Nicolas Bourbaki: Encyclopedia II - Nicolas Bourbaki - Influence on mathematics in general

Nicolas Bourbaki - Influence on mathematics in general

The emphasis on rigour may be seen as a reaction to the work of Jules-Henri Poincaré, who stressed the importance of free-flowing mathematical intuition, at a cost in completeness in presentation. The impact of Bourbaki's work initially was great on many active research mathematicians world-wide.

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 practised.

Bourbaki's influence has decreased over time. 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.

The Bourbaki seminar series founded in post-WWII Paris continues. It is an important source of survey articles, written in a prescribed, careful style.

Adapted from the Wikipedia article "Influence on mathematics in general", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki