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 , 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, 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 ; 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 , including , they are four closely woven strands of 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, 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. I was sur- to Aigrain and said, "What do I do?" Cartier: About 12. It was always a prised to see all these great people I and he said, "Well, of course, you have small, well-delimited group. The semi- had known from a distance. I was ac- to get your degree, but I will teach you nar was different, much more open. cepted very freely. It took three or four physics properly." This shows what the But still, in the 1950s, if you look at the more years before I was formally ac- French university was at the time. In table of contents of the seminar vol- cepted as a member. In the fifties and order to understand the influence of umes, about half the papers were writ- sixties, there was a continuous spec- Bourbaki, you have to understand that. ten by members of Bourbaki; in those trum from the inside core Bourbaki to Bourbaki came into a vacuum. Many days the interaction between the sem- the outside. The work that was printed people have discussed the reasons why inar and the group was very strong.

VOLUME 20, NUMBER 1, 1998 23 Now that's no longer true: it's still a dis- They were the ones to reshuffle mathe- less a group of students of Grothendieck. tinguished series but it's usually written matics. The second generation had al- But at that time Grothendieck had al- by people who have no direct connec- ready been exposed to the new teach- ready left Bourbaki. He belonged to tion with the institution Bourbaki. But ing. My generation, the third generation, Bourbaki for about ten years but he left at that time people published in the did not have to prove that the new in anger. The personalities were very seminar series part of their discoveries, method was better than the old one be- strong at the time. I remember there or preliminary accounts of Bourbaki's cause we were taught with the new were clashes very often. There was ideas that later appeared in the books. method basically. I think I was just on also, as usual, a fight of generations, I was typically a member of the the borderline, because in high school I like in any family. I think a small group third generation. You can say that was still taught in the old method, but like that repeated more or less the psy- there have been four. The first genera- when I went to Paris I was exposed to chological features of a family. So we tion were the fathers: Andr6 Weil, the new thinldng. And so we were less had clashes between generations, Henri Cartan, Claude Chevalley, Jean and less dogmatic, because we didn't clashes between brothers, and so on. Delsarte, and Jean Dieudonn6, people have to prove anything. The core of But they did not distract Bourbaki who founded the group in the thirties. French mathematics had surrendered from his main goal, even though they (Others joined in the beginning, but left to Bourbaki. Bourbaki had already were quite brutal occasionally. At least soon.) Then there was a second gen- seized power, not only in intellectual the goal was clear. There were a few eration, people invited to join people who could not take the during or just after the war: burden of this psychological Laurent Schwartz, Jean-Pierre You can think of the first books of style, for instance Grothendieck Serre, Pierre Samuel, Jean-Louis Bourbaki as an encyclopedia. If you left and also Lang dropped out. Senechal: Did the goals stay Koszul, , Roger consider it as a textbook, it's a Godement, and Sammy clear in people's minds all the Eilenberg. The third generation disaster. time, or were they changing? was Armand Borel, Alexandre Cartier: They changed. The Grothendieck, Franqois Bruhat, first generation had first to cre- myself, , and John Tate. terms but also in academic terms. It ate a project from nothing. They had Senechal: Did these generations dif- was clear that from an institutional to invent a method. Then in the forties fer in their attitudes or outlook? point of view, Bourbaki had won. you can say that the method had Cartier: They were very different. I If you look at the volumes on Lie emerged and Bourbaki knew where to think they became more and more groups, you will see that the later ones go: his goal was to provide the foun- pragmatic, and less and less dogmatic. have chapters that you don't expect in dation for mathematics. They had to Senechal: And how did that show up Bourbaki. It became more and more submit all mathematics to the in Bourbaki's work? explicit; there are tables and drawings. of Hilbert; what van der Waerden had Cartier: From the beginning, the I think this was basically the influence done for algebra would have to be Bourbaki treatise was conceived as of one person, Armand Borel. He was done for the rest of mathematics. What comprising two parts. The first part is fond of quoting Shaw, "It's the Swiss should be included was more or less on foundations and consists of six national character, my dear lady," and clear. The first six books of Bourbaki books, on set theory, algebra, general very often during a discussion he comprise the basic background knowl- topology, elementary calculus, topo- would say, "I'm the Swiss peasant." edge of a modern graduate student. logical vector spaces, and (Lebesgue's) Of course at that time differential The misunderstanding was that integration theory. The last four of geometry was blossoming, and it had al- many people thought that it should be these books give the foundations of ways been a great challenge to Bourbald. taught the way it was written in the analysis, as perceived by Bourbaki, You have to remember that the father of books. You can think of the first books with a strong bias toward functional Henri Caftan was Elie Caftan, the of Bourbaki as an encyclopedia of analysis. The second part, falling short geometer, and the Bourbaki recognized mathematics, containing all the neces- of more ambitious projects, consists of only one godfather, r Caftan, and had sary information. That is a good de- two very successful series, on Lie much dislike for all the other French scription. If you consider it as a text- groups and on . mathematicians of the thirties. Bourbaki book, it's a disaster. Looking back at the list of the Bourbaki came to terms with Poincar6 only after Senechal: Were you aware of that members of the second and third gen- a long struggle. When I joined the group when you were a member of Bourbaki? erations, you realize that some of the in the fifties it was not the fashion to Did people in Bourbaki realize that world's leading experts of the time were value Poincar6 at all. He was old-fash- this was not a textbook? there, and that accounts for the breadth ioned. Of course, the opinion about Cartier: More or less, but not so and depth of the second part of Poincar6 has completely changed. But clearly as now. There was some mis- Bourbaki's work. it's clear that his style and Bourbaki's understanding about that, I suppose The older generation had learned style were totally different. because we didn't have textbooks. I re- mathematics in the old-fashioned way. The fourth generation was more or member very well how I learned alge-

24 THE MATHEMATICAL INTELLIGENCER bra and topology. When I was a stu- In the second generation and third and there is a small volume that tried dent, every time that Bourbaki pub- generation, the two main series were to bridge the gap between probability lished a new book, I would just buy it commutative algebra (with algebraic theory and the way that Bourbaki pre- or borrow it from the library, and learn geometry in the background) on the sented Lebesgue integration theory. it. For me, for people in my generation, one hand, and Lie groups on the other That was an attempt to correct one ob- it was a textbook. But the misunder- hand. And there is an obvious differ- viously mistaken point of view of standing was that it should be a text- ence of style and of emphasis, despite Bourbaki. book for everybody. That was the big the fact that at that time Bourbaki was Senechal: What other areas of math- disaster. really a collective and everyone con- ematics do you see now as having Anyway, by then the scope of the tributed to every book, more or less. been left outside? project was more or less clear. But Serre was a master of both sides; he Cartier: First of all analysis, although what should Bourbaki do after that? was not an expert in Lie groups at first there is an elementary calculus text, a The second generation had an existing but he became one. Serre was the nat- very good book, that was the influence method, and had just to develop a pro- ural leader in the second generation of Jean Delsarte. There is essentially no ject with clearly delineated bound- because, like Weil in the first genera- analysis beyond the foundations: noth- aries. The third generation had to go tion, he was the only one with a really ing about partial differential equations, beyond that, to go into the open world, universal approach to mathematics. nothing about probability. There is also which meant, at that time, geometry in But neither of them was an analyst. nothing about combinatorics, nothing a general way: algebraic geometry, dif- Certainly the contents of Bourbaki about algebraic topology, nothing about ferential geometry, several complex were much more about algebra, alge- concrete geometry. And Bourbaki never variables, Lie groups, moduli spaces, braic geometry, than about analysis. seriously considered logic. Dieudonn~ and so on. By the fourth generation the goal himself was very vocal against logic. I think I'm responsible for the idea was less visible. Grothendieck had de- Anything connected with mathe- that Bourbald should devote a special veloped his own program, outside of matical physics is totally absent from chapter to the geometry of crystallo- Bourbaki, so the need for a Bourbaki Bourbaki's text. In the Bourbaki semi- graphic groups. The reasons for that are was less obvious. And there was also nar, I contributed a long series of pa- clearly stated in the introduction to the some lack of a global understanding of pers with emphasis on questions of series on Lie groups. Coxeter was the mathematics. The members had be- mathematical physics. But I was the first to understand the relation of Lie come more specialized in their inter- only one, and my contributions were groups to the crystallographic groups ests. not always accepted without a fight. and their classification. Certainly the There were various attempts within But even in the areas of mathemat- people who were working on Lie the group to focus on new projects. For ics that were not considered by groups were, by spirit, more geometri- instance, for awhile the idea was that Bourbaki, looking backwards over the cal and more pragmatic than the oth- you should develop the theory of sev- last thirty years, it is obvious that their ers. But I remember that I had to fight eral complex variables, and many development has been very much in- quite hard to convince my colleagues drafts were written. But it never ma- fluenced by the Bourbaki spirit. within Bourbaki that crystallographic tured, I think partly because it was too Senechal: Was there a bias against groups should be given preeminence. late. There were already many good physics, or did Bourbaki just not Senechal: What was Bourbaki's opin- textbooks on several complex vari- think about it? ion of Coxeter? ables in the seventies, by Grauert and Cartier: Well, of course there was a Cartier: I think that by the sixties peo- other people. By the end of the seven- strong bias against it, for most people. ple realized the importance of his ties, the method of Bourbaki had been At the beginning I suppose I was work. Borel had many of the same so well understood that everyone slightly heterodox within the Bourbaki ideas and Jacques Tits also played a knew how to write in this spirit. There group. I had a longstanding interest in role. Tits was much closer in spirit, in was a whole generation of textbooks, mathematical physics. A few years ago, his way of doing mathematics, to and books, which were under his in- in a discussion with Andr~ Weil, just Coxeter than to Bourbaki. He wasn't fluence. Bourbaki was left without a after he published his own memoirs, I formally a member of Bourbaki but he task, and so he decided to devote part said, "You mentioned that in 1926 you had a long collaboration with us. So we of his energy to revising his own books, were at G(ittingen ... in 1926 some- could thank him, in the books, for his the so-called "New Edition." The revi- thing happened in GSttingen." And collaboration without breaking the sion was mostly completed; these were Weil asked, "What did happen in rule of anonymity. Tits was very gen- really thorough revisions. GSttingen?" and I said "Oh! Quantum erous: he supplied us with many of the Senechal: Do the revisions include a mechanics!" And Weil said, "I don't exercises, and many of his results were change of style? know what that is." He was a student published for the first time in Bourbaki Cartier: No, no. But for instance, the of Hilbert in 1926 and Hilbert himself volumes. But of course he had a very section on the topology of metric was interested in quantum mechanics, different way of thinking about math- spaces was much more developed and Max Born was there, Heisenberg was ematics. deepened, the proofs were improved, there, and others, but apparently

VOLUME 20, NUMBER 1, 1998 25 philosophical foundation--that is ,the encyclopedic, or taxonomic part--and a very efficient mathematical tool, to be used in mathematical situations. That set theory and structures are, by con- trast, more rigid can be seen by reading the fmal chapter in Bourbaki set theory, with a monstrous endeavor to formulate categories without categories. It is amazing that category theory was more or less the brainchild of Bourbaki. The two founders were Eilenberg and MacLane. MacLane was never a member of Bourbaki, but Eilenberg was, and MacLane was close in spirit. The first textbook on homo- logical algebra was Cartan-Eilenberg, which was published when both were very active in Bourbaki. Let us also mention Grothendieck, who developed categories to a very large extent. I have been using categories in a conscious or unconscious way in much of my work, and so had most of the Bourbaki mem- bers. But because the way of thinking was too dogmatic, or at least the pre- sentation in the books was too dog- Andr~ Weil didn't pay any attention to "final edition," Bourbaki struggled in matic, Bourbaki could not accommo- it. I recently had an occasion to give a the seventies and the eighties to for- date a change of emphasis, once the public lecture about the philosophy of mulate new directions. I mentioned al- publication process was started. space of Hermann Weyl, so I read the ready a failed project about several I think the eighties were a natural literature about him carefully. There is complex variables. There were at- limit. Under the pressure of Andr~ an obituary of Hermann Weyl written tempts at homotopy theory, at spectral Weil, Bourbaki insisted that every by Chevalley and Well. They praise theory of operators, at the index theo- member should retire at fifty, and I re- him, for good reasons, but there is no rem, at symplectic geometry. But none member that, in my eighties, I said, as mention of his work in physics, not of these projects went beyond a pre- a joke, that Bourbaki should retire even his work in general relativity. By liminary stage. when he reaches fifty. all accounts, the two best books of Bourbaki could not find a new out- Senechal: It seems that this more or Weyl are his book on general relativity let, because they had a dogmatic view less happened. and his book on quantum mechanics! of mathematics: everything should be Cartier: Yes, I think one of the main Senechal: Bourbaki's last publication set inside a secure framework. That reasons is that its stated goal, to pro- was in 1983. Why doesn't it publish was quite reasonable for general topol- vide foundations for all existing math- anything now? ogy and general algebra, which were ematics, was achieved. But also, if you Cartier: There are several reasons for already solidified around 1950. Most have such a rigid format it is very dif- that. First, there was a clash between people agree now that you do need ficult to incorporate new develop- Bourbaki and his publisher, about roy- general foundations for mathematics, ments. If the emphasis doesn't change, alties and translation rights, ending in at least if you believe in the unity of it's still possible. But of course, after a long and unpleasant legal process. mathematics. But I believe now that fifty years, the emphasis had changed. When the matter was settled in 1980, this unity should be organic, while Senechal: Would you say a little more Bourbaki was allowed to make a deal Bourbaki advocated a structural point about that? with a new publisher. Using the ex- of view. Cartier: Andr~ Weil was fond of tensive work done in the seventies to- In accordance with Hilbert's views, speaking of the Zeitgeist, the spirit of wards the revision of the old books, we set theory was thought by Bourbaki to the times. It is no accident that were able to republish them in a new provide that badly needed general Bourbaki lasted from the beginning of edition. We completed the existing se- framework. If you need some logical the thirties to the eighties, while the ries by two or three more volumes, but foundations, categories are a more flex- Soviet system lasted from 1917 to 1989. then.., silence. ible tool than set theory. The point is Andr~ Weil does not like this compar- Beyond the easily stated goal of a that categories offer both a general ison. He says repeatedly, "I've never

26 THE MATHEMATICALINTELLIGENCER been a communist!" There is a joke the communists in the Soviet Union Lagrange proudly stated, in his text- that the 20th century lasted from were claiming the same. We know now book on mechanics, "You will not find Sarajevo 1914 to Sarajevo 1989. The it was a lie, and that the leaders knew any drawing in my book!" The analyti- 20th century, from 1917 to 1989, has at the time they were lying. Certainly cal spirit was part of the French tradi- been a century of ideology, the ideo- Bourbaki was not lying, but still, the tion and part of the German tradition. logical age. spirit was the same. It was the time of And I suppose it was also due to the Senechal: By ideology, do you mean ideology: Bourbaki was to be the New influence of people like Russell, who the idea of a blueprint that can serve Euclid, he would write a textbook for claimed that they could prove every- for all purposes and for all time? the next 2000 years. thing formally--that so-called geomet- Cartier: A final solution. There are Senechal: Why is there a lack of any rical intuition was not reliable in proof. good reasons to hate that expression, kind of visual illustration in most of Again Bourbaki's abstractions and but it was in the people's minds that Bourbaki? disdain for visualization were part of a we could reach a f'mai solution. There Cartier: I think the best answer would global fashion, as illustrated by the ab- is a book by H.G. Wells called A be the description of Chevalley given stract tendencies in the music and the Modern Utopia, which ought to be by his daughter [see insert]. The paintings of that period. reprinted. By chance I was reading it Bourbaki were Puritans, and Puritans Senechal: Did the members of just at the time of the collapse of the are strongly opposed to pictorial rep- Bourbaki appreciate abstract music Soviet system. As you know, H.G. resentations of truths of their faith. and abstract art? Wells was certainly very friendly to- The number of Protestants and Jews in Cartier: I don't think there was much wards the October 1917 revolution, he the Bourbaki group was overwhelm- taste for abstract music or art. You was a friend of the Soviets, admittedly. ing. And you know that the French could say that on the whole they had But he had a very sharp mind and he Protestants especially are very close to standard bourgeois tastes. Educated had such a sharp historical view that Jews in spirit. I have some Jewish bourgeois--not philistine. For in- he could envision developments. Even background and I was raised as a stance, both Cartan and Dieudonn~ though he was excited by this new era, Huguenot. We are people of the Bible, were lovers and practitioners of music, he understood that the final solution of the Old Testament, and many but they were very classical. Caftan doesn't exist and that it was a mistake Huguenots in France are more enam- certainly, in his Protestant education, to consider that you can reach such a ored of the Old Testament than of the was very fond of Bach, and Dieudonn~ state of social historical equilibrium New Testament. We worship Jaweh was quite a good piano player, at an that from then on society will stay as more than Jesus sometimes. amateur level, but quite good, and he it is forever. Wells argued very well So, what were the reasons? The gen- had a fantastic memory. He knew hun- against this idea. If you read his books, eral philosophy as developed by Kant, dreds and hundreds of pages of score you will see that as one of his obses- certainly. Bourbaki is the brainchild of by heart and could follow every single sions. German philosophy. Bourbald was note. I remember I had a few occasions Hilbert, I think, reflected this founded to develop and propagate to go to the concert hall with him. It Zeitgeist. There is one recording of his German philosophical views in science. was fascinating, he would look at the voice; in Constance Reid's book about Andrd Weft has always been fond of score in his hand and exclaim "OH!" if Hilbert there is a floppy disk of it, a German science and he was always a note was missing from the orchestra! record of some speech that Hilbert quoting Gauss. All these people, with He devoted the last six months of his gave in Germany in the thirties. It's their own tastes and their own personal life--when he decided that his mathe- very ideological. At the time Hilbert views, were proponents of German phi- maticai life was finished, he had writ- was aging and so his views were losophy. ten his last book, and he retreated to solidifying. And then there was the idea that his home--to listening to recordings If you put the manifesto of the sur- there is an opposition between art and and following the scores and the notes. realists and the introduction of science. Art is fragile and mortal, be- It's interesting to know that revolu- Bourbaki side by side, as well as other cause it appeals to emotions, to visual tionaries in mathematics were not rev- manifestos of the time, they look very meaning, and to unstated analogies. olutionaries in other things. I suppose similar. My daughter is currently trans- But I think it's also part of the that the only person in the Bourbaki lating a book about the birth of cine- Euclidean tradition. In Euclid, you find group who was really aware of the con- matography, and in a chapter about the some drawings but it is known that nections of the Bourbaki ideology with Italian futurists there is a very similar most of them were added after Euclid, other ideologies was Chevalley. He statement. In science, in art, in litera- in later editions. Most of the drawings was a member of various avant-garde ture, in politics, economics, social af- in the original are abstract drawings. groups, both in politics and in the arts. fairs, there was the same spirit. The You make some reasoning about some As the editor of Chevalley's work, I stated goal of Bourbaki was to create proportions and you draw some seg- have decided, at the urging of his a new mathematics. He didn't cite any ments, but they are not intended to be daughter, to include a special volume other mathematical texts. Bourbaki is geometrical segments, just representa- about his work outside mathematics. self-sufficient. Of course at the time tions of some abstract notions. Also He had written various pamphlets, and

VOLUME 20, NUMBER 1, 1998 27 various notes; Catherine Chevalley will awareness of his own body, the 40 main task of a mathematician was to have to work hard to collect these published volumes. bring order and make a synthesis of ex- things and we will publish them as part And as I said before, Bourbaki was isting material, to create what Thomas of his collected works. more or less like a family. The second Kuhn called norma/ science. Mathe- Chevalley was the only one who or third or fourth generation in any matics, in the forties and fifties, was un- perceived the connection between family or any social group follows def- dergoing what Kuhn calls a solidifica- Bourbaki and the rest, and that may be inite sociological patterns. My own tion period. In a given science there are why, in the seventies, he was more crit- family was typical. My grandfather was times when you have to take all the ex- ical than other people. In the seventies a self-made man, a very successful isting material and create a unified ter- a sensible person could already see the businessman. My father and my uncle minology, unified standards, and train end of a long historical trend, and I went into the business, but they were people in a unified style. The purpose of think he was very sensitive to this. not so devoted to the fight. And peo- mathematics, in the fifties and sixties, Mathematics was the most important ple in my generation--well, I suppose was that, to create a new era of normal part of his life, but he did not draw any I made the right decision not to engage science. Now we are again at the begin- boundary between his mathematics in it. Indeed, people in my generation ning of a new revolution. Mathematics and the rest of his life. Maybe this was who did go into our family business did is undergoing major changes. We don't because his father was an ambassador, not do so well, because they didn't know exactly where it will go. It is not so he had more contact with various have anything to fight for. yet time to make a synthesis of all these people. But these are the inner workings. Of things---maybe in twenty or thirty years Seneehal: Could you state the main course the outside world also has an it will be time for a new Bourbald. I con- reasons for the decline of Bourbaki? influence. That the outside mathemat- sider myself very fortunate to have had Cartier: As I said, in the eighties there ical world has changed is obvious. We two lives, a life of normal science and a was no longer a stated goal, except for all know that what Stalin could never life of scientific revolution. the long legal battle. I think it was one achieve with his army, to conquer the of the cases of the century! We hired a world, the collapse of the Soviet Union BIBLIOGRAPHY famous lawyer who had fought for the has achieved for mathematics. The , Faits et I~gendes, by Michele heirs of Picasso and Fujita. We sur- Russian mathematicians have brought Chouchan, I~ditions du Choix, Argenteuil, vived artificially: we had to win this a different style to the West, a differ- 1995. battle. But it was a pyrrhic victory. As ent way of looking at the problems, a "Nicholas Bourbaki, CollectiveMathematician: an usual in legal battles, both parties lost new blood. interviewwith ClaudeChevalley," by DenisGuedj, and the lawyer got rich. In fame and in It's a different time, with different translated by Jeremy Gray, The Mathematical pocket. values. I have no regrets: I think it was Intelligencer, vol. 7, no. 2, 18-22, 1985. In a sense Bourbaki is like a di- worthwhile to live in the twentieth cen- Les Math~matiques et I'Art, by Pierre Cartier, nosaur, the head too far away from the tury. But now it's finished. Institut des Hautes #tudes Scientifiques tail. When Dieudonn~ was the scribe Seneehal: How would you describe preprint IHES/M/93/33. of Bourbaki, for many many years, your journey with Bourbaki ? "The Continuing Silence of a Poet," by A. B. every printed word came from his pen. Cartier: I have been personally very Yehoshua, in The Continuing Silence of a Poet." Of course there had been many drafts happy, because when I reached the time collected stories, Penguin Books, 1991 .. and preliminary versions, but the of normal retirement from Bourbaki, I printed version was always from the had the very fortunate opportunity to be pen of Dieudonn& And with his fan- asked to deliver the lecture on behalf of tastic memory, he knew every single Vladimir Drinfel'd at the International word. I remember, it was a joke, you Congress of Mathematicians at Berkeley could say, "Dieudonn~, what is this re- in 1986 (Drinfel'd was prevented from sult about so and so?" and he would coming for political reasons). It was a go to the shelf and take down the book great challenge and a great honor for and open it to the right page. After me; his paper is one of the most im- Dieudonn~ (and an interlude by portant papers in the proceedings. Samuel and Dixmier) I was the secre- Overnight that changed my mathemat- tary of Bourbaki, and it was my duty ical life. I said, "This is what I have to to do most of the proofreading, I think do now." Of course I knew the basic I proofread five to ten thousand pages. material but the perspective was new. I have a good visual memory. I would- I cannot claim that within the few n't compare myself with Dieudonn~, hours I had to prepare the lecture I but there was a time when I knew could really master it, but I understood most of the printed material in enough to explain to the people, "This Bourbaki. But no one after me was is new, it is important." able to do this. So Bourbaki lost the When I began in mathematics the

28 THE MATHEMATICALINTELLIGENCER