AChaoticEnd? 241Possibly. The folklore of the group is large, and growing, even though almost all of the founders are
now dead; and there are interviews and more or less gossipy histories (e.g. Mashaal 2002) which
show a series of photographs of young men meeting and presumably arguing about their project
in the sunshine, in the French countryside. The contrast could not be more complete between
the frozen impersonal texts of theÉléments de mathématiquesand the apparently anarchic obscene
atmosphere, part party, part student meeting, in which their content was hammered out.[W]e almost surprised ourselves when for the first time we approved a text as ready to go to press. This was theFascicule
de Résultats[volume of results] of set theory, adopted in its definitive form just before the war. A first text on this theory,
prepared by Cartan, had been read at the ‘Escorial Congress’; Cartan, who had been unable to attend, was informed
by telegram of its rejection: ‘Union intersection partie produit tu es démembré foutu Bourbaki’ [Union intersection subset
product you are dismembered fucked Bourbaki]. (Weil 1992, p. 114)An extract from the introduction to Bourbaki’sAlgebrais provided in Appendix A, to show with
what severity he stated his aims and method. Before analysis could begin, the real numbers had to
be defined; before the real numbers, the elements of topology; before that, the theory of sets. The
student faced a long march before arriving (say) at the least upper bound theorem which we have
seen causing such problems earlier on. The ‘second-generation’ Bourbakist Pierre Cartier provides
a balanced evaluation of strengths and weaknesses of the project:Bourbaki knew where to go: his goal was to provide the foundation for mathematics. They had to submit all math-
ematics to the scheme of Hilbert; what van der Waerden had done for algebra would have to be done for the rest
of mathematics. What should be included was more or less clear. The first six books of Bourbaki comprise the basic
background knowledge of a modern graduate student.
The misunderstanding was that many people thought that it should be taught the way it was written in the
books. You can think of the first books of Bourbaki as an encyclopedia of mathematics, containing all the necessary
information. That is a good description. If you consider it as a textbook, it’s a disaster. (Senechal 1998)This is slightly disingenuous; it is rare to come across an encyclopaedia which is equipped with
a complete set of exercises. All the same, there were few who used the completeÉlémentsas their
textbook. What was much more important about it was that its existence profoundly influenced the
way in which a large number of mathematicians thought about their subject; and some of them
used their thinking to write more readable textbooks of their own, in which the ideas of structures,
the emphasis on the mapping rather than the object, and so on, became foregrounded. The writer
of a textbook can, and usually does pick and choose from available material without necessarily
overt plagiarism. In any case, neat Bourbaki was hard to plagiarize; but a watered down version
became increasingly dominant outside France as well as inside.
The Bourbakists had no particular interest in axiom systems as a means of saving mathematics
from contradiction.^5 They did see axiom systems as the basic tool in defining ‘structures’, which
were to be central to the whole way in which theÉlémentswas presented. As has been pointed
out, the idea of structure was one which, among so many other definitions, was never defined;
but numerous individual structures (group, ring, topological space, uniform space,...) permeated
the text and were central to its particular way of thinking. It became almost a reflex in France, if
one had fallen under the spell of theElements, to speak not of defining a group but of ‘providing
(munir de) a set with a group-structure’.
- Nor were they interested in a number of other things—probability theory, for example, and physics.