Nicolas Bourbaki: Encyclopedia II - Nicolas Bourbaki - The Bourbaki perspective and its limitations

Nicolas Bourbaki - The Bourbaki perspective and its limitations

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 and the German algebraists. 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, with emphasis on formalism and axiomatics. But always through a transforming process of reception and selection—typical of a French salon if more intensive.

Examples of the tendency are the way tensor calculus was renamed multilinear algebra, and the emergence of commutative algebra as independent of elimination theory, which had been a major motivation under its earlier name of ideal theory. Hilbert had already in the 1890s shown a preference for non-constructive methods; these changes of name made visible a definite change of attitude.

The following are now with hindsight (as of 2005) conspicuous in the list of areas where Bourbaki is not 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)
• applications is not covered.

And, it goes without saying, no pictures. In fact geometry as a whole is slighted, where it doesn't reduce to abstract algebra and soft analysis. The Bourbaki approach can be defended on the grounds of effectiveness, rather than elegance: this is the traditional argument against synthetic geometry and not novel with the Bourbaki group. There is a larger historical ebb and flow. Weil discusses in his Collected Works the suspicion that geometric intuition is but a facade. Hilbert did collaborate on the Hilbert-Cohn Vossen book of 'intuitive geometry'. Here Bourbaki is notably selective of the attitudes of its chosen patriarch.

Historical notes accompanied many of the Bourbaki volumes. Mathematicians have always preferred folk-history and anecdotes. Bourbaki's history of mathematics, later gathered as a separate book, suffers in contrast not from lack of scholarship — but from the attitude that history should be written by the victors in the struggle to attain axiomatic clarity. It is inevitably partial, but also partisan.

Adapted from the Wikipedia article "The Bourbaki perspective and its limitations", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki