Editing Nicolas Bourbaki (section)

==Influence and criticism==

{{see also|Structuralism|Structuralism (philosophy of mathematics)}}

Bourbaki was influential in 20th century mathematics and had some interdisciplinary impact on the humanities and the arts, although the extent of the latter influence is a matter of dispute. The group has been praised and criticized for its method of presentation, its working style, and its choice of mathematical topics.

===Influence===

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

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

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

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

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

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

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

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

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

===Praise===

Bourbaki's work has been praised by some mathematicians. In a book review, [[Emil Artin]] described the ''Éléments'' in broad, positive terms:

{{blockquote|Our time is witnessing the creation of a monumental work: an exposition of the whole of present day mathematics. Moreover this exposition is done in such a way that the common bond between the various branches of mathematics become clearly visible, that the framework which supports the whole structure is not apt to become obsolete in a very short time, and that it can easily absorb new ideas.|Emil Artin<ref name="Artin" /> }}

Among the volumes of the ''Éléments'', Bourbaki's work on Lie Groups and Lie Algebras has been identified as "excellent",<ref name="Atiyah" /> having become a standard reference on the topic. In particular, former member Armand Borel described the volume with chapters 4–6 as "one of the most successful books by Bourbaki".{{sfn|Borel|p=379}} The success of this part of the work has been attributed to the fact that the books were composed while leading experts on the topic were Bourbaki members.{{sfn|Senechal|pp=22–28}}{{sfn|Aczel|p=111}}

[[Jean-Pierre Bourguignon]] expressed appreciation for the Séminaire Bourbaki, saying that he'd learned a large amount of material at its lectures, and referred to its printed lecture notes regularly.{{sfn|Mashaal|p=102}} He also praised the ''Éléments'' for containing "some superb and very clever proofs".{{sfn|Mashaal|pp=54–55}}

===Criticism===

Bourbaki has also been criticized by several mathematicians—including its own former members—for a variety of reasons. Criticisms have included the choice of presentation of certain topics within the ''Éléments'' at the expense of others,{{efn|This specific point has itself been criticized.  It has been observed that it is unfair to criticize a work on a given topic for not dealing with other topics.<ref>{{cite web |url=http://www.math.nsc.ru/LBRT/g2/english/ssk/euclid.html |title=Apology of Euclid |author-link=Semën Samsonovich Kutateladze |last=Kutateladze |first=Semën Samsonovich }}</ref>{{sfn|Mashaal|pp=116–18}} }} dislike of the method of presentation for given topics, dislike of the group's working style, and a perceived [[Elitism|elitist]] mentality around Bourbaki's project and its books, especially during the collective's most productive years in the 1950s and 1960s.

Bourbaki's deliberations on the ''Éléments'' resulted in the inclusion of some topics, while others were not treated. When asked in a 1997 interview about topics left out of the ''Éléments'', former member Pierre Cartier replied:

{{blockquote|text=There is essentially no analysis beyond the foundations: nothing about [[partial differential equations]], nothing about [[probability]]. There is also nothing about [[combinatorics]], nothing about [[algebraic topology]],{{efn|Bourbaki has since published a book on algebraic topology.}} nothing about concrete [[geometry]]. And Bourbaki never seriously considered [[mathematical logic|logic]]. Dieudonné himself was very vocal against logic.  Anything connected with [[mathematical physics]] is totally absent from Bourbaki's text.|title=Pierre Cartier{{sfn|Senechal|pp=22–28}} }}

Although Bourbaki had resolved to treat mathematics from its foundations, the group's eventual solution in terms of set theory was attended by several problems. Bourbaki's members were mathematicians as opposed to [[logician]]s, and therefore the collective had a limited interest in [[mathematical logic]].{{sfn|Guedj|p=20}} As Bourbaki's members themselves said of the book on set theory, it was written "with pain and without pleasure, but we had to do it."{{sfn|Mashaal|p=121}} Dieudonné personally remarked elsewhere that ninety-five percent of mathematicians "don't care a fig" for mathematical logic.{{sfn|Mashaal|p=120}} In response, logician Adrian Mathias harshly criticized Bourbaki's foundational framework, noting that it did not take [[Gödel]]'s results into account.{{sfn|Mashaal|pp=120–23}}<ref name="Mathias">{{Cite web|url=https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf|title=The Ignorance of Bourbaki|last=Mathias|first=Adrian|date=August 22, 1990|website=dpmms.cam.ac.uk}}</ref>

Bourbaki also influenced the New Math, a failed{{sfn|Mashaal|p=135}} reform in Western mathematics education at the elementary and secondary levels, which stressed abstraction over concrete examples. During the mid-20th century, reform in basic math education was spurred by a perceived need to create a mathematically literate workforce for the modern economy, and also to compete with the [[Soviet Union]]. In France, this led to the Lichnerowicz Commission of 1967, headed by [[André Lichnerowicz]] and including some (then-current and former) Bourbaki members. Although Bourbaki members had previously (and individually) reformed math instruction at the university level, they had less direct involvement with implementation of the New Math at the primary and secondary levels. New Math reforms resulted in instructional material which was incomprehensible to both students and teachers, failing to meet the [[Cognition|cognitive]] needs of younger students. The attempted reform was harshly criticized by Dieudonné and also by brief founding Bourbaki participant Jean Leray.{{sfn|Mashaal|pp=134–45}} Apart from French mathematicians, the French reforms also met with harsh criticism from Soviet-born mathematician [[Vladimir Arnold]], who argued that in his time as a student and teacher in Moscow, the teaching of mathematics was firmly rooted in analysis and geometry, and interweaved with problems from classical mechanics; hence, the French reforms cannot be a legitimate attempt to emulate Soviet scientific education. In 1997, while speaking to a conference on mathematical teaching in Paris, he commented on Bourbaki by stating: "genuine mathematicians do not gang up, but the weak need gangs in order to survive." and suggested that Bourbaki's bonding over "super-abstractness" was similar to groups of mathematicians in the 19th century who had bonded over anti-Semitism.<ref>{{Cite web|url=https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html|title = V.I. Arnold, on teaching mathematics}}</ref>

[[File:Benoit_Mandelbrot_mg_1804c.jpg|thumb|left|200px|[[Benoit Mandelbrot]] was among Bourbaki's critics]]

Dieudonné later regretted that Bourbaki's success had contributed to a [[snob]]bery for pure mathematics in France, at the expense of [[applied mathematics]]. In an interview, he said: "It is possible to say that there was no serious applied mathematics in France for forty years after Poincaré. There was even a snobbery for pure math. When one noticed a talented student, one would tell him 'You should do pure math.' On the other hand, one would advise a mediocre student to do applied math while thinking, "It's all that he can do! ... The truth is actually the reverse. You can't do good work in applied math until you can do good work in pure math."{{sfn|Mashaal|pp=118–19}} Claude Chevalley confirmed an elitist culture within Bourbaki, describing it as "an absolute certainty of our superiority over other mathematicians."{{sfn|Guedj|p=20}} Alexander Grothendieck also confirmed an elitist mentality within Bourbaki.{{sfn|Corry|2009|pp=38–51}} Some mathematicians, especially geometers and applied mathematicians, found Bourbaki's influence to be stifling.{{sfn|Aubin|p=313}} Benoit Mandelbrot's decision to emigrate to the United States in 1958 was motivated in part by a desire to escape Bourbaki's influence in France.{{sfn|Mashaal|p=130}}

Several related criticisms of the ''Éléments'' have concerned its target audience and the intent of its presentation. Volumes of the ''Éléments'' begin with a note to the reader which says that the series "takes up mathematics at the beginning, and gives complete proofs" and that "the method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular."<ref>''Theory of Sets'', p. v.</ref> Despite the opening language, Bourbaki's intended audience are not absolute beginners in mathematics, but rather undergraduates, graduate students, and professors who are familiar with mathematical concepts.{{sfn|Mashaal|p=54}} Claude Chevalley said that the ''Éléments'' are "useless for a beginner",{{sfn|Guedj|p=22}} and Pierre Cartier clarified that "The misunderstanding was that it should be a textbook for everybody. That was the big disaster."{{sfn|Senechal|pp=22–28}}

The work is divided into two halves. While the first half—the ''Structures fondamentales de l’analyse''—treats established subjects, the second half deals with modern research areas like commutative algebra and spectral theory. This divide in the work is related to a historical change in the intent of the treatise. The ''Éléments'<nowiki/>'' content consists of theorems, proofs, exercises and related commentary, common material in math textbooks. Despite this presentation, the first half was not written as [[original research]] but rather as a reorganized presentation of established knowledge. In this sense, the ''Éléments''' first half was more akin to an [[encyclopedia]] than a textbook series. As Cartier remarked, "The misunderstanding was that many people thought it should be ''taught'' the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics... If you consider it as a textbook, it's a disaster."{{sfn|Senechal|pp=22–28}}

The strict, ordered presentation of material in the ''Éléments''' first half was meant to form the basis for any further additions. However, developments in modern mathematical research have proven difficult to adapt in terms of Bourbaki's organizational scheme. This difficulty has been attributed to the fluid, dynamic nature of ongoing research which, being new, is not settled or fully understood.<ref name="Atiyah" />{{sfn|Borel|pp=377–379}} Bourbaki's style has been described as a particular scientific [[paradigm]] which has been superseded in a [[paradigm shift]]. For example, [[Ian Stewart (mathematician)|Ian Stewart]] cited [[Vaughan Jones|Vaughan Jones']] novel work in [[knot theory]] as an example of topology which was done without dependence on Bourbaki's system.<ref name="Stewart">{{Cite journal |last=Stewart |first=Ian |title=Bye-Bye Bourbaki: Paradigm Shifts in Mathematics |journal=The Mathematical Gazette |date=November 1995 |volume=79 |pages=496–98 |publisher=[[The Mathematical Association]] |doi=10.2307/3618076 |issue=486 |jstor=3618076|s2cid=125418650 }}</ref> Bourbaki's influence has declined over time;<ref name="Stewart" /> this decline has been partly attributed to the absence of certain modern topics—such as category theory—from the treatise.{{sfn|Aczel|p=205}}{{sfn|Mashaal|pp=81–84}}

Although multiple criticisms have pointed to shortcomings in the collective's project, one has also pointed to its strength: Bourbaki was a "victim of its own success"<ref name="Atiyah" /> in the sense that it accomplished what it set out to do, achieving its original goal of presenting a thorough treatise on modern mathematics.{{sfn|Aczel|pp=204–05}}{{sfn|Aubin|p=329}}{{sfn|Borel|p=377}} These factors prompted biographer Maurice Mashaal to conclude his treatment of Bourbaki in the following terms:

{{blockquote|text=Such an enterprise deserves admiration for its breadth, for its enthusiasm and selflessness, for its strongly collective character.  Despite some mistakes, Bourbaki did add a little to 'the honor of the human spirit'.  In an era when sports and money are such great idols of civilization, this is no small virtue.|title=Maurice Mashaal{{sfn|Mashaal|p=153}} }}