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The beginnings of the Bourbaki story are told in the article Bourbaki: the pre-war years.

World War II had intervened at a time when Bourbaki was about to accelerate production and move quickly towards the group's original aim of publishing six books, each with several chapters. The effects of the war were to cause the project to stagger to a halt with only parts of Books 1, 2 and 3 being published. When the project restarted after the war ended, new young mathematicians were recruited. In particular Roger Godement, Pierre Samuel, Jacques Dixmier and Jean-Pierre Serre joined Bourbaki in the late 1940's. Other "second generation" members who carried forward the project in the late 1940s and early 1950s were Samuel Eilenberg, Jean-Louis Koszul, and Laurent Schwartz.

Armand Borel first became acquainted with the Bourbaki team in 1949. He described [11] his first Bourbaki Congress:-

[

Congresses]were private affairs, devoted to the books. A usual session would discuss a draft of some chapter or maybe a preliminary report on a topic under consideration for inclusion, then or later. It was read aloud line by line by a member, and anyone could at any time interrupt, comment, ask questions, or criticize. More often than not, this "discussion" turned into a chaotic shouting match. I had often noticed that Dieudonné, with his stentorian voice, his propensity for definitive statements, and extreme opinions, would automatically raise the decibel level of any conversation he would take part in. Still, I was not prepared for what I saw and heard: "Two or three monologues shouted at top voice, seemingly independently of one another" is how I briefly summarized for myself my impressions that first evening ...

Dieudonné, in [15], gives a similar description of the Congresses and talks about the qualities that members of the team must have:-

During a Congress, the chapters come up in order of the day, in no particular order, and we never know in advance if we shall be doing only differential topology at this Congress, or if at the next one we shall be doing commutative algebra. No everything is mixed ... a Bourbaki member is supposed to take an interest in everything he hears. If he is a fanatical algebraist and says "I am interested in algebra and nothing else" fair enough, but he will never be a member of Bourbaki. One has to take an interest in everything at once. Not to be capable of creating in all fields, that is all right. There is no question of asking everyone to be a universal mathematician; this is reserved for a small number of geniuses. But still, one should take an interest in everything, and be able, when the time comes, to write a chapter of the treatise, even if it is not in one's speciality. This is something which has happened to practically every member, and I think most of them found it extremely beneficial.

The plans at this time were to carry out the aims that Bourbaki had set in 1939, namely to publish six books, each consisting of up to 10 chapters:

Book I. Set Theory

Book II. Algebra

Book III. Topology

Book IV. Functions of One Real Variable

Book V. Topological Vector Spaces

Book VI. Integration

After World War II, publication restarted in 1947 after a break of five years. A description of the chapters which appeared in the five years 1947-52 will illustrate the generality which the authors were achieving as well as the fact that the chapters came out in a somewhat muddled order, certainly not consecutively.

**1947 Book III Topology:** Chapters V, VI, and VII.

Chapter V is concerned with the additive group **R** of real numbers, its subgroups, quotient groups and homomorphisms. Chapters VI and VII are concerned with elementary properties of Euclidean spaces, projective spaces, complex numbers, complex projective spaces, quaternions, etc.

**1947 Book II Algebra:** Chapter II.

Mac Lane writes about this linear algebra chapter:-

The author of these volumes has set himself the task of presenting a systematic and well-balanced exposition of the whole of mathematics, and has properly chosen to achieve the necessary coherence by using a uniform terminology and a conceptual approach, in which special facts and illustrative examples are to be arranged under the appropriate general notions. His treatment of linear algebras follows these principles; bases and matrices are subordinated to left modules and linear mappings. One often needs these concepts in cases when the multiplication of scalars is not commutative; this appropriate measure of generality is here systematically carried out.

**1948 Book II Algebra:** Chapter III.

This chapter on multilinear algebra covers tensor products, tensor spaces, Grassmann algebra and determinants.

**1948 Book III Topology:** Chapter IX.

This chapter looks at the use of the real numbers in general topology. It covers topics such as metric spaces, normed spaces, and Baire spaces.

**1949 Book IV Functions of One Real Variable:** Chapter I.

Bochner writes:-

This instalment of Bourbaki's super-textbook gives a notable account of Rolle's theorem and Taylor's theorem with remainder; of the indefinite integral, as anti-derivative, for a function having only discontinuities of the first kind, such a function being a uniform limit of a function which is constant by intervals, an "interval" being an open interval or a point; of Cauchy limits for such integrals; of integration and differentiation with respect to a parameter under the integral sign; and of the elementary logarithmic, exponential and trigonometric functions. Everything is done for abstractly-valued functions whose values lie in a topologized vector-space, or, at the very least, in a normed ring, but there are many fine features of analysis spread out underneath this superimposed layer of ever-present generalizations.

**1949 Book III Topology:** Chapter X.

This is the final chapter of Book III. The sections are: Uniform structures on functional spaces; Spaces of continuous functions; Equicontinuous sets; Compact sets of continuous functions; and Approximation of real continuous functions.

**1950 Book II Algebra:** Chapters IV, and V.

These two chapters present a carefully worked out development of the theory of polynomials and fields.

**1951 Book IV Functions of One Real Variable:** Chapter IV.

This chapter covers ordinary first order differential equations and systems of such equations, again presented with considerably more generality than previous texts.

**1952 Book II Algebra:** Chapters VI, and VII.

The first of these chapters is concerned with the study of partially ordered semigroups and groups, divisibility in integral domains and similar topics. The second of the two chapters covers the theory of modules over a principal ideal ring.

**1952 Book IV Integration:** Chapters I, II, III, and IV.

These chapters set up the basic properties of integration theory which, as far as Bourbaki is concerned, means integration on locally compact spaces.

The "third generation" of Bourbaki members joined around the time that the project to produce the six books was well on its way to completion. A chapter took from eight to ten years from the time work started on it until final publication. By 1958 publications of the six books was complete so in the mid 1950s only the finishing touches remained to be done. In 1958 Henri Cartan explained why Bourbaki was not growing old [13]:-

Bourbaki has been at work for more than20years. You might think that he has grown old in that time and the dynamism of his members has gradually diminished. But that is not so. Bourbaki will always remain young and lively; like mankind he renews himself constantly. The so-called "founding fathers" ... have stepped down in the course of the years to have their places taken by younger members. These newcomers found their own way to Bourbaki.

Armand Borel, François Bruhat, Pierre Cartier, Alexander Grothendieck, Serge Lang, and John Tate are considered the younger members of this third generation. They did not see things in exactly the same way as the founders, for example Cartier said:-

The misunderstanding was that many people thought that[mathematics]should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopaedia of mathematics, containing all the necessary information. That is a good description. If you consider it as a textbook, it's a disaster.

Clearly this powerful mathematical team did not see their task simple to push the last of the chapters through the publishing process. Not at all - in fact right from the beginning members of Bourbaki had more grandiose ideas than simply the six planned books. Armand Borel explains in [11]:-

In the fifties[the six books]were essentially finished, and it was understood the main energies of Bourbaki would henceforth concentrate on the sequel; it had been in the mind of Bourbaki very early on(after all, there was still no "Traité d'Analyse"). Already in September1940, Dieudonné had outlined a grandiose plan in27books, encompassing most of mathematics. More modest ones, still reaching beyond the "éléments", also usually by Dieudonné, would regularly conclude the Congresses. Also, many reports on and drafts of future chapters had already been written. However, mathematics had grown enormously, the mathematical landscape had changed considerably, in part through the work of Bourbaki, and it became clear we could not go on simply following the traditional pattern. Although this had not been intended, the founding members had often carried a greater weight on basic decisions, but they were now retiring and the primary responsibility was shifting to younger members. Some basic principles had to be re-examined.

These basic principles had required Bourbaki to build everything in a linear fashion without references to outside sources. This clearly would become increasingly difficult as they moved to more advanced topics. Also the idea that every member of Bourbaki should be interested in every topic was one which did indeed require universal mathematicians in the sense which Dieudonné had already made clear they were not:-

There is no question of asking everyone to be a universal mathematician; this is reserved for a small number of geniuses.

Would Bourbaki then split up into smaller units each responsible for an advanced topic? Such questions provoked heated discussion between idealists and realists in the group. Grothendieck, although a third generation member, was fully committed to providing full foundations in accepted "Bourbaki generality" for the advanced topics. He produced a proposal at a 1957 Congress to have:

Book VII: Homological Algebra

Book VIII: Elementary Topology

Book IX: Manifolds

Here the aim was to continue building Bourbaki in the style of the first six books. Grothendieck presented to the next Congress a detailed draft of two chapters, the first containing preliminaries to the book on manifolds and categories of manifolds, while the second contained differentiable manifolds, and the differential formalism [11]:-

[

Grothendieck]warned that much more algebra would be needed, e.g., hyperalgebras. As was often the case with Grothendieck's papers, they were at points discouragingly general, but at others rich in ideas and insights. However, it was rather clear that if we followed that route, we would be bogged down with foundations for many years, with a very uncertain outcome.

There was another problem which constantly bothered Bourbaki members. The original aim was to produce an up-to-date text for students to learn the latest approach to mathematics. But by 1958 when the original six books were completed, the first few of these books were already almost 20 years out of date. It was clearly impossible for the authors of the first few chapters to cover all that was needed in the later texts since mathematics was a rapidly developing subject. How could they make their linear approach consistent with rewriting the earlier books? Grothendieck proposed that Bourbaki now went further in providing foundations than had been done for the first six books and planned foundations:-

... also for future developments to the extent they could be foreseen.

These proposals were, however, not liked by others who favoured a more realistic approach to achieving what they wanted to see. Some wanted to see two parallel approaches, that of Grothendieck in the spirit of the founders, and also advanced texts in areas they wanted to cover. But they just didn't have the manpower for this, however much it might have suited everybody. Although Bourbaki held back from the ultimate goal of writing down the whole of mathematics in their linear development, realising that this was just not feasible, nevertheless they still embarked on a programme which was impossibly ambitious. They planned to rewrite the first six books and publish a "final version" of them. They also planned to give up the totally linear approach but, by way of compromise, to produce a summary text on differential and analytic manifolds which followed the model of their first version of Chapter I of Book I, and give only a list of necessary results to set up enough machinery to push forward on topics they were very keen to cover. They decided on producing advanced texts on commutative algebra, algebraic geometry, Lie groups, global and functional analysis, algebraic number theory, and automorphic forms.

By 1980 they had produced the two planned summary chapters on differential and analytic manifolds, seven chapters on commutative algebra, eight chapters on Lie groups and Lie algebras, and two chapters on spectral theories. They had also produced an English translation of some of the first six books, together with three of the chapters on Lie groups and algebras, and seven of the chapters on commutative algebra. For many the chapters on Lie groups and Lie algebras are Bourbaki's finest achievements. For this Armand Borel must take the bulk of the credit for he was the main driving force behind the style and content of this part.

Despite the achievements after 1960, the project began to falter during the 1970s. Cartier gave what he saw as the reasons in [36]:-

Bourbaki struggled in the seventies and the eighties to formulate new directions.[There was]a failed project about several complex variables. There were attempts at homotopy theory, at spectral theory of operators, at the index theorem, at symplectic geometry. But none of these projects went beyond a preliminary stage. Bourbaki could not find a new outlet, because they had a dogmatic view of mathematics: everything should be set inside a secure framework. That was quite reasonable for general topology and general algebra, which were already solidified around1950. Most people agree now that you do need general foundations for mathematics, at least if you believe in the unity of mathematics. But I believe now that this unity should be organic, while Bourbaki advocated a structural point of view.

There was another problem for Bourbaki in the 1970s which was unrelated to mathematics. They entered into a long legal dispute with their publisher over royalties and translation rights. Cartier said [36]:-

I think it was one of the cases of the century! We hired a famous lawyer who had fought for the heirs of Picasso and Fujita. We survived artificially: we had to win this battle. But it was a pyrrhic victory. As usual in legal battles, both parties lost and the lawyer got rich: In fame and in pocket.

Despite the problems, the project is still alive and new editions are coming out. The founders of Bourbaki had a rule that members should retire at fifty. Cartier has suggested that perhaps it would be appropriate for Bourbaki to retire at fifty. This did not happen and Bourbaki is now seventy. Will there be a resurgence and a sudden return to the passion for change that gripped the founders? Probably not. There may not now be a need for this, and one reason that there is no need is of course the way that Bourbaki has shaken up the world of mathematics. Some love the approach, some hate it, but one cannot deny Bourbaki's influence.

**References (39 books/articles)**

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

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