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Book Review Bourbaki, A Secret Society of Mathematicians and The Artist and the Mathematician Reviewed by

Bourbaki, A Secret Society of Mathematicians Now that we are in the twenty-first century it is Maurice Mashaal perhaps the right time to look back and try to as- AMS, June 2006 sess the overall impact of Bourbaki, before all the US$29.00, 260 pages principal players leave the scene. The basic histori- ISBN-13: 978-0821839676 cal facts are well known and are set out in both

the books under review. France had lost a whole The Artist and the Mathematician: The Story generation of intellectuals in the 1914–18 war, and of Nicolas Bourbaki, the Genius Mathematician the young mathematicians of , in the inter- Who Never Existed war period of the 1920s and 1930s, were looking Amir D. Aczel for new guidance and inspiration. Only Hadamard Thunder’s Mouth Press, August 2006 and Élie Cartan of the older generation still com- US$23.95, 272 pages manded respect. Talented youth, unconstrained by ISBN-13: 978-1560259312 higher authority, is a powerful force and, whatever one’s views about Bourbaki, there is no doubt that All mathematicians of my generation, and the talent was quite exceptional. The list of the even those of subsequent decades, were aware of early members of Bourbaki is truly impressive: Nicolas Bourbaki, the Napoleonic general whose André Weil, , , Jean reincarnation as a radical of young French Dieudonné, … Later recruits were mathematicians was to make such a mark on the of similar calibre: Jean-Pierre Serre, , mathematical world. His memory may now have Alexandre Grothendieck… Harnessing the powers faded, the books are old and yellowed, but his of such a formidable group was not an easy task. influence lives on. Many of us were enthusias- There were fierce debates, some serious quarrels, tic disciples of Bourbaki, believing that he had and much passion. The remarkable fact is that the reinvigorated the mathematics of the twentieth group, by and large, stayed together and kept Bour- century and given it direction. But others believed baki alive and active over several decades. This was that Bourbaki’s influence had been pernicious a tribute to the idealistic vision that they shared, and narrow, confining mathematics behind walls that of remoulding the shape of mathematics in of , and cutting off its external sources of the twentieth century. inspiration. Much of the atmosphere of the early days is brought vividly to life by the many informal pho- Sir Michael Atiyah is Honorary Professor of Mathemat- tographs in the Mashaal book. It is fascinating to ics at the University of Edinburgh. His email address is see pictures of the young André Weil, relaxing [email protected]. in a deck chair, though Henri Cartan was always

1150 Notices of the AMS Volume 54, Number 9 impeccably dressed in jacket and tie, resisting is not responsible for the excesses trendy fashion. perpetrated in the name of Chris- I myself attended a Bourbaki conference in my tianity. youth and can attest to the lively experience of Bourbaki was to some extent debating vigorously (and usually critically) the the victim of its own success. latest version of the next book. Summer sunshine The original aim had been the in the south of France and the friendly and casual modest one of writing a mod- atmosphere did much to prevent arguments devel- ern replacement for Goursat’s oping into armed conflict. To paraphrase Winston Cours d’Analyse but, buoyed up Churchill, “never in the course of human argument by enthusiasm and the success has so much been spoken by so many on so little.” of recruiting many of the leading It appeared a miracle that books, many of them, ac- mathematicians of the time, hori- tually emerged from this process, a result undoubt- zons broadened. All of mathemat- edly due to the diligence and energy of Dieudonné. ics was to be included, analysis, al- If Weil was the prime inspiration behind Bourbaki, gebra, and . For obvious it was Dieudonné who carried it to fruition. reasons algebra lent itself best So what were the basic aims of Bourbaki, and to the Bourbaki treatment. The how much was achieved? Perhaps one can pick out volumes on and particularly two central objectives. One was that mathematics on Lie groups were excellent and became standard needed new and broad foundations, embodied references, due in large part to the personal contri- in a of books that would replace the old- bution of Serre, whose influence and taste guided fashioned textbooks. The other was that the key this whole area. idea of the new foundations lay in the notion of The formal aspects of analysis, as exemplified “structure”, illustrated by the now common word in functional analysis, also had success, though Bourbaki’s treatment of came in for “isomorphism”. severe criticism from the experts who argued that There is no doubt that, with its clear emphasis important parts of the theory were excluded by the on “structure”, Bourbaki produced the right idea restriction to locally compact spaces. A concern for at the right time and changed the way most of us elegance had led to too great a price being paid. thought. Of course it fitted in well with Hilbert’s But this little battle over probability was a mere approach to mathematics and the subsequent sideshow in the Bourbaki approach to analysis, a development of . But structure subject too varied, complex, and untidy to be taken was not confined to algebra, and it was particu- over by Bourbaki. Glimmerings of these problems larly fruitful in and associated areas of already appear in differential geometry, a subject geometry, all of which were to see spectacular at the interface between analysis and geometry, developments in the period following World where structure, though present, is a less dominat- War II. Here the impact of Bourbaki was decisive, ing concept. Though Riemann surface theory, after and, in the hands of Serre and Grothendieck, alge- a century of active development, could conceiv- braic geometry rose to incredible heights. ably be given a coherent Bourbaki treatment, the Laying universal foundations is another mat- same could hardly be said for the current work of ter. Each time it is tried it inevitably gets bogged Thurston-Perelman in three dimensions. Another down by the sheer scale and ambition of the opera- severe limitation of Bourbaki, no doubt conscious, tion. The “ne plus ultra” in this direction was the was the restriction to . Applied Éléments de Géométrie Algébrique of Grothendieck mathematics is too messy and disparate to be and Dieudonné, which expanded voluminously included, and theoretical physics hovers on an both forward and backward and was in danger of uncertain borderline. One distinguishing feature sinking under its own weight. of Bourbaki was the emphasis on clear and unam- Laying ambitious foundations is not only a dan- biguous definitions and on rigorous proofs. This gerous delusion, it can also be a didactic disaster. was, as in algebraic geometry, a reaction against Encyclopaedias are not textbooks, and much of some sloppy treatments of the past, and it served the critique directed against Bourbaki is that it a purpose in creating a firm platform for the fu- was used, or perhaps misused, to reform school ture. Unfortunately, when taken to extremes, the education. This may be unfair, since many of the requirement for total rigour excludes large areas great mathematicians in Bourbaki were excellent of mathematics which are in their early creative lecturers and knew well the difference between stages. Had Euler worried too much about rigour, formal exposition and the conveying of ideas. But, mathematics would have suffered. as so often happens, the disciples are more extreme Over the past thirty years, arguably in the declin- and fanatical than their masters, and education in ing years of Bourbaki, some of the most exciting France and elsewhere suffered from a dogmatic developments in mathematics have arisen from the and ill-informed attempt at reform. Jesus Christ interface with physics and particularly quantum

October 2007 Notices of the AMS 1151 theory. New concepts and the entire scene from mathematics to the social explicit results have emerged sciences. The only place where I can examine the from this interaction, notably evidence for this and make an informed comment, Donaldson’s work on four- is in his treatment of mathematics and the people manifolds, mirror symmetry in in it. Here I have profound misgivings, which relate algebraic geometry, and quan- mainly to Grothendieck, who occupies a central tum cohomology. Much of this place in the author’s pantheon. came directly from very heu- There is no doubt that Grothendieck was an ristic work by physicists such exceptional figure in the mathematical world and as Edward Witten. Most of it, that he deserves a scholarly full-length biography, though by no means all, has preferably written by a mathematician who knew now been given a cloak of re- him personally. I believe such a book is in prepara- spectability involving rigorous tion, and I look forward to reading it. Aczel’s book proofs. does not up to the level of the subject, Clarity and rigour have a vital because of his uncritical acceptance of Grothen- place in mathematics but they dieck as the great prophet, spurned eventually by must not be used as a barrier his people (including Bourbaki). to new ideas from other fields. I knew Grothendieck well when he was in his Free trade is a benefit to us all and should not prime. I greatly admired his mathematics, his pro- be inhibited by excessive attachment to national digious energy and drive, and his generosity with sovereignty. ideas, which attracted a horde of disciples. But his Although Bourbaki recruited most of the fa- main characteristic, both in his mathematics and mous French mathematicians of the time (and sev- in social life, was his uncompromising nature. This eral from outside France), there were some notable was, at the same time, the cause both of his suc- exceptions, the most obvious being cess and of his downfall. No one but Grothendieck (who left very early) and René Thom. In retrospect could have taken on algebraic geometry in the full it is clear that neither fitted the Bourbaki role. The generality he adopted and seen it through to suc- fact that they were also two of the most original cess. It required courage, even daring, total self- mathematicians of the time does perhaps suggest confidence and immense powers of concentration that such originality has difficulty flourishing in a and hard work. Grothendieck was a phenomenon. constrained atmosphere. Both were also closer to But he had his weaknesses. He could navigate than their colleagues. like no one else in the stratosphere, but he was Of the two books under review, the first by Mau- not sure of his ground on earth—examples did rice Mashaal might be described as “authorized”. not appeal to him and had to be supplied by his It has the sanction of the AMS and was first pub- colleagues. lished several years ago in French. It seems clear Aczel is right when he identifies Grothendieck that the author knew many of the French math- as someone who took the new Bourbaki philoso- ematicians personally and derived his information phy seriously and made a tremendous success and in particular the photographs from this source. of it. Where I part company with Aczel is in his It is reliable on the history, the personalities, and assertion that Bourbaki made a fatal mistake in the mathematics. It is also highly readable and not taking Grothendieck’s advice and rewriting noncontroversial. its foundations in the new language of category The other book by is totally dif- theory. Aczel believes that Bourbaki had turned ferent. It has a more ambitious aim, which is to its face away from the future in not following examine the Bourbaki influence on “structure” in Grothendieck. I doubt whether history will come the social sciences. It is also highly controversial in to this verdict. Grothendieck’s own EGA, as well its extensive treatment of the Grothendieck story. as the general fate of over-confident universalists, I was not convinced of the total reliability of its might suggest otherwise. Moreover, given Grothen- sources, nor of its philosophical credentials. dieck’s uncompromising nature and supreme self- Although written in English this book is perme- confidence, it is difficult to see how, with him at ated by French intellectual ideas and will probably the helm, Bourbaki could have continued as a col- seem strange to those not part of that scene. A legial enterprise. slightly tenuous link between André Weil and the Aczel’s total endorsement of Grothendieck sociologist Claude Levi-Strauss is used to claim leads him to make such fatuous statements as: that Bourbaki made a major impact on sociology “Weil was a somewhat jealous person who clearly and related fields such as , anthropol- saw that Grothendieck was a far better mathemati- ogy, and linguistics. This grand aim is clearly cian than he was.” Subtle balanced judgement is set out by the title, and I have no expertise in clearly not Aczel’s forte, and it hardly encourages any of these fields. It may be that the author is a the reader to take seriously his confident and polymath, an intellectual colossus, who straddles sweeping assertions in the social sciences.

1152 Notices of the AMS Volume 54, Number 9