Monday, August 25, 2008

Henri Cartan, RIP

Henri Cartan, the grand old man of French mathematics, has died at 104. He was the founder and leader of the "Bourbaki" movement in the 1930s, a group that tried to redo mathematics as a totally formalized and axiomatized system, following deep intellectual traditions in France of hyper-rationalism that go back to Descartes, if not all the way to Thomas Aquinas, and which in French socio-economics was associated with the social engineering idealism of Saint-Simon and the dirigiste tradition of French indicative central planning. The group operated, at least initially, anonymously, signing their papers with the name of a dead French general, Nicholas Bourbaki. They hoped to clarify mathematics and remove from it any mysticism or fuzzy thinking.

One of Cartan's most assiduous students in the immediate post-WW II period was the late Gerard Debreu, the more mathematical and austere half of Arrow and Debreu, who famously proved the existence of competitive general equilibrium in 1954, both of them eventually receiving the Nobel Prize in economics. Debreu was probably the most articulate and influential spokesman for axiomatizing economic theory, something now very much out of favor for many reasons. Indeed, as Roy Weintraub has noted in his book, _How Economics Became a Mathematical Science_, the trends of uses of math in economics tend to repeat trends in math itself, with the Bourbakist approach having fallen out of fashion within math itself some time ago. Simulation and many other approaches have picked up, and we see this in economics with agent-based modeling and more emphasis on empirics and induction from that now.

Even though Cartan and his allies dominated French mathematics for decades, two of the most interesting mathematicians to come out of France, Rene Thom, the founder of catastrophe theory, and Benoit Mandelbrot, the inventor/discoverer of fractals, were both very bad at proving theorems and following the Bourbakist approach. Indeed their great success, including with influence on economics, was perhaps a sign of the limits of Bourbakism more generally.


Robert Vienneau said...

One aspect of Bourbaki puzzles me. In some ways, they seem to implement Hilbert's program. Yet they did after Godel had showed that Hilbert's program could not work, in some sense. Is this a fair comment? said...


This is a fair comment, and one I have always wondered about. Of course they were aware of Godel's work, which had been published in 1931, before they got going. It would seem that they were making an effort to go as far as they could in laying out clear, axiomatic systems that did not clearly contain any inconsistencies, a kind of "let us see what we can do in spite of Goedel" approach.

There was also an ideological element, that became stronger after WW II, namely a push for something completely abstract and independent of any ideology or mystical viewpoint. One can argue if they succeeded or not, although many consider Arrow-Debreu general equilibrium theory to be an inherently ideological system.

Anonymous said...

V. I. Arnol'd hated the Bourbaki trend in French mathematics, because he thought that it was too divorced from physics, which is the root of mathematics:

In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).

Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the universal hate towards mathematicians - both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users.

As a pure mathematician who defected to applied math for grad school, I have to say that I feel some sympathy towards this point of view. But, on the other hand, all the physicists and engineers in my program were miserable theorem-provers, so it's not clear where the line between "not enough" and "too much" abstraction is. said...


Yes, that line is difficult. V.I. Arnol'd is a dynamicist as is Mandelbrot and was Thom, both of whom were rebels against Bourbakism, or at least did not follow it. Thom's Theorem was proposed by him, but actually proven by someone else. Much of Mandelbrot's work has involved discovering objects, such as the very fractal Mandelbrot set, but not proving much about it, leaving that to others.

Arnol'd has proven lots of stuff, but his spirit is closer to those of Thom and Mandelbrot. I know the Bourbakists did much work in analysis and differential equations, but it seems that some of the more interesting people specializing in those matters tended to be less Bourbakist.