
In 1982, I made my first pilgrimage to the Institute
for Advanced Study in Princetonthe Mecca of modern
mathematics. There, signout 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 theorya theory
that has come to pervade modern economic thoughtwas 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
visitorsmyself includedperpetually 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 centuryand 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
inspiredthe 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 demonstratevia annotated quotations
from the book under reviewthat 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 mid1980'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+2X^{2} a square? One solution is X=2, in which
case 1+2X^{2} is 9, the square of 3. The next
solution is X=12, in which case 1+2X^{2} is 289, the
square of 17.
You can go on to ask other forms of Pell's question: Which
whole numbers make 1+3X^{2} a square? And what
about 1+4X^{2} and 1+5X^{2}, 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+61X^{2} 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 x^{5}
y^{3}=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 conveysin ways that are highly subtle
and not at all obviousinformation 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 visionwhich 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 socalled `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 lowlevel 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 lovearithmetic. 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 Weilin fact it
would be impossible to write about modern
mathematicswithout 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 fullblown 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 purposeto design
better course lecturesquickly evolved into something far
more grandiose. Bourbaki's selfappointed 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.


ANDRE WEIL
