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Nicolas Bourbaki is the pseudonym under which a group of mainly French 20thcentury 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. Overall, the project has produced more than 30 volumes.
The emphasis on rigour, which turned out to be quite influential, can be seen as a reaction to the work of JulesHenri 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 make them unsuitable as text books.
The founding members of the group 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. Other participants were Alexander Grothendieck and Samuel Eilenberg[?].
"Bourbaki" is the name of a French general who was defeated in the FrancoPrussian War.
Public discussion of what Bourbaki 'thinks' (thought?) has in general been through Jean Dieudonné[?], who has written extensively on analysis, and also on other topics mostly connected with algebraic geometry. In a survey of le choix bourbachique he didn't shy away from a hierarchical development of the 'important' mathematics of the time (now two decades ago).
As is noted above, the Bourbaki influence through the books isn't what it was; in fact it may have been at its strongest only when there were few other sources of graduatelevel texts in current pure mathematics, between 1950 and 1960. The seminar series founded immediately postwar in Paris does continue, as a source of survey articles written in a prescribed, careful style.
It is fairly clear, anyway, that the Bourbaki point of view 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. But always through a transforming process of reception.
Conspicuous in the list of areas where Bourbaki is not a neutral:
Bourbaki's history of mathematics suffers not so much from lack of scholarship  mathematicians have always preferred folkhistory and anecdotes  but for the attitude that history should be written by the victors in the struggle to attain axiomatic clarity. Dieudonné managed to state a consistent view, that most workers in mathematics were doing groundclearing work, so that a future Riemann could find the way ahead intuitively open.
In the end the manifesto of Bourbaki has had an influence, and this can be read in detail in parts of this site. It has surely not been the only major influence on mathematics of the twentieth century.
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