Nicolas Bourbaki

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.

Contents

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
II Algebra
III Topology
IV Functions of one real variable
V Topological vector spaces
VI Integration

and later

VII Commutative algebra
VIII Lie groups

A final volume IX on spectral theory from 1983 marked the presumed end of the publishing project; but a further commutative algebra fascicle was produced at the end of the twentieth century.

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 \varnothing; 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.

The group

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. There was a preliminary meeting, the minutes are in the Bourbaki archives [for a full description of the initial meeting consult Liliane Beaulieu in the Mathematical Intellegencer]; besides those already mentioned, René de Possel, Jean Delsarte, Jean Leray and Paul Dubreil were there, but Leray and Dubreil dropped out before the group actually formed. Other notable participants in later days were Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, Samuel Eilenberg, Serge Lang and Roger Godement.

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" was the name of a French general who was defeated in the Franco-Prussian War; it was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue. It was certainly a reference to Greek mathematics, Bourbaki being of Greek extraction. It is a valid reading to take the name as implying a transplantation of the tradition of Euclid to a France of the 1930s, with soured expectations.

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. 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, which these moves made more concrete.

Conspicuous in the list of areas where Bourbaki is not neutral:

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. In particular Serre has often criticised the way the Bourbaki works were written, and has championed in France greater attention to problem-solving, within number theory especially, not an area treated in the main Bourbaki texts.

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. He pointed to the way axiomatic method can be used as a tool for problem-solving, for example by Alexander Grothendieck. Others found him too close to Grothendieck to be an unbiased observer. Comments in Pal Turán's 1970 speech on the award of a Fields Medal to Alan Baker about theory-building and problem-solving were a reply from the traditionalist camp at the next opportunity, Grothendieck having received a Fields Medal in absentia in 1966 and the awards being every four years.

The Bourbachique 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 on the other hand had little to do with that influence. The use of Venn diagrams, for example, 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, internationally rather than for France alone, had possibly been taken over by the programme of the Bonn Arbeitstagung as early as the 1960s.

External links

de:Nicolas Bourbaki fr:Association Bourbaki is:Nicholas Bourbaki [[he:ניקולה בורבאקי]] ja:ニコラ・ブルバキ pl:Nicolas Bourbaki


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