Jeremy Gray has seized the opportunity presented by the recent change of century to revisit David Hilbert's "23 Problems" address in Paris before the International Congress of Mathematicians on August 8, 1900. This valuable book surveys the fate of those problems and the mathematics created to deal with them, probes the lives and milieus of several mathematicians involved with the problems, and includes the full text of Hilbert's printed lecture in English translation. (The lecture was delivered in German and focused on just 10 of the problems, but a printed version with all 23 problems, diplomatically translated into French, was distributed to the audience.)

Chapter 1 ("The future unveiled") summarizes Hilbert's address. Hilbert immediately asserted the importance of problems in advancing mathematics, offering as an example the brachistochrone problem, which though quickly solved left, as Gray says, a "... problem area [that] was still full of life." Hilbert displayed his optimism about the eventual solvability of any mathematical problem with his famous claim that "... in mathematics there is no 'ignorabimus'," then launched into description and analysis of the ten problems he had chosen to discuss in public.

Chapter 2 ("The shaping of a pioneer") offers biographical details of Hilbert's life: from his birth through his university education in Königsberg, and his association there with Hermann Minkowski and Adolf Hurwitz, followed by several productive decades at Göttingen, influenced by and working with Felix Klein. We are led through Hilbert's work in invariant theory, which he eventually dropped in favor of number theory, followed by abandonment of work in that subject to focus on the foundations of geometry. Along the way there are diversions from Hilbert's work to compare his ideas and style with those of Poincaré, to contrast his views on the question of mathematical existence with those of Kronecker, and to outline the contributions of Gauss to the German tradition in number theory. A final section ("The road to Paris") sets the stage for Hilbert's invited address at the 1900 ICM.

Chapter 3 ("The beacons are lit") begins with the audience leaving the Sorbonne auditorium with the 23-problem version of Hilbert's address in hand, then outlines some of the ten problems that had been presented in the talk. We are then given a broad description of the state of mathematics in 1900, with particular attention to serious foundational problems arising in naïve set theory, and what Gray describes as "... the anxiety occasioned by the discovery of non-Euclidean geometry." The next section ("Roads not taken") points out the dearth of topology in most of Hilbert's problems, and tries to imagine what a similar survey address by Poincaré might contain. The chapter concludes, in fact, with an extended discussion of Poincaré's ideas, drawing on his 1897 address to the first ICM in Zurich and his talk in Rome to the 1908 ICM.

Chapter 4 ("The early response: 1900-1914/18"), after describing Hilbert's restraint in suggesting that his students work on the problems, discusses several of the problems in some detail. The 3rd (definition of Euclidean volume), for example, was solved before the address was officially published. The 2nd (completeness and consistency of arithmetic) was presented in such a way that Peano felt Italian work had been slighted. Brouwer's work on the 5th (Lie groups) is described, and there is extended discussion of the problems involving physics, differential equations, or the calculus of variations. After descriptions of results on problems having to do with number theory or geometry, the chapter concludes by observing that the 18th is really three problems: deciding if only a finite number of groups tesselate n-dimensional Euclidean space (solved in the affirmative by Bieberbach in 1910), existence of space-filling polyhedra that are not "fundamental domains" (also solved), and sphere-packing, which Gray says "... even in 1999 ... has still not been proved (although a final proof is thought to be close)." [On this question, however, see the April 2001 Math Horizons and the Hales web site listed there.]

Chapter 5 ("Between the wars: foundations examined") begins with a discussion of the post-war political climate, and then there is an extended analysis of the state of physics in the 1920's. Next, Hilbert's optimism about solvability and decidability is contrasted with the limits imposed by Brouwer's intuitionism (described by Gray as "... an anguished branch of the philosophy of mathematics"), followed by the dashing of Hilbert's hopes by Gödel's incompleteness theorem. There is a section on the precarious positions of some mathematicians in the 1930's Soviet Union, leading to the solution of Problem 7 on the transcendence of a^{b} (a algebraic, b algebraic and irrational) by Gelfond and Schneider, independently. This was surprising to Hilbert, who thought that the Riemann Hypothesis, and perhaps Fermat's Last Theorem, would fall earlier than this problem. We then encounter Pontrjagin and his 1934 solution of Problem 5 for compact locally Euclidean groups, von Neumann's 1933 solution of the same problem for commutative groups is mentioned, and the 1952 solutions of the unrestricted Problem 5 by Montgomery/Zippin and Gleason (independently) are foretold. Mention of the state of knowledge about Problems 15 and 16 is followed by material on Plateau's Problem, contrasting the approaches of Tibor Radó and Jesse Douglas. The chapter concludes with a report on Bieberbach's 1930 analysis of the influence of the 23 problems, and a few paragraphs on the destructive effect of the Nazi regime on German mathematics.

Chapter 6 ("After 1945") is on the legacy of Hilbert (who died in 1943) and his problems. A description of a 1946 Princeton conference on the state of mathematics is followed by discussion of the tension between pure and applied mathematics in the ensuing thirty years or so. Work of a large cast of prominent figures from both camps is described, and the first post-war ICM of 1950 in Cambridge, Massachusetts is mentioned. An interesting sign of the times is that President Truman had to intervene to obtain visas for Laurent Schwartz (described as a former Trotskyist) and for Jacques Hadamard (a non-communist who had visited the Soviet Union on several occasions). There is an extended discussion of the Bourbaki group and its relation to Hilbert's ideas, and particularly of the work of Alexandre Grothendieck. Post-war Soviet mathematics is examined in some detail, and Hilbert-related material at various ICMs is mentioned. The chapter concludes with several pages on mathematical logic, including Paul Cohen's wrapping-up of Problem 1 on the continuum hypothesis.

The concluding Chapter 7 ("Epilogue") has at its core the complete (23-problem) version of Hilbert's 1900 lecture. It is preceded by five pages of rumination by Gray on the meaning and importance of the problems, and two pages of notes on some details of the lecture, and followed by references to the lecture, an appendix summarizing the status of the 23 problems as of 1999, another appendix on logical matters, a brief glossary, and a list of nearly 200 general references. The lecture itself begins with a section called "The Future of Mathematics," consisting of general remarks on the rôle of good problems in the advancement of mathematics, remarks admirably summarized by Gray in Chapter 1. Then the 23 problems are presented, with comments on their place in the general context of mathematical knowledge. The appendix on the fate of the problems shows that nearly all have been solved or partially solved in some sense. Some of the solutions are negative, such as that of Problem 10 (finding an algorithm for solvability of Diophantine equations), which might have disappointed the optimistic Hilbert. The glaring holdout on the list is Problem 8 on the Riemann Hypothesis.

Given the general excellence of this book, it is necessary to wonder about two aspects of its organization and presentation: the function of several explanatory Boxes distributed throughout, and the large number of typographical problems. Both aspects are encountered as early as pages 2 and 3: Box 1.1 on page 3 explains the brachistochrone problem and its solution, part of a cycloid, while on the facing page 2 the curve is described as "... part of a catenary." One wonders if the probable audience for this book would really be unfamiliar with this problem, and why some proofreader did not catch the page 2 error. Some of the boxes are indeed useful, but others seem intended for an unlikely undereducated audience (e.g., one on Cantor's diagonal process, or one giving Euclid's proof of the infinitude of primes), or seem gratuitously placed (e.g., one on the Burali-Forti and Richard paradoxes, which are not mentioned in the text, or one on the word problem interrupting a discussion of Paul Cohen's work on foundations). The lack of adequate proofreading is apparent all too often. Here is a very small sampling of what can be found:

"... one could always find intervals sequences of nested intervals ..."

"... met during in the Franco-Prussian War ..."

"... the second off the five parts ..."

"... called for it to improved upon ..."

"... branch of analysis that it is nowadays known as ..."

"... a group of 3x3 of matrices ..."

and so on. These are the kinds of careless mistakes we all make, but surely a publisher has an obligation to try to minimize them.

Nevertheless, this book is an important contribution to the history of modern mathematics. It is good to have Hilbert's lecture conveniently available (some may wish to read Chapter 7 before going through the rest of the book), and Jeremy Gray's understanding of the problems and comments on them are invaluable.

David Graves ([email protected]) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in cryptology, opera, and history of astronomy as well as the usual run of mathematics courses.