II~'~lvi|,[~]i,~.~|[.-~-nl[,~.],,n,,,,.,nit[:;-1 Marjorie Senechal, Editor ] Nicolas Bourbaki, 1935-???? vided for self-renewal: from time to If you are a mathematician working to- time, younger mathematicians were in- The Continuing day, you have almost certainly been in- vited to join and older members re- fluenced by Bourbaki, at least in style signed, in accordance with mandatory and spirit, and perhaps to a greater ex- "retirement" at age fifty. Now Bourbaki Silence of tent than you realize. But if you are a itself is nearly twenty years older than student, you may never have heard of any of its members. The long-running it, him, them. What or who is, or was, Bourbaki seminar is still alive and well Bourbaki- Bourbaki? and living in Paris, but the voice of Check as many as apply. Bourbakl Bourbaki itself--as expressed through An Interview with is, or was, as the case may be: its books--has been silent for fifteen years. Will it speak again? Can it speak 9the discoverer (or inventor, if you again? prefer) of the notion of a mathemat- Pierre Cartier, Pierre Cartier was a member of ical structure; Bourbaki from 1955 to 1983. Born in 9one of the great abstractionist move- June 18, 1997 Sedan, France in 1932, he graduated ments of the twentieth century; from the l~cole Normale Supdrieure in 9 a small but enormously influential Marjorie Senechal Paris, where he studied under Henri community of mathematicians; Caftan. His thesis, defended in 1958, 9 acollective that hasn't published for was on algebraic geometry; since then fifteen years. This column is a forum for discussion he has contributed to many areas of of mathematical communities The answer is: all of the above, and mathematics, including number theory, they are four closely woven strands of group theory, probability, and mathe- throughout the world, and through all an important chapter in intellectual matical physics. Professor Cartier time. Our definition of "mathematical history. Is it time to write that chapter? taught at Strasbourg for a decade be- community" is the broadest. We include Has the story of Bourbaki come to an ginning in 1961, after which he joined "schools" of mathematics, circles of end? CNRS, the Centre National de la Bourbaki was born in Paris in 1935 Recherche Scientifique. Since 1971 he correspondence, mathematical societies, when a small group of mathematicians has been a professor at IHES (Institut student organizations, and informal at the l~cole Normale Sup~rieure, dis- des Hautes l~tudes Scientifiques) at communities of cardinality greater satisfied with the courses they were Bures-sur Yvette, and has taught at the than one. What we say about the teaching, decided to reformulate them. Most mathematicians have had that ex- communities is just as unrestricted. perience at one time or another, but We welcome contributions from the scope of Bourbaki's dissatisfaction mathematicians of all kinds and in grew quickly and without bound. By all places, and also from scientists, 1939, writing as an anonymous collec- tive under the pseudonym Nicolas historians, anthropologists, and others. Bourbaki, it began to publish a series of books intended to transform the the- ory and practice of mathematics itself. From its beginning, Bourbaki was a fervent believer in the unity and uni- versality of mathematics, and dedi- cated itself to demonstrating both by Please send all submissions to the recasting all of mathematics into a uni- Mathematical Communities Editor, fied whole. Its goals were total for- Marjorie Senechal, Department malization and perfect rigor. In the of Mathematics, Smith College, post-war years, Bourbaki metamor- Northampton, MA 01063, USA; phosed from rebel to establishment. e-maih [email protected] Bourbaki's own rules explicitly pro- Pierre Cartier (photo by Marjorie Senechal). 22 THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER-VERLAG NEW YORK l~cole Polytechnique and at the Ecole in the books, what was reported in the this was so; I don't think this is the Normale, where among other activities seminar, and the work of the students place to discuss it again. But obviously he runs a seminar on epistemology. In were closely linked, and I think that is in the fifties, the early fifties, the teach- 1979 he was awarded the Ampere Prize one of the reasons for the great suc- ing of science was very poor. It took of the French Academy of Sciences. cess of French mathematics at that Bourbaki about five or six years to Professor Cartier has been involved in time. Of course, those times were very subvert the whole system. By 1957 or various programs to help developing different. The scale was much smaller. '58 the subversion had been almost countries, including Chile, Vietnam, and Then there were about ten doctorates complete, in Paris. India, build science at home; he is also a year in mathematics in France (com- Senechal: But Bourbaki began in the an editor of a book about art and math- pared to three hundred today). thirties... ematics. Few people are better qualified At that first meeting I was what they Cartier: The first book was published to discuss the silence of Bourbaki. We call a cobaye, a guinea pig. I was very in 1939, but there was the war, which are grateful to him for agreeing to do so enthusiastic about it. First of all, it was delayed things, and also Andr5 Weil with the readers of The Mathematical the first thing in modern mathematics was in the States, Claude Chevalley Intelligence~: that I saw. I came from a small city, was in the States, and Lanrent from a difficult situation because of Schwartz had to hide during the war The |nte~iew the war. I had been a student in a very because he is a Jew. Bourbaki survived Senechah Please tell us first about provincial, very outdated high school. during the war with only Henri Caftan your own connection to Bourbaki. Some of my teachers were very good and Jean Dieudonn~. But all the work Cartier: As far as I remember, my first but of course they were very far away that had been done in the thirties blos- acquaintance with Bourbaki was in from modern science. The mathemat- somed in the fifties. I remember how June 1951. I was a first-year student at ics I was taught was classical geome- we--the young mathematicians--were the ]~cole Normale, Henri Cartan was try, in the uncultivated, synthetic way. really eager to go to the bookstore to my professor of mathematics there, I did have the luck to have an imagi- buy the new books. And at that time and at his request Bourbaki invited me native teacher in physics, and so at first Bourbaki published at least one or two to join their meeting at Pelvoux, in the I wanted to by a physicist. Then I was volumes every year. Alps. I remember that we discussed a student at the Lyc~e Saint-Louis in When I formally became a member many things, especially a text written Paris before being accepted at the of Bourbaki in 1955, I had to abide by by Laurent Schwartz on the founda- Ecole Normale, and I took private the rule that everyone should leave at tions of Lie groups; it was one of the lessons in physics from a very peculiar 50, and so I left in 1983, when I was al- first drafts in the well-known series of teacher, Pierre Aigrain. (A graduate of most 51. I devoted almost 30 years of Bourbaki on Lie groups. It was not the Naval Academy, he was in 1950 an my life, and at least one third of my many years after Schwartz's invention assistant professor of physics; eventu- work, to Bourbaki. The working habits of distributions, which made him fa- ally he became Secretary of State for of Bourbaki involved very many pre- mous. You have to understand that the science under President Giscard.) liminary drafts of a book before it was mathematics students at l~cole Normale Usually a bright student completes the published. At the time, we had three were all students of both Henri Cag2~ program in two years, but I managed meetings a year, one week in the fall, and Laurent Schwartz (who left Nancy to get through it in one. But both the one week in the spring, and two weeks for Paris in 1952). We attended their mathematics and the physics I was in the summer, which is already one seminars and courses and tried to use taught were totally outmoded at that month of hard work, ten or twelve hours their new tools in all directions. time, totally. I remember that, in a a day. The published books comprised Francois Bruhat and I were among the course called General Physics at the about 10,000 pages, which means ap- first to understand the importance of Sorbonne, the professor made a proximately 1000 to 2000 pages of pre- distributions in the theory of Lie groups solemn declaration: "Gentlemen"--he liminary reports and drafts written every and their representations. Bruhat de- did not mention ladies but there were year. I estimate that I contributed about voted his thesis to these topics and I very few girl students--"in my class 200 pages a year during all this time with published my own contributions only what some people call the 'atomic hy- Bourbaki. much later. pothesis' has no place." That was 1950, Senechal: How many people be- For me, it was very important to be five years after Hiroshima! So I went longed, at that time? exposed from the inside.