Definitions

# Nicolas Bourbaki

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. Bourbaki is a respected name now, but it was initially a clever prank played on the entire scientific establishment. For a few years, people thought that Nicolas Bourbaki existed and admired his talent, which was of course the combined talent of the group.

## Books by Bourbaki

Aiming at a completely self-contained treatment of the core areas of modern mathematics based on set theory, the group produced Elements of Mathematics (Éléments de mathématique) series, which contain the following volumes (with the original French titles in parentheses):

 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) IX Spectral theory (Théories spectrales)

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, some have felt that 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 and a dangerous bend symbol, and the terms injective, surjective, and bijective.

It is frequently claimed that the use of the blackboard bold letters for the various sets of numbers was first introduced by the group. There are several reasons to doubt this claim.

## 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 of 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 practiced.

Bourbaki's direct influence has decreased over time. This is partly because certain concepts which are now important, such as the machinery of category theory, are not covered in the treatise. The completely uniform and essentially linear referential structure of the books became difficult to apply to areas closer to current research than the already mature ones treated in the published books, and thus publishing activity diminished significantly from 1970's . It also mattered that while especially algebraic structures can be naturally defined in Bourbaki's terms, there are areas where the Bourbaki approach was less straightforward to apply.

On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed. This is in particular true in less applied parts of mathematics.

The Bourbaki seminar series founded in post-WWII Paris continues. It is an important source of survey articles, written in a prescribed, careful style. The idea is that the presentation should be on the level of absolute specialists, but for an audience which is not specialized in the particular field.

## 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, towards the end of 1934. 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.

The name "Bourbaki" refers to a French general; 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.

## Criticism 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 German algebraists, like Artin or 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".

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 is a list of some of the criticisms commonly made of the Bourbaki approach:

Furthermore, some have claimed that Bourbaki eschewed all use of pictures, although this is not in fact accurate (see, for example, "Figure 1" of Topologie Générale, Book 1, p. 3). In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.

## 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 did not 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 fulfillment 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! ("Hush, 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 the 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.

## Bourbaki's influence on mathematics education

In the longer term, the manifesto of Bourbaki has had a definite and deep influence, particularly on graduate education in pure mathematics. 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 to its everyday usage owes much to the type of exposition Bourbaki championed. Beforehand Jacques Hadamard despaired of ever getting a clear idea of it.

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