map theorem, which famously asserts that any map can be colored with just four colors (no two adjoining sections may be the same color).
The problem was first stated in 1853 and over the years a number of proofs have been given, all of which turned out to be wrong. In 1976, two mathematicians programmed a computer to exhaustively examine all the possible cases, determining that each case does indeed hold. Many mathematicians, however, have refused to accept this solution because it cannot be verified by hand. In 1996, another group came up with a different (more checkable) computer-assisted proof, and in December this new proof was verified by yet another program. Still, there are skeptics who hanker after a fully human proof.
Both the Poincare Conjecture (which seeks to explain the geometry of three-dimensional spheres) and the Riemann Hypothesis (which deals with prime numbers) are among seven leading problems chosen by the Clay Mathematics Institute for million-dollar prizes. The institute has its own rules for determining whether any one of these problems has been solved and hence whether the prize should be awarded. Critically, the decision is made by a committee, which, Davis said, "comes close to the assertion that mathematics is a socially constructed enterprise."
Another of the institute's million-dollar problems is to find solutions to the Navier-Stokes equations that describe the flow of fluids. Because these equations are involved in aerodynamic drag they have immense importance to the aerospace and automotive industries.
Yacht designers must also wrestle with these legendarily difficult equations. Over lunch, Davis told a story about yacht racing. He had recently talked to an applied mathematician who helped design a yacht that won the America's Cup. This yachtsman couldn't have cared less if the Navier-Stokes equations were solved; what mattered to him was that, practically speaking, he could model the equations on his computer and predict how water would flow around his hull. "Proofs," said Davis, "are just one of the tools that mathematicians now use."
We may never fully solve the Navier-Stokes equations, but according to Davis it will not matter. Like so many other fields, mathematics is becoming less about some Platonic ideal of ultimate answers, and more a functional project of computational simulation and communal negotiation. Dare we say it: Math is becoming postmodern.