Looking Ahead The International Mathematics Union has declared next year World Mathematical Year 2000. In keeping, they called for a compilation of problems to be worked on in the 21st century.

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 G¿del'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 K¿nigsberg, his birthplace, in which he said, "We must know. We shall know"--words that are now on his grave in G¿ttingen. But what G¿del 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 DAVID HILBERT. As one of the greatest mathematicians alive at the last turn of the century, Hilbert challenged his colleagues with 23 problems to be solved before 2000.

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.