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In Search of Solutions

John McLeary

The Honors Class: Hilbert's Problems and Their Solvers. Benjamin H. Yandell. x + 486 pp. A K Peters, 2002. $39.

At the 1900 Paris International Congress of Mathematicians, 38-year old David Hilbert gave an invited address in which he presented a list of problems "from the discussion of which an advancement of the science may be expected." The published list of 23 problems became an icon for the new century of mathematical research, and anyone who solved a Hilbert problem was lionized, entering what Hilbert's student Hermann Weyl called "the honors class of the mathematical community."

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Did the problems generate the advances in mathematics Hilbert envisioned? How was the mathematics of the 20th century affected? These are appropriate questions to ask at the beginning of the 21st century. The mathematical community took stock in 1974 with a symposium and subsequent two-volume retrospective titled Mathematical Developments Arising from Hilbert Problems. Other historical accounts of the problems have appeared in France, Japan and Poland, as well as in Jeremy J. Gray's The Hilbert Challenge (Oxford University Press, 2000). The problems have captured the imagination of several generations of mathematicians, and work done on them constitutes a significant portion of the history of 20th-century mathematics.

Hilbert's optimism, clarity and reputation attracted many mathematicians to the problems. However, some solutions were not what he had expected. Perhaps the most spectacular example is the second problem, which asks for a mathematical proof of the consistency of the foundation of arithmetic. In 1930 the Austrian mathematician Kurt Gödel proved that such a proof of consistency is not possible, bringing to light the existence of statements in arithmetic that can be known to be true but allow no proof.

Benjamin H. Yandell's goal in The Honors Class is to trace the fate of each problem, including the contributions that led up to a solution and the solution itself. His focus, however, is squarely on the solvers—their lives and communities and the world in which their work was done. With this focus, a picture is painted of how modern mathematics developed.

Some of the tragic events of the 20th century play a role in Yandell's stories. Carl Ludwig Siegel, architect of the modern study of transcendental numbers, helped Max Dehn, who solved the third problem, to escape the Nazis. (The third problem was to find two tetrahedra of equal volume for which it would be impossible to cut one into finitely many polyhedral pieces and then rearrange the pieces to make the other.) The ideological battleground of the Stalinist Soviet Union of the 1930s serves as a backdrop for Alexander Osipovich Gelfond's solution of the seventh problem (which asks for a proof of the transcendence of the number 2√2).

Because for much of the book the math problems fade into the background as the solvers' lives are described, The Honors Class can be fairly compared with the perennially popular Men of Mathematics by Eric Temple Bell (who also wrote science fiction under the name John Taine). The books share the goal of making mathematics a living subject whose practitioners are at times heroic, painfully ordinary or even crabby. As a pair they make attractive reading for the potential student of mathematics who wants to know something about the people who made the subject. In contrast to Bell, Yandell has documented his remarks thoroughly. The work of the "honors class" is a good choice of subject, since it offers a broad view of the history of mathematics in the 20th century while bringing it into coherent focus. (Another viewpoint may be found in Michael Monastyrsky's 1997 book, Modern Mathematics in the Light of the Fields Medals.)

At the beginning of the 21st century, the Clay Mathematics Institute announced a list of problems, together with a prize fund for their solution, as part of their mission to spur the advancement and dissemination of mathematics. It can be hoped that these problems will generate the same attention and creative energies that Hilbert's problems have. In the words of Hilbert, "may the new century bring [mathematics] gifted masters and many zealous and enthusiastic disciples."—John McCleary, Mathematics, Vassar College

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