Nicolas Bourbaki

Nicolas Bourbaki (French: nikɔla buʁbaki) is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the Éléments de mathématique…

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Jump to content Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search Appearance Donate Create account Log in Personal tools Donate Create account Log in Contents move to sidebar hide (Top) 1 Background 2 The Bourbaki collective Toggle The Bourbaki collective subsection 2.1 Founding 2.2 World War II 2.3 Postwar until the present 3 Working method Toggle Working method subsection 3.1 Humor 4 Works Toggle Works subsection 4.1 Éléments de mathématique 4.2 Séminaire Bourbaki 4.3 Articles 4.4 La Tribu 5 Membership 6 Influence and criticism Toggle Influence and criticism subsection 6.1 Influence 6.2 Praise 6.3 Criticism 7 See also 8 Notes 9 References 10 Bibliography 11 External links Toggle the table of contents Nicolas Bourbaki 49 languages العربية Asturianu Azərbaycanca Беларуская Български বাংলা Català Čeština Deutsch Español Eesti Euskara فارسی Suomi Français Gaeilge Galego עברית हिन्दी Hrvatski Kreyòl ayisyen Magyar Հայերեն Íslenska Italiano 日本語 Қазақша 한국어 Кыргызча मराठी Nederlands Norsk nynorsk Norsk bokmål Polski Piemontèis Português Română Русский Sicilianu Slovenčina Shqip Svenska ไทย Türkçe Українська Oʻzbekcha / ўзбекча Tiếng Việt 粵語 中文 Edit links Article Talk English Read Edit View history Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikiquote Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia This is an old revision of this page, as edited by Stca74 ( talk | contribs ) at 10:17, 9 March 2024 (Undid revision 1212199334 by 37.174.118.17 -- The plural in the English name is established by the published translations) . The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision . Revision as of 10:17, 9 March 2024 by Stca74 ( talk | contribs ) (Undid revision 1212199334 by 37.174.118.17 -- The plural in the English name is established by the published translations) ( diff ) ← Previous revision | Latest revision ( diff ) | Newer revision → ( diff ) Pseudonym of a group of mathematicians This article is about the group of mathematicians. For the family of French officers, see Bourbaki family (disambiguation) . For the computer scientist, see Nikolaos Bourbakis . Association of Collaborators of Nicolas Bourbaki Association des collaborateurs de Nicolas Bourbaki File:Bourbaki congress1938.png Bourbaki congress at Dieulefit in 1938. From left, Simone Weil , a Charles Pisot , André Weil , Jean Dieudonné (sitting), Claude Chabauty , Charles Ehresmann , and Jean Delsarte . 2 Named after Charles-Denis Bourbaki Formation 10 December 1934 (first unofficial meeting) 10–17 July 1935 (first official, founding conference) Founders Henri Cartan Claude Chevalley Jean Coulomb Jean Delsarte Jean Dieudonné Charles Ehresmann Szolem Mandelbrojt René de Possel André Weil Founded at Latin Quarter , Paris, France (first unofficial meeting) Besse-en-Chandesse , France (first official, founding conference) Type Voluntary association Purpose Publication of textbooks in pure mathematics Headquarters École Normale Supérieure , Paris Members Confidential Official language French Website www .bourbaki .fr Nicolas Bourbaki ( Template:IPA-fr ) is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis . Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics . The series is known collectively as the Éléments de mathématique ( Elements of Mathematics ), the group's central work. Topics treated in the series include set theory , abstract algebra , topology , analysis, Lie groups and Lie algebras . Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg , Henri Cartan complained to his colleague André Weil of the inadequacy of available course material, which prompted Weil to propose a meeting with others in Paris to collectively write a modern analysis textbook. The group's core founders were Cartan, Claude Chevalley , Jean Delsarte , Jean Dieudonné and Weil; others participated briefly during the group's early years, and membership has changed gradually over time. Although former members openly discuss their past involvement with the group, Bourbaki has a custom of keeping its current membership secret. The group's name derives from the 19th century French general Charles-Denis Bourbaki , who had a career of successful military campaigns before suffering a dramatic loss in the Franco-Prussian War . 3 The name was therefore familiar to early 20th-century French students. Weil remembered an ENS student prank in which an upperclassman posed as a professor and presented a "theorem of Bourbaki"; the name was later adopted. The Bourbaki group holds regular private conferences for the purpose of drafting and expanding the Éléments . Topics are assigned to subcommittees, drafts are debated, and unanimous agreement is required before a text is deemed fit for publication. Although slow and labor-intensive, the process results in a work which meets the group's standards for rigour and generality. The group is also associated with the Séminaire Bourbaki , a regular series of lectures presented by members and non-members of the group, also published and disseminated as written documents. Bourbaki maintains an office at the ENS. 4 Nicolas Bourbaki was influential in 20th-century mathematics, particularly during the middle of the century when volumes of the Éléments appeared frequently. The group is noted among mathematicians for its rigorous presentation and for introducing the notion of a mathematical structure , an idea related to the broader, interdisciplinary concept of structuralism . 5 6 7 Bourbaki's work informed the New Math , a trend in elementary math education during the 1960s. Although the group remains active, its influence is considered to have declined due to infrequent publication of new volumes of the Éléments . However, since 2012 the group has published four new (or significantly revised) volumes, the most recent in 2023 (treating spectral theory ). Moreover, at least three further volumes are under preparation. Background Charles-Denis Bourbaki , 19th century general and namesake of the collective Charles-Denis Sauter Bourbaki was a successful general during the era of Napoleon III , serving in the Crimean War and other conflicts. During the Franco-Prussian war however, Charles-Denis Bourbaki suffered a major defeat in which the Armée de l'Est , under his command, retreated across the Swiss border and was disarmed. The general unsuccessfully attempted suicide. The dramatic story of his defeat was known in the French popular consciousness following his death. 8 9 Gaston Julia (right), who was not a member of Bourbaki, lost his nose during World War I. The war created a lost generation of mathematical knowledge, which the Bourbaki founders sought to fill. In the early 20th century, the First World War affected Europeans of all professions and social classes, including mathematicians and male students who fought and died in the front. For example, the French mathematician Gaston Julia , a pioneer in the study of fractals , lost his nose during the war and wore a leather strap over the affected part of his face for the rest of his life. The deaths of ENS students resulted in a lost generation in the French mathematical community; 10 the estimated proportion of ENS mathematics students (and French students generally) who died in the war ranges from one-quarter to one-half, depending on the intervals of time (c. 1900–1918, especially 1910–1916) and populations considered. 11 12 Furthermore, Bourbaki founder André Weil remarked in his memoir Apprenticeship of a Mathematician that France and Germany took different approaches with their intelligentsia during the war: while Germany protected its young students and scientists, France instead committed them to the front, owing to the French culture of egalitarianism . 12 A succeeding generation of mathematics students attended the ENS during the 1920s, including Weil and others, the future founders of Bourbaki. During his time as a student, Weil recalled a prank in which an upperclassman, Raoul Husson fr , posed as a professor and gave a math lecture, ending with a prompt: "Theorem of Bourbaki: you are to prove the following...". Weil was also aware of a similar stunt around 1910 3 in which a student claimed to be from the fictional, impoverished nation of "Poldevia" and solicited the public for donations. 13 14 Weil had strong interests in languages and Indian culture , having learned Sanskrit and read the Bhagavad Gita . 15 16 After graduating from the ENS and obtaining his doctorate, Weil took a teaching stint at the Aligarh Muslim University in India. While there, Weil met the mathematician Damodar Kosambi , who was engaged in a power struggle with one of his colleagues. Weil suggested that Kosambi write an article with material attributed to one "Bourbaki", in order to show off his knowledge to the colleague. 17 Kosambi took the suggestion, attributing the material discussed in the article to "the little-known Russian mathematician D. Bourbaki , who was poisoned during the Revolution." It was the first article in the mathematical literature with material attributed to the eponymous "Bourbaki". 18 19 20 Weil's stay in India was short-lived; he attempted to revamp the mathematics department at Aligarh, without success. 21 The university administration planned to fire Weil and promote his colleague Vijayaraghavan to the vacated position. However, Weil and Vijayaraghavan respected one another. Rather than play any role in the drama, Vijayaraghavan instead resigned, later informing Weil of the plan. 22 Weil returned to Europe to seek another teaching position. He ended up at the University of Strasbourg, joining his friend and colleague Henri Cartan. 23 The Bourbaki collective Bourbaki was founded to produce a text in mathematical analysis , a branch of mathematics entailing calculus Founding During their time together at Strasbourg, Weil and Cartan regularly complained to each other regarding the inadequacy of available course material for calculus instruction. In his memoir Apprenticeship , Weil described his solution in the following terms: "One winter day toward the end of 1934, I came upon a great idea that would put an end to these ceaseless interrogations by my comrade. 'We are five or six friends', I told him some time later, 'who are in charge of the same mathematics curriculum at various universities. Let us all come together and regulate these matters once and for all, and after this, I shall be delivered of these questions.' I was unaware of the fact that Bourbaki was born at that instant." 23 Cartan confirmed the account. 24 The first, unofficial meeting of the Bourbaki collective took place at noon on Monday, 10 December 1934, at the Café Grill-Room A. Capoulade, Paris, in the Latin Quarter . 25 26 27 28 b Six mathematicians were present: Henri Cartan , Claude Chevalley , Jean Delsarte , Jean Dieudonné , René de Possel , and André Weil . Most of the group were based outside Paris and were in town to attend the Julia Seminar, a conference prepared with the help of Gaston Julia at which several future Bourbaki members and associates presented. 30 31 c The group resolved to collectively write a treatise on analysis, for the purpose of standardizing calculus instruction in French universities. The project was especially meant to supersede the text of Édouard Goursat , which the group found to be badly outdated, and to improve its treatment of Stokes' Theorem . 27 35 36 37 The founders were also motivated by a desire to incorporate ideas from the Göttingen school, particularly from exponents Hilbert , Noether and B.L. van der Waerden . Further, in the aftermath of World War I, there was a certain nationalist impulse to save French mathematics from decline, especially in competition with Germany. As Dieudonné stated in an interview, "Without meaning to boast, I can say that it was Bourbaki that saved French mathematics from extinction." 38 Jean Delsarte was particularly favorable to the collective aspect of the proposed project, observing that such a working style could insulate the group's work against potential later individual claims of copyright . 35 39 d As various topics were discussed, Delsarte also suggested that the work begin in the most abstract, axiomatic terms possible, treating all of mathematics prerequisite to analysis from scratch. 41 42 The group agreed to the idea, and this foundational area of the proposed work was referred to as the "Abstract Packet" (Paquet Abstrait). 43 44 45 Working titles were adopted: the group styled itself as the Committee for the Treatise on Analysis , and their proposed work was called the Treatise on Analysis ( Traité d'analyse ). 46 47 In all, the collective held ten preliminary biweekly meetings at A. Capoulade before its first official, founding conference in July 1935. 47 48 During this early period, Paul Dubreil , Jean Leray and Szolem Mandelbrojt joined and participated. Dubreil and Leray left the meetings before the following summer, and were respectively replaced by new participants Jean Coulomb and Charles Ehresmann . 46 49 Sign marking the official founding of Bourbaki in Besse-en-Chandesse The group's official founding conference was held in Besse-en-Chandesse , from 10 to 17 July 1935. 50 51 At the time of the official founding, the membership consisted of the six attendees at the first lunch of 10 December 1934, together with Coulomb, Ehresmann and Mandelbrojt. On 16 July, the members took a walk to alleviate the boredom of unproductive proceedings. During the malaise, some decided to skinny-dip in the nearby Lac Pavin , repeatedly yelling "Bourbaki!" 52 At the close of the first official conference, the group renamed itself "Bourbaki", in reference to the general and prank as recalled by Weil and others. 45 e During 1935, the group also resolved to establish the mathematical personhood of their collective pseudonym by getting an article published under its name. 50 54 A first name had to be decided; a full name was required for publication of any article. To this end, René de Possel's wife Eveline "baptized" the pseudonym with the first name of Nicolas, becoming Bourbaki's "godmother". 50 55 56 57 This allowed for the publication of a second article with material attributed to Bourbaki, this time under "his" own name. 58 Henri Cartan's father Élie Cartan , also a mathematician and supportive of the group, presented the article to the publishers, who accepted it. 54 At the time of Bourbaki's founding, René de Possel and his wife Eveline were in the process of divorcing. Eveline remarried to André Weil in 1937, and de Possel left the Bourbaki collective some time later. This sequence of events has caused speculation that de Possel left the group because of the remarriage, 59 however this suggestion has also been criticized as possibly historically inaccurate, since de Possel is supposed to have remained active in Bourbaki for years after André's marriage to Eveline. 60 World War II Bourbaki's work slowed significantly during the Second World War , though the group survived and later flourished. Some members of Bourbaki were Jewish and therefore forced to flee from certain parts of Europe at certain times. Weil, who was Jewish, spent the summer of 1939 in Finland with his wife Eveline, as guests of Lars Ahlfors . Due to their travel near the border, the couple were suspected as Soviet spies by Finnish authorities near the onset of the Winter War , and André was later arrested. 61 According to an anecdote, Weil was to have been executed but for the passing mention of his case to Rolf Nevanlinna , who asked that Weil's sentence be commuted. 62 However, the accuracy of this detail is dubious. 63 Weil reached the United States in 1941, later taking another teaching stint in São Paulo from 1945 to 1947 before settling at the University of Chicago from 1947 to 1958 and finally the Institute for Advanced Study in Princeton , where he spent the remainder of his career. Although Weil remained in touch with the Bourbaki collective and visited Europe and the group periodically following the war, his level of involvement with Bourbaki never returned to that at the time of founding. Second-generation Bourbaki member Laurent Schwartz was also Jewish and found pickup work as a math teacher in rural Vichy France . Moving from village to village, Schwartz planned his movements in order to evade capture by the Nazis . 64 On one occasion Schwartz found himself trapped overnight in a certain village, as his expected transportation home was unavailable. There were two inns in town: a comfortable, well-appointed one, and a very poor one with no heating and bad beds. Schwartz's instinct told him to stay at the poor inn; overnight, the Nazis raided the good inn, leaving the poor inn unchecked. 65 Meanwhile, Jean Delsarte, a Catholic, was mobilized in 1939 as the captain of an audio reconnaissance battery. He was forced to lead the unit's retreat from the northeastern part of France toward the south. While passing near the Swiss border, Delsarte overheard a soldier say "We are the army of Bourbaki"; 66 67 the 19th-century general's retreat was known to the French. Delsarte had coincidentally led a retreat similar to that of the collective's namesake. Postwar until the present Alexander Grothendieck proposed that Bourbaki revise its foundational basis in terms of category theory as opposed to set theory ; the proposal was not adopted Following the war, Bourbaki had solidified the plan of its work and settled into a productive routine. Bourbaki regularly published volumes of the Éléments during the 1950s and 1960s, and enjoyed its greatest influence during this period. 68 69 Over time the founding members gradually left the group, slowly being replaced with younger newcomers including Jean-Pierre Serre and Alexander Grothendieck . Serre, Grothendieck and Laurent Schwartz were awarded the Fields Medal during the postwar period, in 1954, 1966 and 1950 respectively. Later members Alain Connes and Jean-Christophe Yoccoz also received the Fields Medal, in 1982 and 1994 respectively. 70 The later practice of accepting scientific awards contrasted with some of the founders' views. 71 During the 1930s, Weil and Delsarte petitioned against a French national scientific "medal system" proposed by the Nobel physics laureate Jean Perrin . Weil and Delsarte felt that the institution of such a system would increase unconstructive pettiness and jealousy in the scientific community. 72 Despite this, the Bourbaki group had previously successfully petitioned Perrin for a government grant to support its normal operations. 73 Like the founders, Grothendieck was also averse to awards, albeit for pacifist reason