Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of 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.
Books authored by Bourbaki
Aiming at a completely self-contained treatment of most of modern mathematics based on set theory, the group produced the following volumes:
I Set theory
IV Functions of one real variable
V Topological vector spaces
VII Commutative algebra
VIII Lie groups
A final volume IX on spectral theory from 1983 marked the presumed end of the publishing project.
The emphasis on rigour, which turned out to be quite influential, may be seen as a reaction to the work of Jules-Henri Poincaré, who stressed the importance of free flowing mathematical intuition. The influence of Bourbaki's work has decreased over time, partly because some of their abstractions did not prove as useful as initially thought, and partly because other abstractions which are now considered to be important, such as the machinery of category theory, are not covered.
While several of Bourbaki's books have become standard references in their fields, the austere presentation makes them unsuitable as textbooks. 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.
Notations introduced by Bourbaki include: the symbol ∅ for the empty set, the blackboard bold letters for the various sets of numbers, and the terms injective, surjective, and bijective.
The Bourbaki seminar series founded immediately post-war in Paris does continue, as a source of survey articles written in a prescribed, careful style.
Accounts of the early days vary. The founding members were all connected to the Ecole Normale Supérieure in Paris and included André Weil, Jean Dieudonné, Szolem Mandelbrojt, Claude Chevalley, Henri Cartan; and several other young French mathematicians (amongst them Jean Delsarte, René de Possel). Other notable participants in later days were Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck and Samuel Eilenberg.
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.
"Bourbaki" is the name of a French general who was defeated in the Franco-Prussian War; the name was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue.
The Bourbaki point of view, as non-neutral
It is fairly clear that the Bourbaki point of view, while 'encyclopedic', was never intended as 'neutral'. Quite the opposite, really: more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy of formalism and axiomatics. But always through a transforming process of reception.
Conspicuous in the list of areas where Bourbaki is not a neutral:
algorithmic content is not considered on-topic and is almost completely omitted
problem solving is considered secondary to axiomatics
analysis is treated 'softly', without 'hard' estimates
measure theory is coerced towards Radon measures
combinatorial structure is deemed non-structural
logic is treated minimally (Zorn's lemma to suffice)
And (cela va sans dire) no pictures.
Mathematicians have always preferred folk-history and anecdotes. Bourbaki's history of mathematics suffers not from lack of scholarship - but from the attitude that history should be written by the victors in the struggle to attain axiomatic clarity.
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 didn't shy away from a hierarchical development of the 'important' mathematics of the time.
He also wrote extensive books: on analysis, perhaps in belated fulfilment 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! remarks at the meetings), it may be doubted whether all others agreed with him about mathematical writing and research.
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.
The Bourbakiste influence
In the end the manifesto of Bourbaki has had an influence, particularly on graduate education in pure mathematics. This effect can be read in detail in parts of this site.
The New Maths project of early maths teaching cannot simply be equated with that influence, though. The use for example of Venn diagrams goes back to the pedagogy of the nineteenth century. The furore involved can now be seen as a demarcation dispute along the calculus/discrete maths boundary.
The 'leading role' of Bourbaki, speaking internationally rather than just in terms of France, may have transferred to the programme of the Bonn Arbeitstagung, already by the early 1960s.