Nicolas Bourbaki

The group was founded in 1934–35 by a cohort of young French mathematicians, including André Weil, Henri Cartan, Jean Delsarte, Jean Dieudonné, and Claude Chevalley, initially to write a modern textbook on analysis. They soon expanded the project into a comprehensive treatise, Éléments de mathématique, aiming to provide a rigorous, self-contained foundation for all of mathematics, starting from set theory and using the axiomatic method.¹ The name 'Bourbaki' was borrowed from a French general, Charles-Denis Bourbaki, and the group adopted a fictional biography, complete with a family and a birthday, to maintain anonymity.²

The Éléments is organized into books on set theory, algebra, topology, functions of a real variable, topological vector spaces, and integration, among others. Its hallmark is the 'structuralist' approach, which emphasizes abstract structures (groups, rings, topological spaces) over concrete examples. This approach shaped the 'New Math' movement in school curricula during the 1960s, particularly in France and the United States.³ Bourbaki's notation and terminology, such as the symbols ∅, ℕ, ℤ, ℚ, ℝ, ℂ, and the word 'bijection', became standard. The group's insistence on rigor and generality influenced generations of mathematicians, including Alexander Grothendieck, who was not a member but was deeply affected by Bourbaki's style.⁴

Bourbaki operates as a secret society, with membership by invitation, typically for mathematicians under 50. Members must resign at age 50, a rule that has led to periodic renewals. The group holds intense, often brutal, working sessions where drafts are read aloud and criticized line by line; only texts that survive unanimous approval are published.⁵ The group's collective identity is maintained through a fictional biography: Nicolas Bourbaki was 'born' in 1906, 'died' in 1968, and has a 'daughter' named Betti. The group has also engaged in hoaxes, such as the 1949 letter to the American Mathematical Society announcing a 'proof' of the Riemann hypothesis, and the 1968 'death' announcement.²

Beyond the famous treatises, Bourbaki produced the Éléments d'histoire des mathématiques, a historical survey that, while often criticized for its Whig perspective, contains valuable insights. The group also influenced the development of category theory, as its members, particularly Samuel Eilenberg (an associate), were early adopters of categorical language.⁶ Bourbaki's seminars, held in Paris since 1948, are separate from the group itself but are organized by its members and have introduced many important results. The group's insistence on 'structure' has been debated, with critics like René Thom arguing that it stifled intuition and geometric thinking.⁷ Despite its decline in influence since the 1970s, Bourbaki remains active, with new volumes and revised editions still appearing.

Éléments de mathématique

The multi-volume treatise written by Bourbaki, covering the fundamental structures of mathematics.

Structuralism

An approach that emphasizes abstract structures over concrete examples, central to Bourbaki's philosophy.

Axiomatic method

A method of reasoning that starts from a set of axioms and derives theorems by formal logic.