Bourbaki, Nicolas

by Émilie Richer

The Problem

The devastation of World War I presented a unique challenge to aspiringmathematicians of the mid 1920’s. Among the many casualties of the warwere great numbers of scientists and mathematicians who would at thistime have been serving as mentors to the young students. Whereas othercountries such as Germany were sending their scholars to do scientificwork, France was sending promising young students to the front. A war-timedirectory of the école Normale Supérieure in Paris confirms that about2/3 of their student population was killed in the war.[DJ] Young menstudying after the war had no young teachers, they had no previousgeneration to rely on for guidance. What did this mean? According to JeanDieudonné, it meant that students like him were missing out on importantdiscoveries and advances being made in mathematics at that time. Heexplained : “I am not saying that they (the older professors) did notteach us excellent mathematics (…) But it is indubitable that a 50 yearold mathematician knows the mathematics he learned at 20 or 30, but hasonly 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 fromthe école Normale and I did not know what an ideal was! This gives youand 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 isexactly what a few young Parisian math students set out to do.

The Beginnings

After graduation from the école Normale Supérieure de Paris a group ofabout ten young mathematicians had maintained very close ties.[WA]They had all begun their careers and were scattered across France teachingin universities. Among them were Henri Cartan and André Weil who were bothin charge of teaching a course on differential and integral calculus at theUniversity of Strasbourg. The standard textbook for this class at the time was“Traité d’Analyse” by E. Goursat which the young professors found to beinadequate in many ways.[BA] According to Weil, his friend Cartan wasconstantly asking him questions about the best way to present a given topic tohis class, so much so that Weil eventually nicknamed him “the grandinquisitor”.[WA] After months of persistent questioning, in the winterof 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 atthis moment that Bourbaki was conceived.

The suggestion of writing this treatise spread and very soon a loose circleof friends, including Henri Cartan, André Weil, Jean Delsarte, JeanDieudonné and Claude Chevalley began meeting regularly at the Capoulade,a café in the Latin quarter of Paris to plan it . They called themselvesthe “Committee on the Analysis Treatise”[BL]. According to Chevalleythe project was extremely naive. The idea was to simply write another textbookto replace Goursat’s.[GD] After many discussions over what to include intheir treatise they finally came to the conclusion that they needed to startfrom 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 somesmall part of mathematics but in the greatest possible number of places”.[DJ] Gradually the young men realized that their meetings were notsufficient, and they decided they would dedicate a few weeks in the summerto their new project. The collaborators on this project were not aware ofits enormity, but were soon to find out.

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

The first Bourbaki congress, July 1935. From left toright, back row: Henri Cartan, René de Possel, Jean Dieudonné, AndréWeil, university lab technician, seated: Mirlès, Claude Chevalley, SzolemMandelbrojt.

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

André Weil, Claude Chevalley, Jean Dieudonné, Henri Cartan and JeanDelsarte were among the few present at these first meetings, they were allactive members of Bourbaki until their retirements. Today they areconsidered by most to be the founding fathers of the Bourbaki group.According to a later member they were “those who shaped Bourbaki andgave it much of their time and thought until they retired” he also claimsthat some other early contributors were Szolem Mandelbrojt and René dePossel.[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 andignoring the new. That is why very early on Bourbaki established one itsfirst and only rules : obligatory retirement at age 50. As explained byDieudonné “if the mathematics set forth by Bourbaki no longer correspondto the trends of the period, the work is useless and has to be redone, thisis why we decided that all Bourbaki collaborators would retire at age 50.”[DJ] Bourbaki wanted to create a work that would be an essential toolfor all mathematicians. Their aim was to create something logically ordered,starting with a strong foundation and building continuously on it. Thefoundation that they chose was set theory which would be the first book in aseries of 6 that they named “éléments de mathématique”(with the ’s’dropped from mathématique to represent their underlying belief in the unityof mathematics). Bourbaki felt that the old mathematical divisions were nolonger valid comparing them to ancient zoological divisions. The ancientzoologist would classify animals based on some basic superficial similaritiessuch as “all these animals live in the ocean”. Eventually they realized thatmore complexity was required to classify these animals. Past mathematicianshad 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 thatwhat was done previously has to be thrown overboard.”[DJ] After manyheated 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, thataccording 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 isexcluded(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 ismy picture of mathematics now. It is a ball of wool, a tangled hank whereall mathematics react upon another in an almost unpredictable way. Andthen in this ball of wool, there are a certain number of threads comingout in all directions and not connecting with anything else. Well theBourbaki 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 twoweeks) for Bourbaki “congresses” to work on their books. Their main rule wasunanimity on every point. Any member had the right to veto anything he feltwas inadequate or imperfect. Once Bourbaki had agreed on a topic for a chapterthe 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 atthe next Bourbaki congress. It would be read aloud line by line. Accordingto Dieudonné “each proof was examined point by point and criticizedpitilessly. He goes on “one has to see a Bourbaki congress to realize thevirulence 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 attendthe congress where it would being presented). Bourbaki sent him a telegramsummarizing the congress, it read : “union intersection partie produit tues démembré foutu Bourbaki” (union intersection subset product you aredismembered screwed Bourbaki).[WA] During a congress any member wasallowed to interrupt to criticize, comment or ask questions at any time.Apparently Bourbaki believed it could get better results from confrontationthan from orderly discussion.[BA] Armand Borel, summarized his firstcongress as “two or three monologues shouted at top voice, seeminglyindependent of one another”.[BA]

Bourbaki congress 1951.

After a first draft had been completely reduced to pieces it was the job ofa new collaborator to write up a second draft. This second collaboratorwould use all the suggestions and changes that the group had put forwardduring the congress. Any member had to be able to take on this task becauseone of Bourbaki’s mottoes was “the control of the specialists by thenon-specialists”[BA] i.e. a member had to be able to write a chapterin a field that was not his specialty. This second writer would set out onhis assignment knowing that by the time he was ready to present his draftthe views of the congress would have changed and his draft would also betorn apart despite its adherence to the congress’ earlier suggestions.The same chapter might appear up to ten times before it would finally beunanimously approved for publishing. There was an average of 8 to 12 yearsfrom 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 atuniversities across France. There, they could recruit for Bourbaki, studentsshowing great promise in mathematics. Members would never be replaced formallynor 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 understandeverything, but he would have to actively participate. He also had to showbroad interests and an ability to adapt to the Bourbaki style. If he wassilent he would not be invited again.(A challenging task considering hewould be in the presence of some of the strongest mathematical minds of thetime) Bourbaki described the reaction of certain guinea pigs invited to acongress : “they would come out with the impression that it was a gatheringof madmen. They could not imagine how these people, shouting -sometimes threeor four at a time- about mathematics, could ever come up with somethingintelligent.”[DJ] If a new recruit was showing promise, he would continueto be invited and would gradually become a member of Bourbaki without anyformal announcement. Although impossible to have complete anonymity, Bourbakiwas never discussed with the outside world. It was many years before Bourbakimembers agreed to speak publicly about their story. The following table givesthe 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 accordingto Pierre Cartier)[SM]. Note: There have been a great number of Bourbakicontributors, some lasting longer than others, this table gives the memberslisted by Pierre Cartier. Different sources list different “officialmembers” 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 firstto use an axiomatic presentation. They tried as often as possible to start fromthe general and work towards the particular. Working with the belief thatmathematics are fundamentally simple and for each mathematical question thereis an optimal way of answering it. This required extremely rigid structureand notation. In fact the first six books of “éléments de mathématique”use a completely linearly-ordered reference system. That is, any reference ata given spot can only be to something earlier in the text or in an earlierbook. This did not please all of its readers as Borel elaborates : “I wasrather put off by the very dry style, without any concession to the reader,the apparent striving for the utmost generality, the inflexible system ofinternal references and the total absence of outside ones”. However,Bourbaki’s style was in fact so efficient that a lot of its notation andvocabulary is still in current usage. Weil recalls that his granddaughterwas impressed when she learned that he had been personally responsible forthe symbol $\mathrm{\varnothing }$ for the empty set,[WA] and Chevalley explainsthat to “bourbakise” now means to take a text that is considered screwedup and to arrange it and improve it. Concluding that “it is the notion ofstructure which is truly bourbakique”.[GD]

As well as $\mathrm{\varnothing }$, Bourbaki is responsible for the introduction ofthe $\Rightarrow$ (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 ofa set A), as well as the words bijective, surjective and injective.[DR]

The Decline

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

Book VII Commutative algebra
Book VIII Lie Groups

However Cartier claims that by the end of the seventies, Bourbaki’smethod was understood, and many textbooks were being written in itsstyle : “Bourbaki was left without a task. (…) With their rigidformat they were finding it extremely difficult to incorporate newmathematical developments”[SM] To add to its difficulties, Bourbaki wasnow becoming involved in a battle with its publishing company over royaltiesand translation rights. The matter was settled in 1980 after a “long andunpleasant” legal process, where, as one Bourbaki member put it “bothparties lost and the lawyer got rich”[SM]. In 1983 Bourbaki publishedits last volume : IX Spectral Theory.

By that time Cartier says Bourbaki was a dinosaur, the head too far awayfrom the tail. Explaining : “when Dieudonné was the “scribe of Bourbaki”every printed word came from his pen. With his fantastic memory he knew everysingle 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 rightpage. After Dieudonné retired no one was able to do this. So Bourbaki lostawareness of his own body, the 40 published volumes.”[SM] Now afteralmost twenty years without a significant publication is it safe to say thedinosaur has become extinct? But since Nicolas Bourbaki never infact existed, and was nothing but a clever teaching and research ploy, couldhe ever be said to be extinct?