Departed Glories: BourbakiJanuary 12, 2007
Henri Cartan, a founding member of Bourbaki and one of its most active participants. Photo courtesy of the Russian Academy of Sciences, of which Cartan was a foreign member, and Wikipedia.
Philip J. Davis
The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. By Amir D. Aczel, Thunder's Mouth Press, New York, 2006, 239 pages, $23.95.
At the end of World War II, returning to graduate school to resume my studies and to earn some bread as a "section man" (now known as a TA), I found the mathematical back room abuzz with two topics. The first was the recently published Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern, which, according to the boys in the back room, was going to turn economics on its ear. The second was a series of publications of a certain Nicolas Bourbaki, of which a similar assertion was made for mathematics and mathematical education.
Totally missing from the buzz were the emerging digital computers (tubes, relays, punch cards, etc.) to which von Neumann lent his genius and his authority and which, a half century later, have turned the whole civilized and not so civilized worlds on their ears. One may rightly ask both of game theory and of Bourbakism: What has been the payoff (and I don't mean in the crass sense)? As to the payoff from digital computers, it has been completely engulfing and absolutely staggering. So much for the wisdom of the back room.
But on to Bourbaki. My older colleagues handed around a few of the Bourbaki "fascicules." When I took a look, I was appalled at what I saw: mathematics with all its juices extracted; bare bones, skeletonic, anorexic stuff; Twiggy dressed in the tunic of Euclid. Thankfully, in my graduate years, I was not asked to teach integration by parts à la Monsieur B., nor did any of my courses, taught by men who were his contemporaries, reveal any of the hallmarks of Bourbakism.
Bourbaki had little use for applications or computation, and because that is where over the years my heart and talent lay, I've wasted little love on him. Now, despite my basic antipathy, I must report that Amir Aczel, author of many books on mathematical issues, has written a rip-roaring biography of B., a page-turner and a wonderful read that I can enthusiastically recommend. With very warm feelings toward Bourbakism, Aczel has presented us with a vibrant parade of personalities, ideas, and aspirations that marched through the mathematical landscape for a half century. Truth, they say, is far more interesting than fiction, and this book trumps the math-fi, hoo-hah stuff now produced in great quantities by the mathematical writing community.
It is now seven decades since Bourbaki emerged on the mathematical horizon, and a bit of explanation may be useful for readers who don't recognize the name. Bourbaki is the name of a group of largely French mathematicians, molded into a secret or semi-secret elite society, for the purposes of promulgating a certain "revolutionary" view of what mathematics is all about. The group dates from the end of 1934, when André Weil corralled a bunch of sympathetic souls into a café on the Boulevard Saint-Michel. Weil suggested that the group work out a replacement for the course of calculus/analysis typified by the approach in Goursat's Cours d'Analyse Mathématique, which all present agreed was not only old hat but absolutely stultifying.
From the rejection of Goursat, the group went forward at a remarkable clip---with a considerable hiatus caused by WWII---rewriting material, creating new material, and all the while fostering a characteristic program. This program consisted of axiomatization, extreme rigor, abstraction, super-abstraction, and the attempt to unify mathematics via the identification of certain fundamental deductive structures. With verve and enthusiasm and not a little fanfaronade, the group went from strength to strength, issuing publication after publication (fascicules by the hundreds, it seems), holding conferences galore, and picking up as it rolled onward converts and camp followers among those who had come to see the light.
The group was fluid, with members dropping in, dropping out after quarrels, or leaving mandatorily after passing the age of "creativity." As far as the veil of secrecy adopted by the group, I make no attempt to psychologize but merely note that some of the antics undertaken to preserve the identity of individual members and to try to give flesh to Monsieur Bourbaki (Bourbaki is the name of a valiant French general) strike me as sophomoric.
Aczel has organized his story around a series of biographies of members of the fuzzy set known as Bourbaki. We learn of the careers of André Weil, Laurent Schwartz, Henri Cartan, Claude Chevalley, Jean Delsarte, and Jean Dieudonné, among numerous others. The stories of individual lives during the Hitler period are often gripping but painful to read and contemplate. But biography-wise, Aczel has given pride of place, and space, to the career of Alexandre Grothendieck, a late joiner, who some have shortlisted as the most creative mathematician of the second half of the 20th century. (Lots of competition there, I should think.)
We read of Grothendieck's difficult childhood, his genius, his mathematical accomplishments, and his eminently sensible, if furious, political stance in an age of madness. Ultimately, Grothendieck dropped out not only of Bourbaki but also of the larger mathematical community, retiring as an inaccessible hermit in the Pyrenees Mountains. One often hears that mathematics, especially pure mathematics, can be a psychological refuge from the realities of the world, but apparently in Grothendieck's case it did not suffice.
In explanatory sallies or sidelines, Aczel links the Bourbaki program of abstraction with the rise of abstract art. He links the group's program of unification via deductive structures with the contemporary structuralist movements in linguistics, as exemplified by the ideas of Claude Levi-Strauss, Roman Jakobson, and Roland Barthes, and of the structuralist movement that went on to invade the fields of psychology, psychiatry, and economics. Was abstraction wedded to structuralism simply the Zeitgeist of the '50s, or was there something deeper and less mysterious? Would it be fatuous to suggest that French intellectuals inherited a tendency toward systematization, standardization, and strict organization under unified control that can be traced back to the Encyclopaedists of the Enlightenment?
Bourbaki is no longer alive; its tenets are now located on the palette of available postures that mathematicians might adopt as their personality and occasion warrant. Why did Bourbaki die? From my very limited and somewhat maverick position, I'm glad that it did. I have never believed in the unity of mathematics or in the necessity of providing mathematics with so-called foundations. Previous attempts to do so, whether through logicism, set theory, structuralism, or category theory, have all proved inadequate to epitomize or sum up the rich contents of the intellectual creations and the widespread practices known as mathematics. The interested reader will find an elaborate postmortem in Aczel.
A thought about the legacy of Bourbaki. Among what is left is the strong spotlight directed on such structures as Lie groups or categories, and the contributions of the members as individuals. But there is a downside. Aczel, like the sundial that records only sunny hours, has skipped over what I consider the pernicious effects of Bourbaki. In France, where Bourbaki ruled the roost for a while, applied mathematics received a sharp slap in the face; more than that: a body blow. In the USA, the New Math, located in the high clouds of abstraction, drew its breath of life from Bourbaki. After expending the energies of enthusiasts and spending tens of millions in cash, after abusing the patience of teachers, parents, and a goodly proportion of professional mathematicians, the New Math was finally acknowledged to be an abject and unmitigated failure. (And the failure was predictable, according to some.)
The moral to the rise and fall of Bourbaki, to the extent that there is one, is simple and obvious to those who have read the history of mathematics. No one individual, no one group, no one idea, no one ethnicity can singly bestride and dominate the world of mathematics. This is the sense in which I interpret the oft repeated saying of Georg Cantor that the essence of mathematics lies in its freedom.
I conclude with a quote from the eminent American philosopher William James:
"We feel the pathos of lost causes and misguided epochs even while we applaud what overcame them."---Social Value of the College Bred, 1908.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at firstname.lastname@example.org.