# Nicolas Bourbaki

Nicolas Bourbaki (French pronunciation: [nikɔla buʁbaki]) is the collective pseudonym of a group of (mainly French) mathematicians. Their aim is to reformulate mathematics on an extremely abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strives for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.

Founders Association des collaborateurs de Nicolas Bourbaki Bourbaki congress at Dieulefit in 1938. From left, Simone Weil, Charles Pisot, André Weil, Jean Dieudonné (sitting), Claude Chabauty, Charles Ehresmann, and Jean Delsarte. Voluntary association École Normale Supérieure, Paris Confidential French www.bourbaki.ens.fr Groupe Bourbaki

While there is no one person named Nicolas Bourbaki, the Bourbaki group, officially known as the Association des collaborateurs de Nicolas Bourbaki (Association of Collaborators of Nicolas Bourbaki), has an office at the École Normale Supérieure in Paris.

## The group

In 1934, young French mathematicians from various French universities felt the need to form a group to jointly produce textbooks that they could all use for teaching. André Weil organized the first meeting on 10 December 1934 in the basement of a Parisian grill room, while all participants were attending a conference in Paris.

Accounts of the early days vary, but original documents have now come to light. The founding members were all connected to the École Normale Supérieure in Paris and included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil. There was a preliminary meeting, towards the end of 1934.[1] Jean Leray and Paul Dubreil were present at the preliminary meeting but dropped out before the group actually formed. Other notable participants in later days were Hyman Bass, Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, Jean-Louis Koszul, Samuel Eilenberg, Serge Lang and Roger Godement.

The original goal of the group had been to compile an improved mathematical analysis text; it was soon decided that a more comprehensive treatment of all of mathematics was necessary. There was no official status of membership, and at the time the group was quite secretive and also fond of supplying disinformation. Regular meetings were scheduled (totalling about 4 weeks a year), during which the group would discuss vigorously every proposed line of every book. Members had to resign by age 50, which allegedly resulted in a complete change of personnel by 1958.[2] However, historian Liliane Beaulieu was quoted as never having found written affirmation of this rule.[3]

The atmosphere in the group can be illustrated by an anecdote told by Laurent Schwartz. Dieudonné regularly and spectacularly threatened to resign unless topics were treated in their logical order, and after a while others played on this for a joke. Godement's wife wanted to see Dieudonné announcing his resignation, and so on one occasion while she was there Schwartz deliberately brought up again the question of permuting the order in which measure theory and topological vector spaces were to be handled, to precipitate a guaranteed crisis.

The name "Bourbaki" refers to a French general, Charles Denis Bourbaki;[4] it was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue. It is said that Weil's wife Evelyne supplied Nicolas.[5] This is more or less confirmed by Robert Mainard.[2]

According to biographer Amir Aczel, the collective can be divided into generations.

Bourbaki was always a very small group of mathematicians, typically numbering about twelve people. Its first generation was that of the founding fathers, those who created the group in 1934: Weil, Cartan, Chevalley, Delsarte, de Possel, and Dieudonné. Others joined the group, and others left its ranks, so that some years later there were about twelve members, and that number remained roughly constant. Laurent Schwartz was the only mathematician to join Bourbaki during the war, so his is considered an intermediate generation. After the war, a number of members joined: Jean-Pierre Serre, Pierre Samuel, Jean-Louis Kozul, Jacques Dixmier, Roger Godement, and Sammy Eilenberg. These people constituted the second generation of Bourbaki. In the 1950s, the third generation of mathematicians joined Bourbaki. These people included Alexandre Grothendieck, Francois Bruhat, Serge Lang, the American mathematician John Tate, Pierre Cartier, and the Swiss mathematician Armand Borel.[6]

Aczel also emphasized the importance of Bourbaki's influence on structuralism, a multidisciplinary concept stressing the structural relationships between objects, the objects themselves being incidental. Aczel further emphasized the influence of Bourbaki's work on anthropology via Weil's collaboration with Claude Lévi-Strauss.

### Pranks

The Bourbaki group released a few humorous hoaxes related to the fake life of Nicolas Bourbaki. For example, the group released a wedding announcement, relating the marriage of Betti Bourbaki (daughter of Nicolas) with a certain Hector Pétard (Hector Firecrackers in English). In November 1968, a mock obituary of Nicolas Bourbaki was released during one of the seminars, containing a few mathematical puns.[7] The group is however still active as of 2018, organizing seminars[8] and having released a book in 2016.

## Membership

As of 2000, the Bourbaki collective has had "about forty"[9] historical members. Bourbaki's members have been described in terms of generations. The first generation of founders consisted of a core membership of five, and six others who briefly participated. This was followed by a second generation of another seven core members, and a third generation of another six core members. Roughly twenty later members participated in the collective, not including its current membership. At its founding, Bourbaki's size was limited to nine members. Throughout much of its history, the collective has had about twelve members at any given point.

The group has a custom of keeping its current membership secret, a practice meant to ensure that its output is presented as a collective, unified effort under the Bourbaki pseudonym, not attributable to any one author (e.g. for purposes of copyright or royalty payment).[10] The group's secrecy is also intended as a deterrent against unwanted distraction during its normal operation. However, the group's former members freely discuss its culture and internal practices upon departure.

Prospective members are invited to conferences and styled as "guinea pigs", a process meant to vet the prospective member's mathematical ability. In the event of mutual consent between the prospect and the group, the prospect becomes a full member. Additionally, the group's conferences have regularly been attended by friends, family members, and visiting mathematicians.

Former members of the Nicolas Bourbaki collective[11][lower-alpha 1]
Generation Name Born ENS[lower-alpha 2] Joined Left Died
First[lower-alpha 3] Core members Henri Cartan 1904 1923 1934 2008
Claude Chevalley 1909 1926 1934 1984
Jean Delsarte 1903 1922 1934 1968
Jean Dieudonné 1906 1924 1934 1992
André Weil 1906 1922 1934 1998
Minor members Jean Coulomb 1904 1923 1935 1999
Paul Dubreil 1904 1926 1935 1935 1994
Charles Ehresmann 1905 1924 1935 1979
Jean Leray 1906 1926 1935 1935 1998
Szolem Mandelbrojt 1899 1935 1983
René de Possel 1905 1923 1934 1974
Second[lower-alpha 4] Jacques Dixmier 1924 1942
Samuel Eilenberg 1913 1998
Roger Godement 1921 1940 2016
Jean-Louis Koszul 1921 1940 2018
Pierre Samuel 1921 1940 2009
Laurent Schwartz 1916 1934 2002
Jean-Pierre Serre 1926 1945
Third Armand Borel 1923 2003
François Bruhat 1929 1948 2007
Pierre Cartier 1932 1950
Alexander Grothendieck 1928 2014
Serge Lang 1927 2005
John Tate 1925
Later members[lower-alpha 5] Hyman Bass 1932
Arnaud Beauville 1947 1966
Gérard Ben Arous 1957 1977
Daniel Bennequin 1952 1972
Claude Chabauty 1910 1929 1990
Alain Connes 1947 1966
Michel Demazure 1937 1955
Patrick Gérard 1961 1981
Guy Henniart 1953 1973
Luc Illusie 1940 1959
Pierre Julg 1959 1977
Olivier Mathieu 1960 1980
Joseph Oesterlé 1954 1973
Charles Pisot 1909 1929 1984
Michel Raynaud 1938 1958 2018
Marc Rosso 1962 1982
Georges Skandalis 1955 1975
Bernard Tessier 1945
Jean-Louis Verdier 1937 1955 1989
Jean-Christophe Yoccoz 1957 1975 2016

## Books by Bourbaki

Bourbaki's main work is the Elements of Mathematics (Éléments de mathématique) series. This series aims to be a completely self-contained treatment of the core areas of modern mathematics. Assuming no special knowledge of mathematics, it takes up mathematics from the very beginning, proceeds axiomatically and gives complete proofs.

The dates indicated below are for the first edition of the first chapter of each book. Most of the books were reedited several times (with significant changes between editions), and the books were released in several parts containing different chapters (e.g. Book II, Algebra, was released in five parts, the first in 1942 with chapters 1, 2, and 3, and the last in 1980 containing chapter 10).

• Bourbaki, Nicolas (1939). Livre I: Théorie des ensembles [Book I: Set theory] (in French).[12]
• Bourbaki, Nicolas (1942). Livre II: Algèbre [Book II: Algebra] (in French).[13][14][15]
• Bourbaki, Nicolas (1940). Livre III: Topologie [Book III: Topology] (in French).
• Bourbaki, Nicolas (1949). Livre IV: Fonctions d'une variable réelle [Book IV: Functions of one real variable] (in French).
• Bourbaki, Nicolas (1953). Livre V: Espaces vectoriels topologiques [Book V: Topological vector spaces] (in French).
• Bourbaki, Nicolas (1952). Livre VI: Intégration [Book VI: Integration] (in French).[16][17]
• Bourbaki, Nicolas (1961). Livre VII: Algèbre commutative [Book VII: Commutative algebra] (in French).[18]
• Bourbaki, Nicolas (1960). Livre VIII: Groupes et algèbres de Lie [Book VIII: Lie groups and algebras] (in French).
• Bourbaki, Nicolas (1967). Livre IX: Théories spectrales [Book IX: Spectral theory] (in French).
• Bourbaki, Nicolas (1967). Livre X: Variétés différentielles et analytiques [Book X: Differentiable and analytic manifolds] (in French).
• Bourbaki, Nicolas (2016). Livre XI: Topologie algébrique [Book XI: Algebraic topology] (in French).[19]

The book Variétés différentielles et analytiques was a fascicule de résultats, that is, a summary of results, on the theory of manifolds, rather than a worked-out exposition. The (still incomplete) volume on spectral theory (Théories spectrales) from 1967 was for almost four decades the last new book to be added to the series. After that several new chapters to existing books as well as revised editions of existing chapters appeared until the publication of chapters 8 and 9 of Commutative Algebra in 1983. A long break in publishing activity followed, leading many to suspect the end of the publishing project. However, chapter 10 of Commutative Algebra appeared in 1998, and after another long break a completely re-written and expanded chapter 8 of Algèbre was published in 2012. More importantly, the first four chapters of a completely new book on algebraic topology were published in 2016. The new material from 2012 and 2014 address some references to forthcoming books in the book on Lie Groups and Algebras; there remain other such references (some very precise) to expected additional chapters of the book spectral theory.

Besides the Éléments de mathématique series, lectures from the Séminaire Bourbaki also have been periodically published in monograph form since 1948.

## Influence

Notations introduced by Bourbaki include the symbol ${\displaystyle \varnothing }$ for the empty set and a dangerous bend symbol ☡, and the terms injective, surjective, and bijective.[20]

The emphasis on rigour may be seen as a reaction to the work of Henri Poincaré,[21] who stressed the importance of free-flowing mathematical intuition, at a cost of completeness in presentation. The impact of Bourbaki's work initially was great on many active research mathematicians worldwide. For example:

Our time is witnessing the creation of a monumental work: an exposition of the whole of present day mathematics. Moreover this exposition is done in such a way that the common bond between the various branches of mathematics become clearly visible, that the framework which supports the whole structure is not apt to become obsolete in a very short time, and that it can easily absorb new ideas.

Emil Artin (Bull.AMS 59 (1953), 474–479)

It provoked some hostility, too, mostly on the side of classical analysts; they approved of rigour but not of high abstraction. Around 1950, also, some parts of geometry were still not fully axiomatic — in less prominent developments, one way or another, these were brought into line with the new foundational standards, or quietly dropped. This led to a gulf with the way theoretical physics was practiced.[22]

Bourbaki's direct influence has decreased over time.[22] This is partly because certain concepts which are now important, such as the machinery of category theory, are not covered in the treatise. The completely uniform and essentially linear referential structure of the books became difficult to apply to areas closer to current research than the already mature ones treated in the published books, and thus publishing activity diminished significantly from the 1970s.[23] It also mattered that, while especially algebraic structures can be naturally defined in Bourbaki's terms, there are areas where the Bourbaki approach was less straightforward to apply.

On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed.[24] This is particularly true for the less applied parts of mathematics.

The Bourbaki seminar series founded in post-WWII Paris continues; it has been going on since 1948, and contains more than 1000 items. It is an important source of survey articles, with sketches (or sometimes improvements) of proofs. The topics range through all branches of mathematics, including sometimes theoretical physics. The idea is that the presentation should be on the level of specialists, but should be tailored to an audience which is not specialized in the particular field.

## Appraisal of the Bourbaki perspective

The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Göttingen school, particularly Hilbert and the modern algebra school of Noether, Artin and van der Waerden. It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics. But always through a transforming process of reception and selection—their ability to sustain this collective, critical approach has been described as "something unusual".[25]

The following is a list of some of the criticisms commonly made of the Bourbaki approach. Pierre Cartier, a Bourbaki member between 1955 and 1983, said that:[26]

essentially no analysis beyond the foundations: nothing about partial differential equations, nothing about probability. There is also nothing about combinatorics, nothing about algebraic topology, nothing about concrete geometry. And Bourbaki never seriously considered logic. Dieudonné himself was very vocal against logic. Anything connected with mathematical physics is totally absent from Bourbaki's text.

In addition, algorithms are considered off-topic and almost completely omitted.[27] Analysis is treated 'softly', without 'hard' estimates.[28] Measure theory is developed from a functional analytic perspective. Taking the case of locally compact measure spaces as fundamental focuses the presentation on Radon measures and leads to an approach to measurable functions that is cumbersome, especially from the viewpoint of probability theory.[29] However, the last chapter of the book addresses limitations, especially for use in probability theory, of the restriction to locally compact spaces. Logic is treated minimally.[30][31]

Furthermore, Bourbaki makes only limited use of pictures in their presentation. Pierre Cartier is quoted as later saying: "The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith."[26] In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.[32]

While several of Bourbaki's books have become standard references in their fields, some have felt that the austere presentation makes them unsuitable as textbooks.[33] The books' influence may have been at its strongest when few other graduate-level texts in current pure mathematics were available, between 1950 and 1960.[34]

In the longer term, the manifesto of Bourbaki has had a definite and deep influence. In secondary education the new math movement corresponded to teachers influenced by Bourbaki. In France the change was secured by the Lichnerowicz Commission.[35]

## Dieudonné as speaker for Bourbaki

Public discussion of, and justification for, Bourbaki's thoughts has in general been through Jean Dieudonné (who initially was the 'scribe' of the group) writing under his own name. In a survey of le choix bourbachique written in 1977, he did not shy away from a hierarchical development of the 'important' mathematics of the time.

He also wrote extensively under his own name: nine volumes on analysis, perhaps in belated fulfillment of the original project or pretext; and also on other topics mostly connected with algebraic geometry. While Dieudonné could reasonably speak on Bourbaki's encyclopedic tendency and tradition, it may be doubted—after innumerable frank tais-toi, Dieudonné! ("Hush, Dieudonné!") remarks at the meetings—whether all others agreed with him about mathematical writing and research. In particular Serre has often championed greater attention to problem-solving, within number theory especially, not an area treated in the main Bourbaki texts.

Dieudonné stated the view that most workers in mathematics were doing ground-clearing work, in order that a future Riemann could find the way ahead intuitively open. He pointed to the way the axiomatic method can be used as a tool for problem-solving, for example by Alexander Grothendieck. Others found him too close to Grothendieck to be an unbiased observer. Comments in Pál Turán's 1970 speech on the award of a Fields Medal to Alan Baker about theory-building and problem-solving were a reply from the traditionalist camp at the next opportunity,[36] Grothendieck having received the previous Fields Medal in absentia in 1966.

People

## Notes

1. By custom, the group keeps its current membership secret. However, former members regularly discuss their past experience with the collective.
2. Dates refer to entrance into the university, not graduation.
3. The collective's founding generation included a core group of five who led its activities and established its norms, remaining active for several years. Another six minor members participated on shorter-term bases during its earliest days, ranging from a few months to a few years.
4. Aczel described Schwartz as an inter-generational member, the only one to join during the Second World War. However Schwartz did not participate in the group's founding.
5. Most other members were born after the above three generations and were therefore active in the group at later dates. However, two were born contemporaries of the founding generation: Charles Pisot in 1909, and Claude Chabauty in 1910.

## References

• Luca Vercelloni, Filosofia delle strutture, La Nuova Italia, Firenze, 1989
• Maurice Mashaal (2006). Bourbaki: A Secret Society of Mathematicians. American Mathematical Society. ISBN 0-8218-3967-5.
• Amir Aczel (2007). The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. High Stakes Publishing, London. ISBN 1-84344-034-2.
1. The minutes are in the Bourbaki archives — for a full description of the initial meeting consult Liliane Beaulieu in the Mathematical Intelligencer.
2. Mainard, Robert (October 21, 2001). "Le Mouvement Bourbaki" (PDF). afb.31.free.fr. Retrieved October 29, 2018.
3. Aubin, David (1997). "The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Structuralism, and the Oulipo in France". Science in Context. Cambridge University Press. 10 (2): 297–342. doi:10.1017/S0269889700002660.
4. Weil, André (1992). The Apprenticeship of a Mathematician. Birkhäuser Verlag. pp. 93–122. ISBN 978-3764326500.
5. McCleary, John (December 10, 2004). "Bourbaki and Algebraic Topology" (PDF). math.vassar.edu. Archived from the original (PDF) on October 30, 2006.
6. Aczel, 108-109.
7. "according to Groth. IV.22". Retrieved 2018-10-24.
8. "Association des collaborateurs de Nicolas Bourbaki". www.bourbaki.ens.fr (in French). Retrieved 2018-10-29.
9. Mashaal, p. 18.
10. Mashaal, p. 14.
11. Michon, Gérard P. "The Many Faces of Nicolas Bourbaki". Numericana.
12. Bagemihl, F. (1958). "Review: Théorie des ensembles (Chapter III)" (PDF). Bull. Amer. Math. Soc. 64 (6): 390–391. doi:10.1090/s0002-9904-1958-10248-7.
13. Artin, E. (1953). "Review: Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I–VII" (PDF). Bull. Amer. Math. Soc. 59 (5): 474–479. doi:10.1090/s0002-9904-1953-09725-7.
14. Rosenberg, Alex (1960). "Review: Éléments de mathématiques by N. Bourbaki. Book II, Algèbre. Chapter VIII, Modules et anneaux semi-simples" (PDF). Bull. Amer. Math. Soc. 66 (1): 16–19. doi:10.1090/S0002-9904-1960-10371-0.
15. Kaplansky, Irving (1960). "Review: Formes sesquilinéairies et formes quadratiques by N. Bourbaki, Éléments de mathématique I, Livre II" (PDF). Bull. Amer. Math. Soc. 66 (4): 266–267. doi:10.1090/s0002-9904-1960-10461-2.
16. Halmos, Paul (1953). "Review: Intégration (Chap. I-IV) by N. Bourbaki" (PDF). Bull. Amer. Math. Soc. 59 (3): 249–255. doi:10.1090/S0002-9904-1953-09698-7.
17. Munroe, M. E. (1958). "Review: Intégration (Chapter V) by N. Bourbaki" (PDF). Bull. Amer. Math. Soc. 64 (3): 105–106. doi:10.1090/s0002-9904-1958-10176-7.
18. Nagata, M. (1985). "Éléments de mathématique. Algèbre commutative, by N. Bourbaki, Chapitres 8 et 9" (PDF). Bull. Amer. Math. Soc. (N.S.). 12 (1): 175–177. doi:10.1090/s0273-0979-1985-15338-8.
19. Bourbaki, Nicolas. "Topologie Algébrique, Chapitres 1 à 4". springer.com. Springer. Retrieved 2016-02-08.
20. Gunderman, David. "Nicolas Bourbaki: The greatest mathematician who never was". The Conversation. Retrieved 2019-12-14.
21. Bourbaki came to terms with Poincaré only after a long struggle. When I joined the group in the fifties it was not the fashion to value Poincaré at all. He was old-fashioned. Pierre Cartier interviewed by Marjorie Senechall. "The Continuing Silence of Bourbaki". Mathematical Intelligencer. 19: 22–28. 1998.
22. Stewart, Ian (November 1995). "Bye-Bye Bourbaki: Paradigm Shifts in Mathematics". The Mathematical Gazette. The Mathematical Association. 79 (486): 496–498. doi:10.2307/3618076. JSTOR 3618076.
23. Borel, Armand (March 1998). "Twenty-Five Years with Nicolas Bourbaki, (1949-1973)" (PDF). Notices Amer. Math. Soc. 45 (3): 373–380.
24. Guedj, Denis (1985). "Nicholas Bourbaki, collective mathematician : An interview with Claude Chevalley". Math. Intelligencer. 7 (2): 18–22. doi:10.1007/BF03024169.
25. Hector C. Sabelli, Louis H. Kauffman, BIOS (2005), p. 423.
26. "The Continuing Silence of Bourbaki". ega-math.narod.ru. Retrieved 2018-10-29.
27. "Bourbaki Nicolas". publimath.irem.univ-mrs.fr. Retrieved 2018-10-29.
28. Carleson, Lennart (August 2006). "Interview with Lennart Carleson" (PDF). matematikkforeningen.no. Archived from the original (PDF) on 2007-09-28.
29. König, Heinz. "Stochastic Processes on the Basis of New Measure Theory". math.tu-dresden.de. Archived from the original on 2007-03-04.
30. Mathias, Adrien (August 22, 1990). "The Ignorance of Bourbaki" (PDF). dpmms.cam.ac.uk.