Mathematics is often perceived as an established body of knowledge, passed on to us by the Greeks and a few outstanding figures of the 17th and 18th centuries. Recent popularisations of mathematics have gone some way towards changing that perception. What is the nature of progress in mathematics? Is mathematics driven by theories or problems? Are there fashions in mathematics? Such questions have to some extent entered the general intellectual consciousness, and they form a central theme in The Hilbert Challenge.
In 1900, David Hilbert (with Henri Poincaré the leading mathematician of his time) presented a lecture at the International Congress of Mathematicians in Paris on ``The Future of Mathematics''. He tried to predict the development of mathematics in the 20th century and to encourage research in certain directions. To that end, he gave a list of 23 problems ``from the discussion of which an advancement of science can be expected.'' These problems include, at least implicitly, such famous ones as Fermat's last theorem and the Goldbach conjecture. Others were a lot more technical to state, and the solutions -- where they exist -- do not lend themselves easily to popularisation.
Jeremy Gray describes how Hilbert arrived at this list of problems and how attempts to earn a reputation by solving one of them have shaped 20th century mathematics. Gray is at his best outlining general trends in mathematics. The history of the various mathematical ``schools'' after Hilbert's death in 1943 is a particularly good read. Whether they shared or disputed Hilbert's views of what is important in mathematics, they all had to acknowledge the significance of those views.
A major flaw in the book is that Gray seems unable to decide for whom he is writing. He makes the very laudable attempt of introducing the lay reader to some of the mathematical concepts required to appreciate the significance of Hilbert's lecture. These introductions are given in separate boxes so as not to disrupt the flow of the main text, but they are often unenlightening. The main text, by contrast, contains passages dense with jargon incomprehensible to the non-mathematician, yet befuddling to the mathematician in their superficiality. It would have been better for Gray to focus on just a few problems.
But my main complaint is that the book is in many ways sloppily produced. The number of misprints (especially in German references) is intolerable, the mathematical notation often curious, and many references and allusions are obscure. For instance, we learn that in his report on the then-unsolved sphere-packing problem, ``Milnor called the situation scandalous.'' The report is not in the references, the reader is not told who Milnor is, and the ``scandalous'' remains a mystery; and the recent solution of the sphere-packing problem by Thomas Hales is unmentioned. (When it was announced, a greengrocer was interviewed on television news. He was clearly bewildered that mathematicians had taken hundreds of years to prove what was obvious to him: the usual way of stacking oranges is the best.)
The philosopher Edmund Husserl is mentioned only once. He is reported to have met Hilbert when he (Husserl) gave a lecture at Göttingen in 1901. In fact, Husserl was a colleague of Hilbert at Göttingen for some 15 years, and Husserl's departure to Freiburg was not unconnected to quarrels he had with Hilbert. Surely this is relevant?
The mathematician Ludwig Bieberbach is introduced as ``opportunistic and power-seeking''. We never really learn what justifies these damning attributes. Scheveningen, where Hilbert spent a summer holiday, is described as ``now close to the main airport for the Netherlands, but then a seaside resort.'' Well, it still is a seaside resort, and it is about as close to the airport as half of the Netherlands. And what does the airport have to do with Hilbert?
Finally, a factual slip: Hilbert's Paris lecture is mistakenly described as a ``plenary lecture''. It was given in the section of the Congress on bibliography and history, surprising as this may be in view of Hilbert's standing at the time and of the lecture's lasting influence.