Editing Nicolas Bourbaki (section)

===Influence===

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Bourbaki introduced several mathematical notations which have remained in use. Weil took the letter {{large|[[Ø]]}} of the Norwegian alphabet and used it to denote the [[empty set]], {{large|&empty;}}.<ref>[http://jeff560.tripod.com/set.html Earliest Uses of Symbols of Set Theory and Logic.]</ref> This notation first appeared in the Summary of Results on the ''Theory of Sets'',<ref name="set theory">{{cite book |last=Bourbaki |first=Nicolas |title=Theory of Sets |publisher=Springer |isbn=9783540225256 |pages=72, 349 |year=2004}}</ref> and remains in use. The words [[injective]], [[surjective]] and [[Bijection|bijective]] were introduced to refer to [[Function (mathematics)|functions]] which satisfy certain properties.<ref>''Theory of Sets'', p. 84.</ref><ref name="Gunderman">{{Cite web|url=http://theconversation.com/nicolas-bourbaki-the-greatest-mathematician-who-never-was-122845|title=Nicolas Bourbaki: The greatest mathematician who never was|last=Gunderman|first=David|website=The Conversation|date=7 December 2019 |language=en|access-date=2019-12-14}}</ref> Bourbaki used simple language for certain geometric objects, naming them ''pavés'' ([[paving stone]]s) and ''boules'' ([[Ball (mathematics)|balls]]) as opposed to "[[Parallelohedron|parallelotopes]]" or "[[Hypersphere|hyperspheroids]]".{{sfn|Mashaal|p=105}} Similarly in its treatment of topological vector spaces, Bourbaki defined a [[Barrelled space|barrel]] as a set which is [[convex set|convex]], [[balanced set|balanced]], [[absorbing set|absorbing]], and [[closed set|closed]].<ref name="tvs">{{cite book |last=Bourbaki |first=Nicolas |translator-last1=Eggleston |translator-first1=H.G. |translator-last2=Madan |translator-first2=S. |title=Topological Vector Spaces: Chapters 1-5 |year=1987 |publisher=Springer |isbn=9783540423386}} Chapter III, p. 24.</ref> The group were proud of this definition, believing that the shape of a [[wine barrel]] typified the mathematical object's properties.{{sfn|Beaulieu|1999|p=228}}{{sfn|Mashaal|pp=107–08}} Bourbaki also employed a "[[Bourbaki dangerous bend symbol|dangerous bend]]" symbol {{large|☡}} in the margins of its text to indicate an especially difficult piece of material. Bourbaki enjoyed its greatest influence during the 1950s and 1960s, when installments of the ''Éléments'' were published frequently.

Bourbaki had some interdisciplinary influence on other fields, including [[anthropology]] and [[psychology]]. This influence was in the context of [[structuralism]], a school of thought in the [[humanities]] which stresses the relationships between objects over the objects themselves, pursued in various fields by other French intellectuals. In 1943, André Weil met the anthropologist [[Claude Lévi-Strauss]] in New York, where the two undertook a brief collaboration. At Lévi-Strauss' request, Weil wrote a brief appendix describing marriage rules for four classes of people within [[Aboriginal Australian]] society, using a [[mathematical model]] based on [[group theory]].{{sfn|Aczel|pp=129–48}}{{sfn|Aubin|pp=308–11}} The result was published as an appendix in Lévi-Strauss' [[Claude Lévi-Strauss#Expatriation|''Elementary Structures of Kinship'']], a work examining family structures and the [[incest taboo]] in human cultures.<ref name="kinship">{{cite book |title=The Elementary Structures of Kinship |via=[[Internet Archive]] |last=Weil |first=André |chapter=Chapter XIV: Appendix to Part One |editor-last=Lévi-Strauss |editor-first=Claude |chapter-url=https://archive.org/details/TheElementaryStructuresOfKinshipLeviStrauss |year=1971 |pages=[https://archive.org/details/TheElementaryStructuresOfKinshipLeviStrauss/page/n255 221–29]}}</ref> In 1952, Jean Dieudonné and [[Jean Piaget]] participated in an interdisciplinary conference on mathematical and mental structures. Dieudonné described mathematical "mother structures" in terms of Bourbaki's project: composition, neighborhood, and order.{{sfn|Aczel|pp=161–64}} Piaget then gave a talk on children's mental processes, and considered that the psychological concepts he had just described were very similar to the mathematical ones just described by Dieudonné.{{sfn|Aczel|p=162}}{{sfn|Mashaal|p=73}} According to Piaget, the two were "impressed with each other".{{sfn|Aubin|p=318}} The psychoanalyst [[Jacques Lacan]] liked Bourbaki's collaborative working style and proposed a similar collective group in psychology, an idea which did not materialize.{{sfn|Aczel|p=169}}

Bourbaki was also cited by [[Post-structuralism|post-structuralist]] philosophers. In their joint work ''[[Anti-Oedipus]]'', [[Gilles Deleuze]] and [[Félix Guattari]] presented a [[criticism of capitalism]]. The authors cited Bourbaki's use of the axiomatic method (with the purpose of establishing truth) as a distinct counter-example to [[management]] processes which instead seek [[economic efficiency]]. The authors said of Bourbaki's axiomatics that "they do not form a Taylor system", inverting the phrase used by Dieudonné in "The Architecture of Mathematics".{{sfn|Bourbaki 1950|p=227}}<ref name="oedipus">{{cite book |last1=Deleuze |first1=Gilles |last2=Guattari |first2=Félix |title=Anti-Oedipus |url=https://archive.org/details/antioedipuscapit00dele_367 |url-access=limited |date=1972 |publisher=[[University of Minnesota Press]] |isbn=978-0816612253 |page=[https://archive.org/details/antioedipuscapit00dele_367/page/n270 251] }}</ref> In ''[[The Postmodern Condition]]'', [[Jean-François Lyotard]] criticized the "legitimation of knowledge", the process by which statements become accepted as valid. As an example, Lyotard cited Bourbaki as a group which produces knowledge within a given system of rules.{{sfn|Aubin|pp=332–33}}<ref name="Lyotard">{{cite book |last=Lyotard |first=Jean-François |title=The Postmodern Condition: A Report on Knowledge |publisher=University of Minnesota Press |series=Theory and History of Literature |volume=10 |isbn=978-0816611737 |pages=43, 57–60 |year=1984 }}</ref> Lyotard contrasted Bourbaki's hierarchical, "structuralist" mathematics with the [[catastrophe theory]] of [[René Thom]] and the fractals of [[Benoit Mandelbrot]],{{efn|Mandelbrot was the nephew of Bourbaki founder Szolem Mandelbrojt.{{sfn|Beaulieu|1993|p=31}}<ref>{{cite web |url=http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/mandelbrot-benoit.pdf |title=Benoit B. Mandelbrot, 1924–2010: A Biographical Memoir by Michael Frame |last=Frame |first=Michael |date=2014 |publisher=[[National Academy of Sciences]] |website=nasonline.org |page=2}}</ref>  Like early Bourbaki associate Gaston Julia, Mandelbrot also worked on fractals.}} expressing preference for the latter "postmodern science" which problematized mathematics with "fracta, catastrophes, and pragmatic paradoxes".{{sfn|Aubin|pp=332–33}}<ref name="Lyotard" />

Although biographer [[Amir Aczel]] stressed Bourbaki's influence on other disciplines during the mid-20th century, Maurice Mashaal moderated the claims of Bourbaki's influence in the following terms:

{{Blockquote|text=While Bourbaki's structures were often mentioned in social science conferences and publications of the era, it seems that they didn't play a real role in the development of these disciplines.  David Aubin, a science historian who analyzed Bourbaki's role in the structuralist movement in France, believes Bourbaki's role was that of a "cultural connector".{{sfn|Aubin|p=297}}  According to Aubin, while Bourbaki didn't have any mission outside of mathematics, the group represented a sort of link between the various cultural movements of the time.  Bourbaki provided a simple and relatively precise definition of concepts and structures, which philosophers and social scientists believed was fundamental within their disciplines and in bridges among different areas of knowledge.  Despite the superficial nature of these links, the various schools of structuralist thinking, including Bourbaki, were able to support each other.  So, it is not a coincidence that these schools suffered a simultaneous decline in the late 1960s. |title=Maurice Mashaal, citing David Aubin{{sfn|Mashaal|p=73}}{{efn|Maurice Mashaal and Amir Aczel each wrote separate biographies on Bourbaki, both published in 2006.  In a review of both books, [[Michael Atiyah]] wrote that "the basic historical facts are well known and are set out in both the books under review".  However Atiyah identified Mashaal's book as the better of the two and criticized Aczel's book, writing: "I was not convinced of the total reliability of its (Aczel's) sources, nor of its philosophical credentials."  Atiyah also wrote that the collaboration between Weil and Lévi-Strauss was a "slightly tenuous link" which Aczel used to make "grand" claims on the scale of Bourbaki's interdisciplinary influence.<ref name="Atiyah">{{cite web |url=https://www.ams.org/notices/200709/tx070901150p.pdf |title=Book Review: Bourbaki, A Secret Society of Mathematicians and The Artist and the Mathematician, Reviewed by Michael Atiyah |last=Atiyah |first=Michael |publisher=American Mathematical Society |website=ams.org}}</ref>}}{{efn|In a 2011 letter to the ''Mathematical Intelligencer'', the mathematician [[:de:Jean-Michel Kantor|Jean-Michel Kantor]]<small>[[:de:Wikipedia:Hauptseite|[de]]]</small> was harshly critical of the notion that Bourbaki's mathematical structures had anything to do with the structuralism of the humanities, rejecting the connections made by Aczel in 2006.<ref name="Kantor">{{cite journal |last=Kantor |first=Jean-Michel |title=Bourbaki's Structures and Structuralism |journal=Mathematical Intelligencer |volume=33 |issue=1 |page=1 |date=2011 |url=https://www.researchgate.net/publication/251293966 |doi=10.1007/s00283-010-9173-4 |doi-access=free }}</ref>  Kantor observed that the two versions of structuralism had developed independently of one another, and that Lévi-Strauss' conception of structure had derived from the [[Prague linguistic circle|Prague circle]] of linguistics, not from Bourbaki.  On the other hand, Aczel had already acknowledged the linguistic origins of the structuralism of the humanities.{{sfn|Aczel|pp=149–59}}  In 1997 David Aubin had pre-emptively moderated both extremes, observing that the two schools of thought had distinct origins, but also had certain interactions and "common features".  Aubin also cited Lévi-Strauss to show that the latter had reached certain conclusions in anthropology independently of Weil's mathematical help, although Weil's help provided confirmation of Lévi-Strauss' conclusions.{{sfn|Aubin|p=311}}  This undermined Aczel's argument that mathematics and Bourbaki played an important role in the development of structuralism in the humanities, although Aubin also stressed that the two schools had some collaboration.}} }}

The impact of "structuralism" on mathematics itself was also criticized. The mathematical historian Leo Corry argued that Bourbaki's use of mathematical structures was unimportant within the ''Éléments'', having been established in ''Theory of Sets'' and cited infrequently afterwards.<ref name="Corry1992">{{cite journal |last=Corry |first=Leo |s2cid=16981077 |author-link=Leo Corry |title=Nicolas Bourbaki and the concept of Mathematical Structure |url=https://www.tau.ac.il/~corry/publications/articles/bourbaki-structures-synthese.html |journal=Synthese |volume=92 |issue=3 |pages=328–32 |date=September 1992 |doi=10.1007/BF00414286 }}</ref><ref name="Corry2001">{{cite book |last=Corry |first=Leo |title=Changing Images in Mathematics: From the French Revolution to the New Millennium |chapter=Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics |editor-last1=Bottazzini |editor-first1=Umberto |editor-last2=Dalmedico |editor-first2=Amy Dahan |url=https://www.tau.ac.il/~corry/publications/articles/images-structures.html |publisher=Routledge |pages=1–3, 17–18 |date=2001 |isbn=978-0415868273 }}</ref>{{sfn|Corry|2004|p=338}}{{sfn|Corry|2009|pp=25–31}} Corry described the "structural" view of mathematics promoted by Bourbaki as an "image of knowledge"—a conception about a scientific discipline—as opposed to an item in the discipline's "body of knowledge", which refers to the actual scientific results in the discipline itself.<ref name="Corry2001" />

Bourbaki also had some influence in the arts. The literary collective [[Oulipo]] was founded on 24 November 1960 under circumstances similar to Bourbaki's founding, with the members initially meeting in a restaurant. Although several members of Oulipo were mathematicians, the group's purpose was to create [[experimental literature]] by playing with language. Oulipo frequently employed mathematically-based [[constrained writing]] techniques, such as the [[Oulipo#Constraints|S+7 method]]. Oulipo member [[Raymond Queneau]] attended a Bourbaki conference in 1962.{{sfn|Mashaal|p=73}}{{sfn|Aczel|pp=173–82}}

In 2016, an anonymous group of economists collaboratively wrote a note alleging academic misconduct by the authors and editor of a paper published in the ''[[American Economic Review]]''.<ref name="Bearbaki">{{cite web | url=https://mpra.ub.uni-muenchen.de/71699/ | title=A Comment on "Family Ruptures, Stress, and the Mental Health of the Next Generation" | last=Nicolas |first= Bearbaki |date=June 4, 2016 | access-date = February 1, 2021}}</ref><ref>{{cite web | url=https://retractionwatch.com/2016/05/26/economists-go-wild-over-overlooked-citations-in-preprint-on-prenatal-stress/ | date=May 26, 2016| title = Economists go wild over overlooked citations in preprint on prenatal stress|publisher=Retraction Watch|access-date=February 1, 2021}}</ref> The note was published under the name Nicolas Bearbaki in homage to Nicolas Bourbaki.<ref>{{cite web | url=https://statmodeling.stat.columbia.edu/2016/09/23/why-doesnt-this-apparent-case-of-plagiarism-bother-me-at-a-gut-level/ | title= Andrew Gelman is not the plagiarism police because there is no such thing as the plagiarism police. | last=Andrew |first= Gelman |date= September 23, 2016 | access-date = February 1, 2021}}</ref>

In 2018, the American musical duo [[Twenty One Pilots]] released a [[concept album]] named ''[[Trench (album)|Trench]]''. The album's conceptual framework was the mythical city of "Dema" ruled by nine "bishops"; one of the bishops was named "Nico", short for Nicolas Bourbaki.  Another of the bishops was named Andre, which may refer to André Weil.  Following the album's release, there was a spike in internet searches for "Nicolas Bourbaki".<ref name="numericana" />{{efn|Similarly, Bourbaki created nicknames for its members.  Jean Delsarte was referred to as "bishop", which may have been a reference to his Catholicism.{{sfn|Mashaal|p=111}} }}