The Sayings of the Fathers---And Several MothersJuly 17, 2000
Was Hilbert's list of problems, historically, the engine that drove 20th-century mathematics?
Philip J. Davis
Mathematics: Frontiers and Perspectives. By V. Arnold, M. Atiyah, P. Lax, and B. Mazur (editors) American Mathematical Society, Providence, RI, 1999, xi + 459 pages, $49.00.
"This volume, commissioned by the International Mathematical Union, is part of the activities celebrating 2000 as the Year of Mathematics. It is inspired by the famous list of problems that Hilbert proposed 100 years ago, but it has the more general purpose of describing the state of mathematics at the end of the 20th century."
So states the first sentence of the preface written by Sir Michael Atiyah. My eyebrows shot up immediately. The notion that the state of mathematics can be described in 29 articles, even when written by influential and internationally prominent specialists, blows my mind. I have piled up on my desk publishers' brochures that list at least a hundred twenty-nine advanced monographs in pure and applied mathematics, all of whose authors feel, I'm sure, in their heart of hearts, that what they have produced is the cat's pajamas and the gateway to the future. And keep in mind that not all mathematics is created by mathematicians.
Following in Hilbert's footsteps, many of the authors present their personal lists of unsolved, difficult, significant, inaccessible, intractable problems. (All these adjectives are used.) Thus, for example, Richard Stanley came up with 25 problems, and Steve Smale with 18; Peter Sarnak contributed 14 conjectures, and there are 10 problems from V.F.R. Jones. It is an easy guess that at least a hundred excellent papers and not a few mathematical cottage industries will emerge from these lists.
One might come away from these lists with the feeling that the Hilbert list, historically, was the engine that drove 20th-century mathematics. But this is denied:
"The influence of Hilbert's problems on 20th Century mathematics can be exaggerated. . . . It was Hilbert's own work that had much more influence" (Atiyah).
V.I. Arnold goes further:
"Hilbert tried to predict the future development of mathematics and to influence it by his problems. . . . The influence of H. Poincaré and of H. Weyl . . . was much deeper."
But let me push beyond this "mirror, mirror, on the wall" talk. What struck me in scanning the book, most of which is devoted to "hard core" mathematics, was how few of the core articles I could understand on even a superficial level. Two, three, perhaps, and even they would require a fair amount of study. As regards the others, I wouldn't know where to start. This being the case, I limit my comments in this review to the few immediately understandable sentences or paragraphs written "in English" (i.e., the side comments, not theorematic material) that a few of the authors have seen fit to include. With one exception (Grattan-Guinness), my clips are all from the book. I understand that in the formative stages of the book, Arnold, on behalf of the IMU, wrote to a select group of mathematicians and physicists soliciting their contributions. He suggested that, as one possible theme, they might formulate their lists of great problems for the next century. He also expressed the hope that the main theme of the book would be the fundamental unity of mathematics.
Even with the number of specialties growing by the dozens, producing practitioners who communicate only imperfectly with each other, the mathematical community worries about the unity of its subject. Unity is one of its major ideals, and the existence of unity is one of its principal myths.
Atiyah cites the long continuity of the subject, although with some uncertainty:
"If Newton, Gauss or even Archimedes were to return, I believe that after a short course to learn the new jargon, they would understand and even approve of the progress that has been made. . . . Can mathematics continue at this ever-increasing rate and still remain the subject we love?"
Physics is enlisted in the service of unity:
"I believe one major change that will occur in the next century is a unification of some existing branches of mathematics" [with the unification to come via physics] (Cumrun Vafa).
Barry Mazur quotes an assertion of André Weil that gives unity a spin from Weil's Vedantic philosophy:
"The yoked theories reveal their common source before disappearing. As the GITA teaches, one achieves knowledge and indifference at the same time."
My experience as regards unity is otherwise and I would cite as evidence my inability to understand "the state of mathematics at the end of the 20th Century" as presented in this book. I agree with the distinguished historian of science Ivor Grattan-Guinness when he writes:
"The many branches of mathematics, whether of ancient origins or not, exhibit very different histories, as does any one branch in different countries or cultures. Mathematics is rich, even dense, with interconnections but it exhibits no unity." (From The Norton History of the Mathematical Sciences, 1998; I.G.-G.'s italics.)
W.T. Gowers considers disunity of a different type when he discusses the inner tension-occasionally serious-between the "problem solvers" and the "theory builders." Although some observers assert that the digital computer is the most significant mathematical achievement of the 20th century (at least as regards its impact on society), Frontiers and Perspectives contains relatively few allusions to computation. Among those included are assertions of the superiority of the human brain:
"I would be skeptical about the use of Gödel's incompleteness theorem (as in Penrose, 1991) for arguing the limitations of any kind of intelligence" (Smale).
"Mathematics is an enormously deeper activity than what computers have been able to do so far. . . ." (Smale).
"Computing is entering into center stage of the sciences; Nobel prizes have been awarded for work whose main methodology is computational" (Lax).
Computers have also generated abstract toughies, e.g., the P versus NP problem, which turns up in Smale's list as a reformulation of a problem about Hilbert's Nullstellensatz. Prediction is an important and necessary part of any enterprise: It gives direction to the present and will inspire laughter in future generations. Here are some of the contributors' predictions:
"Number theory and algebraic geometry hold the key to much of the future evolution of mathematics" (A. Baker and G. Wüstholz).
"A major part of differential geometry in the 21st Century should be Riemann-Finsler geometry" (Shiing-Shen Chern).
"One change in number theory over the past 20 years is that it has become an applied subject" [through cryptography] (Andrew Wiles).
"I predict that in the next century, we will witness deep applications of number theory in fundamental physics" (Vafa).
Wonderful! The vision of Pythagoras stands reanimated: All is number.
"There is one rather safe, though perhaps seemingly provocative, prediction about twenty-first century mathematics: trying to come to grips with quantum field theory will be one of the main themes" (Edward Witten).
Perhaps the most revolutionary of the predictions is that of David Mumford:
"I believe stochastic methods will transform pure and applied mathematics in the beginning of the third millennium. Probability and statistics will come to be viewed as the natural tools to use in mathematical as well as scientific modeling. The intellectual world as a whole will come to view logic as a beautiful elegant idealization but to view statistics as the standard way in which we reason and think."
Logic and statistics were, of course, among the first of the breakaway mathematical disciplines, having organized their own departments and societies and formulated their own agendas many years ago. Now, it would seem that the roost is about to go after the chickens.
The psychology of the mathematician is not greatly stressed, but I found an appeal to the esthetic sense arising from the perception of the occasional cross-connection:
"Nothing is more fruitful . . . than those obscure analogies, those disturbing reflections of one theory on another; those furtive caresses, those inexplicable discords; nothing gives more pleasure to the researcher" (André Weil, quoted by Mazur).
Although the authors were given carte blanche, few chose to discuss mathematics and society. Among those who did was Dusa McDuff:
"I am hopeful that in the not-too-distant future, the gender of a mathematician will simply no longer be an issue."
But McDuff makes no conjecture, as some feminist writers have, as to how an influx of women into the field might or should change it.
Frances Kirwan, paraphrasing Paul Halmos, writes:
"To be a mathematician you must love mathematics more than family, religion, or any other interest."
She goes on to say:
"Perhaps before my children were born, I was a mathematician in Halmos' sense, and possibly once my children have grown up and left home I will be so again, but at present I am not."
Shall we now add The Market to Halmos' list of distractions? But mathematics doesn't have an exclusive grip on passionate devotion. I'm reminded of the old quip by the poet Wordsworth that a botanist is someone who pipes and botanizes on his mother's grave.
Atiyah raises the question of how mathematics might change when it is produced by the myriads of oriental mathematicians now entering the field: "The scene will be barely recognizable." He also gives one cheer for the philosophy of social constructivism (i.e., the idea that mathematical concepts reflect particular cultures and times more than platonists like to admit).
Yu. I. Manin raises the question of whether mathematics can or should go on and on producing more and more material:
"It is easy to list intellectual arguments against this enchantment [with science and mathematics] but all too often they fall into the void. They did not help Alexandre Grothendieck, one of the most creative mathematical minds of this century, who more acutely than most of us, felt the dangers of unsustainable development [and pulled out of it]."
In a rather depressing introduction to his article, Arnold makes the claim, reminiscent of Jean Jacques Rousseau's prize-winning essay of 1750, that historically the impetus for mathematics derives from unpleasant things. I have sympathy with this claim but in my writings in that vein have always hoped to stimulate the questioning of the social impact of mathematics.
The book---99% of it---is written by the inside for the inside. One could not plunk it down on the desk of a congressman's aide and ask for support on the basis of its contents. There is hardly anything in the "English" portions of the book that tells the reader why society should allocate any of its resources to the support of this kind of thing. I wonder what sense the average reader would make of Attiya's assertion (quoted by Gowers):
"The ultimate justification for doing mathematics is intimately related to its overall unity."
Nonetheless, with the exception of a few dark passages, the articles are upbeat, enthusiastic. As one contributor wrote, we can anticipate "exciting times ahead."
The core mathematics in this book will be a guide to researchers and graduate students for many years to come, and in the easily understood clips, such as those I have accumulated here, there is enough for philosophers and historians of science to chew on for an equal period.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.