Web edition: August 29, 2008
In the 1930s, a group of young French mathematicians led an uprising that revolutionized mathematics. France had lost most of a generation in the First World War, so the emerging hotshots in mathematics had few elders to look up to. And when these radicals did look up, they didn’t like what they saw. The practice of mathematics at the time was dry, scattered and muddled, they believed, in need of reinvention and invigoration.
So they took up arms: pens and typewriters. Using the nom de plume “Nicolas Bourbaki” (after a dead Napoleonic general), they wrote a series of textbooks laying out mathematics the right way. Though the young mathematicians started out only intending to write a good textbook for analysis (essentially an advanced form of calculus), they ended up creating dozens of volumes which formed a manifesto for a new philosophy of mathematics.
The last of the founders of Bourbaki, Henri Cartan, died August 13 at age 104. In addition to his work in Bourbaki, Cartan made groundbreaking contributions to a wide array of mathematical fields, including complex analysis, algebraic topology and homological algebra. He received the Wolf Prize in 1980, one of the highest honors in mathematics, for his work on the theory of analytic functions. Two of his students won the Fields medal, sometimes considered equivalent to the Nobel Prize in mathematics, one won the Nobel Prize in physics and another won the economics Nobel.
“He had a talent for seeing what was most interesting in mathematics at a given time,” says Luc Illusie of Paris South University, a former student of Cartan. “He changed the way we were thinking about mathematics.” Christian Houzel of Paris North University calls Cartan “one of the greatest mathematicians of the 20th century.”
Cartan — son of another eminent mathematician, Élie Cartan — met many of his future Bourbakian cofounders while a student at the Ecole Normale Supérieure in Paris and the rest just a few years later. The renegades were too young to have proven many big theorems when they began the project in 1934, but they developed into some of the greatest mathematicians of their time: Cartan, André Weil, Jean Dieudonné, Szolem Mandelbrojt, Claude Chevalley, René de Possel, Jean Coulomb, Charles Ehresmann and Jean Delsarte.
Their discussions appeared to be “a gathering of madmen,” Dieudonné remarked. When Armand Borel first visited the group in 1949, he described “two or three monologues shouted at top voice, seemingly independently of one another.” The group refused to appoint a leader and insisted on unanimous decision-making. Drafts were revised time and time again, often even simply thrown out. “Why such a cumbersome process did converge was somewhat of a mystery even to the founding members,” Borel said.
According to Borel, “For us H. Cartan was the most striking illustration, almost an incarnation, of Bourbaki.”
Bourbaki’s guiding principles were extreme rigor, abstractness and unification. Because mathematics is so broad, researchers tend to specialize within a particular subfield. The result is that different subfields often developed similar ideas and results without realizing it, using different notation or approaches. “Bourbaki knew enough mathematics to try to unify this,” says Jean-Pierre Serre of the Collège de France. They sought the most powerful, direct, economical expression of each theorem, stated in the greatest generality so that it would be useful to as many subfields as possible.
The result was austere books with almost no examples, guide for intuition or pictures. Philip Davis of Brown University described them in an article in SIAM News as “mathematics with all its juices extracted; bare bones, skeletonic, anorexic stuff; Twiggy dressed in the tunic of Euclid.” Michael Atiyah of the University of Edinburgh says: “They’re not designed to be read. They’re designed to set out a thesis for how mathematics ought to be done.” That thesis had tremendous influence.
In the 1950s, Ralph Boas of Northwestern University wrote an article for the Encyclopaedia Britannica on Bourbaki, explaining that it was the pseudonym for a consortium of French mathematicians. The editors of the encyclopedia soon received a scalding letter signed by Nicolas Bourbaki himself, declaring that he would not allow anyone to question his right to exist. In revenge, Bourbaki began spreading the rumor that Ralph Boas himself didn’t exist, and that B.O.A.S was an acronym of a group of American mathematicians.
The group also wrote a fanciful article describing Bourbaki’s origins. Bourbaki himself was a Russian mathematician, they claimed, who had done brilliant work but fallen on hard times. The young French mathematicians had chanced to meet him in Paris, realized they had a common cause and embarked on the joint publishing project with him.
“Nicolas Bourbaki, who became somewhat of a misanthropist following his misfortunes, refuses to see anyone except the collaborators he has chosen for himself,” they wrote. “This gave rise to the legend that he is merely a pseudonym, but anyone who has met him knows the strength and vitality of his extraordinary personality, to which even his collaborators are prone to ascribe somewhat mysterious powers. This is what causes, in the midst of discussions in which Bourbaki does not even take part, all the present members to simultaneously have sudden flashes of inspiration that show them the solutions to particularly thorny problems whose solutions had been elusive until then.”
Cartan himself retired from Bourbaki after he turned 50, as the group required. Its members believed that as mathematicians aged, they were less able to adopt new ideas and might hold back progress.
Cartan began to work to protect dissidents, particularly mathematicians in the Soviet Union. He collected the signatures of more than a thousand mathematicians advocating for the release of the mathematician Leonid Plyushch from a “special” psychiatric hospital in Russia. Plyushch’s release in 1976, after two years, was seen as a major success for Cartan’s work.
For photos of Henri Cartan: