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| {{About|the group of mathematicians named Nicolas Bourbaki|the family of French officers named Bourbaki|Bourbaki family (disambiguation)|the computer scientist|Nikolaos Bourbakis}}
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| '''Nicolas Bourbaki''' is the collective [[pseudonym]] under which a group of (mainly [[France|French]]) 20th-century [[mathematician]]s wrote a series of books presenting an exposition of modern advanced [[mathematics]], beginning in 1935. With the goal of founding 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 discussed.
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| [[File:Bourbaki congress1938.png|thumb|Bourbaki congress, 1938]]
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| While there is no 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]].
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| ==Books by Bourbaki==
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| [[File:Bourbaki, Theorie des ensembles maitrier.jpg|thumb|First volume of ''[[Éléments de mathématique]]'', 1970 edition]]
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| Bourbaki's main work is the ''[[Éléments de mathématique|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 tries to take up mathematics from the very beginning, proceed axiomatically and give complete proofs.
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| <ol type="I">
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| <li> [[Set theory]] (''Théorie des ensembles'') </li>
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| <li> [[Algebra]] (''Algèbre'') </li>
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| <li> [[Topology]] (''Topologie générale'') </li>
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| <li> [[real analysis|Functions of one real variable]] (''Fonctions d'une variable réelle'') </li>
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| <li> [[Topological vector space]]s (''Espaces vectoriels topologiques'') </li>
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| <li> [[Integral|Integration]] (''Intégration'') </li>
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| </ol>
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| and later | |
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| <ol type="I" start="7">
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| <li> [[Commutative algebra]] (''Algèbre commutative'') </li>
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| <li> [[Lie theory]] (''Groupes et algèbres de Lie'') </li>
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| <li> [[Spectral theory]] (''Théories spectrales'') </li>
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| </ol>
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| The book ''Variétés différentielles et analytiques'' was a ''fascicule de résultats'', that is, a summary of results, on the theory of [[manifold]]s, rather than a worked-out exposition. A final volume IX on [[spectral theory]] (''Théories spectrales'') from 1983 marked the presumed end of the publishing project; but a further commutative algebra [[Fascicle (book)|fascicle]] was produced in 1998 and the eighth chapter of Algèbre was published in 2012.
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| Besides the ''[[Éléments de mathématique]]'' series, lectures from the [[Séminaire Nicolas Bourbaki|Séminaire Bourbaki]] also have been periodically published in book form since 1948.
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| ==Influence on mathematics in general==
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| Notations introduced by Bourbaki include the symbol <math> \varnothing </math> for the [[empty set]] and a [[Bourbaki dangerous bend symbol|dangerous bend symbol]], and the terms ''[[injective]]'', ''[[surjective]]'', and ''[[bijective]]''.{{citation needed|date=December 2011}}
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| The emphasis on [[rigour]] may be seen as a reaction to the work of [[Henri Poincaré]],<ref>''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. {{Cite journal| title=The Continuing Silence of Bourbaki | journal=[[Mathematical Intelligencer]] | year=1998 | volume=19 | pages=22–28}} [http://www.ega-math.narod.ru/Bbaki/Cartier.htm]</ref> 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 world-wide. As an example here is a quotation from the great number-theorist Emil Artin (Bull.AMS 1953, 474-479): "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 ..." | |
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| It provoked some hostility, too, mostly on the side of [[classical analysis|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 undoubtedly led to a gulf with the way [[theoretical physics]] is practiced.<ref name=ByeByeBourbaki>>{{Cite journal| title= Bye-Bye Bourbaki: Paradigm Shifts in Mathematics | author=Ian Stewart | journal=The Mathematical Gazette| year=1995 | month=11 | volume=79 | pages=496–498 | publisher=[[The Mathematical Association]] | doi= 10.2307/3618076 | issue=486 | jstor= 3618076}}</ref>
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| Bourbaki's direct influence has decreased over time.<ref name=ByeByeBourbaki/> This is partly because certain concepts which are now important, such as the machinery of [[category theory]], are not covered in the treatise. {{Citation needed|date=January 2014}} 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.<ref>Borel (1998)</ref> It also mattered that, while especially [[algebraic structure]]s can be naturally defined in Bourbaki's terms, there are areas where the Bourbaki approach was less straightforward to apply.{{Citation needed|date=March 2009}}
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| 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.<ref>Chevalley in Guedj (1985)</ref> This is particularly true for the less applied parts of mathematics.
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| 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 article]]s, 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.
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| ==The group==
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| [[image:Weil.jpg|thumb|170px|[[André Weil]], de facto early leader of the group]]
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| 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.<ref>The minutes are in the Bourbaki archives — for a full description of the initial meeting consult Liliane Beaulieu in the ''[[Mathematical Intelligencer]]''.</ref> [[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]].
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| 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.<ref>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.</ref>
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| 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 space]]s were to be handled, to precipitate a guaranteed crisis.
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| The name "Bourbaki" refers to a French general [[Charles Denis Bourbaki]];<ref>A. Weil: The Apprenticeship of a Mathematician, Birkhäuser Verlag 1992, pp 93-122.</ref> it was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue.<ref>It is said that Weil's wife Evelyne supplied ''Nicolas''. ([http://www.math.vassar.edu/faculty/mccleary/Bourbaki.pdf Mentioned by McCleary (PDF)]. This is more or less confirmed by [http://www.academie-stanislas.org/TomeXIII/Mainard98.pdf 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.</ref>
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| ==Appraisal of the Bourbaki perspective==
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| The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the [[University of Göttingen|Göttingen]] school, particularly [[David Hilbert|Hilbert]] and the modern algebra school of [[Emmy Noether]], [[Emil Artin|Artin]] and [[Bartel van der Waerden|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".<ref>Hector C. Sabelli, [[Louis H. Kauffman]], BIOS (2005), p. 423.</ref>
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| The following is a list of some of the criticisms commonly made of the Bourbaki approach. [[Pierre Cartier (mathematician)|Pierre Cartier]], a Bourbaki member 1955–1983, commented explicitly on several of these points:<ref>[http://ega-math.narod.ru/Bbaki/Cartier.htm ''The Continuing Silence of Bourbaki'', article from the ''Mathematical Intelligencer'']</ref> ''...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.
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| * [[algorithm]]ic content is not considered on-topic and is almost completely omitted<ref>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''[http://publimath.irem.univ-mrs.fr/glossaire/BO025.htm]</ref>
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| * [[problem solving]], in the sense of [[heuristics]], receives less emphasis than axiomatic theory-building<ref>[[Tim Gowers]] discusses at length the ''distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories'' in his [http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf ''The Two Cultures of Mathematics'' (PDF)].</ref>
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| * [[Mathematical analysis|analysis]] is treated 'softly', without 'hard' estimates<ref>[[Lennart Carleson]] spoke of this in an interview ([http://www.matematikkforeningen.no/INFOMAT/06/0608.pdf ''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.''</ref>
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| * [[Measure theory]] is developed from a [[functional analysis|functional analytic]] perspective. Taking the case of [[locally compact space|locally compact]] measure spaces as fundamental focuses the presentation on [[Radon measure]]s and leads to an approach to measurable functions that is cumbersome, especially from the viewpoint of probability theory.<ref>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.''[http://www.math.tu-dresden.de/~pos_iv/Abstracts/koenig_abstract/index.html]</ref> However, the last chapter of the book addresses limitations, especially for use in [[probability theory]], of the restriction to locally compact spaces.
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| * [[combinatorics]] is not discussed
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| * [[Mathematical logic|logic]] is treated minimally<ref>Discussed by the set theorist Adrian Mathias ([http://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf ''The Ignorance of Bourbaki'' (PDF)]). See also Mashaal (2006), p.120, "Lack of interest in foundations".</ref>
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| * [[applied mathematics|applications]] are not covered.
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| Furthermore, Bourbaki make no 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]].<ref>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."[http://www.apmep.asso.fr/spip.php?article210]</ref>
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| ==Dieudonné as speaker for Bourbaki==
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| 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.
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| He also wrote extensively under his own name: nine volumes on [[mathematical analysis|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.
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| Dieudonné stated the view that most workers in mathematics were doing ground-clearing work, in order that a future [[Bernhard Riemann|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 (mathematician)|Alan Baker]] about theory-building and problem-solving were a reply from the traditionalist camp at the next opportunity<ref>[http://links.jstor.org/sici?sici=0022-4812(197209)37%3A3%3C606%3AOTWOAB%3E2.0.CO%3B2-9 On the Work of Alan Baker]</ref>{{Failed verification|date=July 2008}}, Grothendieck having received the previous Fields Medal ''in absentia'' in 1966.
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| ==Bourbaki's influence on mathematics education==
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| 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.<ref>{{Cite journal| last=Hewitt | first=Edwin | title=Review: Espaces vectoriels topologiques | journal=[[Bulletin of the American Mathematical Society]] | volume=62 | year=1956 | pages=507–508 | doi=10.1090/S0002-9904-1956-10042-6| issue=5 }} [http://www.ams.org/bull/1956-62-05/S0002-9904-1956-10042-6/home.html]</ref> 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.<ref>http://turnbull.mcs.st-and.ac.uk/~history/PrintHT/Bourbaki_2.html</ref>
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| 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 [[Andre Lichnerowicz|Lichnerowicz Commission]].<ref>Mashaal (2006) Ch.10: New Math in the Classroom</ref>
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| The influence on graduate education in pure mathematics is perhaps most noticeable in the treatment now current of [[Lie group]]s and [[Lie algebra]]s. 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.{{citation needed|date=March 2013}}{{vague|date=June 2013}}
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| ==See also==
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| *[[Bourbaki–Witt theorem]]
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| *[[Jacobson–Bourbaki theorem]]
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| *[[Bourbaki dangerous bend symbol]]
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| *[[Arthur Besse]]
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| *[[G. W. Peck]]
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| *[[John Rainwater]]
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| ==Notes==
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| {{Reflist|colwidth=45em}}
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| ==References==
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| *{{Cite journal
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| | last =Borel
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| | first = Armand
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| | authorlink = Armand Borel
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| | coauthors =
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| | title = Twenty-Five Years with Nicolas Bourbaki, (1949-1973)
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| | journal = Notices Amer. Math. Soc.
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| | volume = 45
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| | issue = 3
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| | pages = 373–380
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| | publisher =
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| | location =
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| | year = 1998
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| | url = http://www.ams.org/notices/199803/borel.pdf
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| | doi =
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| | id =
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| | accessdate = }}
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| *{{Cite journal
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| | last =Guedj
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| | first = D
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| | authorlink =
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| | coauthors =
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| | title = Nicholas Bourbaki, collective mathematician : An interview with Claude Chevalley
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| | journal = Math. Intelligencer
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| | volume = 7
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| | issue = 2
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| | pages = 18–22
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| | publisher =
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| | location =
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| | year = 1985
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| | url =
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| | doi = 10.1007/BF03024169
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| | id =
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| | accessdate = }}
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| * {{cite journal | last1 = Aubin | first1 = David | author-separator =, | author-name-separator= | year = 1997 | title = The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics | url = http://www.institut.math.jussieu.fr/~daubin/publis/1997.pdf | format = PDF | journal = Science in Context | volume = 10 | issue = | pages = 297–342 }}
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| * Luca Vercelloni, Filosofia delle strutture, La Nuova Italia, Firenze, 1989
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| * Maurice Mashaal (2006). ''Bourbaki: A Secret Society of Mathematicians''. [[American Mathematical Society]]. ISBN 0-8218-3967-5.
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| * [[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.
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| ==External links==
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| *[http://www.bourbaki.ens.fr/ Official Website of ''L'Association des Collaborateurs de Nicolas Bourbaki''] {{fr icon}}
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| *[http://mathdoc.emath.fr/archives-bourbaki/ Archives of the association] {{fr icon}}
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| *[http://planetmath.org/node/34161 A long article about Nicolas Bourbaki], from [[PlanetMath]]
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| *[http://www.ega-math.narod.ru/Bbaki/Bourb3.htm 25 Years with Bourbaki], by [[Armand Borel]]
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| * {{MacTutor Biography|id=Bourbaki}}
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| *[http://www.math.nsc.ru/LBRT/g2/english/ssk/euclid.html Apology of Euclid], by [[S. Kutateladze]]
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| {{DEFAULTSORT:Bourbaki, Nicolas}}
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| [[Category:Nicolas Bourbaki]]
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| [[Category:Collective pseudonyms]]
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| [[Category:French mathematicians]]
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| [[Category:Collegiate secret societies]]
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| [[Category:Pseudonymous mathematicians]]
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| [[Category:Large-scale mathematical formalization projects]]
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