Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way. The Bourbaki family is a French family of Greek extraction, that produced two prominent officers: Colonel Constantin Denis Bourbaki, a hero of the Greek War of Independence. ... A pseudonym (Greek: , pseudo + -onym: false name) is an artificial, fictitious name, also known as an alias, used by an individual as an alternative to a persons legal name. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... 1935 (MCMXXXV) was a common year starting on Tuesday (link will display full calendar). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Look up Rigour in Wiktionary, the free dictionary. ...

While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki ("association of collaborators of Nicolas Bourbaki"), which has an office at the École Normale Supérieure in Paris. Bourbaki is a respected name now, but it was initially a clever prank played on the entire scientific establishment. For a few years, people thought that Nicolas Bourbaki existed and admired his talent, which was of course the combined talent of the group. See also École Normale de Musique de Paris. ... This article is about the capital of France. ...

Books by Bourbaki

Aiming at a completely self-contained treatment of the core areas of modern mathematics based on set theory, the group produced Elements of Mathematics (Éléments de mathématique) series, which contain the following volumes (with the original French titles in parentheses):

The book Variétés différentielles et analytiques was a fascicule de résultats, that is, a summary of results, on the theory of manifolds, rather than a worked-out exposition. A final volume IX on spectral theory (Théories spectrales) from 1983 marked the presumed end of the publishing project; but a further commutative algebra fascicle was produced at the end of the twentieth century. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... This article is about the branch of mathematics. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... This article is about the concept of integrals in calculus. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ... The word fascicle derives from the Latin fascis (bundle). Fascicles are the sections of a book, usually a reference work, that because of its length, is issued in parts so that the information may be made available to the public as soon as possible rather than waiting years or decades... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ...

While several of Bourbaki's books have become standard references in their fields, some have felt that the austere presentation makes them unsuitable as textbooks^{[citation needed]}. The books' influence may have been at its strongest when few other graduate-level texts in current pure mathematics were available, between 1950 and 1960.^[1]

Notations introduced by Bourbaki include: the symbol for the empty set and a dangerous bend symbol, and the terms injective, surjective, and bijective. The empty set is the set containing no elements. ... The dangerous bend symbol was created by Bourbaki and appears in the margins of mathematics books written by the group. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

It is frequently claimed that the use of the blackboard bold letters for the various sets of numbers was first introduced by the group. There are several reasons to doubt this claim.^[2] An example of blackboard bold letters. ... For other uses, see Number (disambiguation). ...

Influence on mathematics in general

The emphasis on rigour may be seen as a reaction to the work of Jules-Henri Poincaré ^{[citation needed]}, who stressed the importance of free-flowing mathematical intuition, at a cost of completeness in presentation. The impact of Bourbaki's work initially was great on many active research mathematicians world-wide. Look up Rigour in Wiktionary, the free dictionary. ... Jules TuPac Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ...

It provoked some hostility, too, mostly on the side of classical analysts; they approved of rigour but not of high abstraction. Around 1950, also, some parts of geometry were still not fully axiomatic — in less prominent developments, one way or another, these were brought into line with the new foundational standards, or quietly dropped. This undoubtedly led to a gulf with the way theoretical physics is practiced.^{[citation needed]} Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... For other uses, see Geometry (disambiguation). ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ...

Bourbaki's influence has decreased over time.^{[citation needed]} This is partly because some of the abstractions did not prove as useful as initially thought, and partly because other concepts which are now important, such as the detailed machinery of category theory, are not covered. Algebraic structure can reasonably be defined, in Bourbakiste terms; but mathematical structure is not an idea exhausted by infinitary algebraic structures, as might appear from the books. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... In mathematics or logic, a finitary operation is one, like those of arithmetic, that take a number of input values to produce an output. ...

The Bourbaki seminar series founded in post-WWII Paris continues. It is an important source of survey articles, written in a prescribed, careful style. The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. ... In academia, a survey article is a paper that is a work of synthesis, published through the usual channels (a learned journal or collective volume, such as conference proceedings or collection of essays). ...

The group

Accounts of the early days vary, but original documents have now come to light. The founding members were all connected to the Ecole Normale Supérieure in Paris and included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil. There was a preliminary meeting, towards the end of 1934.^[3] Jean Leray and Paul Dubreil were present at the preliminary meeting but dropped out before the group actually formed. Other notable participants in later days were Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, Samuel Eilenberg, Serge Lang and Roger Godement. The quadrangle at the main ENS building on rue dUlm is known as the Cour aux Ernests – the Ernests being the goldfish in the pond. ... This article is about the capital of France. ... Henri Cartan (born July 8, 1904) is a son of Élie Cartan, and is, as his father was, a distinguished and influential French mathematician. ... Claude Chevalley (11 February 1909 - 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry and the theory of algebraic groups. ... Jean Frédéric Auguste Delsarte (October 19, 1903 - November 28, 1968) was a French mathematician. ... Jean-Alexandre-Eugène Dieudonné (July 1, 1906 - November 29, 1992) was a French mathematician, known for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a... Charles Ehresmann (1905-1979) was a French mathematician who worked on differential topology and category theory. ... Lucien Alexandre Charles René Possel (1905 - 1974) was a French mathematician, one of the founders of the Bourbaki group, and later a pioneer computer scientist, working in particular on optical character recognition. ... Szolem Mandelbrojt (1899 – 1983) was a French mathematician, from a Polish-Lithuanian Jewish background. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... Jean Leray (7 November 1906-10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. ... Paul Dubreil (March 1, 1904 - March 9, 1994) was a French mathematician. ... Laurent Schwartz (5 March 1915 – 4 July 2002 in Paris) was a French mathematician. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ... Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ... Serge Lang (May 19, 1927–September 12, 2005) was a French-born American mathematician. ... Roger Godement is a French mathematician, known for his work in functional analysis, and also his expository books. ...

The original goal of the group had been to compile an improved mathematical analysis text; it was soon decided that a more comprehensive treatment of all of mathematics was necessary. There was no official status of membership, and at the time the group was quite secretive and also fond of supplying disinformation. Regular meetings were scheduled, during which the whole group would discuss vigorously every proposed line of every book. Members had to resign by age 50.^[4] Analysis has its beginnings in the rigorous formulation of calculus. ...

The atmosphere in the group can be illustrated by an anecdote told by Laurent Schwartz. Dieudonné regularly and spectacularly threatened to resign unless topics were treated in their logical order, and after a while others played on this for a joke. Godement's wife wanted to see Dieudonné announcing his resignation, and so on one occasion while she was there Schwartz deliberately brought up again the question of permuting the order in which measure theory and topological vector spaces were to be handled, to precipitate a guaranteed crisis. In mathematics, a measure is a function that assigns a number, e. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...

The name "Bourbaki" refers to a French general;^[5] it was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue. It was certainly a reference to Greek mathematics, Bourbaki being of Greek extraction. It is a valid reading to take the name as implying a transplantation of the tradition of Euclid to a France of the 1930s, with soured expectations.^[6] Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... For other uses, see Euclid (disambiguation). ...

Criticism of the Bourbaki perspective

The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Göttingen school and the German algebraists^{[citation needed]}. It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics. But always through a transforming process of reception and selection—typical of a French salon if more intensive ^{[citation needed]}. The Georg-August University of Göttingen (Georg-August-Universität Göttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... A Salon of Ladies by Abraham Bosse A salon is a gathering of people under the roof of an inspiring hostess or host, partly to amuse one another and partly to refine their taste and increase their knowledge through conversation and readings, often consciously following Horaces definition of the...

Examples of the tendency are the way "tensor calculus" was renamed multilinear algebra, and the emergence of commutative algebra as independent of elimination theory, which had been a major motivation under its earlier name of ideal theory. Hilbert had already in the 1890s shown a preference for non-constructive methods; these changes of name made visible a definite change of attitude. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... In algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables. ... In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. ...

The following is a list of some of the criticisms commonly made of the Bourbaki approach:^[7]

• algorithmic content is not considered on-topic and is almost completely omitted^[8]
• problem solving, in the sense of heuristics, receives less emphasis than axiomatic theory-building^[9]
• analysis is treated 'softly', without 'hard' estimates^[10]
• measure theory is referred to as 'integration theory'; by taking the case of locally compact spaces as fundamental, the centre of the theory becomes Radon measures^[11]
• combinatorics is not discussed
• logic is treated minimally ^[12]
• applications are not covered.

Furthermore, some have claimed that Bourbaki eschewed all use of pictures,^[13] although this is not in fact accurate (see, for example, "Figure 1" of Topologie Générale, Book 1, p. 3). In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.^[14] In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... Problem solving forms part of thinking. ... For heuristics in computer science, see heuristic (computer science) Heuristic is the art and science of discovery and invention. ... Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, a measure is a function that assigns a number, e. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... For other uses, see Geometry (disambiguation). ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

Dieudonné as speaker for Bourbaki

Public discussion of, and justification for, Bourbaki's thoughts has in general been through Jean Dieudonné (who initially was the 'scribe' of the group) writing under his own name. In a survey of le choix bourbachique written in 1977, he did not shy away from a hierarchical development of the 'important' mathematics of the time. Jean-Alexandre-Eugène Dieudonné (July 1, 1906 - November 29, 1992) was a French mathematician, known for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a... Also: 1977 (album) by Ash. ...

He also wrote extensively under his own name: nine volumes on analysis, perhaps in belated fulfillment of the original project or pretext; and also on other topics mostly connected with algebraic geometry. While Dieudonné could reasonably speak on Bourbaki's encyclopedic tendency, and tradition (after innumerable frank tais-toi, Dieudonné! ("Hush, Dieudonné!") remarks at the meetings), it may be doubted whether all others agreed with him about mathematical writing and research. In particular Serre has often criticised the way the Bourbaki works were written, and has championed in France greater attention to problem-solving, within number theory especially, not an area treated in the main Bourbaki texts. Analysis has its beginnings in the rigorous formulation of calculus. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Dieudonné stated the view that most workers in mathematics were doing ground-clearing work, in order that a future Riemann could find the way ahead intuitively open. He pointed to the way the axiomatic method can be used as a tool for problem-solving, for example by Alexander Grothendieck. Others found him too close to Grothendieck to be an unbiased observer. Comments in Pal Turán's 1970 speech on the award of a Fields Medal to Alan Baker about theory-building and problem-solving were a reply from the traditionalist camp at the next opportunity^[15], Grothendieck having received a Fields Medal in absentia in 1966 and the awards being every four years. Bernhard Riemann. ... Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ... Paul (Pál) Turán (August 28, 1910–September 26, 1976) was a Hungarian mathematician who made contributions in number theory and group theory. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... Alan Baker (born on August 19, 1939) is an English mathematician. ...

The Bourbachique influence: education, institutions, trends

In the longer term, the manifesto of Bourbaki has had a definite and deep influence, particularly on graduate education in pure mathematics. This effect can be read in detail in parts of this encyclopedia. It is perhaps most noticeable in the treatment now current of Lie groups and Lie algebras. Dieudonné at one point said 'one can do nothing serious without them', for which he was reproached; but the change in Lie theory to its everyday usage owes much to the type of exposition Bourbaki championed. Beforehand Jacques Hadamard despaired of ever getting a clear idea of it. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... This page is a candidate for speedy deletion. ...

The leading role of Bourbaki, internationally rather than for France alone, had possibly been taken over by the programme of the Bonn Arbeitstagung as early as the first years of the 1960s. Another point representing a turn of the tide in mathematics can be identified in 1959, when Jean-Pierre Serre and Armand Borel ran a seminar on complex multiplication. This was a key classical theory — a remark attributed to Hilbert made it 'the most beautiful part of mathematics' — but in doctrinaire Bourbakiste terms excluded, like much of number theory, from the 'core topics'. The Mathematische Arbeitstagung taking place annually in Bonn since 1957, and founded by Friedrich Hirzebruch, was an international meeting of mathematicians intended to act in clearing-house fashion, by disseminating current research ideas; and, at the same time, to bring mathematics in West Germany back into its place in European... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Armand Borel (21 May 1923 –11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. ... In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

See also

• Bourbaki–Witt theorem

In mathematics, the Bourbaki–Witt theorem in order theory , named after Nicolas Bourbaki and Ernst Witt, is a basic fixed-point theorem for partially ordered sets. ...

Notes

Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common amongst such journals. ... Charles Denis Sauter Bourbaki ( April 22, 1816 - September 27, 1897) was a French general. ... Combatants Allies: Second French Empire British Empire Ottoman Empire Kingdom of Sardinia Russian Empire Bulgarian volunteers Casualties 90,000 French 35,000 Turkish 17,500 British 2,194 Sardinian killed, wounded and died of disease ~134,000 killed, wounded and died of disease The Crimean War (1853–1856) was fought... Combatants Second French Empire North German Confederation allied with South German states (later German Empire) Commanders Napoleon III François Achille Bazaine Patrice de Mac-Mahon, duc de Magenta Otto von Bismarck Helmuth von Moltke the Elder Strength 400,000 at wars beginning 1,200,000 Casualties 150,000... Pierre Cartier (born in Sedan, France in 1932) is a mathematician - more specifically, a category theorist. ... William Timothy Gowers (born November 20, 1963, Wiltshire, United Kingdom) is a British mathematician. ... Lennart Carleson (b. ... New math is a term referring to a brief dramatic change in the way mathematics was taught in American grade schools during the 1960s. ...

References

David Aubin, The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics. Science in Context, 10 (1997), p. 297-342. [5]

External links

• Official Website of L'Association des Collaborateurs de Nicolas Bourbaki (French)
• A long article about Nicolas Bourbaki, from PlanetMath
• 25 Years with Bourbaki, by Armand Borel
• O'Connor, John J; Edmund F. Robertson "Nicolas Bourbaki". MacTutor History of Mathematics archive.