Editing Nicolas Bourbaki (section)

==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.<ref name="archiver">{{cite web |url=http://archives-bourbaki.ahp-numerique.fr/elements-mathematique |title=Éléments de Mathématique |website=Archives Bourbaki}}</ref>{{sfn|Mashaal|pp=108–09}}

===''Éléments de mathématique''===
{{main|Éléments de mathématique}}
{{Quote box
 | quote  = 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]].<ref>{{cite journal |last1=Mathias |first1=A. R. D. |title=A Term of Length 4 523 659 424 929 |journal=Synthese |date=2002 |volume=133 |issue=1/2 |pages=75–86 |doi=10.1023/A:1020827725055 |jstor=20117295 |url=https://www.jstor.org/stable/20117295 |access-date=5 January 2024 |issn=0039-7857 |quote=ABSTRACT. Bourbaki suggest that their definition of the number 1 runs to some tens of thousands of symbols. We show that that is a considerable under-estimate, the true number of symbols being that in the title, not counting 1 179 618 517 981 links between symbols that are needed to disambiguate the whole expression.}}</ref>
 | source = Michael Barany<ref>{{cite journal |last1=Barany |first1=Michael |title=The Mathematical Pranksters behind Nicolas Bourbaki |journal=JSTOR Daily |date=24 March 2021 |url=https://daily.jstor.org/the-mathematical-pranksters-behind-nicolas-bourbaki/ |access-date=5 January 2024}}</ref>
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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.<ref name="archiver" /> 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 {{year}}.

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''.{{sfn|Mashaal|p=11}} The unusual, singular "Mathematic" was meant to connote Bourbaki's belief in the unity of mathematics.{{sfn|Aczel|pp=99–100}}{{sfn|Borel|p=374}}{{sfn|Mashaal|p=55}}  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.<ref>''Theory of Sets'', pp. v-vi.</ref>  This first half of the work bore the subtitle ''Les structures fondamentales de l’analyse'' (''Fundamental Structures of Analysis''),<ref name="archiver" />{{sfn|Mashaal|p=83}}<ref name="Boyer">{{cite book |author-link1=Carl Benjamin Boyer |last1=Boyer |first1=Carl B. |author-link2=Uta Merzbach |last2=Merzbach |first2=Uta C. |contribution=foreword |contributor-link=Isaac Asimov |contributor-last=Asimov |contributor-first=Isaac |title=A History of Mathematics |date=20 March 1991 |edition=Second |publisher=Wiley |isbn=9780471543978 |page=629}}</ref> 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).{{efn|The [[History of mathematics|mathematical historian]] [[Leo Corry]] also observed that the phrase "Summary of Results" is a misleading one for a distinct reason, instead referring to the content of the ''Éléments'' rather than the publication history of its volumes.{{sfn|Corry|1992|p=326}}{{sfn|Corry|2004|p=320}} }} The first volume of Bourbaki's ''Éléments'' to be published was the Summary of Results in the ''Theory of Sets'', in 1939.{{sfn|Senechal|pp=22–28}}<ref name="archiver" />{{sfn|Mashaal|p=52}} 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.

[[File:Bourbaki, Theorie des ensembles maitrier.jpg|thumb|First book of the ''[[Éléments de mathématique]]'', 1970 edition]]

{| class="wikitable sortable" style="text-align: center"
|+''Éléments de mathématique''<ref name="archiver" />{{efn|Years refer to the date of publication of each book's first volume, which also contains its first proper chapter.  There are two exceptions: the first published installment of the ''Theory of Sets'' was a summary of results in 1939, and its first proper chapter did not appear until 1954.  For ''Differential and Analytic Manifolds'', only a two-volume summary of results was published in 1967 and 1971, with no proper chapters appearing.}}
!Year
!Book
!References
|-
|1954
|''Theory of Sets''
|<ref>{{cite journal |last=Bagemihl |first=Frederick |author-link=Frederick Bagemihl |title=Review: ''Théorie des ensembles'' (Chapter III) |journal=[[Bulletin of the American Mathematical Society]] |year=1958 |volume=64 |issue=6 |pages=390–91 |url=https://www.ams.org/journals/bull/1958-64-06/S0002-9904-1958-10248-7/S0002-9904-1958-10248-7.pdf |doi=10.1090/s0002-9904-1958-10248-7 |doi-access=free }}</ref>
|-
|1942
|''Algebra''
|<ref name="Artin">{{cite journal |last=Artin |first=Emil |author-link=Emil Artin |title=Review: ''Éléments de mathématique'', by N. Bourbaki, Book II, ''Algebra''. Chaps. I–VII |journal=Bulletin of the American Mathematical Society |year=1953 |volume=59 |issue=5 |pages=474–79 |url=https://www.ams.org/journals/bull/1953-59-05/S0002-9904-1953-09725-7/S0002-9904-1953-09725-7.pdf |doi=10.1090/s0002-9904-1953-09725-7 |doi-access=free }}</ref><ref>{{cite journal |last=Rosenberg |first= Alex |author-link=Alex F. T. W. Rosenberg |title=Review: ''Éléments de mathématiques'' by N. Bourbaki. Book II, Algèbre. Chapter VIII, ''Modules et anneaux semi-simples'' |journal=Bulletin of the American Mathematical Society |year=1960 |volume=66 |issue=1 |pages=16–19 |url=https://www.ams.org/journals/bull/1960-66-01/S0002-9904-1960-10371-0/S0002-9904-1960-10371-0.pdf |doi=10.1090/S0002-9904-1960-10371-0 |doi-access=free }}</ref><ref>{{cite journal |last=Kaplansky |first=Irving |author-link=Irving Kaplansky |title=Review: ''Formes sesquilinéairies et formes quadratiques'' by N. Bourbaki, ''Éléments de mathématique'' I, Livre II |journal=Bulletin of the American Mathematical Society |year=1960 |volume=66 |issue=4 |pages=266–67 |url=https://www.ams.org/journals/bull/1960-66-04/S0002-9904-1960-10461-2/S0002-9904-1960-10461-2.pdf|doi=10.1090/s0002-9904-1960-10461-2 |doi-access=free }}</ref>
|-
|1940
|''General Topology''
|
|-
|1949
|''Functions of a Real Variable''
|
|-
|1953
|''Topological Vector Spaces''
|
|-
|1952
|''Integration''
|<ref>{{cite journal |last=Halmos |first=Paul |author-link=Paul Halmos |title=Review: ''Intégration'' (Chap. I–IV) by N. Bourbaki |journal=Bulletin of the American Mathematical Society |year=1953 |volume=59 |issue=3 |pages=249–55 |url=https://www.ams.org/journals/bull/1953-59-03/S0002-9904-1953-09698-7/S0002-9904-1953-09698-7.pdf |doi=10.1090/S0002-9904-1953-09698-7 |doi-access=free }}</ref><ref>{{cite journal |last=Munroe |first=M. E. |title=Review: ''Intégration'' (Chapter V) by N. Bourbaki |journal=Bulletin of the American Mathematical Society |year=1958 |volume=64 |issue=3 |pages=105–06 |url=https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10176-7/S0002-9904-1958-10176-7.pdf |doi=10.1090/s0002-9904-1958-10176-7 |doi-access=free }}</ref>
|-
|1960
|''Lie Groups and Lie Algebras''
|
|-
|1961
|''Commutative Algebra''
|<ref>{{cite journal |last=Nagata |first=Masayoshi |author-link=Masayoshi Nagata |title=''Éléments de mathématique. Algèbre commutative'', by N. Bourbaki, Chapitres 8 et 9 |journal=Bulletin of the American Mathematical Society |series=New Series |year=1985 |volume=12 |issue=1 |pages=175–77 |url=https://www.ams.org/journals/bull/1985-12-01/S0273-0979-1985-15338-8/S0273-0979-1985-15338-8.pdf |doi=10.1090/s0273-0979-1985-15338-8 |doi-access=free }}</ref>
|-
|1967
|''Spectral Theory''
|
|-
|1967
|''Differential and Analytic Manifolds''
|
|-
|2016
|''Algebraic Topology''
|<ref>{{Cite book |url=https://www.springer.com/us/book/9783662493601|title=Topologie Algébrique, Chapitres 1 à 4 |last=Bourbaki |first=Nicolas |year=2016 |publisher=Springer |doi=10.1007/978-3-662-49361-8 |isbn=978-3-662-49360-1 |access-date=2016-02-08 }}</ref>
|-
|1960
|''Elements of the History of Mathematics''
|
|}

===Séminaire Bourbaki===
{{main|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 {{year}} the Séminaire Bourbaki has run to over a thousand recorded lectures in its written incarnation, denoted chronologically by simple numbers.<ref name="seminar">{{cite web |url=http://www.bourbaki.ens.fr/Editeurs.html |title=Éditeurs du Séminaire |publisher=Association des collaborateurs de Nicolas Bourbaki}}</ref> 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.{{sfn|Mashaal|pp=98–99}}

===Articles===

[[File:Kosambi-dd.jpg|thumb|right|150px|[[Damodar Dharmananda Kosambi|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<ref name="archiver" /> 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]].{{sfn|Aubin|pp=305–08}}{{sfn|Corry|1997|pp=272–73}} 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.{{sfn|Corry|2004|pp=303–05}} 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 (mathematics)|group]], a specific kind of ([[algebraic structure|algebraic]]) structure.{{sfn|Bourbaki 1950|pp=224–26}} Dieudonné described the [[axiomatic method]] as "the '[[Taylor system]]' for mathematics" in the sense that it could be used to solve problems efficiently.{{sfn|Bourbaki 1950|p=227}}{{efn|Dieudonné immediately qualified the comparison as "a very poor analogy", continuing: "the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition, which is not the popular sense-intuition, but rather a kind of direct divination... of the normal behavior... of mathematical beings."{{sfn|Bourbaki 1950|p=227}} }} Such a procedure would entail identifying relevant structures and applying established knowledge about the given structure to the specific problem at hand.{{sfn|Bourbaki 1950|p=227}}
* {{cite journal |last=Kosambi |first=D.D. |title=On a Generalization of the Second Theorem of Bourbaki |journal=Bulletin of the Academy of Sciences of the United Provinces of Agra and Oudh, Allahabad, India |volume=1 |pages=145–47 |date=1931 |isbn=978-81-322-3674-0 }} Reprinted in {{cite book|title= D.D. Kosambi: Selected Works in Mathematics and Statistics|publisher=Springer|year=2016|doi=10.1007/978-81-322-3676-4_6|editor-first=Ramakrishna|editor-last=Ramaswamy|pages=55–57}} Kosambi attributed material in the article to "D. Bourbaki", the first mention of the eponymous Bourbaki in the literature.
* {{cite journal |url=https://gallica.bnf.fr/ark:/12148/bpt6k31534/f1311.image |title=Sur un théorème de Carathéodory et la mesure dans les espaces topologiques |last=Bourbaki |first=Nicolas |journal=[[Comptes rendus de l'Académie des Sciences]] |volume=201 |pages=1309–11 |date=1935}} Presumptive author: André Weil.
* {{cite journal |author-mask=2 |url=https://gallica.bnf.fr/ark:/12148/bpt6k3158p/f1701.image |title=Sur les espaces de Banach |last=Bourbaki |first=Nicolas |journal=Comptes rendus de l'Académie des Sciences |volume=206 |pages=1701–04 |date=1938}} Presumptive author: Jean Dieudonné.
* {{cite journal |author-mask=2 |title=Note de tératopologie II |last1=Bourbaki |first1=Nicolas |last2=Dieudonné |first2=Jean |journal=Revue scientifique (Or, "Revue rose") |pages=180–81 |date=1939}} Presumptive author: Jean Dieudonné. Second in a series of three articles.
* {{cite journal |author-mask=2 |url=https://gallica.bnf.fr/ark:/12148/bpt6k3164q/f215.image |title=Espaces minimaux et espaces complètement séparés |last=Bourbaki |first=Nicolas |journal=Comptes rendus de l'Académie des Sciences |volume=212 |pages=215–18 |date=1941}} Presumptive author: Jean Dieudonné or André Weil.
* {{cite book    |author-mask=2 |title=Les grands courants de la pensée mathématique |last=Bourbaki |first=Nicolas |chapter=L'architecture des mathématiques |editor-last=[[François Le Lionnais|Le Lionnais]] |editor-first=François |publisher=Actes Sud |pages=35–47 |date=1948}} Presumptive author: Jean Dieudonné.
* {{cite journal |author-mask=2 |title=Foundations of Mathematics for the Working Mathematician |last=Bourbaki |first=Nicolas |journal=[[Journal of Symbolic Logic]] |volume=14 |issue=1 |pages=1–8 |date=1949|jstor=2268971 |doi=10.2307/2268971 |s2cid=26516355 }} Presumptive author: André Weil.
* {{cite journal |author-mask=2 |title=Sur le théorème de Zorn |last=Bourbaki |first=Nicolas |s2cid=117826806 |journal=[[Archiv der Mathematik]] |volume=2 |issue=6 |pages=433–37 |date=1949|doi=10.1007/BF02036949 }} Presumptive author: Henri Cartan or Jean Dieudonné.
* {{cite journal |author-mask=2 |title=The Architecture of Mathematics |last=Bourbaki |first=Nicolas |journal=[[American Mathematical Monthly]] |volume=57 |issue=4 |pages=221–32 |date=1950|jstor=2305937 |doi=10.1080/00029890.1950.11999523 }} Presumptive author: Jean Dieudonné. Authorized translation of the book chapter ''L'architecture des mathématiques'', appearing in English as a journal article.
* {{cite journal |author-mask=2 |url=http://www.numdam.org/item/?id=AIF_1950__2__5_0 |title=Sur certains espaces vectoriels topologiques |last=Bourbaki |first=Nicolas |journal=[[Annales de l'Institut Fourier]] |volume=2 |pages=5–16 |date=1950 |doi = 10.5802/aif.16 |doi-access=free }} 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.{{sfn|Mashaal|pp=108–11}} Member [[Pierre Samuel]] wrote the newsletter's narrative sections for several years.{{sfn|Beaulieu|1999|p=234}} Early editions of ''La Tribu'' and related documents have been made publicly available by Bourbaki.<ref name="barchive" />

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.{{sfn|Beaulieu|1999|p=224}} 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.{{sfn|Beaulieu|1999|pp=231–32}} 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.{{sfn|Beaulieu|1999|p=226}}