Tuesday, June 05, 2007

Doing a number on Bourbaki

Religion minus insanity equals epsilon

It helped that the book was short. Otherwise, I might never have made it to the end of Amir Aczel's The Artist and the Mathematician. Although it's short, it's also repetitive, which served to make it boring, too.

A pity, because there's room for a book like this, a pseudo-biography of a pseudo-mathematician. You can't dabble in math very long before you meet a mathematician who never existed. “Nicholas Bourbaki” was the pen name of a group of iconoclastic mathematicians, most of them French, who were surprised by the success of their pseudonymous attempt to refashion mathematics. The Bourbaki name appeared on a series of influential books, each one a tactical maneuver to advance the overarching plan to rebuild mathematics on a rigorous foundation of set theory and logic. The vision of the Bourbaki group came to represent a broad mainstream current in modern mathematics, although most would agree that its influence has waned as the founding members of Bourbaki have departed from the scene. The group still exists, but as a shadow of its former self.

Aczel narrates a few well-known anecdotes about Bourbaki and adds a handful of interesting details, but a better source on the nonexistent mathematician is Armand Borel's memoir, published in the Notices of the American Mathematical Society or the detailed account at PlanetMath.org. Of somewhat greater interest to me than the hashed-over Bourbaki story was Aczel's disorganized attempt to discuss Lévi-Strauss's structuralism in the context of Bourbaki's formalism.

I've discussed this famous anthropologist before and have neither qualification nor inclination to evaluate his seminal work on kinship or his establishment of structural anthropology. I have, however, opined that some of his quasi-mathematical formulations seem more of a mésalliance of symbols and symbolism than a rigorous formalization (as his literal formulas would imply). Aczel tells us that Lévi-Strauss was influenced in his approach to anthropology by an encounter with André Weil, a founding member of Bourbaki.
In 1943, Claude Lévi-Strauss met André Weil in New York, and the exchange of ideas between them was what eventually led to Bourbaki's ideas being introduced into anthropology. The first important example of the use of mathematical principles in anthropology was to be the solution of the difficult marriage-rules problem in tribes of Australian aborigines studied by Lévi-Strauss. This problem was solved by André Weil using purely abstract algebraic methods. Weil was so proud of his mathematical solution of Lévi-Strauss's problem, and the connection forged between “pure” mathematics and applied science, that he continued to tell the story about this cooperation between practitioners of different fields until his death.
In brief, Lévi-Strauss's fascination with structure—and his development of structuralism—sprang from the abstract structures of abstract algebra. Perhaps Aczel is right. Certainly the collaboration between Lévi-Strauss and Weil was fruitful for the former and delightful for the latter.

Bourbaki now bestrode the mathematical world like a colossus, so a proper concern for the needs of melodrama required the production of an arch-rival. To heighten the thrill, it would be ideal if the nemesis were from within. History dutifully provided this person, who figures as Aczel's hero in his account of Bourbaki. Enter Alexander Grothendieck, a citizenless World War II waif whose burgeoning mathematical genius brought him to the attention of the Bourbaki group. Grothendieck spent approximately ten years in Bourbaki, until he decamped amidst a welter of disagreement and dissatisfaction. While Bourbaki's ranks were replenished with some of his students, Grothendieck pursued his interests in new mathematical directions and his life moved into ever more isolated eccentricity. And bizarre religious beliefs.

Grothendieck's descent into a deeply wacky religiosity is one of the most puzzling things about a man who was capable of grasping the concept of proof at its most fundamental level. Why would a mathematician embrace fanciful faith? We know that there are plenty of smart people—even professional scientists—who see no apparent contradiction in mixing a career of reason with a personal life of dogmatic belief. It depends in part on the scope of one's initial axioms. If you decide that you want the most parsimonious set possible of initial assumptions, you might embrace a doctrine of proof: I'll believe it if it can be demonstrated. If you're willing to expand the axiomatic base by adjoining the culturally popular acceptance of deity and revelation (you get to pick which book is the True book, of course, out of many rivals), then you might end up with the point of view that many people have: I'll believe it if it's in the Book. (This is also known in bumper-sticker theology as “God said it. I believe it. That settles it!”)
On January 10, 1988, Grothendieck retired officially from all his positions. In August 1991 he suddenly left his home and disappeared into the Pyrenees, where he is believed to be hiding from the world. Several people have attempted to find him over the years, but he remains elusive. People who managed to see him have reported that he is obsessed with the devil. Grothendieck believes that the devil is constantly working to destroy the harmony of the world. Among the many bad things the devil has done are the creation of pollution, the destruction of the environment, and the promotion of war and destruction.

When he was last seen ten years ago, Grothendieck was obsessed with the meter. Throughout his life, he shunned physics and related sciences, seeing in them instruments of evil: physics brought us the nuclear bomb and other bad things. In his isolation, Grothendieck apparently began to contemplate the meter, and he came to the conclusion that here, too, the devil is at play. The devil, he believes, has maliciously replaced the nice whole number 300,000 km/second by the ugly number 299,887 km/second as the speed of light.
Where, exactly, did God announce that nature's constants were supposed to be round numbers? I forget.

Some people think that Grothendieck has really lost it (if, indeed, he is even still alive). His theory of the devil, however, is no worse than anyone else's. I mean, we can't ask him to prove it, so it's a matter of faith. He at least chose nuclear weapons as evidence of the devil's wickedness as opposed to, say, evolution or homosexuality or women in the workplace. Yes, Grothendieck's demons are as good as anyone else's. Perhaps better than most. You can call it insanity, but it's really much more charitable to call it religion.

1 comment:

John Armstrong said...

Grothendieck was indeed still alive a few years ago. One visitor to our department told us of a friend of his who was a documentary filmmaker. This filmmaker heard of Grothendieck through our visitor and decided to seek him out. He actually managed to track down Grothendieck's whereabouts and drove up to his farmhouse. Grothendieck was on the porch, shotgun in hand, as the filmmaker drove up. When the man stepped from his pickup truck, Grothendieck started firing until the man scrambled back into the truck and peeled out.