It can be hard to chart real progress in any scientific field--but perhaps in none more so than mathematics. During the past century, steps forward in physics have often come in the form of newly found particles; in engineering, more complex devices; in astronomy, farther planets and stars; in biology, rarer genes; and in chemistry, more useful materials and medications. But in math, there are no obvious markers. The significance--or simplicity, for that matter--of an individual problem is frequently not clear from the start.
So on the eve of the millennium, how do we judge the last 100 years of mathematical advances? Fortunately, at the International Congress of Mathematicians in Paris in 1900, one of the greatest among them at the time--David Hilbert--set forth 23 problems that he felt his colleagues should solve over the course of the 20th century. In fact, most of Hilbert's problems have now been resolved. But the exceptions--and those solutions that didn't meet expectations--are of continuing interest.
What may have most surprised Hilbert himself were Kurt Gdel's resolutions in the 1930s of problem 1, the Continuum Hypothesis, and problem 2, regarding whether the axioms of arithmetic are consistent. He had not anticipated a speedy answer to the Continuum Hypothesis, nor probably the result that the axioms of arithmetic--and indeed any theory--cannot prove their own consistency. Upon his retirement in 1930, he gave a speech to the Society of German Scientists and Physicians in Knigsberg, his birthplace, in which he said, "We must know. We shall know"--words that are now on his grave in Gttingen. But what Gdel showed was precisely that we cannot always know.
Some of Hilbert's problems were very straightforward, including problem 3, which collapsed almost as soon as he drew attention to it. In 1902, Max Dehn demonstrated that two tetrahedra of equal base and altitude do not necessarily have the same volume. But others were too broad to perhaps ever be finished. For instance, in problem 6, Hilbert--who said on another occasion that "physics is becoming too difficult for the physicists"--called for the "mathematical treatment of the axioms of physics." John von Neumann and others deserve credit for axiomatizing quantum mechanics, but the problem may always stay open-ended.
Image: University of St. Andrews
Other problems fell more quickly than Hilbert had likely guessed. For instance, problem 7 is easily stated two ways: If a is an algebraic value and b is an irrational value, is a b trancendental? Geometrically, the question is if, in an isosceles triangle, the ratio of the base angle to the angle at the vertex is algebraic but not rational, whether the ratio between the base and side is always transcendental. Hilbert noted, "In spite of the simplicity of this statement..., I consider the proof of this theorem very difficult." But by 1934, Alekander Gelfond had proved that the answer is yes. In contrast, problem 18, otherwise known as the sphere-packing problem dating back to Kepler's day, seemed direct, but it was put to rest only two years ago.
Perhaps most interesting are those Hilbert problems that have yet to be solved. Among the most important may be number 8, proving the Riemann Hypothesis, which pertains to the distribution of prime numbers (those divisible only by one and themselves). Hilbert allegedly once said that if he could wake from the dead in 1,000 years, his first question would be whether the Riemann Hypothesis, named after Georg Riemann, had been proved. Part of its significance comes from other more general questions it leads into--including Goldbach's Conjecture that every integer is expressible as the sum of two positive prime numbers and whether there are an infinite number of pairs of prime numbers with the difference of 2. The good news is that rumors abound and some scholars think a solution is near.
Possibly even closer at hand is a proof of the Taniyama-Shimura Conjecture, which describes the topology of algebraic curves and surfaces that Hilbert called for in problem 16<. The first step was what has been to date perhaps the most famous solution of the 20th century: Andrew Wiles's 1995 proof of Fermat's Last Theorem, which represents a special case of the Taniyama-Shimura Conjecture. And in the July 9th issue of Science, a story reported that Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor, Wiles's co-author when tackling Fermat, have a proof of all cases. (Hilbert, when asked why he hadn't attempted to solve Fermat's Last Theorem, commented, "Before beginning I should put in three years of intensive study, and I haven't that much time to squander on probable failure.")
So what comes next? One potentially profitable problem for would-be solvers is a broader generalization of Fermat's Last Theorem, called Beal's Conjecture, which never made Hilbert's list. In 1997, Dallas banker Andrew Beal first announced the conjecture, which states that if Ax + By = Cz, then A, B and C have a common factor (when A, B and C are whole numbers and x, y and z are greater than 2). At the time, he offered a $50,000 prize for a proof or counterexample. The pot is now up to $75,000.
And other candidates for a 21st-century version of Hilbert's list are under review. On May, 6th, 1992, the International Mathematical Union--then meeting in Rio de Janeiro, declared next year World Mathematical Year 2000. In keeping, they appointed a committee, headed by Jacob Palis of the Institute for Pure and Applied Mathematics in Rio, to come up with a collection of unsolved problems, Hilbert-style. The IMU is also coordinating the publication of Mathematics Tomorrow, containing articles by prominent mathematicians on the prospects of math in the next century.
How will these new challenges compare to Hilbert's? Maybe we're not entirely ready to leave Hilbert behind. In his 1900 address, he noted that "an old French mathematician said: 'A mathematical theory is not to be considered complete until you have made it so clear that you can explain to the first man whom you meet on the street.'" By those standards, mathematicians still have a ways to go.