Nicolas Bourbaki

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Nicolas Bourbaki is the collective pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an 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.

While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki ("association of collaborators of Nicolas Bourbaki"), which has an office at the École Normale Supérieure in Paris.

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 (with the original French titles in brackets):

I Set theory (Théorie des ensembles)
II Algebra (Algèbre)
III Topology (Topologie générale)
IV Functions of one real variable (Fonctions d'une variable réelle)
V Topological vector spaces (Espaces vectoriels topologiques)
VI Integration (Intégration)

and later

VII Commutative algebra (Algèbre commutative)
VIII Lie groups (Groupes et algèbres de Lie)

The book Variétés différentielles et analytiques was a fascicule de résultats, that is, a summary of results, on the theory of manifolds, rather than a worked-out exposition. A final volume IX on spectral theory (Théories spectrales) 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.

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 <math> \varnothing <math> for the empty set, the blackboard bold letters for the various sets of numbers, and the terms injective, surjective, and bijective.

Impact 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 immediately post-war in Paris continues. It is an important source of survey articles, written in a prescribed, careful style.

The group

Accounts of the early days vary, but original documents have now come to light. The founding members were all connected to the Ecole Normale Supérieure in Paris and included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil. 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 Intelligencer]; Jean Leray and Paul Dubreil were present at the preliminary meeting but 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.

The atmosphere in the group can be illustrated by an anecdote told by Laurent Schwartz. Dieudonné regularly and spectacularly threatened to resign unless topics were treated in their logical order, and after a while others played on this for a joke. Godement's wife wanted to see Dieudonné announcing his resignation, and so on one occasion while she was there Schwartz deliberately brought up again the question of permuting the order in which measure theory and topological vector spaces were to be handled, to precipitate a guaranteed crisis.

"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 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 enough 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:

And (cela va sans dire) 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.

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 extensively under his own name: nine volumes 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: education, institutions, trends

In the longer term, the manifesto of Bourbaki has had a definite and deep influence, particularly on graduate education in pure mathematics. This effect can be read in detail in parts of this encyclopedia. It is perhaps most noticeable in the treatment now current of Lie groups and Lie algebras. Dieudonné at one point said 'one can do nothing serious without them', for which he was reproached; but the change in Lie theory from something of which Jacques Hadamard could despair of getting a clear idea, to its everyday usage, owes much the type of exposition Bourbaki championed.

The New Maths project of early maths teaching, on the other hand, had little directly to do with Bourbaki. It is certainly true, though, that the inception of the New Math coincided with the height of Bourbaki's prestige, and the feeling that mathematics could simply be 'modernised', by being 'structuralised'. The use of Venn diagrams, for example, goes back to the pedagogy of the nineteenth century, rather than connecting with the École Normale Supérieure. The furore involved might 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 first years of the 1960s. Another point representing a turn of the tide in mathematics can be identified in 1959, when Jean-Pierre Serre and Armand Borel ran a seminar on complex multiplication. This was a key classical theory — a remark attributed to Hilbert made it 'the most beautiful part of mathematics' — but in doctrinaire Bourbakiste terms excluded, like much of number theory, from the 'core topics'.

External links

de:Nicolas Bourbaki fr:Association Bourbaki it:Nicolas Bourbaki is:Nicholas Bourbaki he:ניקולה בורבאקי ja:ニコラ・ブルバキ pl:Nicolas Bourbaki ru:Бурбаки, Николя zh:尼古拉·布尔巴基

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