VI.96 Nicolas Bourbaki

b. Paris, 1935; d. —

Set theory; algebra; topology; foundations of mathematics; analysis; differential and algebraic geometry; integration theory; spectral theory; Lie algebras; commutative algebras; history of mathematics

Bourbaki is a pseudonym chosen in 1935 by a group of French mathematicians, including Henri Cartan, Jean Dieudonné, and ANDRÉ WELL [VI.93]. Under this nom de plume, several generations of mostly French mathematicians conceived, wrote, and published a series of treatises under the general title Éléments de Mathématique. The uncommon use of the singular “mathématique” underscored a strong commitment to the unity of mathematics that is one of the chief characteristics of the group. Together with the “Bourbaki Seminar,” this monumental work promoted a unified, axiomatic, and structural view of pure mathematics that has exerted a strong influence on teaching and research since World War II, especially in France.

Charles Denis Sauter Bourbaki was a French general who fought in the Franco-Prussian war in 1870–71. A hoax lecture given by students at the École Normale Supérieure to the entering class in 1923 culminated with a “Bourbaki theorem.” In 1935, a group of mathematicians, many of whom had taken part in that lecture, as either audience or pranksters, decided to adopt that name for the fictive author of the modern treatise of analysis they were planning to write.

Their first meeting had taken place in Paris on December 10, 1934. In addition to Cartan, Dieudonné, and Weil, other young university professors of mathematics were present: Claude Chevalley, Jean Delsarte, and René de Possel. Agreeing that analysis textbooks available in French (such as Édouard Goursat’s Cours d’ Analyse) were outdated, they decided to write a book, collectively, to replace them. Having been in touch with modern German mathematics, especially at HILBERT’s [VI.63] Göttingen, and influenced in particular by Barteel van der Waerden’s Moderne Algebra, they thought that their large treatise should begin with an “abstract packet” summarizing in axiomatic form basic general notions such as sets, groups, and fields. Soon after this, Szolem Mandelbrojt joined the group. Paul Dubreil and Jean Leray took part in just a few of the original meetings, and were replaced by Charles Ehresmann and the physicist Jean Coulomb.

In July 1935, the group had its first “congress” (as its annual summer meetings would later be called) in Besse-en-Chandesse, Auvergne, where the pen name “N. Bourbaki” was definitively adopted (the first name, Nicolas, was chosen later). Settling on working procedures, they drew up the general outline of the planned treatise. The members of the group worked collectively following certain ritual rules. They co-opted new collaborators, kept membership secret, and refused to acknowledge individual contributions. During the three or four working sessions they held every year, each contribution prepared in advance by one of them was read line by line, discussed, and severely criticized by the others. Up to ten successive drafts and several years of work by various authors were often needed before a final version was unanimously adopted.

The first booklet—a digest of results in set theory—was dated 1939 but issued in 1940. Despite the difficult working conditions during World War II, this was soon followed in the 1940s by several booklets dealing mostly with general topology and algebra. Today, the Elements of Mathematics consists of several books: Theory of Sets, Algebra, General Topology, Real-Variable Functions, Topological Vector Spaces, Integration, Commutative Algebra, Differential and Analytic Manifolds, Lie Groups and Lie Algebra, Spectral Theories, and Elements of the History of Mathematics. Many of them have been extensively revised over the years and translated into several languages, including English and Russian.

The first six books formed a tight linear exposition entitled “The fundamental structures of analysis.” When they first appeared, they were striking for the logical organization of the topics covered. The axiomatic method was used systematically, and great effort was made to ensure a global unity of style, notation, and terminology. The avowed ambition was to take mathematics from its very start and, proceeding from the general toward the particular, write a unified survey of most of modern mathematics.

Several generations of mathematicians were co-opted into the “Association of Bourbaki’s Collaborators,” as the group is now officially known. After World War II, Samuel Eilenberg, Laurent Schwartz, Roger Godement, Jean-Louis Koszul, and Jean-Pierre Serre, among others, took part in the writing of the treatise. Later, Armand Borel, John Tate, François Bruhat, Serge Lang, and Alexander Grothendieck also joined. Although its frequency of publication has now slowed to a trickle, the group is still functioning in the first decade of the twenty-first century.

Notwithstanding the number of collaborators involved and the extensiveness of the work they published, Bourbaki’s vision of mathematics was, and has remained, surprisingly coherent. Most of the crucial mathematical choices, which would come to have a huge impact on the structural image of mathematics that the group would later vigorously promote, were made in the late 1930s. In the following decades, many mathematicians shared a conviction that a tight axiomatic refoundation of their research domains would help overcome current blockages. This was felt, for example, in probability theory, model theory, algebraic geometry and topology, commutative algebra, Lie groups, and Lie algebras.

After World War II, as the notoriety of both the group and its individual members steadily grew, Bourbaki’s public image soon encompassed more than just the treatise. At the level of mathematical research, the Bourbaki Seminar was a prestigious outlet established in Paris in 1948, and it has met three times a year ever since. Members of Bourbaki selected speakers who usually summarized someone else’s work, and supervised the publication of their talks. The topics selected emphasized specific domains of mathematics, such as algebraic and differential geometry, at the expense of others, such as probability theory or applied mathematics.

Bourbaki’s views on the philosophy of mathematics were always clear, especially after two articles published in the late 1940s under that name argued for a complete reorganization of mathematics, eschewing older classification schemes in favor of fundamental structures (sometimes called “mother-structures” and supposedly closer to the deep mental structures of humans) meant to underscore the organic unity of mathematics. Bourbaki’s public image was echoed by structuralists in the human sciences as well as artists and philosophers, and it was invoked by radical reformers of mathematical education from kindergarten to university—although actual members of Bourbaki were rarely involved directly.

From the late 1960s, Bourbaki’s critics became louder on two counts: they took issue with the Bourbaki approach to the logical foundations of mathematics and they found gaps in the group’s encyclopedic objectives. CATEGORY THEORY [III.8] developed by Saunders Mac Lane and Samuel Eilenberg was found to offer a more fruitful foundational framework than Bourbaki’s structures. It also became clear that whole branches of mathematics—probability theory, geometry, and, to a lesser extent, analysis and logic—were to remain absent from the treatise, their very place in the grand architecture of Bourbakist mathematics left unclear. For a new generation of mathematicians, it was Bourbaki’s elitist contempt for applications that was especially damaging.

Bourbaki’s impact on mathematics was profound: despite its excesses, Bourbaki’s unified, structural, rigorous image of mathematics is still with us. But it was those very characteristics that led to a feeling that Bourbaki was corseting mathematical research. The backlash seems to be abating somewhat nowadays, but no new Bourbaki is in view.

Further Reading

Beaulieu, L. 1994. Questions and answers about Bourbaki’s early work (1934–1944). In The Intersection of History and Mathematics, edited by S. Chikara et al., pp. 241–52. Basel: Birkhäuser.

Corry, L.1996. Modern Algebra and the Rise of Mathematical Structures. Basel: Birkhäuser.

Mac Lane, S. 1996. Structures in mathematics. Philosophia Mathematica 4:174–86.

David Aubin