# Bourbaki: the pre-war years

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*Traité d'Analyse*Ⓣ, and has been suggesting to him better ways to introduce various concepts in the calculus. Today, however, he comes up with a new idea. "Let's talk to our friends when we next go up to Paris", Weil suggests, "about writing a new analysis textbook."

Henri Cartan and André Weil are regularly in Paris. They usually go there every second Monday to attend the Séminaire de mathématiques in the Institut Henri-Poincaré. This not only gives them the chance to visit bookstores and libraries, but also to meet up with other former students of the École Normale Supérieur. They meet regularly for lunch with their friends at the Café Capoulade in Boulevard Saint-Michel near the Luxembourg Gardens. In Café Capoulade, Weil talks excitedly about his idea for a new analysis text asking his friends if they think it is a good idea and whether they would be interested in contributing. They agree to meet at noon on Monday 10 December 1934 to formally discuss Weil's idea.

The mathematicians Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, René de Possel and André Weil who meet at Café Capoulade are well aware of the basic problem facing French mathematics. France kept rigorously to its belief in equality during World War I and as a consequence academics were as likely to have been killed in the trenches as anyone else. In other countries involved in the war the academics undertook special war duties which used their special skills, and as a consequence they were much safer. The young French mathematicians of the 1930s were being taught by lecturers who were nearing the end of their careers. There was a missing generation of lecturers in France. Weil enthusiastically made his ambitious proposal that they aim:-

The other members of the group meeting in Café Capoulade are as enthusiastic as the proposer. They talk about writing a book of 1000 pages to be published within six months. It is important to them at this stage that the work be available quickly. They decide that they will meet regularly at Café Capoulade and set themselves the goal that they must have agreed the syllabus of the book by the summer of 1935. They begin a lively discussion about the topics to be covered and the order in which they should appear. This at least makes them realise the magnitude of the task and they also discover that it will not be easy to reach a consensus but there is no following the majority - all must agree every decision. Choosing for themselves the title of "Committee for the Analysis Treatise", they set the date of the next meeting. At this second meeting the Committee decided to limit membership to nine mathematicians, the six we listed above and Paul Dubreil, Jean Leray and Szolem Mandelbrojt. Dubreil and Leray, however, did not participate for long. Charles Ehresmann was asked to join in place of Leray while Jean Coulomb, who was a physics colleague of Mandelbrojt and de Possel at Clermont-Ferrand, was asked to join in place of Dubreil.... to define for25years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis. Of course, this treatise will be as modern as possible.

At this stage the task of constructing a list of topics for the Analysis Treatise was divided up between various subcommittees. One decision, which would become a fundamental part of their philosophy, was that there had to be non-specialists on each subcommittee, most of which had three members. A large number of subcommittees were formed, given the size of the group, and these were to cover the following topics: algebra, analytic functions, integration theory, differential equations, existence theorems for differential equations, partial differential equations, differentials and differential forms, calculus of variations, special functions, geometry, Fourier series, and representations of functions. Soon a topology subcommittee was added.

By the summer of 1935 the group had decided that they would write under the name Nicolas Bourbaki. It is rather surprising that members of the group seemed a little unsure of exactly where the name had come from. Certainly the name came from General Charles Soter Bourbaki was a French general who had fought in the Franco-Prussian war of 1870-71. When André Weil was a first year student at the École Normale Supérieur a lecture was announced which all first year students were encouraged to attend. The lecturer was Raoul Husson, a senior student, who disguised himself as a distinguished venerable mathematician. To appear the part he put on a false beard and delivered the lecture in a heavy foreign accent. He presented a series of theorems, all completely wrong, each attributed to a different fake mathematician. The names for the theorems were taken from French generals, and the final and most ridiculous theorem he presented he had named "Bourbaki's theorem", taking the name from General Bourbaki. The humour of this was so enjoyed by all members of the group designing the Analysis Treatise that they adopted the name Bourbaki. It would appear that Nicolas was a classical reference to an ancient Greek hero from whom General Bourbaki was descended.

Another illustration of the humour of the members, which relates to the pseudonym they had chosen, is their reaction to an article written by R P Boas, the executive editor of *Mathematical Reviews*. Boas explained to his readers that Nicolas Bourbaki was the pseudonym for a group of young French mathematicians. Soon after, the publisher of Boas's article received a strongly worded letter from Nicolas Bourbaki objecting violently that his right to exist had been questioned. Further, Bourbaki's letter put forward the theory that BOAS was simply a pseudonym for the editors of *Mathematical Reviews*. See [10] for Boas's own account of these events.

The first Bourbaki Congress was held at Besse-en-Chandesse in July 1935. It was the first of what would become regular meetings, usually three per year, most lasting a week but some lasting two weeks. Descriptions of these congresses by founder members of Bourbaki are fascinating. Dieudonné wrote that anyone attending for the first time would [15]:-

André Weil wrote in [39] that the anarchic character of the congresses, which led to the shouting, was really by design:-... always come out with the impression that it is a gathering of madmen. They could not imagine how these people, shouting -- sometimes three or four at the same time -- could ever come up with something intelligent. It is perhaps a mystery but everything calms down in the end.

[There were, however, some clear decisions taken by the Bourbaki group on how to present mathematics which set the pattern for how the whole work would develop. First they decided to base their presentation of mathematics on the axiomatic method. Henri Cartan [13] explains the consequences of this:-We maintained]in our discussions a carefully disorganized character. In a meeting of the group, there has never been a president. Anyone speaks who wants to and everyone has the right to interrupt him ... The anarchic character of these discussions has been maintained throughout the existence of the group ... A good organization would have no doubt required that everyone be assigned a topic or a chapter, but the idea to do this never occurred to us ... What is to be learned concretely from that experience is that any effort at organization would have ended up with a treatise like any other ...

For example there are algebraic structures, order structures, and topological structures. All three of these structures are present in the concept of the real numbers, for example, and certainly not in an independent way but interlinked in a complex fashion. Also it was decided that Bourbaki would never generalise from special cases but would always deduce special cases from the most general. The consequence of this approach was a strong logical ordering on the way that the mathematical building was constructed. To many this was a major strength of the highly logical approach but to others it was a major weakness in that real numbers, which seem of fundamental importance, could not be introduced until vast areas of algebra and topology had been set up, of course always in the most general form possible [13]:-Bourbaki's decision to use the axiomatic method throughout brought with it the necessity of a new arrangement of mathematics' various branches. It proved impossible to retain the classical division into analysis, differential calculus, geometry, algebra, number theory, etc. Its place was taken by the concept of structure, which allowed definition of the concept of isomorphism and with it the classification of the fundamental disciplines within mathematics.

Henri Cartan fully understood the dangers of their approach [13]:-... the construction of real numbers starting from the rationals is for[Bourbaki]a special case of a more general construction: the completion of a topological group(Chapter3Book III). And this completion is itself based on the theory of the completion of a uniform space(Chapter2Book III).

Already in 1935 Bourbaki had taken the decision to produce a series of books which were linearly ordered in the sense that no reference could be made except to books earlier in the linear progression. Also no references could be made to material outside Bourbaki, for the group wanted to construct mathematics from scratch within their work. The only exception to this was in the historical notes which they included. They set out their plan of six books, each with several chapters. The books were:Bourbaki's method of going from the general to the specific is, of course, a bit dangerous for a beginner whose store of concrete problems is limited, since he could be led to believe that the generality is a goal for itself. But that is not Bourbaki's intention. For Bourbaki, a general concept is useful only if it is applicable to a number of more special problems and really saves time and effort. Such savings have become a necessity today. If Bourbaki members considered it their duty to work out everything from the ground up, they did so with the hope of placing in the hands of future mathematicians an instrument that would make their work easier and enable them to advance further.

Book II. Algebra

Book III. Topology

Book IV. Functions of One Real Variable

Book V. Topological Vector Spaces

Book VI. Integration

The first publication was a book containing the notation of set theory and included the formulas that were then available for subsequent work but without giving proofs which would be completed much later. Eilenberg reviewed thisThe title "Éléments de Mathématique" was chosen in1938. It is worth noting that they chose "Mathématique" rather than the much more usual "Mathématiques". The absence of the "s" was of course quite intentional, one way for Bourbaki to signal its belief in the unity of mathematics.

*Fascicule de résultats*Ⓣ of Chapter 1 of Book I and wrote:-

In fact the completed version of Chapter 1 of Book I, with full proofs, appeared very much later as the seventeenth volume that Bourbaki published. The second Bourbaki publication came in 1940 when Chapters 1 and 2 of Book III appeared:-Bourbaki is a pen name of a group of younger French mathematicians who set out to publish an encyclopaedic work covering most of modern mathematics. This issue is devoted to set theory and is only a digest of the proper volume. The purpose is to give the reader interested in one of the further volumes the necessary set theoretic preparation without bothering with a rigorous axiomatic approach and proofs; actually the material is arranged so excellently that most of the proofs can be easily completed. ... The last section outlines an interesting method of treating structures, such as order, topology, group, ring, etc., on a general basis and having concepts like isomorphism defined quite generally.

Two more publications appeared in 1942, namely Chapters 3 and 4 of Book III, and Chapter 1 of Book II. World War II, however, was having a major impact on the project with members of Bourbaki such as André Weil and Claude Chevalley being in the United States and only Henri Cartan and Jean Dieudonné active in continuing the development.The first chapter entitled "Topological Structures" is devoted to the study of topological and Hausdorff spaces. ... Chapter two is devoted to uniform structures which are the modern substitute for metric spaces. With the use of filters an exceedingly elegant treatment is presented. ... The notations and terminology are rigorous and thoroughly consistent.

After World War II ended it took a little time for France to recover sufficiently for the work of Bourbaki to move forward again but, once it had recruited new members, it went forward with renewed vigour. We discuss this second phase in the article Bourbaki: the post-war years.

**Article by:** *J J O'Connor* and *E F Robertson*

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