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Prove and Conjecture!

Joel Spencer

Mathematics: Frontiers and Perspectives. Edited by V. I. Arnold, Michael Atiyah, Peter D. Lax and Barry Mazur. xi + 459 pp. American Mathematical Society, 2000. $49.

At the close of the 19th century, David Hilbert, the leading mathematician of his time, addressed the Second International Congress of Mathematics in Paris. He gave a list of 23 problems, covering much of the mathematical landscape of his day. These "Paris problems" have well withstood the test of time. As the current century draws to a close, there have naturally been attempts to emulate Hilbert's feat.

It is not an easy task.

In this volume, 30 extraordinary mathematicians examine their own specialties to tell us where we are and what the future holds. Some list what they consider to be the most vital open questions. Some discuss relationships between mathematics and other disciplines, and a few even dare to take on the relationship between mathematics and society. Nearly all attempt at least some rumination.

Much of this volume is not an easy read. Only the true polymath will follow Andrew Majda on applied mathematics, Gregory Margulis on rigidity theory, Barry Mazur on p-adic variation, Dusa McDuff on symplectic geometry and Shigefumi Mori on algebraic varieties. The average reader will gain most from a nonlinear perusal of this work—selecting a few pieces for careful study and skimming the remainder for more general commentary.

"The Two Cultures of Mathematics," a splendid piece by Tim Gowers, will reward both scrutinizers and skimmers. The two cultures examined by Gowers are the "theory builders" and the "problem solvers." He uses the late Paul Erd"os to exemplify the latter. Erd"os's dedication, as illustrated by his constant admonition, "Prove and conjecture!," was inspirational. But his place in the 20th-century mathematical pantheon is a matter of some controversy because he resolutely concentrated on particular theorems and conjectures throughout his illustrious career. Gowers gives a spirited defense of the problem solver and, as a corollary, of the modus operandi of my own specialty, discrete mathematics. According to Gowers, "The important ideas of combinatorics do not usually appear in the form of precisely stated theorems but more often as general principles of wide applicability."

As a perfect example Gowers takes Erdös's three-page 1947 gem giving a lower bound on the Ramsey function. To simplify only slightly, Erdös proves that you can have a group of one million people in which no 40 all like each other and no 40 all do not like each other. To show this, we consider a gedankenexperiment, in which each pair of people flips a coin to decide whether they like each other. The experiment fails if the 780 coin flips between some 40 people all come up the same way; the probability of this happening is roughly 10–234. Each group of 40 gives another chance of failure; however, there are "only" some 10192 such groups—hence the total probability of failure is less than one. The experiment succeeds with positive probability. But then, the Erdös magic: It follows that the existence of a set of successful relationships among the one million is absolutely assured. The specific result has been shown using a general principle that has proved to have great and far-reaching power.

Why does mathematics go in the directions it does? In the preface, Michael Atiyah states that "The idea that science, including mathematics, is a 'social construct' has gained much notoriety and can be pernicious, but it contains a grain of truth."

Atiyah himself follows his own muse. Gowers quotes from a 1984 interview with him: "I have practically never started out with any idea of what I'm going to be doing or where it's going to go. I'm interested in mathematics. I talk, I learn, I discuss, and then interesting questions simply emerge."

Formalized mathematics was a key issue at the end of the 19th century. Hilbert wanted a clear (what we would now call recursive) set of axioms from which the truths of mathematics would inexorably follow. Russell and Whitehead's Principia Mathematica was probably the high-water mark in this direction. The formalistic view of mathematics is striking in this volume for its absence. David Ruelle's article contains the following characterization:

Human mathematics consists . . . in talking about formal proofs, and not actually performing them. . . . Human mathematics is a sort of dance around an unwritten formal text, which if written would be unreadable. This may not seem very promising, but human mathematics has in fact been prodigiously successful.

For V. I. Arnold, "Proofs are to mathematics what spelling (or even calligraphy) is to poetry."

Proofs are the sine qua non of mathematics. But an overemphasis on the place of proof is sophomoric—every decent mathematician knows what a proof is, and the true breakthroughs generally (with exciting exceptions) lie at another level—what Arnold calls the complexifications and quaternionizations of different mathematical objects. Barry Mazur quotes André Weil's meditation on the place of analogies:

Nothing is more fruitful—all mathematicians know it—than those obscure analogies, those disturbing reflections of one theory on another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when this illusion dissolves: the presentiment turns into certainty; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time.

Although this is an official book, commissioned by the International Mathematical Union, its coverage of mathematics is far from uniform. Mathematical physics—with contributions by Roger Penrose, Ed Witten and Cumrun Vafa—is particularly well served. Discrete mathematics—with contributions by Richard Stanley, Tim Gowers, Jean Bourgain and K. F. Roth—has no cause for complaint. Number theory, everyone's favorite, is well represented by Alan Baker and Gisbert Wüstholz, Peter Sarnak and, yes, Andrew Wiles. Peter Lax discourses on the increasing role of computing. Algebra, topology and applied mathematics are all here.

Of course it is impossible to cover everything; nonetheless, I have a complaint. Theoretical computer science (the wonderful mathematical problems that have come out of computers, as opposed to the use of computers in mathematical research) is missing. True, Steve Smale (whose problem list best captures Hilbert's spirit) calls P = NP? "a gift to mathematics from computer science." But this is only the Holy Grail. How wonderful it would have been to have had Avi Wigderson, Richard Karp or, best of all, Donald Knuth giving their views!

Mathematics is so beautiful that all of its practitioners would be wise to heed the warning given a half-century ago by John von Neumann (and quoted by Atiyah in the preface):

There is a grave danger that [mathematics] will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganised mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up.

This collection demonstrates well that mathematics is alive and vital and that von Neumann's concerns have not materialized. The reason is that although mathematical truth has a Platonic reality, the direction of mathematical inquiry is guided by less ethereal sources, such as physics, biology and psychology. Today the computer is omnipresent, and as a material object it has led to new mathematical analyses of computing itself. Can a class of mathematical problems (linear programming, graph planarity, sorting, satisfiability, etc.) be solved efficiently? Robust concepts for efficiency (P, NC, etc.) have been developed for a variety of models of computation. We are witnessing in theoretical computer science the rise of a new area of mathematics, comparable to the emergence of algebra and topology not too long ago. In the nascent century it will be accorded equal standing; that is my conjecture.

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