Andre Weil

by Steven E. Landsburg                                           

In 1982, I made my first pilgrimage to the Institute for Advanced Study in Princeton---the Mecca of modern mathematics. There, sign-out cards tucked into the backs of old library volumes still bore the signatures of Albert Einstein, Kurt Godel, and other illustrious former members. (The "members'' are the Institute's faculty.) Recent Ph.D.s like myself, who were visiting for a semester or a year, were usually assigned offices in the "ECP'' building, where John von Neumann had directed the Electronic Computer Project and built the world's first modern programmable computer. (This accomplishment, like his invention of game theory---a theory that has come to pervade modern economic thought---was a relatively minor episode in von Neumann's brilliant career.) A bust of the mathematical titan Hermann Weyl, also late of the Institute, guarded the entrance to the dining hall.

But the glories of the Institute were not only in its past. Then as now, the permanent members of the Institute included a substantial fraction of the finest mathematical minds on earth. In that atmosphere suffused with intellectual ferment, it was intoxicating just to breathe.

The constant proximity to greatness left many young visitors---myself included---perpetually in a state that combined awe, exhilaration, and terror. In the rare moments when we weren't talking about mathematics, we used to talk about these feelings quite freely. I remember one of my colleagues saying the great men were father figures, and he felt tremendous anxiety about making a mistake in front of his father. He was, I thought, close to the mark but not right on target. We didn't think of the permanent members as fathers; we thought of them as gods.

Of all my heady moments in that enchanted time and place, one is most vivid in my memory. I had arrived early for a lecture, and found the room empty except for a gaunt, wizened old man hunched quietly in the front row. I took a seat a few rows behind him; he briefly turned around; we nodded greetings as strangers do. Then we both returned to silence, waiting for the speaker and the rest of the audience to arrive. To pass the time, my companion reached into the inside pocket of his rumpled sports jacket and extracted the morning's mail. I snuck a peek over his shoulder. The envelopes were addressed to "Dr. Andre Weil''.

I suppose I should not have been so astonished by seeing that name attached to a living breathing human being. The legendary Weil had retired from the Institute seven years earlier, at the age of 70, but he continued to live on the grounds and I knew that he was a frequent presence at both seminars and social events. But somehow I had failed to anticipate that he might be made of flesh and blood, or that I would be able sit within ten feet of him (though not, surely, to converse with him, which would have required something like composure).

It was one thing to have come to Mt. Olympus; quite another to be in the presence of Zeus. What was it about Weil that inspired such reverence? First and foremost, it was the depth and influence of his life's work, which surely established him as one of the great mathematicians of the twentieth century---and therefore, given the extraordinary mathematical achievements of the twentieth century, one of the great mathematicians of all time. When the French mathematician Jean Dieudonne compiled a "Panorama of Pure Mathematics'' in 1982, he listed the major areas of mathematics and the men and women who had made either "major'' or "significant'' contributions to those areas since the beginning of time. With 11 major contributions to his credit, Weil's name appeared more often than any other.

But the aura that surrounded Weil was based on more than raw achievement. His profound grasp of mathematical history made him seem all the more a part of that history; he was the natural heir to the tradition he cherished. In paper after paper, Weil exhibited his own ideas as natural extensions of the foundations long since laid by great masters like Fermat, Euler, and Gauss in the 17th, 18th and 19th centuries.

Steeped in the history of mathematics and the history of civilization, he was thoroughly a scholar. He spoke and read multiple languages (besides his native French, Weil was comfortable in Sanskrit, Latin, Greek, English, German, Portuguese and probably more), wrote poetry and literary criticism, mastered the Bhaghavad Gita and the Upanishads, and was renowned for the clarity and directness of his prose. He spoke incisively and knowledgeably about philosophy, painting, music and architecture.

Weil's presence was enhanced, as is the case with many great geniuses, by his personal eccentricities and the legends they inspired---the strangely guttural French accent, the acerbic wit, the exacting standards, the complete inability to tolerate any form of stupidity (quite a burden for a man compared to whom almost everyone else in the world was basically a dunce), and the mischievous vanity. These traits live on in his writings and in the oral history that is lovingly preserved by mathematicians worldwide.



Not a fool could call him friend. In 1973, an associate professor at Princeton University had the temerity to write a biography of Weil's revered Fermat, and the bad luck to draw Weil as a reviewer. Without a doubt, it was the most devastating book review in the history of literature. Weil begins by reminding us that "in order to write even a tolerably good book about Fermat, a modicum of abilities is required''. He then lists these abilities: (a) Ordinary accuracy. (b) The ability to express simple ideas in plain English. (c) Some knowledge of French. (d) Some knowledge of Latin. (e) Some historical sense. (f) Some familiarity with the work of Fermat's contemporaries and of his successors. (g) Knowledge and sensitivity to mathematics. He then proceeds to consider these requisites one by one, and to demonstrate---via annotated quotations from the book under review---that the author apparently possesses none of them.

Such irreverence was typical for Weil, who once described the Taj Mahal as a "bastardized offspring of Italian baroque grafted onto the ostentatious whims of a Mughal despot'' (though he could just as easily wax rhapsodic in the presence of genuine beauty). And I was an eyewitness to this one: When told that a certain mathematician had proposed a certain theorem, Weil dismissed the subject by saying, "That can't be true. Because if it were true, he wouldn't know it.''

Weil had a profound sense both of his place in history and of his intimidating effect on others, in which he took a roguish delight. In the mid-1980's, he gave a series of lectures on "Pell's Equation'', which is named for an English mathematician who had absolutely nothing to do with it. By all rights it should be called "Fermat's Equation''. Nevertheless, said Weil, he would bow to common usage and call it "Pell's equation''. "This has happened many times in mathematics'', he explained in accented English. "For example, I live on Von Neumann Circle. I live there...but still it is called...Von Neumann Circle''. With a shrug and a barely perceptible twinkle in his eye, he turned to the mathematics.

Pell's Equation, which I will rephrase as "Pell's question'' (though it should really be called "Fermat's question'') begins by asking: Which whole numbers X make 1+2X² a square? One solution is X=2, in which case 1+2X² is 9, the square of 3. The next solution is X=12, in which case 1+2X² is 289, the square of 17.

You can go on to ask other forms of Pell's question: Which whole numbers make 1+3X² a square? And what about 1+4X² and 1+5X², and so on? In principle, some of these questions might have no answers at all. It's by no means obvious, for example, that any value of X will make 1+61X² a square. You certainly won't find a solution to that one by simple trial and error, because the smallest is X=226153980. But Fermat devised a general method that allowed him to find such solutions, and his method always works. Using Fermat's method, you can generate any number of solutions to any form of Pell's equation.

Pell's equation is an example of what mathematicians call a "Diophantine problem'' (after the 3rd century mathematician Diophantos), meaning that it concerns itself only with the simple arithmetic of whole numbers (as opposed to, say, fractions). Such questions are often easy to state but notoriously difficult to answer.

The essence of Weil's great vision was that Diophantine problems, although they appear to concern only the ancient subject of pure arithmetic, are inextricably linked to problems in geometry and topology, many of which can be stated only in the language of twentieth century mathematics. High school seniors know that the germ of this idea goes back to Fermat's contemporary Descartes, who discovered that by "graphing'', you can translate equations into geometry. But that translation is too crude to tell you very much about Diophantine questions. You can plot a curve that represents all the solutions to an equation like x⁵- y³=31, but no matter how long you stare, you'll never be able to discern which points on that curve represent whole number solutions. (One solution is x=2 and y=1. How can you tell whether this is the only whole number solution? Or one of many? Or one of an infinitude?)

So it's natural to guess that if you're interested in whole numbers, geometry won't be much help. But thanks largely to Weil (and others including L.J. Mordell and Carl Ludwig Siegel), we now know that guess to be the exact opposite of the truth. Weil was able to prove that the geometric structure of a curve conveys---in ways that are highly subtle and not at all obvious---information about the arithmetic of the associated equation. From there, he articulated a grand vision of how arithmetic and geometry should be linked in far more general circumstances. This grand vision---which became known as the "Weil conjectures''---was formulated in 1948 and soon became the Holy Grail of algebraic geometry. Throughout the 1960's, a team comprising several of the world's very best mathematicians, and led by the charismatic and indefatigable Alexandre Grothendieck, developed the machinery that made it possible, in 1973, for Pierre Deligne to prove the Weil conjectures and justify the audacious courage that had allowed Weil to suggest that such an extraordinary set of statements might actually be true.

Nowadays, it would be unthinkable to work on problems in arithmetic without exploiting the power of geometry. To a large extent, it was Weil's prescience that made this development inevitable.

But that gets slightly ahead of the story. Before you can apply geometry to arithmetic, you need proper foundations for geometry. When Weil was doing his most important work in the 1940's, those foundations did not exist. For several decades, algebraic geometry had been dominated by the traditions of the "Italian school''---traditions which included a somewhat breezy attitude toward the details of proofs. There was a vast literature full of beautiful results, but it had become essentially impossible to tell which had been proven true and which had only been proven plausible.

The only remedy was to rebuild algebraic geometry from the ground up. Weil felt a particular urgency about this, because he needed a rigorous version of geometry to continue his work in arithmetic. This inspired him to write what he called "the indispensable key to my later work'', his book on Foundations of Algebraic Geometry. With the appearance of this book in 1946, the methods of the Italians were finally legitimized. In the process, Weil had to introduce new ideas and a new language, but characteristically he emphasized the continuity between his own work and the masters of the past. "Nor should one forget'', he wrote, "when discussing such subjects as algebraic geometry and in particular the work of the Italian school, that the so-called `intuition' of earlier mathematicians, reckless as their use of it may sometimes appear to us, often rested on a most painstaking study of numerous special examples, from which they gained an insight not always found among modern exponents of the axiomatic creed...Our wish and aim must be to return at the earliest possible moment to the palaces which are ours by birthright, to consolidate shaky foundations, to provide roofs where they are missing, to finish, in harmony with the portions already existing, what has been left undone.''

Within a few decades, Weil's rebuilt palaces were no longer the foundation of geometry, but the foundation of the foundation. In the 1960's, Grothendieck and his school used the palaces themselves as the groundwork for fantastic modern skyscrapers, reworking every assumption and expanding the realm of geometry to unimaginable heights. From these heights the Weil conjectures were eventually conquered. Grothendieck's project was one of the most remarkable episodes in the history of mathematics. Weil's conjectures made that project necessary, and Weil's foundations made it possible. If Weil had never lived, I cannot imagine what modern geometry would even be about.

None of this work was produced in some luxurious ivory tower. In 1939, Weil was arrested in Finland on the (apparently spurious) charge of spying for France. The day before his scheduled execution, the chief of police happened to mention to the Finnish mathematician Nevanlinna that "tomorrow we are executing a spy who says he knows you''. Nevanlinna intervened and Weil was deported instead. On his return to France, he was jailed for draft evasion and eventually released on condition that he join a combat unit. Following the war, Weil came to the United States, where European expatriate scientists were a dime a dozen. He held a series of jobs that were beneath him, including one particularly frustrating low-level teaching job at Lehigh University, where he was unappreciated, overworked and poorly paid. It was under these trying circumstances that modern algebraic geometry was born.

Had the Foundations of Algebraic Geometry been the culmination of his career, Weil would be remembered as one of the most influential mathematicians of his generation. But for him, the Foundations were only a necessary distraction from his true love---arithmetic. It has been said that mathematics rules the sciences and arithmetic rules mathematics. In his lifetime, Andre Weil ruled arithmetic.

It would be impossible to write about Andre Weil---in fact it would be impossible to write about modern mathematics---without mentioning the remarkable Nicholas Bourbaki. Like Weil, Bourbaki has been one of the most influential mathematicians of the century. Like Weil, he has taken responsibility for consolidating vast literatures and solidifying their foundations so that future researchers can build on them with confidence. Like my original vision of Weil, but unlike the Weil who really lived, Bourbaki was never made of flesh and blood.

In 1934, Bourbaki sprang full-blown from the head of Andre Weil. Weil was teaching at Strasbourg and engaged in endless discussions with his colleague Henri Cartan about the "right'' way to present various mathematical concepts to students. It occurred to him these discussions were probably being duplicated by his friends in other universities all over France. Weil proposed that they all meet to settle these questions once and for all. "Little did I know'', wrote Weil, "that at that moment Bourbaki was born''.

Nicholas Bourbaki was the name the discussion group adopted for its collective identity. The surname was that of Charles Bourbaki, the Napoleonic general who had suffered one of the most humiliating defeats in French history. The given name Nicholas was bestowed by Weil's wife Eveline, for reasons no longer remembered. Bourbaki's initial purpose---to design better course lectures---quickly evolved into something far more grandiose. Bourbaki's self-appointed task was to rework the foundations of all the major areas of mathematics, with particular attention to the notion of mathematical "structure'' as a unifying theme for the entire subject. Bourbaki was given a personality, a unique prose style, and even a biography: He was born in the mythical country of Poldavia. Decades later, Weil's official Institute biography omitted mention of his many awards and honors, listing him only as a "Member, Poldavian Academy of Sciences''.

Bourbaki soon began producing a series of encyclopedic volumes that synthesized the content of one mathematical subject after another. Those volumes left mathematics indelibly changed. Their births were excruciating: One member was assigned to write a draft, which was presented at a meeting and criticized mercilessly. Then the draft was discarded, and another member was assigned to write a new draft from scratch, making use of what he had learned from the first author's mistakes. The process was repeated until a draft was unanimously deemed worthy of publication. Each member had veto power, and a veto meant that the manuscript was discarded in its entirety.

Bourbaki survives, a living extension of Weil's extraordinary influence. New members are occasionally added, and an invitiation to join is one of the highest honors a mathematician can receive. The identities of the members are in principle kept secret. There is mandatory retirement at age 50, in accordance with the founders' wishes.

I saw him once in Princeton, about a mile from the Institute. There was snow on the ground, and he was walking down a wide path toward home, with his back to me. He was bent and leaned on a walking stick. Tall trees towered over him. Yet he dominated the landscape, an embodiment of the highest ideals of civilization. I wish I'd had a camera.

Andre Weil died in Princeton on August 8 at the age of 92, having looked almost every day of his life on Beauty Bare. With his vision to guide me, I've been grateful to catch an occasional glimpse.