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Bourbaki: A Secret Society of Mathematicians

Maurice Mashaal
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Already when I was a student in the 1970s and 1980s it was clear to me, largely due to the attitudes of my professors, that Bourbaki was a lot like grand opera: either you love it or you hate it (or, indeed, can’t abide it in the least). Perhaps the analogy is even better if we restrict our attention to Wagner, seeing that, perhaps like Bourbaki, he sought to revolutionize and modernize a vast genre and experts differ sharply on whether the result was a net win or a net loss. For what it’s worth, just as I love grand opera and have long adored the music of Wagner, I have always enthusiastically supported Bourbaki’s philosophy and its opera omnia (in statu quo).

On the other hand, for the last decade or so my musical taste has, shall we say, evolved, in the sense that as far as opera goes, for me, the center of gravity has shifted from Beyreuth to La Scala and, parallel to this, as I’ve had more and more occasion to study works by members of Bourbaki, I’ve found myself subjected to an irresistible pull in the opposite direction, away from ”the Spirit of Bourbaki.” I propose that my experience is really very widespread and perhaps even qualifies as a natural reaction to what Bourbaki intended at its inception and carried out so successfully.

It all started when André Weil (1906–1998) and Henri Cartan (1904– ), as young academics in French provincial universities and recent graduates of l'École Normale Supérieure (ENS), agreed that it would be a very good thing indeed if the teaching of analysis at their universities — and, by analytic continuation, throughout France — were uniformized or normalized. (In homage to the men I’m writing about I’ll continue to pun egregiously, and unapologetically — when the occasion arises!) Presently a quintette of founders of Bourbaki, Weil and Cartan, abetted by Claude Chevalley (1909–1984), Jean Dieudonné (1906–1992) and Jean Delsarte (1903–1968), all Normaliens, got their project off the ground, replete with sets of rules governing election to membership, the frequency of meetings (and the way these should be conducted), and a mandatory retirement age.

Bourbaki was born on December 10, 1934, in a café in Paris’ Latin Quarter, and, while its output and activity may have slowed over recent years, it’s still around now. Several generations of French mathematicians (and fellow travelers, e.g. Samuel Eilenberg (1913–1998)) have been members of Bourbaki — with their work stamped accordingly — but many others have in fact opposed the Society, usually on strong ideological grounds. But the list of prominent members of Bourbaki includes Jean-Pierre Serre, Laurent Schwartz, Jean-Cristophe Yoccoz, Alain Connes, and Alexander Grothendieck: all winners of the Fields Medal. This said, it’s “clear as vodka” (what movie?) that the Society succeeded spectacularly in its charter of raising French mathematical scholarship to incomparable heights. The famous Séminaire Bourbaki (discussed at some length in Mashaal’s book) takes a lot of the credit for this, of course, given the magnificent forum it supplies to its participants for the dissemination of avant garde mathematics.

What, then, could be said against Bourbaki? Simply put, albeit in the form of another (rhetorical) question: is there such a thing as too much abstraction? To many, the answer is an emphatic yes! Consider, e.g., Paul Halmos’ 1953 report on the first four chapters of Bourbaki’s book on integration: “[the answer to] whether the book’s perspective was the one that would best help a student understand the material and broaden his interests would be no” (Mashaal, p. 59). Or consider Edwin Hewitt (1956): “The presentation is austere and monolithic. A huge number of definitions are strewn throughout the book, and many are left unmotivated. There is a continuous stream of exercises that are tiresome to complete. The reader must be prepared to refer constantly to the author’s other works.” Additionally, as far as research is concerned the area of logic and foundations is notoriously absent from Bourbaki’s agenda (and always has been), as is the case for e.g. mathematical physics. But such critiques are of course well-known by now and do not detract from the appraisal that Bourbaki’s influence is nothing if not sweeping and dramatic.

All the preceding (mixed?) motifs are faithfully represented in Maurice Mashaal’s eminently readable book — especially so if you enjoy mathematical gossip, innuendo even, and canular — and there’s a lot more there, much of a very diverting variety. There are marvelous biographical essays (the one on André Weil being my favorite, though the material on Claude Chevalley is also irresistible); a fine account of the actual historical figure, Nicholas Bourbaki, including a discussion of a putative “visit” by the man himself; a series of polemical discussions concerning the roles played and influence wielded by Bourbaki; and (last and certainly least) a sequence of ten short sections, presented in bijective correspondence to ten books authored by Bourbaki (“Set Theory” to “Spectral Theory”), attempting to explain to outsiders what some of the respective mathematics is about. More than a Herculean task, this last undertaking is perhaps best characterized as misbegotten: those who need it won’t get it, those who get it won’t need it. But this is certainly eclipsed by the wealth of highly appealing stuff in Mashaal’s book, and a simply ridiculous quantity of fantastic pictures (to paraphrase Littlewood [or was it Pólya] concerning pictures of G. H. Hardy (“Pólya [or was it Halmos?] has more than exist…”), Mashaal includes more pictures of X than exist, where X Є {Weil, Chevalley, Grothendieck, Dieudonné, &c.}. It’s great fun.

The reader is particularly urged to check out p.79: is Pierre Cartier framing a question, or is he about to jump to an unwarranted conclusion?

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

  • A group forms
  • The story of a name
  • Young turks against stubborn priests
  • Bourbaki's \textit{Éléments de Mathématique}
  • Towards axioms and structures
  • A snapshot of Bourbaki's work: Filters
  • The Bourbaki seminar
  • Subtle and austere schoolboys
  • "For the honor of the human spirit"
  • New math in the classroom
  • An immortal mathematician?
  • Acknowledgments
  • Bibliography
  • Captions
  • Supplementary material