Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.
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.
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.
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.
- Set theory (Théorie des ensembles) 
- Algebra (Algèbre) 
- Topology (Topologie générale)
- Functions of one real variable (Fonctions d'une variable réelle)
- Topological vector spaces (Espaces vectoriels topologiques)
- Integration (Intégration) 
- Commutative algebra (Algèbre commutative) 
- Lie groups and algebras (Groupes et algèbres de Lie)
- Spectral theory (Théories spectrales)
- Algebraic topology (Topologie Algébrique)
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 8-9 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.
Influence on mathematics in general
The emphasis on rigour may be seen as a reaction to the work of Henri Poincaré, 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.
Bourbaki's direct influence has decreased over time. 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. 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. 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.
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. 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.
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; it was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue.
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".
The following is a list of some of the criticisms commonly made of the Bourbaki approach. Pierre Cartier, a Bourbaki member 1955–1983, commented explicitly on several of these points:
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.
- Algorithmic content is not considered on-topic and is almost completely omitted.
- Analysis is treated 'softly', without 'hard' estimates.
- 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. However, the last chapter of the book addresses limitations, especially for use in probability theory, of the restriction to locally compact spaces.
- Combinatorics is not discussed.
- Logic is treated minimally.
- Applications are not covered.
Furthermore, Bourbaki makes only limited use of pictures in their presentation. Pierre Cartier, in the article cited above, is quoted as later saying: "The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith." In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.
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, Grothendieck having received the previous Fields Medal in absentia in 1966.
Bourbaki's influence on mathematics education
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. 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.
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.
The influence on graduate education in pure mathematics is perhaps most noticeable in the treatment now current of Lie groups and Lie algebras. Dieudonné at one point said 'one can do nothing serious without them', for which he was reproached; but the change in Lie theory to its everyday usage owes much to the type of exposition Bourbaki championed. Beforehand Jacques Hadamard despaired of ever getting a clear idea of it.[vague]
- Bourbaki–Witt theorem
- Jacobson–Bourbaki theorem
- Bourbaki dangerous bend symbol
- Arthur Besse
- G. W. Peck
- John Rainwater
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- "Topologie Algébrique, Chapitres 1 à 4 | N. Bourbaki | Springer". www.springer.com. Retrieved 2016-02-08.
- 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. 
- 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.
- Borel (1998)
- Chevalley in Guedj (1985)
- The minutes are in the Bourbaki archives — for a full description of the initial meeting consult Liliane Beaulieu in the Mathematical Intelligencer.
- This resulted in a complete change of personnel by 1958; see Robert Mainard paper cited below. However, the Aubin paper cited below quotes the historian Liliane Beaulieu as never having found written affirmation of this rule.
- A. Weil: The Apprenticeship of a Mathematician, Birkhäuser Verlag 1992, pp 93-122.
- It is said that Weil's wife Evelyne supplied Nicolas. (Mentioned by McCleary (PDF) Archived 2006-10-30 at the Wayback Machine.. This is more or less confirmed by Robert Mainard((PDF), a long article in French, which gives numerous further details: why N?, and the prank lecture of Raoul Husson in a false beard that gave rise to Bourbaki's theorem). They married in 1937, she having previously been with de Possel; who then unsurprisingly left the group.
- Hector C. Sabelli, Louis H. Kauffman, BIOS (2005), p. 423.
- The Continuing Silence of Bourbaki, article from the Mathematical Intelligencer
- This is one of the reasons for diminishing influence: Le développement des mathématiques dites appliquées, de la statistique et des probabilités, des théories liées à l'informatique a diminué l'influence de Bourbaki
- Lennart Carleson spoke of this in an interview (Infomat August 2006 (PDF)): ...that book [from 1968] was written mostly as a way to encourage the teachers to stay with established values. That was during the Bourbaki and New Math period and mathematics was really going to pieces, I think. The teachers were very worried and they had very little backing.
- Heinz König: The traditional abstract measure theory which emerged from the achievements of Borel and Lebesgue in the first two decades of the 20th century is burdened with its total limitation to sequential procedures and its neglect of regularity. The alternative theory due to Bourbaki which arose in the middle of the century was able to relieve these burdens, but produced new ones. In particular its fundamental turn to inner regularity, based on the profound role of compactness, was done with the inappropriate weapons from the outer arsenal, which subsequently enforced that unfortunate construction named the essential one. All this produced serious obstacles against a unified theory of measure and integration, for example for the notion of signed measures, the formation of products and for the representation theorems of Daniell-Stone and Riesz types.
- Discussed by the set theorist Adrian Mathias (The Ignorance of Bourbaki (PDF)). See also Mashaal (2006), p.120, "Lack of interest in foundations".
- In the French context it has been said that geometry was in effect exiled from secondary teaching: Pour ce qui est des années 1960, l’effet de la réforme dite des mathématiques modernes sur l’enseignement de la géométrie est bien connu : si Dieudonné, comme Bourlet finalement, lance "A bas Euclide", le résultat n’est pas l’élaboration d’une géométrie plus expérimentale, plus intuitive. C’est l’effacement de la géométrie derrière l’algèbre linéaire et la quasi-disparition de l’enseignement de la géométrie élémentaire au collège et au lycée pour une dizaine d’années.—"As for the 1960s, the effect of this reform of modern mathematics on the teaching of geometry is well-known: if Dieudonné, like Bourlet finally, says "push Euclid back," the result is not the development of a geometry that is more experimental, more intuitive. It's the erasure of geometry behind linear algebra, and the quasi-disappearance of the teaching of elementary geometry in high school, for ten years."
- Church, Alonzo (1 January 1972). "Review of On the Work of ., Paul Turán; Effective Methods in the Theory of Numbers., Alan Baker". 37 (3): 606–606. doi:10.2307/2272765. JSTOR 2272765.
- Hewitt, Edwin (1956). "Review: Espaces vectoriels topologiques". Bulletin of the American Mathematical Society. 62 (5): 507–508. doi:10.1090/S0002-9904-1956-10042-6. 
- Mashaal (2006) Ch.10: New Math in the Classroom
- Borel, Armand (1998). "Twenty-Five Years with Nicolas Bourbaki, (1949-1973)" (PDF). Notices Amer. Math. Soc. 45 (3): 373–380.
- Guedj, D. (1985). "Nicholas Bourbaki, collective mathematician : An interview with Claude Chevalley". Math. Intelligencer. 7 (2): 18–22. doi:10.1007/BF03024169.
- Aubin, David (1997). "The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics" (PDF). Science in Context. 10: 297–342. doi:10.1017/s0269889700002660.[permanent dead link]
- 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.
- André Weil (1992). ``The Apprenticeship of a Mathematician``. Birkhauser. ISBN 0-8176-2650-6. pp. 100–105.
- Official Website of L'Association des Collaborateurs de Nicolas Bourbaki (in French)
- Archives of the association (in French)
- A long article about Nicolas Bourbaki, from PlanetMath
- 25 Years with Bourbaki, by Armand Borel
- O'Connor, John J.; Robertson, Edmund F., "Nicolas Bourbaki", MacTutor History of Mathematics archive, University of St Andrews.
- Apology of Euclid, by Semën Samsonovich Kutateladze