The Honors Class: Hilbert's Problems and their Solvers

At the dawn of the twentieth century, David Hilbert challenged the mathematicians of the world with twenty-three problems. These problems encompassed a wide breadth of mathematics and stimulated mathematicians for decades to come. Yandell's book is a (relatively) leisurely stroll through the people and the mathematics associated with these problems. I say "relatively" here because while a deep mathematical background is not necessary to enjoy this book, some mathematical sophistication will add to the reader's appreciation.

We can find other descriptions of Hilbert's 1900 lecture and subsequent paper that listed these problems. For example, the two volumes entitled Mathematical developments arising from Hilbert problems [1] contains papers from a 1974 symposium sponsored by the American Mathematical Society. This presents the mathematics behind the problems quite thoroughly, but it is not meant to be casual reading. Jeremy Gray's recent text The Hilbert Challenge [2] would also be a nice companion to this volume. To learn more about Davis Hilbert himself, the best reference is Contance Reid's classic biography entitled simply Hilbert [3].

Yandell's book includes Hilbert's 1900 lecture to the International Congress of Mathematicians in Paris and statements of all twenty-three problems in the Appendix. This is not to say that the book treats the problems themselves as an afterthought. On the contrary, the book is a refreshing visit with the people behind the mathematics as well as an introduction to the mathematics. We see that there are faces and personalities behind the mathematicians. Recent popular media has attempted to do this in special cases. Consider the Oscar-winning movie A Beautiful Mind and the Pulitzer Prize- and Tony-winning play Proof. In Yandell's book we see mathematicians at their best and their worst. I mention a couple of the profiles in particular.

Before a discussion of the second problem we are greeted with a vivid portrait of Kurt Gödel (1906-1978). We read of his personal life as well as his friendship with Albert Einstein. Before we see his contribution to Hilbert's second problem, we see Gödel as a person. We are even treated to a wedding photo (page 53). Gödel's story offers a glimpse of the world during his time.

This happens again later in the text where we are told of the colorful life of Carl L. Siegel (1896-1981). The account of Siegel's life further illustrates the effects of events outside of mathematics (world wars) have of mathematics and men.

This personalization of the mathematics and mathematicians is one of the strongest points of this book. This is something we should strive for in our mathematics classroom — point out that mathematicians are people, too.

I also liked the fact that Yandell assembles his discussion of the problems according to subject area rather than sticking to the more typical numerical order. In this way, someone whose particular interests lie in number theory for example can immediately turn to those problems associated with that area (7,8,9,11,12) while the analyst can turn to the material devoted to the analysis questions (13,19,20,21,22,23). Yandell's exposition is clear enough that readers will appreciate and understand all areas covered.

I recommend this book for reading by mathematicians and mathematics students at all levels.

References:

[1] Browder, Felix (ed.), Mathematical Developments Arising from Hilbert Problems (2 volumes), Proceedings of Symposia in Pure Mathematics Vol. XXVIII, American Mathematical Society, 1976

[2] Gray, Jeremy, The Hilbert Challenge, Oxford University Press, 2000

[3] Reid, Constance, Hilbert, Springer-Verlag, New York, 1970

Herb Kasube (hkasube@hilltop.bradley.edu) is Associate Professor of Mathematics at Bradley University in Peoria, Illinois. A member of both CRAFTY and CUPM his mathematical interests lie in the history of mathematics.