The Bourbaki School of Mathematics

Overview

Nicolas Bourbaki is the pen name of a group of mathematicians, most of them French, who have undertaken the writing of a definitive treatise of modern mathematics. The Bourbaki volumes emphasize the highest degree of mathematical rigor and the structures common to different areas of mathematics. The group perpetuates itself by continually electing new members and requiring that current members must leave the group at age 50.

Background

Nicolas Bourbaki is the invention of two French mathematicians, Claude Chevalley (1909-1984) and André Weil (1906-1998), who decided to write a more modern calculus text for French-speaking students than the ones that were typically used. The choice of the Bourbaki name may have had its origin in a student prank. At some time in the 1930s, students at the Ecole Normale Supérieure in Paris were invited to a lecture by a famous mathematician named Nicholas Bourbaki. The lecturer was really a student in disguise, and the lecture given was pure double-talk. There was, however, an important Bourbaki in French history, General Charles Denis Sauter Bourbaki, a French military officer of Greek descent who experienced a decisive defeat in the Franco-Prussian war of 1870. It may be that the choice of name honors the Greek roots of mathematics, but it may also simply reflect the memory of an unusual name, memorialized in a statue of General Bourbaki at Nancy, France, where some of the Bourbaki group had taught.

A new calculus book was needed because of the many developments that had occurred in mathematics over the preceding half century. In particular, important insights had been gained into the system of real numbers and the process of taking limits that are fundamental to the calculus. Calculus deals with derivatives, which are the limits of the ratios of small quantities as the quantities become infinitesimally small, and integrals, which are the limits of sums of quantities where the number of terms in the sum becomes infinitely large while the terms themselves become infinitely small.

Chevalley and Weil invited a number of other prominent French mathematicians to join in their project. The first Bourbaki group thus came to include Henri Cartan (1904- ), Jean Dieudonné (1906-1992), Szolem Mandelbrojt, and Rene de Possel. Although the group intended to finish its writing project in two years, they soon realized that to provide the detailed and rigorous treatment that they felt the subject required would take many years. The group thus made provisions for new members to be elected and agreed upon a rule that members would retire from the group at the age of 50. This rule reflected a consensus that mathematicians are at their most creative in their twenties and thirties, a belief widely held even today, despite some important examples to the contrary. Membership has primarily been drawn from French mathematicians, and the group has numbered between 10 and 20 members at a time. A very few American mathematicians have been members, however. Initially the group met monthly in Paris. Soon they changed to a slightly less demanding schedule, with Bourbaki congresses held three times a year.

The main product of the Bourbaki group is a set of volumes, the Eléments de mathématique, published beginning in 1939 and now amounting to more than 30 volumes, many of them revised more than once. The title is undoubtedly meant as a reference to Euclid's Elements, the first attempt to summarize all mathematical knowledge. The fiction of a single author is maintained throughout. There was no public acknowledgment of the multiple authorship, though the fact became public knowledge in 1954, after an article on Bourbaki by the American mathematician Paul R. Halmos (1916- ) appeared in the magazine Scientific American.

Most of the published volumes of the Elements come from the first part of the series, The Fundamental Structures of Analysis, which will consist of six "books" of from four to ten chapters each. A number of chapters of an unnamed second part have also appeared. While the contents of the first part are arranged from most fundamental to more specialized, the chapters are not being published in order. The first published chapters, appearing in 1940, are the first two in Book Three, dealing with general topology. The first two sequential chapters, on formal mathematics and the theory of sets, did not appear until 1954. This is not an illogical procedure, however, as the writers needed to decide how they would present the more advanced topics and then arrange the fundamentals accordingly.

For the Bourbaki group the foundation of mathematics is set theory. The original formulation of set theory, that of German mathematician Georg Cantor (1845-1918), was plagued by paradoxes resulting from the existence of infinite sets and the possibility of sets belonging to themselves. To avoid these problems, the Bourbaki group adopted the axiomatic form of set theory devised by Ernst Zermelo (1871-1953) and Adolf Abraham Fraenkel (1891-1965). Among these axioms is a rule that no set can belong to itself as well as the axiom of choice, which allows one to construct a set containing exactly one member from any collection of non-empty sets. Within the axiomatic approach, the earlier troubling paradoxes can be avoided. The Bourbaki writers do not claim, however to have settled the question of the foundations of mathematics permanently; rather, they believe it will be possible to make adjustments as the need arises, without having to discard 2,500 years of mathematical thinking.

Impact

Like the Elements of Euclid (c. 335-270 b.c.), the Elements of Bourbaki is not so much one of new creation as much as it is one of synthesis and systematic exposition. The Bourbaki writers are involved in solidifying the mathematical insights of their times, and thus the conclusions obtained in the Elements are often those of other mathematicians. The Bourbaki approach is thus not limited to the group. When features found in the Bourbaki approach are reflected in the work of other mathematical schools or in mathematics education, it is difficult to tell whether Bourbaki has originated the trend or is just following it.

There has been a persistent controversy over the proper place of rigor in mathematics education throughout the twentieth century. The Bourbaki members do not claim that their writings should be the basis for elementary mathematics instruction or even the university education of scientists and engineers who will use mathematics as a tool without attempting to add to mathematical knowledge. Since the 1950s there have been a number of efforts by mathematicians in the United States to base general mathematical education on notions from set theory and on the laws obeyed by the common arithmetic operations of addition and multiplication. These attempts, called the "new math" in the elementary grades, have been generally disappointing and have largely been abandoned.

The Bourbaki group's practice in accepting communal credit for their work is quite unusual in modern times. For most mathematicians and scientists, the assignment of credit for new discoveries is a serious matter. This was not always so. In late classical times a writer might have attached the name of an earlier philosopher to his work to give it more credibility. In the Middle Ages authors might have preferred to see their works remain anonymous as an act of humility. With the invention of printing and the publication of the first scientific journals, priority became important. When English physicist Robert Hooke (1635-1703) discovered the basic law governing the motion of elastic bodies, he first made it known as an anagram, a letter puzzle. Thus Hooke was able to prove his own priority and yet keep the discovery to himself for a while. The great English physicist and mathematician Isaac Newton (1642-1727) became engaged in a bitter dispute with German mathematician and philosopher Gottfried Wilhelm Leibniz (1646-1716) over credit for the invention of the calculus. In contrast to this overall trend, the willingness of the Bourbaki members to share or forego credit for a substantial amount of work is truly remarkable. The Bourbaki practice may, however, have anticipated some of the issues facing "big science" towards the end of the twentieth century, in which the number of creative contributors to a short research paper may number in the hundreds.

Scholars make a distinction between the primary literature of a field and the secondary literature. Since the appearance of the first scientific journals in the seventeenth century, the primary literature has consisted primarily of research papers, written immediately after each new discovery or observation. The secondary literature includes monographs, encyclopedias, and textbooks, in which current understanding of a field is systematized for the benefits of workers in the field. Most works in the secondary literature merely summarize the current status of a field, often with many authors each writing a specific chapter or section. The Bourbaki attempt to synthesize, rather than summarize, such an extensive body of knowledge is unique.

DONALD R. FRANCESCHETTI