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Wednesday, January 16, 2008
Reading List: Bourbaki: A Secret Society of Mathematicians
 Mashaal, Maurice. Bourbaki: A Secret Society of Mathematicians. Translated by Anna Pierrehumbert. Providence, RI: American Mathematical Society, [2002] 2006. ISBN 9780821839676.

In 1934,
André
Weil and
Henri Cartan,
both young professors of mathematics at the University of
Strasbourg, would frequently, when discussing the calculus
courses they were teaching, deplore the textbooks available,
all of which they considered antiquated and inadequate.
Weil eventually suggested getting in touch with several
of their fellow alumni of the
École
Normale Supérieure who were teaching similar courses
in provincial universities around France, inviting them to collaborate
on a new analysis textbook. The complete work was expected to
total 1000 to 1200 pages, with the first volumes ready about
six months after the project began.
Thus began one of the most flabbergasting examples of “mission creep” in human intellectual history, which set the style for much of mathematics publication and education in subsequent decades. Working collectively and publishing under the pseudonym “Nicolas Bourbaki” (after the French general in the FrancoPrussian War Charles Denis Bourbaki), the “analysis textbook” to be assembled by a small group over a few years grew into a project spanning more than six decades and ten books, most of multiple volumes, totalling more than seven thousand pages, systematising the core of mathematics in a relentlessly abstract and austere axiomatic form. Although Bourbaki introduced new terminology, some of which has become commonplace, there is no new mathematics in the work: it is a presentation of preexisting mathematical work as a pedagogical tool and toolbox for research mathematicians. (This is not to say that the participants in the Bourbaki project did not do original work—in fact, they were among the leaders in mathematical research in their respective generations. But their work on the Bourbaki opus was a codification and grand unification of the disparate branches of mathematics into a coherent whole. In fact, so important was the idea that mathematics was a unified tree rooted in set theory that the Bourbaki group always used the word mathématique, not mathématiques.)
Criticisms of the Bourbaki approach were many: it was too abstract, emphasised structure over the content which motivated it, neglected foundational topics such as mathematical logic, excluded anything tainted with the possibility of application (including probability, automata theory, and combinatorics), and took an eccentric approach to integration, disdaining the Lebesgue integral. These criticisms are described in detail, with both sides fairly presented. While Bourbaki participants had no ambitions to reform secondary school mathematics education, it is certainly true that academics steeped in the Bourbaki approach played a part in the disastrous “New Math” episode, which is described in chapter 10.
The book is extravagantly illustrated, and has numerous boxes and marginal notes which describe details, concepts, and the dramatis personæ in this intricate story. An appendix provides English translations of documents which appear in French in the main text. There is no index.
La version française reste disponible.
Posted at January 16, 2008 23:43