Bourbaki, Nicolas

by Émilie Richer

The Problem

The devastation of World War I presented a unique challenge to aspiring mathematicians of the mid 1920’s. Among the many casualties of the war were great numbers of scientists and mathematicians who would at this time have been serving as mentors to the young students. Whereas other countries such as Germany were sending their scholars to do scientific work, France was sending promising young students to the front. A war-time directory of the école Normale Supérieure in Paris confirms that about 2/3 of their student population was killed in the war.[DJ] Young men studying after the war had no young teachers, they had no previous generation to rely on for guidance. What did this mean? According to Jean Dieudonné, it meant that students like him were missing out on important discoveries and advances being made in mathematics at that time. He explained : “I am not saying that they (the older professors) did not teach us excellent mathematics (…) But it is indubitable that a 50 year old mathematician knows the mathematics he learned at 20 or 30, but has only notions, often rather vague, of the mathematics of his epoch, i.e. the period of time when he is 50.” He continued : “I had graduated from the école Normale and I did not know what an ideal was! This gives you and idea of what a young French mathematician knew in 1930.”[DJ] Henri Cartan, another student in Paris shortly after the war affirmed : “we were the first generation after the war. Before us there was a vide, a vacuum, and it was necessary to make everything new.”[JA] This is exactly what a few young Parisian math students set out to do.

The Beginnings

After graduation from the école Normale Supérieure de Paris a group of about ten young mathematicians had maintained very close ties.[WA] They had all begun their careers and were scattered across France teaching in universities. Among them were Henri Cartan and André Weil who were both in charge of teaching a course on differential and integral calculus at the University of Strasbourg. The standard textbook for this class at the time was “Traité d’Analyse” by E. Goursat which the young professors found to be inadequate in many ways.[BA] According to Weil, his friend Cartan was constantly asking him questions about the best way to present a given topic to his class, so much so that Weil eventually nicknamed him “the grand inquisitor”.[WA] After months of persistent questioning, in the winter of 1934, Weil finally got the idea to gather friends (and former classmates) to settle their problem by rewriting the treatise for their course. It is at this moment that Bourbaki was conceived.

The suggestion of writing this treatise spread and very soon a loose circle of friends, including Henri Cartan, André Weil, Jean Delsarte, Jean Dieudonné and Claude Chevalley began meeting regularly at the Capoulade, a café in the Latin quarter of Paris to plan it . They called themselves the “Committee on the Analysis Treatise”[BL]. According to Chevalley the project was extremely naive. The idea was to simply write another textbook to replace Goursat’s.[GD] After many discussions over what to include in their treatise they finally came to the conclusion that they needed to start from scratch and present all of essential mathematics from beginning to end. With the idea that “the work had to be primarily a tool, not usable in some small part of mathematics but in the greatest possible number of places”. [DJ] Gradually the young men realized that their meetings were not sufficient, and they decided they would dedicate a few weeks in the summer to their new project. The collaborators on this project were not aware of its enormity, but were soon to find out.

In July of 1935 the young men gathered for their first congress (as they would later call them) in Besse-en-Chandesse. The men believed that they would be able to draft the essentials of mathematics in about three years. They did not set out wanting to write something new, but to perfect everything already known. Little did they know that their first chapter would not be completed until 4 years later. It was at one of their first meetings that the young men chose their name : Nicolas Bourbaki. The organization and its membership would go on to become one of the greatest enigmas of 20th century mathematics.

The first Bourbaki congress, July 1935. From left to right, back row: Henri Cartan, René de Possel, Jean Dieudonné, André Weil, university lab technician, seated: Mirlès, Claude Chevalley, Szolem Mandelbrojt.

André Weil recounts many years later how they decided on this name. He and a few other Bourbaki collaborators had been attending the école Normale in Paris, when a notification was sent out to all first year science students : a guest speaker would be giving a lecture and attendance was highly recommended. As the story goes, the young students gathered to hear, (unbeknownst to them) an older student, Raoul Husson who had disguised himself with a fake beard and an unrecognizable accent. He gave what is said to be an incomprehensible, nonsensical lecture, with the young students trying desperately to follow him. All his results were wrong in a non-trivial way and he ended with his most extravagant : Bourbaki’s Theorem. One student even claimed to have followed the lecture from beginning to end. Raoul had taken the name for his theorem from a general in the Franco-Prussian war. The committee was so amused by the story that they unanimously chose Bourbaki as their name. Weil’s wife was present at the discussion about choosing a name and she became Bourbaki’s godmother baptizing him Nicolas.[WA] Thus was born Nicolas Bourbaki.

André Weil, Claude Chevalley, Jean Dieudonné, Henri Cartan and Jean Delsarte were among the few present at these first meetings, they were all active members of Bourbaki until their retirements. Today they are considered by most to be the founding fathers of the Bourbaki group. According to a later member they were “those who shaped Bourbaki and gave it much of their time and thought until they retired” he also claims that some other early contributors were Szolem Mandelbrojt and René de Possel.[BA]

Reforming Mathematics : The Idea

Bourbaki members all believed that they had to completely rethink mathematics. They felt that older mathematicians were holding on to old practices and ignoring the new. That is why very early on Bourbaki established one its first and only rules : obligatory retirement at age 50. As explained by Dieudonné “if the mathematics set forth by Bourbaki no longer correspond to the trends of the period, the work is useless and has to be redone, this is why we decided that all Bourbaki collaborators would retire at age 50.” [DJ] Bourbaki wanted to create a work that would be an essential tool for all mathematicians. Their aim was to create something logically ordered, starting with a strong foundation and building continuously on it. The foundation that they chose was set theory which would be the first book in a series of 6 that they named “éléments de mathématique”(with the ’s’ dropped from mathématique to represent their underlying belief in the unity of mathematics). Bourbaki felt that the old mathematical divisions were no longer valid comparing them to ancient zoological divisions. The ancient zoologist would classify animals based on some basic superficial similarities such as “all these animals live in the ocean”. Eventually they realized that more complexity was required to classify these animals. Past mathematicians had apparently made similar mistakes : “the order in which we (Bourbaki) arranged our subjects was decided according to a logical and rational scheme. If that does not agree with what was done previously, well, it means that what was done previously has to be thrown overboard.”[DJ] After many heated discussions, Bourbaki eventually settled on the topics for “éléments de mathématique” they would be, in order:

I Set theory
II Algebra
III Topology
IV Functions of one real variable
V Topological vector spaces
VI Integration

They now felt that they had eliminated all secondary mathematics, that according to them “did not lead to anything of proved importance.”[DJ] The following table summarizes Bourbaki’s choices.

What remains after cutting the loose threads	What is excluded(the loose threads)
Linear and multilinear algebra	Theory of ordinals and cardinals
A little general topology the least possible	Lattices
Topological vector Spaces	Most general topology
Homological algebra	Most of group theory finite groups
Commutative algebra	Most of number theory
Non-commutative algebra	Trigonometrical series
Lie groups	Interpolation
Integration	Series of polynomials
Differentiable manifolds	Applied mathematics
Riemannian geometry

Dieudonné’s metaphorical ball of yarn: “here is my picture of mathematics now. It is a ball of wool, a tangled hank where all mathematics react upon another in an almost unpredictable way. And then in this ball of wool, there are a certain number of threads coming out in all directions and not connecting with anything else. Well the Bourbaki method is very simple-we cut the threads.”[DJ]

Reforming Mathematics : The Process

It didn’t take long for Bourbaki to become aware of the size of their project. They were now meeting three times a year (twice for one week and once for two weeks) for Bourbaki “congresses” to work on their books. Their main rule was unanimity on every point. Any member had the right to veto anything he felt was inadequate or imperfect. Once Bourbaki had agreed on a topic for a chapter the job of writing up the first draft was given to any member who wanted it. He would write his version and when it was complete it would be presented at the next Bourbaki congress. It would be read aloud line by line. According to Dieudonné “each proof was examined point by point and criticized pitilessly. He goes on “one has to see a Bourbaki congress to realize the virulence of this criticism and how it surpasses by far any outside attack.” [DJ] Weil recalls a first draft written by Cartan (who has unable to attend the congress where it would being presented). Bourbaki sent him a telegram summarizing the congress, it read : “union intersection partie produit tu es démembré foutu Bourbaki” (union intersection subset product you are dismembered screwed Bourbaki).[WA] During a congress any member was allowed to interrupt to criticize, comment or ask questions at any time. Apparently Bourbaki believed it could get better results from confrontation than from orderly discussion.[BA] Armand Borel, summarized his first congress as “two or three monologues shouted at top voice, seemingly independent of one another”.[BA]

Bourbaki congress 1951.

After a first draft had been completely reduced to pieces it was the job of a new collaborator to write up a second draft. This second collaborator would use all the suggestions and changes that the group had put forward during the congress. Any member had to be able to take on this task because one of Bourbaki’s mottoes was “the control of the specialists by the non-specialists”[BA] i.e. a member had to be able to write a chapter in a field that was not his specialty. This second writer would set out on his assignment knowing that by the time he was ready to present his draft the views of the congress would have changed and his draft would also be torn apart despite its adherence to the congress’ earlier suggestions. The same chapter might appear up to ten times before it would finally be unanimously approved for publishing. There was an average of 8 to 12 years from the time a chapter was approved to the time it appeared on a bookshelf. [DJ] Bourbaki proceeded this way for over twenty years, (surprisingly) publishing a great number of volumes.

Bourbaki congress 1951.

Recruitment and Membership

During these years, most Bourbaki members held permanent positions at universities across France. There, they could recruit for Bourbaki, students showing great promise in mathematics. Members would never be replaced formally nor was there ever a fixed number of members. However when it felt the need, Bourbaki would invite a student or colleague to a congress as a “cobaye” (guinea pig). To be accepted, not only would the guinea pig have to understand everything, but he would have to actively participate. He also had to show broad interests and an ability to adapt to the Bourbaki style. If he was silent he would not be invited again.(A challenging task considering he would be in the presence of some of the strongest mathematical minds of the time) Bourbaki described the reaction of certain guinea pigs invited to a congress : “they would come out with the impression that it was a gathering of madmen. They could not imagine how these people, shouting -sometimes three or four at a time- about mathematics, could ever come up with something intelligent.”[DJ] If a new recruit was showing promise, he would continue to be invited and would gradually become a member of Bourbaki without any formal announcement. Although impossible to have complete anonymity, Bourbaki was never discussed with the outside world. It was many years before Bourbaki members agreed to speak publicly about their story. The following table gives the names of some of Bourbaki’s collaborators.

$1^{st}$ generation (founding fathers)	$2^{nd}$ generation (invited after WWII)	$3^{rd}$ generation
H. Cartan	J. Dixmier	A. Borel
C. Chevalley	R. Godement	F. Bruhat
J. Delsarte	S. Eilenberg	P. Cartier
J. Dieudonné	J.L. Koszul	A. Grothendieck
A. Weil	P. Samuel	S. Lang
	J.P Serre	J. Tate
	L. Shwartz

3 Generations of Bourbaki (membership according to Pierre Cartier)[SM]. Note: There have been a great number of Bourbaki contributors, some lasting longer than others, this table gives the members listed by Pierre Cartier. Different sources list different “official members” in fact the Bourbaki website lists J.Coulomb, C.Ehresmann, R.de Possel and S. Mandelbrojt as $1^{st}$ generation members.[BW]

Bourbaki congress 1938, from left to right: S. Weil, C. Pisot, A. Weil, J. Dieudonné, C. Chabauty, C. Ehresmann, J. Delsarte.

The Books

The Bourbaki books were the first to have such a tight organization, the first to use an axiomatic presentation. They tried as often as possible to start from the general and work towards the particular. Working with the belief that mathematics are fundamentally simple and for each mathematical question there is an optimal way of answering it. This required extremely rigid structure and notation. In fact the first six books of “éléments de mathématique” use a completely linearly-ordered reference system. That is, any reference at a given spot can only be to something earlier in the text or in an earlier book. This did not please all of its readers as Borel elaborates : “I was rather put off by the very dry style, without any concession to the reader, the apparent striving for the utmost generality, the inflexible system of internal references and the total absence of outside ones”. However, Bourbaki’s style was in fact so efficient that a lot of its notation and vocabulary is still in current usage. Weil recalls that his granddaughter was impressed when she learned that he had been personally responsible for the symbol $\mathrm{\varnothing }$ for the empty set,[WA] and Chevalley explains that to “bourbakise” now means to take a text that is considered screwed up and to arrange it and improve it. Concluding that “it is the notion of structure which is truly bourbakique”.[GD]

As well as $\mathrm{\varnothing }$, Bourbaki is responsible for the introduction of the $\Rightarrow$ (the implication arrow), $\mathbb{N}$, $ℝ$, $ℂ$, $\mathbb{Q}$ and $\mathbb{Z}$ (respectively the natural, real, complex, rational numbers and the integers) $C_A$ (complement of a set A), as well as the words bijective, surjective and injective. [DR]

The Decline

Once Bourbaki had finally finished its first six books, the obvious question was “what next?”. The founding members who (not intentionally) had often carried most of the weight were now approaching mandatory retirement age. The group had to start looking at more specialized topics, having covered the basics in their first books. But was the highly structured Bourbaki style the best way to approach these topics? The motto “everyone must be interested in everything” was becoming much more difficult to enforce. (It was easy for the first six books whose contents are considered essential knowledge of most mathematicians) Pierre Cartier was working with Bourbaki at this point. He says “in the forties you can say that Bourbaki know where to go: his goal was to provide the foundation for mathematics”.[12] It seemed now that they did not know where to go. Nevertheless, Bourbaki kept publishing. Its second series (falling short of Dieudonné’s plan of 27 books encompassing most of modern mathematics [BA]) consisted of two very successful books :

Book VII Commutative algebra
Book VIII Lie Groups

However Cartier claims that by the end of the seventies, Bourbaki’s method was understood, and many textbooks were being written in its style : “Bourbaki was left without a task. (…) With their rigid format they were finding it extremely difficult to incorporate new mathematical developments”[SM] To add to its difficulties, Bourbaki was now becoming involved in a battle with its publishing company over royalties and translation rights. The matter was settled in 1980 after a “long and unpleasant” legal process, where, as one Bourbaki member put it “both parties lost and the lawyer got rich”[SM]. In 1983 Bourbaki published its last volume : IX Spectral Theory.

By that time Cartier says Bourbaki was a dinosaur, the head too far away from the tail. Explaining : “when Dieudonné was the “scribe of Bourbaki” every printed word came from his pen. With his fantastic memory he knew every single word. You could say “Dieudonné what is the result about so and so?” and he would go to the shelf and take down the book and open it to the right page. After Dieudonné retired no one was able to do this. So Bourbaki lost awareness of his own body, the 40 published volumes.”[SM] Now after almost twenty years without a significant publication is it safe to say the dinosaur has become extinct? But since Nicolas Bourbaki never in fact existed, and was nothing but a clever teaching and research ploy, could he ever be said to be extinct?

Title	Bourbaki, Nicolas
Canonical name	BourbakiNicolas
Date of creation	2013-03-22 13:33:23
Last modified on	2013-03-22 13:33:23
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	17
Author	Daume (40)
Entry type	Biography
Classification	msc 01A60
Synonym	Bourbaki
Related topic	AlexanderGrothendieckABiographyOf