Mathematicians Face Uncertainty

This will surely be remembered as the year mathematicians finally had to agree that their prized notion of “absolute proof” is an unattainable ideal—an excellent goal to strive for, but achievable only in relatively simple cases. Moreover, they were forced to make this adjustment under the harsh glare of the media, following three major news stories about so-called mathematical proofs.

Early in the year, American mathematician Daniel Goldston and his Turkish colleague Cem Yildirim announced a proof of the twin prime conjecture, which says there are an infinite number of prime numbers differing by two, such as 3 and 5, or 11 and 13. Although experts around the world initially agreed that the new proof was correct, a few weeks later an insurmountable error was discovered.

In late 2002 the Russian mathematician Grigori Perelman posted on the Internet what he claimed was an outline for a proof of the Poincaré conjecture, a famous century-old problem of the branch of mathematics known as topology. If Perelman is correct, he will pocket a $1 million prize offered for a solution by the Clay Mathematics Institute. But after months of examining the argument, mathematicians are still unsure whether it is right or not.

Never mind a delay of weeks or months—pity poor Thomas Hales, an American mathematician who has been waiting for five years to hear whether the mathematical community has accepted his 1998 proof of astronomer Johannes Kepler’s 390-year-old conjecture that the most efficient way to pack equal-size spheres (such as cannonballs on a ship, which is how the question arose) is to stack them in the familiar pyramid fashion that greengrocers use to stack oranges on a counter. After examining Hales’s argument for five years, in the spring of 2003 a review panel of world experts appointed by the prestigious journal Annals of Mathematics finally declared that, whereas they had not found any irreparable error in the proof, they were still not sure that it was correct. The journal agreed to publish Hales’s proof, but only with a disclaimer saying they were not sure that it was right.

What all three episodes reflect is the complexity and abstraction of many modern proofs. Even the experts find it almost impossible to be sure if some arguments are correct.

So where does all this leave the field of mathematics? Ever since the idea of proof was introduced by the ancient Greeks around 600 B.C., it has played a major role in the subject. In his mammoth work Elements, written around 300 B.C., Euclid began by writing down axioms—basic assumptions whose truth was assumed to be self-evident—and using them, with logically sound arguments, to deduce the theorems of geometry. As mathematicians from the 19th century onward ventured into ever greater heights of abstraction, the axiom-proof approach became an indispensable tool for handling concepts that were frequently counterintuitive.

So what exactly is a proof? There are two very different answers. One answer, which I will call the right-wing (the “right or wrong” or “rule of law”) definition, is that a proof is a logically correct argument that establishes the truth of a given statement. The other answer, the left-wing definition (fuzzy, democratic, and human centered), is that a proof is an argument that convinces a typical mathematician.

The right-wing concept of a proof is valid in an idealistic sense. The problem is that, except for trivial cases, it is not clear that anyone has ever seen such a thing. The examples familiar to most people from their high school math classes are the geometric arguments Euclid presented in Elements. But as the German mathematician David Hilbert pointed out in the late 19th century, many of those arguments are logically incorrect. Euclid made repeated use of axioms that he had not stated, without which his arguments are not logically valid.

It took some effort—following a delay of more than 2,000 years—for Hilbert to present correct proofs. Today most mathematicians regard his arguments as valid right-wing proofs, myself included. But if you push me to say how I know this, I will end up mumbling that his arguments convince me and have convinced all the other mathematicians I know. But that’s the left-wing definition of proof, not the right-wing one. After all, like everyone else I thought Euclid’s proofs were correct until I learned otherwise.

And there’s the problem. The right-wing definition of mathematical proof is an unrealistic ideal, unattainable in the real world. Actual mathematical proofs are all left wing. An argument becomes a proof when the mathematical community agrees it is such. But at what point does that happen? The three cases I started with highlight the problem. All three arguments are far too long and complicated for anyone to seriously believe these are anything more than left-wing proofs. But are they even that? How can we ever be sure?

Mathematicians have always said that proofs are to mathematics what experiments are to scientists: the way of telling right from wrong. After the events of 2003, it is clear that the comparison is closer than most mathematicians would care to admit. Just as all scientists must live with the ever-present possibility of experimental error, so too mathematicians, like our judicial system, may sometimes have to settle not for absolute proof but rather proof beyond a reasonable doubt.

—Keith Devlin