Nicolas Bourbaki - Appraisal of The Bourbaki Perspective

Appraisal of The Bourbaki Perspective

The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Göttingen school, particularly Hilbert and the modern algebra school of Emmy Noether, Artin and van der Waerden. It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics. But always through a transforming process of reception and selection — their ability to sustain this collective, critical approach has been described as "something unusual".

The following is a list of some of the criticisms commonly made of the Bourbaki approach. Pierre Cartier, a Bourbaki member 1955–1983, commented explicitly on several of these points : ...essentially no analysis beyond the foundations: nothing about partial differential equations, nothing about probability. There is also nothing about combinatorics, nothing about algebraic topology, nothing about concrete geometry. And Bourbaki never seriously considered logic. Dieudonné himself was very vocal against logic. Anything connected with mathematical physics is totally absent from Bourbaki's text.

• algorithmic content is not considered on-topic and is almost completely omitted
• problem solving, in the sense of heuristics, receives less emphasis than axiomatic theory-building
• analysis is treated 'softly', without 'hard' estimates
• Measure theory is developed from a functional analytic perspective. Taking the case of locally compact measure spaces as fundamental focuses the presentation on Radon measures and leads to an approach to measurable functions that is cumbersome, especially from the viewpoint of probability theory. However, the last chapter of the book addresses limitations, especially for use in probability theory, of the restriction to locally compact spaces.
• combinatorics is not discussed
• logic is treated minimally
• applications are not covered.

Furthermore, Bourbaki make no use of pictures in their presentation. Pierre Cartier, in the article cited above, is quoted as later saying The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith. In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.