Bourbaki’s many achievements include laying the foundation for the “new math” approach instituted in the United States at midcentury. In a sense Bourbaki was able to redo mathematics, as “he” had intended, restructuring the basics of the discipline. Its foundation was set theory—the discipline initiated by Georg Cantor a few decades earlier.
The series of books published by Nicolas Bourbaki was called Elements of Mathematics, recalling Euclid’s Elements—the first, ancient Greek foundation of mathematics. The Bourbaki books recast much of mathematics in a new structural sense that demands logic and strict proof of all results. In helping to make mathematics more rigorous, more precise, and more proof-based, Bourbaki essentially tore down an edifice and built it back up from scratch. Of course, this was the view of Bourbaki and its adherents. Other mathematicians downplay its importance.
Bourbaki’s Théorie des Ensembles (Theory of Sets), volume 1 of their six-volume series Elements of Mathematics, was first published in France in 1938. This photograph shows the cover of the first edition. Note that Bourbaki uses the singular word “mathématique”—not “mathématiques”—to emphasize the unity of mathematics.
After one of the Bourbaki summer meetings in 1939, André Weil became worried about the prospects of war in Europe. Technically an officer in the French army since his graduation from the university, Weil would have been expected to remain in France to await his call-up. But in an effort to avoid military service, he abruptly left for Finland with his wife, Éveline, and spent some time by a lake not far from the Russian border. People became suspicious, and eventually a police inquiry was made. The police found in his possession documents bearing the name of Nicolas Bourbaki, including invitations for Betti Bourbaki’s wedding. They assumed he was a spy. Éveline returned home, and Weil’s wartime misadventures began, including an arrest, a deportation to France, imprisonment, and conscription as private (after he was stripped of his officer rank for leaving). He then deserted from the front and escaped to the United States. Thereafter, Bourbaki’s leader would remain based in America, at the University of Chicago and later at Princeton, but he continued to attend meetings in France.4
The Bourbaki organization still exists, but its influence is now diminished. There is still a Bourbaki Seminar in Paris that meets regularly, and technically there is still a membership in the organization, which includes about forty people. But by the 1960s, the Bourbaki group had done much of what it had set out to do—it had revolutionized the mathematics curriculum and published its Elements. It was now time for new ideas in mathematics.5
ALEXANDER GROTHENDIECK
While the Bourbaki group was most active, Weil’s leadership would often be challenged by an unusual individual: the immensely brilliant mathematician Alexander Grothendieck (b. 1928), a one-time member of the secret group but more often a lone genius who changed much of modern mathematics and redid the theory of algebraic geometry.
What we know about Grothendieck, who lives in an unknown location, comes mostly from his ruminative autobiography, Recoltes et Semailles (Harvesting and Sowing), which includes writings on mathematics and other topics. The book has not been published, but was circulated in manuscript form among a number of his friends. In early 2010 Grothendieck sent a letter from his hiding place to a friend in Paris, demanding that all his writings be removed from circulation. Consequently, a Web site called the Grothendieck Circle, maintained by some of the mathematician’s admirers, pulled all electronic copies of the manuscript out of cyberspace, along with other writings by Grothendieck. This move was only the latest in the bizarre story of the most mysterious mathematician of our time.
Very few other writings on the life of Grothendieck exist, but the most important is a paper by a member of Bourbaki who knew him well. Therefore, much of the information on Grothendieck in this chapter is drawn from the article “A Mad Day’s Work: From Grothendieck to Connes and Kontsevich—The Evolution of Concepts of Space and Symmetry” by Pierre Cartier, which is based almost exclusively on what Grothendieck had recounted to Cartier.6 Other sources include articles by mathematics professor Winfried Scharlau7 and American Mathematical Society editor Allyn Jackson.8
Alexander Grothendieck’s father, Alexander “Sacha” Shapiro, was born on October 11, 1889, in a town near the meeting point of the borders of Russia, Belarus, and Ukraine. It was a region within the larger “pale of settlement” of Imperial Russia, where Jews were allowed to live. This smaller Russian-speaking borderland has changed hands over time, and in it a large Jewish community flourished until the Second World War, when most of its members were exterminated.
Cartier claims, based on Grothendieck’s recollections, that all Shapiros (regardless of variations in the spelling of the name) came from this limited geographical region, and that Shapiro’s father, Grothendieck’s paternal grandfather, belonged to the Hasidic community there. At some point, his son Sacha shed the restrictions of Orthodox Judaism and, politically, began to embrace revolutionary ideas.
Shapiro took part in the ill-fated revolt against the tsar in 1905, when he was only sixteen years old. He spent a dozen years in jail in Siberia and was released in 1917—just in time for the Russian Revolution against the same tsar, Nicholas II. In February of that year, the so-called Menshevik Revolution erupted in Saint Petersburg, and in October the Bolshevik Revolution began. After much bloodshed, the Russian monarchy was abolished, and Shapiro assumed a leadership position in the Socialist-Revolutionary Party of the Left. At first his party was allied with the Bol-sheviks, but it soon went its own way and became opposed to Lenin, who then took his revenge on its members. During this turbulent period, Shapiro’s actions as a revolutionary leader earned him a place in John Reed’s famous book about the Russian Revolution, Ten Days That Shook the World.
Mathematician Alexander Grothendieck’s father, Sacha Shapiro, took part in the Russian Revolution of 1917, when street protests like this one—in which a Bolshevik regiment marches through the streets of St. Petersburg—were a familiar sight. Shapiro was a well-known leader of a leftist political party at the time.
Like these female soldiers, Alexander Grothendieck’s mother, Hanka, fought with the Republican forces against Francisco Franco in the Spanish Civil War.
According to Cartier, Shapiro went on to take part in the Béla Kun revolution in Hungary, followed by uprisings in Berlin and Munich. He also joined anarcho-communist guerrilla leader Nestor Makhno’s Black Army in Ukraine against the Red Bolsheviks and the White Tsarists. He seemed to go wherever revolutions were erupting, but Russian authorities were bent on capturing him. He was arrested once again by the Russian police, and this time condemned to death. Just before his execution, he managed to escape from prison but as a result lost an arm. As a wanted person, he had no choice but to flee Russia in the middle of the night. He did so with the help of a woman named Lia, who, with him, slipped across the border into Poland. Sacha Shapiro, a one-armed person on the run, changed his name to Alexander Taranoff. For the rest of his life, he would be a stateless person.
Alexander and Lia crossed borders as if they didn’t exist, hiding in France, Belgium, and Germany. Shapiro/Taranoff became an ardent anarchist and had a string of relationships with women. While still a Russian revolutionary, he had secretly married a Jewish woman named Rachel and fathered a son, Dodek. In Western Europe he had other relationships while living with Lia. At some point the couple moved to Paris, where they stayed for two years.
Through his anarchist connections, Sacha met many people who were involved in the social upheavals of the time. He left Lia and traveled to Germany, where he became deeply involved with the local anarchists and met a young fellow activist named Hanka Grothendieck. Born in Hamburg in 1900, she wrote for a local newspaper about the German sex trade, interviewing girls and women who were being exploited. Sacha and Hanka moved in together, even though she was then married to a journalist named Alf Raddatz and had a daughter by him. (Her husband was continually traveling in
connection with his anarchist activities, and the couple had separated.) On March 28, 1928, Sacha and Hanka had a son they named Alexander. Since they were, in fact, married to other people, and since Sacha was living under an assumed name, the baby was given his mother’s married name, Raddatz, which was changed to Grothendieck after her divorce.
Sacha and Hanka’s time together did not last long. When Alexander was only five years old, his father had to leave. Hitler had come to power in Germany, and an anarchist Jew like Sacha was doubly threatened. Thus Sacha escaped to Paris, where he worked as a street photographer. For a one-armed man, this was a difficult job, and he barely made a living. After some time, Hanka placed her son in foster care at the home of a Lutheran pastor named Wilhelm Heydorn, who was fervently anti-Nazi. Her daughter was placed in an institution, and she joined Sacha in Paris.
In 1936—the year of the Spanish Civil War—no anarchist in Europe could resist the urge to join the Spanish Republicans in their war against Generalissimo Francisco Franco. Sacha and Hanka were among thousands who went to fight. When Franco won, thousands of Republican volunteers again infiltrated the border across the Pyrenees back into France. Many of them were caught as they crossed and sent to internment camps in the foothills of the mountains, but Hanka and Sacha managed to avoid capture. Sacha found his way back to Paris, while Hanka fled to the southern French city of Nîmes, where she found temporary employment as a teacher until she rejoined Sacha in Paris. All through this time, Alexander received no visits from his mother’s many relatives in Germany, and no news about his parents.
In 1939 the Heydorns decided it was too dangerous for them to continue keeping a boy they thought “looked Jewish,” so just as World War II began, eleven-year-old Alexander was sent alone by train to Paris, where he was reunited with both his parents. But again, alas, their time together did not last long. French authorities were bent on arresting anar-chists and revolutionaries, whom they viewed as a danger to the security of the country, and eventually the family was caught. Sacha was then sent to the notoriously brutal camp Vernet, near the Pyrenees, and from there to Auschwitz, where he died in 1942. Hanka Grothendieck and her son were moved from camp to camp. Shortly after World War II was over, she died from tuberculosis contracted at the camps.
ALEXANDER GROTHENDIECK, a child living with his mother in hunger and privation in wartime concentration camps, taught himself mathematics. Children in the camps were often provided with some education, but it was sporadic and of generally low quality. It is therefore extremely surprising that a boy who grew up in such dire circumstances rediscovered by himself the entire mathematical theory of measure.
When he visited Paris in 1947, Alexander was told by the professional mathematicians he met at the university that the theory he had developed so amazingly well all by himself had already been discovered in 1905 by the French mathematician Henri Lebesgue. It was a major accomplishment by an untrained young genius, but it was only the first of many.
At war’s end, Grothendieck was finally able to pursue his education at the university and catch up on the mathematics others had learned in school. Possessing a vision in mathematics that is unparalleled, he quickly caught up and surpassed everyone else. At advanced seminars in Paris later on, the young maverick would often ask provocative questions that older and far more experienced mathematicians could not answer. It became clear to many excellent mathematicians in Paris—the center of the world of mathematics—that Grothendieck was a rising leader in the field.
This photograph of the mysterious and elusive mathematician Alexander Grothendieck was taken in Montreal in 1970.
Grothendieck developed a strong interest in topological groups. These are the groups of abstract algebra that have a topological structure, meaning that they admit notions of continuity, as do lines, planes, and surfaces. In order to learn about them, he attended a lecture by Charles Ehresmann, a member of the Bourbaki group. Ehresmann was a recognized world expert on topological groups, but the young and naive Grothendieck did not know that. Right after the talk, he rushed to the speaker and asked, “Are you an expert on topological groups?” Ehresmann, a modest man, replied, “Well, I know some-thing about topological groups.” Grothendieck, turning to walk away, said, “But I need a real expert in this field!”9
Grothendieck received his formal education in mathematics at the University of Montpellier in southern France. As Cartier describes it, he was an outstanding student at the university: “It has often been said that he was ahead of his teachers and that he was already exhibiting a taste for extreme generalization in mathematics.”10 In 1948, having obtained his degree, Grothendieck arrived in Paris with a letter of recommendation from one of his professors in his hand; it was addressed to the great French geometer Élie Cartan. By then Cartan was quite old and in poor health, so Grothendieck was taken under the care of Élie’s son, Henri Cartan, a member of the Bourbaki group. Eventually, Grothendieck’s doctoral adviser would be Laurent Schwartz, another Bourbaki member, who, coincidentally, had visited the camp at which Grothendieck was interned during the war and gave lectures to the few students there. Under Schwartz, Grothendieck wrote a brilliant dissertation on topological tensor products for a doctorate from the University of Nancy. It was published in 1955.
Grothendieck went on to develop many new concepts in mathematics, including the idea of a sheaf and many other topics in algebraic geometry. He also changed the way we view space, as Cartier explains:
Our definition of points in space goes back directly to the ancient Greeks. Euclid defined a point as something that has no extent at all. In modern particle physics, for example, an elementary particle such as the electron is a “point particle,” with no extent or internal structure, although it has mass. In the seventeenth century, Leibniz extended this ancient Greek idea to things that were either physical or spiritual. To Leibniz, the basis of the universe of all things—material or not—was the monad, a “windowless” element that has no internal structure. Leibniz considered the mutual relationships among these monads to hold the secret to all structures of creation.11
Cartier then invokes Bourbaki’s work on the foundation of mathematics, quoting what he calls “his” (i.e., Bourbaki’s) definition of set theory:
A set is composed of elements capable of having certain properties and having certain relations among themselves or with elements of other sets.
These points, or elements, are preexisting, and their combinations and mutual relations create structure in the physical universe, in Leibniz’s spiritual realm, and in pure mathematics. The idea that lies at the very heart of the effort by the Greeks, followed by Descartes, Newton, Leibniz, and then by Bourbaki, is to capture the most essential properties of the physical universe and to abstract them and turn them into the basic elements of all of mathematics. It is an immensely deep and ambitious idea.
A point in Greek geometry is the intersection of two lines, and a line is the intersection of two planes. The great nineteenth-century German mathematician Bernhard Riemann, who, as mentioned earlier, created Riemannian geometry (later used by Einstein, but also extremely important in pure mathematics), proposed the idea of a surface that is stacked over a plane. Grothendieck was inspired by this idea and went a step further: he replaced the system of “open sets” used in topology—basically extensions of the idea of open intervals of numbers, such as the set of all points between 0 and 1—into something new. He defined spaces stacked over a given space.
Grothendieck then proposed an even more abstract concept: he defined a topos as the ultimate generalization of a space. This idea transcended the usual mathematics of the time by creating something far more abstract and general. Grothendieck allowed his topos to solve a major problem in mathematics: the nonexistence of a set containing all sets. In fact, the category of all sets constitutes a topos. In the words of Cartier, “Grothendieck claimed the right to transcribe mathematics into any topos whatever.” This was the work, and a reflection of the immense co
nfidence, of a grand master.
Grothendieck met many members of Bourbaki, and even joined the group for a time, but he worked much better in a milieu where he was the leader; so he moved to an institution the French founded almost exclusively for him: the Institut des Hautes Études Scientifiques (IHES), outside Paris. There, in the late 1950s and 1960s, he almost single-handedly redid the theory of algebraic geometry.
Descartes had first wed algebra with geometry in the seventeenth century, showing how equations could be precisely associated with geo-metric figures using his Cartesian coordinate system. In showing that each equation really describes a trajectory in space, Descartes laid the foundation for the field of algebraic geometry. Grothendieck extended this nascent field much further than anyone could imagine by generalizing the idea of an algebraic variety—i.e., the set of solutions to a system of equations—into something he called a scheme, based on an idea given to him by Pierre Cartier. The theory of schemes developed by Grothendieck weds arithmetic with geometry, realizing the century-old dream of such a unification by Cantor’s nemesis, Leopold Kronecker, who once suggested this union might someday be possible.
Grothendieck is a man who thinks in great generalities and doesn’t seem to care at all about details. During a lecture he once gave, in which he was making an argument based in part on prime numbers, a listener raised his hand to ask him a question.
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