But Noether’s work went far beyond mathematical physics. She made important contributions to Galois theory, to many other areas of abstract algebra, and to topology. Noether was, in fact, the greatest algebraist of her time. She worked prodigiously, having little social life other than close contact with her students. In her lectures she talked very fast, a reflection of her very quick thought processes. She also suggested many research topics to her students, and many of her solutions to key problems in abstract algebra appeared in print as the work of her students, even though she had played a role in deriving the results.
Unfortunately, when Hitler came to power in 1933, Noether was summarily dismissed from her position so that the university could comply with the newly passed race law forbidding Jews from holding academic jobs in Germany. After failing to obtain a position at Moscow State University in the Soviet Union that year, she accepted a visiting professorship at Bryn Mawr College in Pennsylvania and came to the United States. Two years later she died from an infection following an operation to remove a tumor. Meanwhile, Emmy’s brother Fritz Noether—also a mathematics professor dismissed by the Nazi regime—obtained a position at the University of Tomsk in Siberia, but his tenure didn’t last. In 1937 the Soviet government falsely accused him of being a German spy and sent him to a Soviet camp, where he was executed in 1941.
Emmy Noether expanded the role of women in the mathematical community. She advanced the field of algebra, and used algebraic methods to address key problems in physics. Utilizing Lagrangian methods and Euler’s discovery of the calculus of variation, she tied together the work of the algebraists with calculus and topology to form a framework for explaining the workings of the physical world. Likewise, Ramanujan’s prolific, unconventional results inspired a great deal of research in the twentieth century, finding application in the work of André Weil, who, at the time of Noether’s death, was engineering one of the biggest pranks in mathematical history.
The Chapelle de St. Eutrope, or Chapel of St. Eutropius, in Castanet-le-Haut, France, is situated in the Languedoc-Roussillon region, not far from the Pyrenees, where some suspect that the mathematician Alexander Grothendieck lives in hiding. The village of Andabre can be seen in the valley at right.
FIFTEEN
THE STRANGEST
WILDERNESS
After Germany’s defeat in the First World War, the world center of mathematics returned to France, and Paris became home to more import-ant working mathematicians than any other city. These “new” mathematicians, politically disillusioned by the Great War (and certainly constituting part of Gertrude Stein’s génération perdue, or Lost Generation), were highly cynical and angry. They hated the “establishment,” both political and educational. And they disparaged the older mathematicians, whom they viewed—perhaps unfairly at times—as morally corrupt. This new sociopolitical milieu in France made fertile ground for the greatest revolution in mathematics—and the invention of a person who never existed.
NICOLAS BOURBAKI
There was once, in French history, a general of Greek origin named Charles Bourbaki, who in 1871 led French forces to one of the most humiliating defeats in the war against the Prussians, and later attempted suicide—unsuccessfully. His story captured the imagination of Raoul Husson, a history buff who also happened to be a third-year math student at the École Normale Supérieure (ENS), which, since its acceptance of Évariste Galois a century earlier, had become a very prestigious university in Paris. Every year, the third-year students at ENS would play a trick on the entering freshmen in mathematics, and in 1923, it was Husson’s turn to plan and carry out the annual prank.
Husson, wearing a long fake beard and pretending to be a professor, walked into a room full of freshman and wrote on the board: “Theorem of Bourbaki. Prove the following …” The “theorem” was completely nonsensical, and the poor souls sat there for an hour scratching their heads and worrying that they were failing their very first assignment in a mathematics course at this august university.
The pseudonym Nicolas Bourbaki originated with student pranks carried out during the early twentieth century at Paris’s École Normale Supérieure, the courtyard of which is seen here.
At the same time, other students played tricks on passersby. Below the university area, south of the Luxembourg Gardens and toward the Paris Observatory, passes one of the major boulevards of Paris, the Boulevard du Montparnasse. Long before the boulevard was built, this area took its name from “Mont Parnasse,” the facetious nickname that French students in the seventeenth century gave to the large garbage heap that festered there. (“Mont Parnasse” is the French version of Mount Par-nassus, the sacred peak near Delphi in central Greece, where Apollo was worshipped in ancient times.)
On a sidewalk of the Boulevard du Montparnasse, a student stood on a podium and lectured to the crowd about the poor conditions in the (fictitious) nation of Poldevia. Other students would tell passersby that they were collecting money to aid the people of Poldevia, who were so poor that they couldn’t even afford trousers. As they said this, a student appeared from behind the podium and was introduced as the prime minister of Poldevia. He was wearing only underwear.
André Weil (1906–98) loved these two student tricks. He came from a wealthy Jewish family in Alsace who had chosen French citizenship when the residents of Alsace-Lorraine were given a choice between becoming citizens of France or Germany. His father was a surgeon who had worked for the military, and the family owned a beautiful apartment on the Left Bank in Paris, right next to the beautiful Luxembourg Gardens. It was there that André and his sister, Simone—who would become a famous Catholic philosopher—grew up.
Weil was extremely smart, but perhaps not a genius on the level of Cantor or Galois. A precocious child, he found that school was too easy—and besides, he was more interested in having a good time. He was especially keen on playing tricks on unsuspecting people. Immedi-ately after earning his doctorate in mathematics from the École Normale Supérieure, the young man was sent to India to become the chair of the department of mathematics of one of the state universities—a position customarily given to a senior member of the faculty, not a brash young Ph.D. transplanted from Europe. Weil was still in his early twenties, and the accolades went to his head.
One day in 1930, Weil suggested to his Indian friend Damodar D. Kosambi, who had just obtained his doctorate in mathematics from Harvard and returned to his home country, that he write a completely nonsensical paper and see if he could get it accepted by a professional mathematics journal. He also told him the stories about Bourbaki and Poldevia from his student days in Paris. Kosambi wrote some mathematical nonsense and titled the paper “On a Generalization of the Second Theorem of Bourbaki,” the inside joke being that the first theorem of Bourbaki had been written on the blackboard at ENS in Paris some years earlier, supposedly to be proved by the hapless entering students. In the introduction to the paper, Kosambi wrote that the theorem was attributed to “the little-known Russian mathematician D. Bourbaki, a member of the Academy of Sciences of Poldevia, who was poisoned during the Russian Revolution.” He and Weil had a good laugh, and then he sent it to the Bulletin of the Academy of Sciences of the Provinces of Agra and Oudh Allahabad for review. To their surprise the journal accepted the paper!
In this mood of irreverence, trickery, and contempt for prevailing academic systems, Weil returned to France some years later and, in 1933, took a teaching position at the University of Strasbourg. There he reunited with Henri Cartan, whom he had known during his student days in Paris. Cartan, now a fellow faculty member, kept complaining to Weil that he hated the textbooks they were expected to use in their classes, which were chosen by the Ministry of Education. Weil, with Cartan and with four other disgruntled young mathematicians from various French universities, decided to meet at a trendy café at the corner of the Boulevard Saint-Michel and rue Soufflot called A. Capoulade. Weil’s autobiography, The Apprenticeship of a Mathematician (1992), describes how t
his happened:
Several members of the Bourbaki group and their friends posed for this photograph in 1938. From left to right, they are: Simone Weil, Charles Pisot, André Weil, Jean Dieudonné, Claude Chabauty, Charles Ehresmann, and Jean Delsarte. Not present are Henri Cartan, René de Possel, and Claude Chevalley.
One winter day toward the end of 1934, I came upon a great idea that would put an end to these ceaseless interrogations by my comrade [Cartan]. “We are five or six friends,” I told him some time later, “who are in charge of the same mathematics curriculum at various universities. Let us all come together and regulate these matters once and for all. And after this I shall be delivered of these questions.” I was unaware of the fact that Bourbaki was born at that instant.1
Meeting at the café, and in spirit continuing Weil’s long string of pranks, the six friends—Henri Cartan, Jean Dieudonné, René de Possel, Claude Chevalley, Jean Delsarte, and André Weil—founded a secret group. They assumed the collective pseudonym Nicolas Bourbaki, and even faked a birth certificate for this individual. They also gave him a godfather: the prominent French mathematician Jacques Hadamard (1865–1963). They also pretended that Nicolas Bourbaki had a daughter, Betti, whose marriage they announced with printed wedding invitations. A nonexistent person thus became “real.”
HENRI POINCARÉ VERSUS
JACQUES HADAMARD
Henri Poincaré was undoubtedly one of the greatest mathematicians of his time, and as noted earlier, he made important contributions to many parts of mathematics. These include the invention of algebraic topology, discussed in his book Analysis Situs (1895). The celebrated Poincaré conjecture, mentioned earlier and proved by Grigori Perelman in 2003, is a statement in topology. It says that every closed three-dimensional manifold that is homotopy-equivalent to a three-dimensional sphere is, indeed, a sphere. The three-dimensional sphere is a generalization to one more spatial dimension of the usual (two-dimensional) surface of a ball (which is what we call a sphere). In simple terms, what the Poincaré conjecture says is that such a three-dimensional sphere is the only kind of bounded (i.e., not extending to infinity) three-dimensional object that has no holes in it (i.e., it is continuous, and has no tears or any kind of discontinuity in it).
Poincaré introduced the fundamental group in topology—an algebraic device for studying topological properties. His work was thus somewhat related to that of Felix Klein at Göttingen, who studied groups in geometry. The two men were, in fact, on friendly terms until Klein objected to Poincaré’s high opinion of the work of the Prussian mathematician Lazarus Fuchs. Poincaré worked on functions he called Fuchsian, since Fuchs had done some related work, but today we call these mathematical objects automorphic functions. Poincaré made a stunning connection between these functions and non-Euclidean geometry. The story Poincaré told in his book Science and Method (1908) about making his discovery is one of the most interesting tales about how some minds make mathematical discovery:
At that moment I left Caen, where I then lived, to take part in a geological expedition organized by the École des Mines. The circumstances of the journey made me forget my mathematical work. Arrived at Coutances, we boarded an omnibus for I don’t know what journey. At the moment when I put my foot on the step, the idea came to me, without anything in my previous thoughts having prepared me for it: that the transformations I had made use of to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify this, I did not have time for it, since scarcely had I sat down in the bus than I resumed the conversation already begun, but was entirely certain at once. On returning to Caen I verified the result at leisure to salve my conscience.
Poincaré worked in complex function theory, differential equations, probability, and mathematical physics. He also founded chaos theory when he discovered chaotic dynamics while working on the celebrated three-body problem. Poincaré seemed to favor intuition over rigorous mathematical derivations and proofs, and he wrote very popular books about science and mathematics.
Thus, although he was a truly great mathematician, the post–Great War generation of young French mathematicians—including the founders of the Bourbaki group—disapproved of his approach, viewing it as inexact and unrigorous. The Bourbaki mathematicians wanted as their role model a mathematician who stressed proofs and rigor; hence, they chose as their “godfather” a prominent French mathematician other than Poincaré (who had, in fact, died in 1912 at the age of fifty-eight). Not surprisingly, André Weil played the deciding role in this choice.
This photograph of the mathematician Henri Poincaré served as the frontispiece to the 1913 edition of his book Last Thoughts.
André Weil’s dissertation adviser at the École Normale Supérieure in Paris was the mathematician Jacques Hadamard (1865–1963). Hadamard was born in Versailles to a family of Jewish descent and received his doctorate from the École Normale Supérieure, where he later became a professor. In addition to Weil, Hadamard also directed the doctoral work of three other French mathematicians who would become well known in the field: Paul Lévy (one of the “fathers” of modern probability theory), Maurice Fréchet (who did key work in analysis, topology, and abstract spaces), and Szolem Mandelbrojt (who later became a member of the Bourbaki group and whose nephew, Benoit Mandelbrot, discovered fractals).
Hadamard’s work spanned many areas, including complex function theory, analysis, number theory, probability, functional analysis, the calculus of variations, differential equations, and differential geometry. The eponymous Hadamard inequality and Hadamard matrices attest to his important work in linear algebra. In number theory Hadamard proved a major theorem called the prime number theorem, which, as we recall, was proposed by Gauss and says that the number of prime numbers less than x tends to infinity at the same rate as the function x/lnx. This theorem was independently proved by another French mathematician, Charles de la Valée Poussin. The proof of this theorem had been attempted but not completed by Bernhard Riemann. In related work, Hadamard won the Grand Prix des Sciences Mathématiques for work associated with the very famous, still-open problem in mathematics called the Riemann hypothesis (mentioned earlier), which is a statement about the zeros of Riemann’s zeta function arising in number theory within the context of the complex plane.
In the 1890s a young French military officer of Jewish descent was falsely accused of treason and sentenced to life imprisonment on Devil’s Island in French Guiana. Hadamard, whose wife was a relative of the accused, Alfred Dreyfus, became personally involved in what would become known as the Dreyfus affair. He wrote in defense of Dreyfus—an act of courage that united him with Henri Poincaré, who had done the same—and became a leading proponent of the young officer’s innocence. Thanks in part to his efforts, the false charges were ultimately dismissed and Dreyfus was named an officer in the Legion of Honor in 1906. The celebrated French author Émile Zola was not as fortunate. After his criticism of the sham Dreyfus trial appeared in the press in the form of his now-famous article, J’Accuse, he fled France for Britain to avoid a jail sentence (which was later commuted).
Hadamard succeeded Poincaré as the chair of mathematics at the French Academy of Sciences and spent much time organizing and editing the mathematical work left behind by Poincaré after the mathematician’s untimely death in 1912. Four years later, at Verdun, during the height of World War I, Hadamard lost both his older sons to the war (a third son would die in World War II).
There is a little-known story involving Weil, Hadamard, and the famous French anthropologist Claude Lévi-Strauss. With the Nazi conquest of Paris in 1940, these three Jewish men found it extremely dangerous to stay in France (Weil had also deserted the French army), so they traveled to New York. Hadamard had a visiting professorship at Columbia University, and Lévi-Strauss was trying to make sense of the complicated marriage laws of the aboriginal Murngin tribe of northern Australia. At some point he came to the realization that the problem was highly mathematical, so he visited
Hadamard at Columbia and asked him for help. Hadamard listened to him sympathetically, and then replied, “Mathematics has four operations: addition, subtraction, multiplication, and division—marriage is not one of them.” Lévi-Strauss was disappointed by Hadamard’s response, but he didn’t give up. Some days later he found Weil, Hadamard’s former student. Weil studied the Murngin marriage rules the anthropologist showed him, and he found that, indeed, the problem was deeply mathematical and very complicated. (According to Murngin marriage laws, a man must marry one kind of cousin, if she exists, but is absolutely forbidden to marry a woman who happens to be another kind of cousin. Similar rules hold for women. This leads to the existence of sets of people within the tribe who, in turn, are either must-marry or taboo.) Weil was intrigued, and he ended up solving the problem (of determining the long-term structure of a society that follows these intricate marriage laws) using the abstract mathematics of group theory. It was an applied piece of work he remained very proud of throughout his life, even though he was otherwise a pure mathematician.2
The six young men of Bourbaki—among the best mathematicians of France at that time—continued pursuing practical jokes, and would also use them to promote their goals. But they also had a serious common purpose: to overthrow the stagnant educational regime of the time. They wanted mathematical education to be completely revamped and all the old textbooks thrown away. And beyond this, they had a loftier and far more ambitious goal: to redo all of mathematics.
Nicolas Bourbaki began publishing mathematical papers and textbooks, which were intended to replace the old and ineffectual ones. These were all high-quality publications, and the group, which grew over time and in later years included the noted French mathematicians Jean-Pierre Serre and Pierre Cartier, held regular meetings in French resort towns. Bourbaki became so convinced of “his” existence that he once wrote a letter to the American Mathematical Association requesting membership. But the secretary of the AMA at the time, Ralph P. Boas, was no fool. He wrote back saying, “I understand that this is not an application for membership from an individual,” adding that if Mr. Bourbaki wanted to become a member of the American Mathematical Association, he would have to reapply as an association (and pay the much higher membership fee!).3
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