Nicolas Bourbaki

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
Bourbaki congress at Dieulefit in 1938. From left, Simone Weil, Charles Pisot, André Weil, Jean Dieudonné (sitting), Claude Chabauty, Charles Ehresmann, and Jean Delsarte.
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
Membership	Confidential
Official language	French
Website	www.bourbaki.fr
Formerly called	Committee for the Treatise on Analysis

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 (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. 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.

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. 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.

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; 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. 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.

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, 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 in which a student claimed to be from the fictional, impoverished nation of "Poldevia" and solicited the public for donations. Weil had strong interests in languages and Indian culture, having learned Sanskrit and read the Bhagavad Gita. 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. 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". Weil's stay in India was short-lived; he attempted to revamp the mathematics department at Aligarh, without success. 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. Weil returned to Europe to seek another teaching position. He ended up at the University of Strasbourg, joining his friend and colleague Henri Cartan.

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." Cartan confirmed the account.

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. 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. 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. 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."

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. 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. The group agreed to the idea, and this foundational area of the proposed work was referred to as the "Abstract Packet" (Paquet Abstrait). 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). In all, the collective held ten preliminary biweekly meetings at A. Capoulade before its first official, founding conference in July 1935. 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.

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. 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!" 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. During 1935, the group also resolved to establish the mathematical personhood of their collective pseudonym by getting an article published under its name. 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". This allowed for the publication of a second article with material attributed to Bourbaki, this time under "his" own name. Henri Cartan's father Élie Cartan, also a mathematician and supportive of the group, presented the article to the publishers, who accepted it.

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, 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.

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. 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. However, the accuracy of this detail is dubious. 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. 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.

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"; 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. 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.

The later practice of accepting scientific awards contrasted with some of the founders' views. 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. Despite this, the Bourbaki group had previously successfully petitioned Perrin for a government grant to support its normal operations. Like the founders, Grothendieck was also averse to awards, albeit for pacifist reasons. Although Grothendieck was awarded the Fields Medal in 1966, he declined to attend the ceremony in Moscow, in protest of the Soviet government. In 1988, Grothendieck rejected the Crafoord Prize outright, citing no personal need to accept prize money, lack of recent relevant output, and general distrust of the scientific community.

Born to Jewish anarchist parentage, Grothendieck survived the Holocaust and advanced rapidly in the French mathematical community, despite poor education during the war. Grothendieck's teachers included Bourbaki's founders, and so he joined the group. During Grothendieck's membership, Bourbaki reached an impasse concerning its foundational approach. Grothendieck advocated for a reformulation of the group's work using category theory as its theoretical basis, as opposed to set theory. The proposal was ultimately rejected in part because the group had already committed itself to a rigid track of sequential presentation, with multiple already-published volumes. Following this, Grothendieck left Bourbaki "in anger". Biographers of the collective have described Bourbaki's unwillingness to start over in terms of category theory as a missed opportunity. However, Bourbaki has in 2023 announced that a book on category theory is currently under preparation (see below the last paragraph of this section).

During the founding period, the group chose the Parisian publisher Hermann to issue installments of the Éléments. Hermann was led by Enrique Freymann, a friend of the founders willing to publish the group's project, despite financial risk. During the 1970s, Bourbaki entered a protracted legal battle with Hermann over matters of copyright and royalty payment. Although the Bourbaki group won the suit and retained collective copyright of the Éléments, the dispute slowed the group's productivity. Former member Pierre Cartier described the lawsuit as a pyrrhic victory, saying: "As usual in legal battles, both parties lost and the lawyer got rich." Later editions of the Éléments were published by Masson, and modern editions are published by Springer. From the 1980s through the 2000s, Bourbaki published very infrequently, with the result that in 1998 Le Monde pronounced the collective "dead".

However, in 2012 Bourbaki resumed the publication of the Éléments with a revised chapter 8 of algebra, the first 4 chapters of a new book on algebraic topology, and two volumes on spectral theory (the first of which is an expanded and revised version of the edition of 1967 while the latter consist of three new chapters). Moreover, the text of the two latest volumes announces that books on category theory and modular forms are currently under preparation (in addition to the latter part of the book on algebraic topology).

Working method

At Armand Borel's urging, Bourbaki's treatment of Lie groups and Lie algebras included uncharacteristic illustrations, such as graphs of finite Coxeter systems

Bourbaki holds periodic conferences for the purpose of expanding the Éléments; these conferences are the central activity of the group's working life. Subcommittees are assigned to write drafts on specific material, and the drafts are later presented, vigorously debated, and re-drafted at the conferences. Unanimous agreement is required before any material is deemed acceptable for publication. A given piece of material may require six or more drafts over a period of several years, and some drafts are never developed into completed work. Bourbaki's writing process has therefore been described as "Sisyphean". Although the method is slow, it yields a final product which satisfies the group's standards for mathematical rigour, one of Bourbaki's main priorities in the treatise. Bourbaki's emphasis on rigour was a reaction to the style of Henri Poincaré, who stressed the importance of free-flowing mathematical intuition at the cost of thorough presentation. During the project's early years, Dieudonné served as the group's scribe, authoring several final drafts which were ultimately published. For this purpose, Dieudonné adopted an impersonal writing style which was not his own, but which was used to craft material acceptable to the entire group. Dieudonné reserved his personal style for his own work; like all members of Bourbaki, Dieudonné also published material under his own name, including the nine-volume Éléments d'analyse, a work explicitly focused on analysis and of a piece with Bourbaki's initial intentions.

Most of the final drafts of Bourbaki's Éléments carefully avoided using illustrations, favoring a formal presentation based only in text and formulas. An exception to this was the treatment of Lie groups and Lie algebras (especially in chapters 4–6), which did make use of diagrams and illustrations. The inclusion of illustration in this part of the work was due to Armand Borel. Borel was minority-Swiss in a majority-French collective, and self-deprecated as "the Swiss peasant", explaining that visual learning was important to the Swiss national character. When asked about the dearth of illustration in the work, former member Pierre Cartier replied:

The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith. The number of Protestants and Jews in the Bourbaki group was overwhelming. And you know that the French Protestants especially are very close to Jews in spirit.

— Pierre Cartier

The conferences have historically been held at quiet rural areas. These locations contrast with the lively, sometimes heated debates which have occurred. Laurent Schwartz reported an episode in which Weil slapped Cartan on the head with a draft. The hotel's proprietor saw the incident and assumed that the group would split up, but according to Schwartz, "peace was restored within ten minutes." The historical, confrontational style of debate within Bourbaki has been partly attributed to Weil, who believed that new ideas have a better chance of being born in confrontation than in an orderly discussion. Schwartz related another illustrative incident: Dieudonné was adamant that topological vector spaces must appear in the work before integration, and whenever anyone suggested that the order be reversed, he would loudly threaten his resignation. This became an in-joke among the group; Roger Godement's wife Sonia attended a conference, aware of the idea, and asked for proof. As Sonia arrived at a meeting, a member suggested that integration must appear before topological vector spaces, which triggered Dieudonné's usual reaction.

Despite the historical culture of heated argument, Bourbaki thrived during the middle of the twentieth century. Bourbaki's ability to sustain such a collective, critical approach has been described as "something unusual", surprising even its own members. In founder Henri Cartan's words, "That a final product can be obtained at all is a kind of miracle that none of us can explain." It has been suggested that the group survived because its members believed strongly in the importance of their collective project, despite personal differences. When the group overcame difficulties or developed an idea that they liked, they would sometimes say l'esprit a soufflé ("the spirit breathes"). Historian Liliane Beaulieu noted that the "spirit"—which might be an avatar, the group mentality in action, or Bourbaki "himself"—was part of an internal culture and mythology which the group used to form its identity and perform work.

Humor

Humor has been an important aspect of the group's culture, beginning with Weil's memories of the student pranks involving "Bourbaki" and "Poldevia". For example, in 1939 the group released a wedding announcement for the marriage of "Betti Bourbaki" (daughter of Nicolas) to one "H. Pétard" (H. "Firecrackers" or "Hector Pétard"), a "lion hunter". Hector Pétard was itself a pseudonym, but not one originally coined by the Bourbaki members. The Pétard moniker was originated by Ralph P. Boas, Frank Smithies and other Princeton mathematicians who were aware of the Bourbaki project; inspired by them, the Princeton mathematicians published an article on the "mathematics of lion hunting". After meeting Boas and Smithies, Weil composed the wedding announcement, which contained several mathematical puns. Bourbaki's internal newsletter La Tribu has sometimes been issued with humorous subtitles to describe a given conference, such as "The Extraordinary Congress of Old Fogies" (where anyone older than 30 was considered a fogy) or "The Congress of the Motorization of the Trotting Ass" (an expression used to describe the routine unfolding of a mathematical proof, or process).

During the 1940s–1950s, the American Mathematical Society received applications for individual membership from Bourbaki. They were rebuffed by J.R. Kline who understood the entity to be a collective, inviting them to re-apply for institutional membership. In response, Bourbaki floated a rumor that Ralph Boas was not a real person, but a collective pseudonym of the editors of Mathematical Reviews with which Boas had been affiliated. The reason for targeting Boas was because he had known the group in its earlier days when they were less strict with secrecy, and he'd described them as a collective in an article for the Encyclopædia Britannica. In November 1968, a mock obituary of Nicolas Bourbaki was released during one of the seminars.

The group developed some variants of the word "Bourbaki" for internal use. The noun "Bourbaki" might refer to the group proper or to an individual member, e.g. "André Weil was a Bourbaki." "Bourbakist" is sometimes used to refer to members but also denotes associates, supporters, and enthusiasts. To "bourbakize" meant to take a poor existing text and to improve it through an editing process.

Bourbaki's culture of humor has been described as an important factor in the group's social cohesion and capacity to survive, smoothing over tensions of heated debate. As of 2024, a Twitter account registered to "Betty_Bourbaki" provides regular updates on the group's activity.

Works

Bourbaki's work includes a series of textbooks, a series of printed lecture notes, journal articles, and an internal newsletter. The textbook series Éléments de mathématique (Elements of mathematics) is the group's central work. The Séminaire Bourbaki is a lecture series held regularly under the group's auspices, and the talks given are also published as lecture notes. Journal articles have been published with authorship attributed to Bourbaki, and the group publishes an internal newsletter La Tribu (The Tribe) which is distributed to current and former members.

Éléments de mathématique

Main article: Éléments de mathématique

Like those before him, Bourbaki insisted on setting mathematics in a “formalized language” with crystal-clear deductions based on strict formal rules. When Bertrand Russell and Alfred North Whitehead applied this approach at the turn of the twentieth century, they famously filled over 700 pages with formal symbols before establishing the proposition usually abbreviated as 1+1=2. Bourbaki's formalism would dwarf even this, requiring some 4.5 trillion symbols just to define the number 1.

Michael Barany

The content of the Éléments is divided into books—major topics of discussion, volumes—individual, physical books, and chapters, together with certain summaries of results, historical notes, and other details. The volumes of the Éléments have had a complex publication history. Material has been revised for new editions, published chronologically out of order of its intended logical sequence, grouped together and partitioned differently in later volumes, and translated into English. For example, the second book on Algebra was originally released in eight French volumes: the first in 1942 being chapter 1 alone, and the last in 1980 being chapter 10 alone. This presentation was later condensed into five volumes with chapters 1–3 in the first volume, chapters 4–7 in the second, and chapters 8–10 each remaining the third through fifth volumes of that portion of the work. The English edition of Bourbaki's Algebra consists of translations of the three volumes consisting of chapters 1–3, 4–7 and 8, with chapters 9 and 10 unavailable in English as of 2024.

When Bourbaki's founders began working on the Éléments, they originally conceived of it as a "treatise on analysis", the proposed work having a working title of the same name (Traité d'analyse). The opening part was to comprehensively deal with the foundations of mathematics prior to analysis, and was referred to as the "Abstract Packet". Over time, the members developed this proposed "opening section" of the work to the point that it would instead run for several volumes and comprise a major part of the work, covering set theory, abstract algebra, and topology. Once the project's scope expanded far beyond its original purpose, the working title Traité d'analyse was dropped in favor of Éléments de mathématique. The unusual, singular "Mathematic" was meant to connote Bourbaki's belief in the unity of mathematics. The first six books of the Éléments, representing the first half of the work, are numbered sequentially and ordered logically, with a given statement being established only on the basis of earlier results. This first half of the work bore the subtitle Les structures fondamentales de l’analyse (Fundamental Structures of Analysis), covering established mathematics (algebra, analysis) in the group's style. The second half of the work consists of unnumbered books treating modern areas of research (Lie groups, commutative algebra), each presupposing the first half as a shared foundation but without dependence on each other. This second half of the work, consisting of newer research topics, does not have a corresponding subtitle.

The volumes of the Éléments published by Hermann were indexed by chronology of publication and referred to as fascicules: installments in a large work. Some volumes did not consist of the normal definitions, proofs, and exercises in a math textbook, but contained only summaries of results for a given topic, stated without proof. These volumes were referred to as Fascicules de résultats, with the result that fascicule may refer to a volume of Hermann's edition, or to one of the "summary" sections of the work (e.g. Fascicules de résultats is translated as "Summary of Results" rather than "Installment of Results", referring to the content rather than a specific volume). The first volume of Bourbaki's Éléments to be published was the Summary of Results in the Theory of Sets, in 1939. Similarly one of the work's later books, Differential and Analytic Manifolds, consisted only of two volumes of summaries of results, with no chapters of content having been published.

Later installments of the Éléments appeared infrequently during the 1980s and 1990s. A volume of Commutative Algebra (chapters 8–9) was published in 1983, and no other volumes were issued until the appearance of the same book's tenth chapter in 1998. During the 2010s, Bourbaki increased its productivity. A re-written and expanded version of the eighth chapter of Algebra appeared in 2012, the first four chapters of a new book treating Algebraic Topology was published in 2016, and the first two chapters of a revised and expanded edition of Spectral Theory was issued in 2019 while the remaining three (completely new) chapters appeared in 2023.

First book of the Éléments de mathématique, 1970 edition

Éléments de mathématique

Year	Book	References
1954	Theory of Sets
1942	Algebra
1940	General Topology
1949	Functions of a Real Variable
1953	Topological Vector Spaces
1952	Integration
1960	Lie Groups and Lie Algebras
1961	Commutative Algebra
1967	Spectral Theory
1967	Differential and Analytic Manifolds
2016	Algebraic Topology
1960	Elements of the History of Mathematics

Séminaire Bourbaki

Main article: Séminaire Bourbaki

The Séminaire Bourbaki has been held regularly since 1948, and lectures are presented by non-members and members of the collective. As of 2024 the Séminaire Bourbaki has run to over a thousand recorded lectures in its written incarnation, denoted chronologically by simple numbers. At the time of a June 1999 lecture given by Jean-Pierre Serre on the topic of Lie groups, the total lectures given in the series numbered 864, corresponding to roughly 10,000 pages of printed material.

Articles

Damodar Kosambi authored the first article attributing material to "Bourbaki"

Several journal articles have appeared in the mathematical literature with material or authorship attributed to Bourbaki; unlike the Éléments, they were typically written by individual members and not crafted through the usual process of group consensus. Despite this, Jean Dieudonné's essay "The Architecture of Mathematics" has become known as Bourbaki's manifesto. Dieudonné addressed the issue of overspecialization in mathematics, to which he opposed the inherent unity of mathematic (as opposed to mathematics) and proposed mathematical structures as useful tools which can be applied to several subjects, showing their common features. To illustrate the idea, Dieudonné described three different systems in arithmetic and geometry and showed that all could be described as examples of a group, a specific kind of (algebraic) structure. Dieudonné described the axiomatic method as "the 'Taylor system' for mathematics" in the sense that it could be used to solve problems efficiently. Such a procedure would entail identifying relevant structures and applying established knowledge about the given structure to the specific problem at hand.

• Kosambi, D.D. (1931). "On a Generalization of the Second Theorem of Bourbaki". Bulletin of the Academy of Sciences of the United Provinces of Agra and Oudh, Allahabad, India. 1: 145–47. ISBN 978-81-322-3674-0. Reprinted in Ramaswamy, Ramakrishna, ed. (2016). D.D. Kosambi: Selected Works in Mathematics and Statistics. Springer. pp. 55–57. doi:10.1007/978-81-322-3676-4_6. Kosambi attributed material in the article to "D. Bourbaki", the first mention of the eponymous Bourbaki in the literature.
• Bourbaki, Nicolas (1935). "Sur un théorème de Carathéodory et la mesure dans les espaces topologiques". Comptes rendus de l'Académie des Sciences. 201: 1309–11. Presumptive author: André Weil.
• —— (1938). "Sur les espaces de Banach". Comptes rendus de l'Académie des Sciences. 206: 1701–04. Presumptive author: Jean Dieudonné.
• ——; Dieudonné, Jean (1939). "Note de tératopologie II". Revue scientifique (Or, "Revue rose"): 180–81. Presumptive author: Jean Dieudonné. Second in a series of three articles.
• —— (1941). "Espaces minimaux et espaces complètement séparés". Comptes rendus de l'Académie des Sciences. 212: 215–18. Presumptive author: Jean Dieudonné or André Weil.
• —— (1948). "L'architecture des mathématiques". In Le Lionnais, François (ed.). Les grands courants de la pensée mathématique. Actes Sud. pp. 35–47. Presumptive author: Jean Dieudonné.
• —— (1949). "Foundations of Mathematics for the Working Mathematician". Journal of Symbolic Logic. 14 (1): 1–8. doi:10.2307/2268971. JSTOR 2268971. S2CID 26516355. Presumptive author: André Weil.
• —— (1949). "Sur le théorème de Zorn". Archiv der Mathematik. 2 (6): 433–37. doi:10.1007/BF02036949. S2CID 117826806. Presumptive author: Henri Cartan or Jean Dieudonné.
• —— (1950). "The Architecture of Mathematics". American Mathematical Monthly. 57 (4): 221–32. doi:10.1080/00029890.1950.11999523. JSTOR 2305937. Presumptive author: Jean Dieudonné. Authorized translation of the book chapter L'architecture des mathématiques, appearing in English as a journal article.
• —— (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier. 2: 5–16. doi:10.5802/aif.16. Presumptive authors: Jean Dieudonné and Laurent Schwartz.

La Tribu

La Tribu is Bourbaki's internal newsletter, distributed to current and former members. The newsletter usually documents recent conferences and activity in a humorous, informal way, sometimes including poetry. Member Pierre Samuel wrote the newsletter's narrative sections for several years. Early editions of La Tribu and related documents have been made publicly available by Bourbaki.

Historian Liliane Beaulieu examined La Tribu and Bourbaki's other writings, describing the group's humor and private language as an "art of memory" which is specific to the group and its chosen methods of operation. Because of the group's secrecy and informal organization, individual memories are sometimes recorded in a fragmentary way, and may not have significance to other members. On the other hand, the predominantly French, ENS background of the members, together with stories of the group's early period and successes, create a shared culture and mythology which is drawn upon for group identity. La Tribu usually lists the members present at a conference, together with any visitors, family members or other friends in attendance. Humorous descriptions of location or local "props" (cars, bicycles, binoculars, etc.) can also serve as mnemonic devices.

Membership

As of 2000, Bourbaki has had "about forty" members. Historically the group has numbered about ten to twelve members at any given point, although it was briefly (and officially) limited to nine members at the time of founding. Bourbaki's membership has been described in terms of 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 Koszul, 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, François Bruhat, Serge Lang, the American mathematician John Tate, Pierre Cartier, and the Swiss mathematician Armand Borel.

After the first three generations there were roughly twenty later members, not including current participants. Bourbaki 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). This secrecy is also intended to deter unwanted attention which could disrupt normal operations. However, former members freely discuss Bourbaki's internal practices upon departure.

Prospective members are invited to conferences and styled as guinea pigs, a process meant to vet the newcomer's mathematical ability. In the event of agreement between the group and the prospect, the prospect eventually becomes a full member. The group is supposed to have an age limit: active members are expected to retire at (or about) 50 years of age. At a 1956 conference, Cartan read a letter from Weil which proposed a "gradual disappearance" of the founding members, forcing younger members to assume full responsibility for Bourbaki's operations. This rule is supposed to have resulted in a complete change of personnel by 1958. However, historian Liliane Beaulieu has been critical of the claim. She reported never having found written affirmation of the rule, and has indicated that there have been exceptions. The age limit is thought to express the founders' intent that the project should continue indefinitely, operated by people at their best mathematical ability—in the mathematical community, there is a widespread belief that mathematicians produce their best work while young. Among full members there is no official hierarchy; all operate as equals, having the ability to interrupt conference proceedings at any point, or to challenge any material presented. However, André Weil has been described as "first among equals" during the founding period, and was given some deference. On the other hand, the group has also poked fun at the idea that older members should be afforded greater respect.

Bourbaki conferences have also been attended by members' family, friends, visiting mathematicians, and other non-members of the group. Bourbaki is not known ever to have had any female members.

Jean Dieudonné, founding member

Jean-Pierre Serre, second generation member

Alexander Grothendieck, third generation member, left Bourbaki largely over a disagreements about incorporation of category theory in the treatise

Арманд Борель , член третьего поколения

Хайман Басс , позже участник

Former members of the Nicolas Bourbaki collective

Generation		Name	Born	ENS	Joined	Left	Died
First	Core members	Henri Cartan	1904	1923	1934	c. 1956–58	2008
		Claude Chevalley	1909	1926	1934	c. 1956–58	1984
		Jean Delsarte	1903	1922	1934	c. 1956–58	1968
		Jean Dieudonné	1906	1924	1934	c. 1956–58	1992
		André Weil	1906	1922	1934	c. 1956–58	1998
	Minor members	Jean Coulomb	1904	1923	1935	1937	1999
		Paul Dubreil	1904	1923	1935	1935	1994
		Charles Ehresmann	1905	1924	1935	1950	1979
		Jean Leray	1906	1926	1935	1935	1998
		Szolem Mandelbrojt	1899	—	1935	—	1983
		René de Possel	1905	1923	1934	—	1974
Second		Jacques Dixmier	1924	1942	—	—	—
		Samuel Eilenberg	1913	—	c. 1951	1966	1998
		Roger Godement	1921	1940	—	—	2016
		Jean-Louis Koszul	1921	1940	—	—	2018
		Pierre Samuel	1921	1940	1947	1971	2009
		Laurent Schwartz	1916	1934	—	—	2002
		Jean-Pierre Serre	1926	1945	—	—	—
Третий		Armand Borel	1923	—	c. 1953	1973	2003
		François Bruhat	1929	1948	—	—	2007
		Pierre Cartier	1932	1950	1955	1983	—
		Alexander Grothendieck	1928	1945	1955	1960	2014
		Серж Ланг	1927	—	—	—	2005
		Джон Тейт	1925	—	—	—	2019
Более поздние участники		Хайман Басс	1932	—	—	—	—
		Арно Бовиль	1947	1966	—	1997	—
		Жерар Бен Арус	1957	1977	—	—	—
		Дэниел Беннекин	1952	1972	—	—	—
		Клод Шаботи	1910	1929	—	—	1990
		Ален Конн	1947	1966	—	—	—
		Мишель Демазюр	1937	1955	—	в. 1985 год	—
		Адриан Дуади	1935	1954	—	—	2006
		Патрик Жерар [ фр ]	1961	1981	—	—	—
		Гай Хенниарт	1953	1973	—	—	—
		Люк Иллюзия	1940	1959	—	—	—
		Пьер Джадж	1959	1977	—	—	—
		Жиль Лебо	1954	1974	—	—	—
		Андре Мартино	1930	1949	—	—	1972
		Оливье Матье	1960	1980	1989	в. 2010 год	—
		Луи Буте де Монвель	1941	1960	1971	1991	2014
		Джозеф Остерле	1954	1973	—	—	—
		Шарль Писо	1909	1929	—	—	1984
		Мишель Рейно	1938	1958	—	—	2018
		Марк Россо	1962	1982	—	—	—
		Жорж Скандалис	1955	1975	—	—	—
		Бернар Тейсье	1945	—	—	—	—
		Жан-Луи Вердье	1937	1955	—	—	1989
		Жан-Кристоф Йоккоз	1957	1975	в. 1995 год	в. 1995 год	2016

Влияние и критика

Смотрите также: Структурализм и Структурализм (философия математики)

Бурбаки оказал влияние на математику 20-го века и оказал некоторое междисциплинарное влияние на гуманитарные науки и искусство, хотя степень последнего влияния является предметом споров. Группу хвалили и критиковали за метод презентации, стиль работы и выбор математических тем.

Влияние

Бурбаки ввел обозначения для пустого множества , а также символ опасного изгиба, обозначающий трудный материал.

Бурбаки ввел несколько математических обозначений, которые до сих пор используются. Вейль взял букву Ø норвежского алфавита и использовал ее для обозначения пустого множества . $\varnothing$. Впервые это обозначение появилось в «Сводке результатов по теории множеств» . и продолжает использоваться. Слова инъективный , сюръективный и биективный были введены для обозначения функций , которые удовлетворяют определенным свойствам. Бурбаки использовал простой язык для обозначения некоторых геометрических объектов, называя их паве ( брусчаткой ) и булями ( шарами ) в отличие от « параллелотопов » или « гиперсфероидов ». Точно так же в своей трактовке топологических векторных пространств Бурбаки определил бочку как множество, которое является выпуклым , сбалансированным , поглощающим и замкнутым . Группа гордилась этим определением, полагая, что форма винной бочки символизирует свойства математического объекта. Бурбаки также использовал символ « опасного изгиба » ☡ на полях текста, чтобы указать на особенно трудный фрагмент материала. Наибольшее влияние Бурбаки пользовался в 1950-е и 1960-е годы, когда отрывки из «Элементов» часто публиковались .

Бурбаки оказал междисциплинарное влияние на другие области, включая антропологию и психологию . Это влияние было в контексте структурализма , школы мысли в гуманитарных науках , которая подчеркивает отношения между объектами над самими объектами, преследуемой в различных областях другими французскими интеллектуалами. В 1943 году Андре Вейль встретился с антропологом Клодом Леви-Стросом в Нью-Йорке, где они некоторое время сотрудничали. По просьбе Леви-Стросса Вейль написал краткое приложение, описывающее правила брака для четырех классов людей в аборигенов Австралии обществе , используя математическую модель, основанную на теории групп . Результат был опубликован в качестве приложения к книге Леви-Стросса « Элементарные структуры родства » , работе, исследующей семейные структуры и табу на инцест в человеческих культурах. В 1952 году Жан Дьедонне и Жан Пиаже приняли участие в междисциплинарной конференции по математическим и ментальным структурам. Дьедонне описал математические «материнские структуры» в терминах проекта Бурбаки: композиция, соседство и порядок. Затем Пиаже выступил с докладом о психических процессах детей и считал, что только что описанные им психологические понятия очень похожи на математические понятия, только что описанные Дьедонне. По словам Пиаже, они были «впечатлены друг другом». Психоаналитику Жаку Лакану понравился стиль совместной работы Бурбаки, и он предложил аналогичную коллективную группу в психологии, но эта идея не была реализована.

Бурбаки также цитировали философы -постструктуралисты . В своей совместной работе «Анти-Эдип» Жиль Делёз и Феликс Гваттари представили критику капитализма . Авторы привели использование Бурбаки аксиоматического метода (с целью установления истины) как явный контрпример процессам управления , которые вместо этого стремятся к экономической эффективности . Авторы сказали об аксиоматике Бурбаки, что «они не образуют систему Тейлора», перевернув фразу, использованную Дьедонне в «Архитектуре математики». В «Состоянии постмодерна » Жан-Франсуа Лиотар раскритиковал «легитимацию знания», процесс, посредством которого утверждения принимаются как действительные. В качестве примера Лиотар привел Бурбаки как группу, производящую знания в рамках заданной системы правил. Лиотар противопоставлял иерархическую, «структуралистскую» математику Бурбаки теории катастроф Рене Тома и фракталам Бенуа Мандельброта . выражая предпочтение последней «постмодернистской науке», которая проблематизировала математику с помощью «фрактов, катастроф и прагматических парадоксов».

Хотя биограф Амир Аксель подчеркивал влияние Бурбаки на другие дисциплины в середине 20-го века, Морис Машаал смягчал утверждения о влиянии Бурбаки следующими словами:

Хотя структуры Бурбаки часто упоминались на конференциях и публикациях по общественным наукам того времени, похоже, что они не сыграли реальной роли в развитии этих дисциплин. Дэвид Обен, историк науки, анализировавший роль Бурбаки в структуралистском движении во Франции, считает, что роль Бурбаки заключалась в «культурном связующем звене». По словам Обена, хотя у Бурбаки не было никакой миссии за пределами математики, группа представляла собой своего рода связующее звено между различными культурными движениями того времени. Бурбаки дал простое и относительно точное определение понятий и структур, которое, по мнению философов и социологов, имело фундаментальное значение в их дисциплинах и служило связующим звеном между различными областями знаний. Несмотря на поверхностный характер этих связей, различные школы структуралистского мышления, в том числе и Бурбаки, смогли поддержать друг друга. Поэтому не случайно в конце 1960-х годов эти школы пережили одновременный упадок.

— Морис Машал, цитируя Дэвида Обена

Критике подверглось и влияние «структурализма» на саму математику. Историк математики Лео Корри утверждал, что использование Бурбаки математических структур не имело значения в « Элементах» , поскольку оно было установлено в «Теории множеств» и впоследствии нечасто цитировалось. Корри описал «структурный» взгляд на математику, продвигаемый Бурбаки, как «образ знания» — концепцию научной дисциплины — в отличие от элемента «корпуса знаний» дисциплины, который относится к фактическим научным результатам в сама дисциплина.

Бурбаки также имел некоторое влияние в искусстве. Литературный коллектив Oulipo был основан 24 ноября 1960 года при обстоятельствах, аналогичных основаниям Бурбаки: участники первоначально встретились в ресторане. Хотя некоторые члены Улипо были математиками, целью группы было создание экспериментальной литературы , играя с языком. Улипо часто использовал математически обоснованные методы письма с ограничениями , такие как метод S + 7 . Член Улипо Раймон Кено присутствовал на конференции Бурбаки в 1962 году.

В 2016 году анонимная группа экономистов совместно написала заметку, в которой обвиняла авторов и редактора в академических нарушениях статьи, опубликованной в American Economic Review . Заметка была опубликована под именем Николя Беарбаки в честь Николя Бурбаки.

В 2018 году американский музыкальный дуэт One Pilots выпустил концептуальный альбом Trench Twenty . Концептуальной основой альбома стал мифический город «Дема», которым управляют девять «епископов»; одного из епископов звали «Нико», сокращенно от Николя Бурбаки. Другого епископа звали Андре, что может относиться к Андре Вейлю. После выпуска альбома в Интернете резко возросло количество поисковых запросов по запросу «Николя Бурбаки».

Хвалить

Некоторые математики высоко оценили работу Бурбаки. В рецензии на книгу Эмиль Артин описал Éléments в общих и позитивных терминах:

Наше время является свидетелем создания монументального труда: изложения всей современной математики. Более того, это изложение сделано таким образом, чтобы стала ясно видна общая связь между различными разделами математики, чтобы структура, поддерживающая всю структуру, не устаревала за очень короткое время и могла легко поглощать новые знания. идеи.

— Эмиль Артин

Среди томов «Элементов » работа Бурбаки по группам Ли и алгебрам Ли была отмечена как «превосходная». став стандартным справочником по этой теме. В частности, бывший участник Арман Борель охарактеризовал том с главами 4–6 как «одну из самых успешных книг Бурбаки». Успех этой части работы объясняется тем, что книги были написаны, когда ведущими экспертами по этой теме были члены Бурбаки.

Жан-Пьер Бургиньон выразил признательность Семинару Бурбаки, заявив, что он изучил на его лекциях большой объем материала и регулярно ссылался на его печатные конспекты лекций. Он также похвалил « Элементы» за «несколько превосходных и очень умных доказательств».

Критика

Бурбаки также подвергался критике со стороны нескольких математиков, в том числе его бывших членов, по разным причинам. Критика включала выбор представления одних тем в рамках Éléments за счет других, неприязнь к методу изложения определенных тем, неприязнь к стилю работы группы и воспринимаемый элитарный менталитет вокруг проекта Бурбаки и его книг, особенно в самые продуктивные годы коллектива в 1950-х и 1960-х годах.

Обсуждения Бурбаки « Элементов» привели к включению некоторых тем, а другие не рассматривались. Когда в интервью 1997 года его спросили о темах, не включенных в Éléments , бывший участник Пьер Картье ответил:

По сути, нет никакого анализа, выходящего за рамки основ: ничего об уравнениях в частных производных , ничего о вероятности . Там также нет ничего о комбинаторике , ничего об алгебраической топологии , ничего о конкретной геометрии . А Бурбаки никогда серьезно не задумывался о логике . Сам Дьедонне резко выступал против логики. все, что связано с математической физикой . В тексте Бурбаки полностью отсутствует

— Пьер Картье

Хотя Бурбаки решил рассматривать математику с ее основ, окончательное решение группы в терминах теории множеств сопровождалось несколькими проблемами. Члены Бурбаки были математиками, а не логиками , и поэтому коллектив имел ограниченный интерес к математической логике . Как говорили сами бурбаки о книге по теории множеств, она была написана «с болью и без удовольствия, но нам пришлось это сделать». Дьедонне лично заметил в другом месте, что девяносто пять процентов математиков «наплевать» на математическую логику. В ответ логик Адриан Матиас резко раскритиковал основополагающую концепцию Бурбаки, отметив, что она не принимает во внимание результаты Гёделя .

Бурбаки также повлиял на «Новую математику», провалившуюся реформа западного математического образования на начальном и среднем уровнях, в которой упор делался на абстракцию, а не на конкретные примеры. В середине 20-го века реформа базового математического образования была стимулирована осознанной необходимостью создать математически грамотную рабочую силу для современной экономики, а также для конкуренции с Советским Союзом . Во Франции это привело к созданию Комиссии Лихнеровича 1967 года, которую возглавил Андре Лихнерович и в которую входили некоторые (действующие и бывшие) члены Бурбаки. Хотя члены Бурбаки ранее (и индивидуально) реформировали преподавание математики на университетском уровне, они имели менее непосредственное участие во внедрении Новой математики на уровнях начальной и средней школы. Новые реформы математики привели к созданию учебного материала, который был непонятен как ученикам, так и учителям и не отвечал познавательным потребностям младших школьников. Попытка реформы подверглась резкой критике со стороны Дьедонне, а также со стороны краткого участника Бурбаки Жана Лере. Помимо французских математиков, французские реформы также встретили резкую критику со стороны математика советского происхождения Владимира Арнольда , который утверждал, что в его время, когда он был студентом и учителем в Москве, преподавание математики было прочно основано на анализе и геометрии и переплеталось с задачи классической механики; следовательно, французские реформы не могут быть законной попыткой подражать советскому научному образованию. В 1997 году, выступая на конференции по преподаванию математики в Париже, он прокомментировал Бурбаки, заявив: «Настоящие математики не объединяются в банды, но слабым нужны банды, чтобы выжить». и предположил, что объединение Бурбаки из-за «суперабстрактности» было похоже на объединение групп математиков XIX века, объединенных антисемитизмом.

Бенуа Мандельброт был среди критиков Бурбаки.

Позже Дьедонне сожалел, что успех Бурбаки способствовал снобизму в отношении чистой математики во Франции в ущерб прикладной математике . В интервью он сказал: «Можно сказать, что в течение сорока лет после Пуанкаре во Франции не было серьезной прикладной математики. Был даже снобизм по отношению к чистой математике. Когда замечали талантливого студента, ему говорили: «Вы должен заниматься чистой математикой». С другой стороны, можно было бы посоветовать посредственному студенту заниматься прикладной математикой, думая: «Это все, на что он способен! ... На самом деле все наоборот. Вы не сможете хорошо работать в прикладной математике, пока не сможете хорошо работать в чистой математике». Клод Шевалле подтвердил наличие элитарной культуры внутри Бурбаки, охарактеризовав ее как «абсолютную уверенность в нашем превосходстве над другими математиками». Александр Гротендик также подтвердил элитарный менталитет Бурбаки. Некоторые математики, особенно геометры и прикладные математики, сочли влияние Бурбаки удушающим. Решение Бенуа Мандельброта эмигрировать в Соединенные Штаты в 1958 году было частично мотивировано желанием избежать влияния Бурбаки во Франции.

Несколько связанных критических замечаний в адрес Éléments касались его целевой аудитории и цели его представления. Тома « Элементов» начинаются с примечания к читателю, в котором говорится, что серия «сначала изучает математику и дает полные доказательства» и что «выбранный нами метод изложения является аксиоматическим и абстрактным и обычно исходит из общих к особенностям». Несмотря на вступительный язык, предполагаемой аудиторией Бурбаки являются не абсолютные новички в математике, а скорее студенты, аспиранты и профессора, знакомые с математическими концепциями. Клод Шевалле сказал, что Éléments «бесполезны для новичка». а Пьер Картье пояснил: «Недоразумение заключалось в том, что это должно быть учебником для всех. Это было большой катастрофой».

Работа разделена на две половины. В то время как первая половина — «Основные структуры анализа» — посвящена устоявшимся предметам, вторая половина посвящена современным областям исследований, таким как коммутативная алгебра и спектральная теория. Этот раскол в работе связан с историческим изменением цели трактата. Содержание Éléments состоит из теорем, доказательств, упражнений и соответствующих комментариев — обычного материала в учебниках по математике. Несмотря на эту презентацию, первая половина была написана не как оригинальное исследование , а скорее как реорганизованное изложение устоявшихся знаний. В этом смысле первая половина «Элементов» была больше похожа на энциклопедию , чем на серию учебников. Как заметил Картье: «Недоразумение заключалось в том, что многие люди считали, что ее следует преподавать так, как написано в книгах. Первые книги Бурбаки можно воспринимать как энциклопедию математики... Если рассматривать ее как учебник, это катастрофа».

Строгое и упорядоченное изложение материала в первой половине «Элементов» должно было стать основой для любых дальнейших дополнений. Однако развитие современных математических исследований оказалось трудно адаптировать к организационной схеме Бурбаки. Эту трудность объясняют изменчивым и динамичным характером текущих исследований, которые, будучи новыми, не урегулированы и не полностью поняты. Стиль Бурбаки был описан как особая научная парадигма , которая была заменена в результате смены парадигмы . Например, Ян Стюарт процитировал Воана Джонса новую работу по теории узлов как пример топологии, которая была создана без зависимости от системы Бурбаки. Влияние Бурбаки со временем снизилось; этот спад частично объясняется отсутствием в трактате некоторых современных тем, таких как теория категорий.

Хотя многочисленные критические замечания указывали на недостатки проекта коллектива, одна из них также указала на его силу: Бурбаки стал «жертвой собственного успеха». в том смысле, что он выполнил то, что намеревался сделать, достигнув своей первоначальной цели — представить подробный трактат по современной математике. Эти факторы побудили биографа Мориса Машаля завершить свое рассмотрение Бурбаки следующими словами:

Такое предприятие заслуживает восхищения своим размахом, энтузиазмом и самоотверженностью, своим сильным коллективным характером. Несмотря на некоторые ошибки, Бурбаки все-таки добавил немного «чести человеческого духа». В эпоху, когда спорт и деньги являются такими великими идолами цивилизации, это немалое достоинство.

— Морис Машал

См. также

• Теорема Бурбаки – Витта
• Теорема Джейкобсона – Бурбаки.
• Тайное общество

Другие коллективные математические псевдонимы

• Артур Бесс
• Бланш Декарт
• Джон Рейнуотер
• Г.В. Пек

Библиография

• Аксель, Амир Д. (2006). Художник и математик: история Николя Бурбаки, гениального математика, которого никогда не существовало . Пресс Громовой Пасты. ISBN  978-1560259312 .
• Обен, Дэвид (1997). «Увядающее бессмертие Николя Бурбаки: культурный соединитель на стыке математики, структурализма и Улипо во Франции» (PDF) . Наука в контексте . 10 (2). Издательство Кембриджского университета : 297–342. дои : 10.1017/S0269889700002660 . S2CID   170683589 .
• Болье, Лилиан (1993). «Парижское кафе и десять встреч протобурбаков (1934–1935)» . Математический интеллект . 15 (1): 27–35. дои : 10.1007/BF03025255 . S2CID   189888171 .
• Болье, Лилиан (1999). «Искусство памяти Бурбаки» (PDF) . Осирис . 14 : 219–51. дои : 10.1086/649309 . S2CID   143559711 .
• Борель, Арманд (март 1998 г.). «Двадцать пять лет с Николя Бурбаки (1949–1973)» (PDF) . Уведомления Американского математического общества . 45 (3): 373–80.
• Бурбаки, Николя (1950). «Архитектура математики». Американский математический ежемесячник . 57 (4): 221–32. дои : 10.1080/00029890.1950.11999523 . JSTOR   2305937 . Предполагаемый автор: Жан Дьедонне. Авторизованный перевод главы книги L'architecture des mathématiques , опубликованной на английском языке в виде журнальной статьи.
• Корри, Лео (2004). «Николя Бурбаки: Теория структур». Современная алгебра и возникновение математических структур . Спрингер. стр. 289–338. ISBN  978-3764370022 .
• Корри, Лео (2009). «Написание идеального учебника математики: «Элементы математики » Николя Бурбаки ». В Робсоне, Элеонора; Стедалл, Жаклин (ред.). Оксфордский справочник по истории математики . Издательство Оксфордского университета . стр. 565–87. ISBN  978-0199213122 .
• Гедж, Денис (1985). «Николя Бурбаки, коллективный математик: интервью с Клодом Шевалле» (PDF) . Математический интеллект . 7 (2). Перевод Грея, Джереми : 18–22. дои : 10.1007/BF03024169 . S2CID   123548747 .
• Кремер, Ральф (2006). «Основана ли «машина Гротендика» только на метаматематических терминах? Бурбаки и категории в 1950-е годы» . Журнал истории математики . 12 :119–162. дои : 10.24033/rhm.38 . Збл   1177.01034 .
• Машаал, Морис (2006). Бурбаки: Тайное общество математиков . Американское математическое общество . ISBN  978-0821839676 .
• Сенешаль, Марджори (1998). «Продолжающееся молчание Бурбаки: интервью с Пьером Картье, 18 июня 1997 года» . Математический интеллект . 20 : 22–28. дои : 10.1007/BF03024395 . S2CID   124159858 .

Внешние ссылки

• Официальный сайт L'Association des Collaborateurs de Nicolas Bourbaki (на французском языке)
• Архивы ассоциации (на французском языке)