Twenty-three Hard Ones

June 3, 2002

The problems proposed by David Hilbert in 1900 at the International Congress of Mathematicans in Paris immediately took on a life and a notoriety of their own. Photograph from The Honors Class.

Book Review
Philip J. Davis

The Honors Class: Hilbert's Problems and their Solvers. Benjamin H. Yandell, AK Peters, Natick, Massachusetts, 2002, 486 pages, $39.00

The turn of a century has proved to be a particularly good time to consider the past and to conjecture about the future. The world of mathematics knows that in 1900, in a lecture delivered at the International Congress of Mathematicians in Paris, David Hilbert proposed 23 mathematical problems of major interest and of major difficulty.

Hilbert was not the only mathematician of the first rank who had a personal vision of where mathematics ought to go. Henri Poincaré, delayed by illness, put forth his own suggestions for future research at the 1908 Congress in Rome. Poincaré's suggestions were presented in a less specific fashion than Hilbert's, and as a group they are perhaps somewhat less known to the mathematician-on-the-street.

At the turn of the millennium in 2000, a number of books containing far more than 23 "problems for the future" were published. Two of these books, Mathematics Unlimited---2001 and Beyond (The Power of Numerics, September 2001) and Mathematics: Frontiers and Perspectives (The Sayings of the Fathers---and Several Mothers, July/August 2000) have been reviewed in these pages.

Returning to 1900, Hilbert's problems immediately took on a life and a notoriety of their own within the mathematical community. Over the past hundred years, one by one, solutions have been wrested from the storehouse of mathematical knowledge and the invention of brilliant investigators. The panache associated with the solution of a Hilbert problem, or of any of a myriad of recently assembled problems for which pots of gold have been placed at the end of the mathematical rainbow, is great. The news of solutions sometimes hits the front pages of The New York Times or the Frankfurter Allgemeine Zeitung and might even be converted into a Broadway play.

As I write, three problems remain unsolved. For aficionados of the Hilbert problems, they are the problems numbered 6, 8, and 16. Problem 6 calls for the axiomatization of physics. Problem 8 is the Riemann hypothesis, and problem 16 demands a qualitative but exhaustive description of the kinds of curves that can arise from real algebraic functions. The Honors Class, which follows the history of the Hilbert problems, is a remarkable book, and I take my hat off to its author. Benjamin Yandell has worked assiduously and intensely. He seems to have used every known method of communication, except possibly carrier pigeon, to get in touch with those who could cast light on the individual problems, on how their solutions came about, and on the lives of those people who have had a hand (or even a finger) in their solution. He has absorbed and extracted the pith of perhaps 300 reference works, and has produced a multilayered book whose individual layers can be read with enjoyment by a number of different constituencies.

A few words about Yandell are in order here. I have heard that he is a person who lives by his own drumbeat. He did a good mathematics major at Stanford but did not want to pursue an academic career. For a while, he wrote poetry and supported himself by fixing TV sets. He has a substantial circle of friends, including scientists and some journalists. (His wife is a copy editor for the Los Angeles Times.) His ability to hold his own in discussions with professional mathematicians is impressive. Until the present work, he was not a science writer, but I hope that in the future, he will embark on other projects of a similar nature.

Now to the book itself. Firstly, the original article by Hilbert is reprinted in an appendix. This article contains not only the problems, most of them stated succinctly, but also Hilbert's background commentary. The body of Yandell's book presents "biographies" of the problems, so to speak, grouped more or less according to the mathematical areas in which they are located.

For those whose mathematical knowledge is limited, Yandell begins with simple explanations of the problems. This is accompanied by discussions of what led to the problems, early steps toward their solution, and a few indications of what happened afterward. Interwoven is biographical material related to the solvers and to "the giants upon whose shoulders" the solvers stood. Photographs of many of these individuals are interspersed throughout the book; many of them, I suspect, are published here for the first time, having been loosened from family albums by the author's persuasive efforts.

In this one book, then, we have at least three interrelated and intertwined books:

1. A popularized explanation of the problems. This first "book" requires a knowledge of mathematics at, say, the level of an undergraduate major, occasionally a bit less. It could very well stimulate budding young mathematicians; interest often springs from early exposure to material of this sort.

2. A much more technical, almost day-to-day diary, sometimes in excruciating detail, of who told what to whom, what theorems were useful or suggestive, and how the final clincher was wrung from the plethora of partial results. The second "book" is for professionals. Since most pros are specialists working in pods of similarly concerned specialists, they can learn here what goes on in other pods. It amazes me that an author who is not a professional mathematician has been able to digest and present this large variety of technical material.

3. Biographical material. The biographies of some of the involved solvers are quite extensive; others consist of just a fragment or two. There are descriptions of the historical milieus in which the almost 500(!) mathematicians mentioned worked. Here the reader will find a rich collection of stories (often dramatic), shop talk, gossip, and much more. This portion of The Honors Class exhibits mathematicians working in a hothouse atmosphere of collaboration and competition and will be accessible to the (we hope) wide readership that is not frightened off by a term like "rational number."

There has long been a feeling that brilliant mathematicians are all asocial eccentrics with more or less serious mental problems. In recent years, that feeling has been driven home by jokes, by plays, novels, biographies, and movies---I'm sure I needn't provide names and titles---all of which have reinforced the impression that you don't have to be crazy to be a mathematician, but it helps.

The biographical material presented here dampens this notion considerably. Mathematical "brains" may be wired differently than the average brain, but an examination of the lives written up here reveals sanity, conventional social behavior and its opposites, joy, suffering, tragedy (often overcome)---in other words, the full plate of what life offers. It is to the credit of the author that he has detailed the sufferings of individual European mathematicians as a result of the Nazi persecutions during the years from 1933 to 1945.


It is difficult to arrive at a firm notion of the importance of the Hilbert (and Poincaré) problems and their solutions for the development of 20th-century mathematics. The lists are small and the authority of the list-makers is great. Contemporary authorities hedge. Thus, Michael Atiyah, as quoted in Mathematics: Frontiers and Perspectives:

"The influence of Hilbert's problems on 20th Century mathematics can be exaggerated. . . . It was Hilbert's own work that had much more influence."

And, in the same book, V.I. Arnold:

"Hilbert tried to predict the future development of mathematics and to influence it by his problems. . . . The influence of H. Poincaré and H. Weyl . . . was much deeper."

What is clear is that the sequelae, in terms of sheer numbers of printed pages, have been enormous. Entire conferences and summaries have been put together to describe this fallout. The proceedings of one such conference, Mathematical Developments Arising from Hilbert's Problems (1976; Felix Browder, ed.), is a 600-page volume.

What is also clear is that the emphasis has been on pure and not on applied mathematics. Yet Problem 6, the axiomatization of physics (a problem proposed before relativity and quantum physics were born and considered impossible by many observers), elicited a ninety-page writeup in Browder's anthology from physicist A.S. Wightman.

There is a limit to human prescience. According to Felix Browder, writing in the 1976 proceedings:

"We should not believe that anyone, even Hilbert, could see the mysteries of the future in terms of new discoveries and new turns of interest."

What is eminently clear is that both Hilbert and Poincaré missed the most significant mathematical development of the 20th century: the digital computer. I call the computer a mathematical development because lurking behind every PC is an enormous amount of invisible mathematics, both pure and applied.

Martin Davis's fine The Universal Computer: The Road from Leibniz to Turing (James Case's review of the recently issued paperback version appears in this issue), which "is about the underlying concepts on which our modern computers are based . . . ," celebrates the work of relevant logicians, e.g., Boole, Turing, Church. Being himself a logician (and one of the Hilbert solvers), Martin Davis can honorably plug his field while forgetting about the makers of mechanical automata from the medieval and industrial revolution periods---Jacquard et al.---and all the electrical and electronic inventors and engineers without whom abstract concepts would have given birth only to additional abstract concepts.

The computer has been significant in two ways. Its effects in terms of many aspects of the lives we lead, even the lives of the most mathematically innumerate, have been revolutionary. It has been significant also in changing the way mathematicians conceptualize and operate within their subject.

Many straws in the wind lead me to conjecture that in the not too distant future, mathematicians will have split into two weakly intercommunicating groups: the computer-oriented, for whom the type of question and the type of methodology implicit in the Hilbert problems will be antiquated and irrelevant, and the "Old Believers," who will carry on proudly and magnificently as before. Such a split would be disastrous for the discipline. The procession from logic to hardware to the supermarket to the Web to who knows what next proves the point.

The Honors Class provides a vivid description of how mathematics at the highest level was pursued in the 20th century in the absence of this conjectured split.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at [email protected].

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