The Honors Class; Hilbert’s Problem and Their Solvers by Benjamin H. Yandell, A.K. Peters, 2002, 496 pp, US $39.00, £28.00, Euro 46.00, ISBN 1-56881-1410-1

The key to enjoying this book is to realise that the title is correct. It is about the honours class, the people who solved, or in some significant way contributed to the solution of, one of Hilbert’s 23 problems. There are fascinating accounts here of the lives of a number of the 20th Century’s greatest mathematicians, notably but by no means exclusively Kurt Gödel, Paul Cohen, Julia Robinson. Yuri Matjasevich, Dehn, Gleason, Siegel, Gelfond, Takagi, Emil Artin, Poincaré, and Kolmogorov. The lengthy discussion of Kolmogorov’s life and work is among the best, but it is typical in focussing on the man and his contributions, and not confining itself to his work on Hilbert problems alone ? considerable though that was. When he has used written sources, Yandell has looked around with care (perhaps more could have been done with Poincaré) and he has been particularly sensitive to the quality of information about mathematicians from the former Soviet Union, which has improved markedly in recent years. But a welcome and novel feature of the book is that he has talked at length with a number of the mathematicians involved, and in this way learned a lot about their lives, their attraction to this or that part of mathematics, and their memories of the personalities involved.

The problems and their solutions are also described, and often very well. There is almost no problem in the list that requires less than a graduate course for its proper understanding, so a lower bound for a full treatment of Hilbert’s problems is 23 volumes (let us ignore the fact that Hilbert wrapped up several problems in one on some occasions). If one adds that, outside the honours class itself, few mathematicians can feel equally comfortable with the different techniques involved in the different domains of mathematics, it is clear that both size and complexity are against the author and his readers here. Yandell’s solution has been to try and make the central difficulty clear, sometimes to indicate the techniques involved, and then to try and explain what the solution actually means.

This works very well with the first few problems. Perhaps because Yandell lives in Pasadena and has relatively easy access to the community of Californian logicians including Martin Davis and Paul Cohen, he gives a good account of forcing, and a fine insight into Julia Robinson’s approach to the tenth problem. It is bound up with the social history in a most illuminating way. But there are other good examples along the way: a nice sketch of the argument that Liouville originally gave for the existence of transcendental numbers is a case in point. Sometimes Yandell pitches his tents too far from the summits, in the hope that the tourist can at least appreciate something of the view. The reader who needs to be reminded what a prime number is will not get close to understanding the Riemann zeta function and the 8th problem, nor will someone who needs a quick course in high school algebra get close to the fourteenth problem. This is not a criticism. There are whole books on the Riemann zeta function which presume a fair amount of familiarity with undergraduate complex function theory. Yandell wants to reach a more elementary audience, and I think he is unquestionably right.

Over the years, the various Hilbert problems have led different lives. The strong start to this book reflects the vigorous life of the foundations of mathematics in the 20th Century, and the sense we have that Hilbert’s problems in this area have had definitive answers. The five last problems are in analysis, most but not all in differential equations. The consensus is that this is a highly technical area in which enormous progress has been made, perhaps to the point of leaving Hilbert behind altogether. Several of these problems get treated rather abruptly. And some, numbers 14 to 18, are often thought to be rather obscure, which may be why they are dispatched in the shortest chapter in the book. Yandell’s compromises are most painful here, but after all, the modern consensus, if there is one, is that these priorities are the views of the members of the honours class. However, one misses a sense of what made Hilbert choose these problems, and how and why some of them are more significant than the treatment here might suggest.

Quite deliberately, the book is impressionistic. There are occasions when some crucial distinctions are missing (first versus second order logic, for example) and a few technical terms fly by undefined (homology manifold, for example). I invite readers who find this sort of thing inexcusable to write politely to the author pointing out what has to be done to put matters right; the book merits a second edition. But any young mathematician who wants to know what the leading figures in the mathematics community were like in the 20th Century now has an excellent chance to find out, and to find out what is so exciting about the Hilbert problems and indeed mathematics itself.

Jeremy Gray Open University