Bourbaki, Nicolas

Bourbaki is the collective pseudonym of an influential group of mathematicians, almost all French, who since the late 1930’s have been engaged in writing what is intended to be a definitive survey of all of mathematics, or at least of all those parts of the subject which Bourbaki considers worthy of the name. The work appears in installments that are usually from 100 to 300 pages long. The first appeared in 1939 and the thirty-third in 1967; many intervening installments have been extensively revised and reissued. The selection of topics is very different from that in a traditional introduction to mathematics. In Bourbaki’s arrangement, mathematics begins with set theory, which is followed, in order, by (abstract) algebra, general topology, fonctions of a real variable (including ordinary calculus), topological vector spaces, and the general theory of integration. To some extent the order is forced by the logical dependence of each topic on its predecessors. Bourbaki has not yet reached the other parts of mathematics. Although the work as a whole is called Elements of Mathematics, no one could read it without at least two years of college mathematics as preparation, and further mathematical study would be an advantage.

The exact composition of the Bourbaki group varies from year to year and has been deliberately kept mysterious. The project was begun by a number of brilliant young mathematicians who had made important contributions to mathematics in their own right. At the beginning they made no particular attempt at secrecy. With the passage of time, however, they seem to have become more and more enamored of their joke, and have often tried to persuade people that there is indeed an individual named N. Bourbaki, who writes the books. Indeed, Bourbaki once applied for membership in the American Mathematical Society, but was rejected on the ground that he was not an individual. The original group included H. Cartan, C. Chevalley, J. Dieudonné, and A. Weil (all of whom are among the most eminent mathematicians of their generation). Many younger French mathematicians have joined the group, which is understood to have ten to twenty members at any one time and has included two or three Americans. The founding members are said to have agreed to retire at the age of fifty, and are believed to have done so, although with some reluctance.

The origin of the name Nicolas Bourbaki is obscure. The use of a collective pseudonym was presumably intended to obviate title pages with long and changing lists of names and to provide a simple way of referring to the project. The family name appears to be that of General Charles-Denis-Sauter Bourbaki (1816–1897), a statue of whom stands in Nancy, where several members of the group once taught. Possibly the Christian name was supposed to suggest St. Nicholas bringing presents to the mathematical world.

In the early days Bourbaki published articles in mathematical journals, as any mathematician would. He soon gave that up, however, and his reputation rests on his books. People who are unsympathetic to the “new mathematics” introduced into the schools since 1960 accuse Bourbaki of having inspired that movement. The accusation is probably unjustified, although aspects of his work bear a superficial resemblance to less attractive aspects of new mathematics. Bourbaki himself does not intend his approach to be used even in college teaching. Rather, it is meant to improve a mathematician’s understanding of his subject after he has learned the fundamentals and to serve as a guide to research.

The most obvious aspects of Bourbaki’s work are his insistence on a strict adherence to the axiomatic approach to mathematics and his use of an individual and (originally) unconventional terminology (much of which has since become widely accepted). The former is the more important. Any mathematical theory starts, in principle, from a set of axioms and deduces consequences from them (although many subjects, such as elementary algebra, are rarely presented to students in this way). In classical axiomatic theories, such as Euclidean geometry or Peano’s theory of the integers, one attempts to find a set of axioms that precisely characterize the theory. Such an axiomatization is valuable in showing the logical arrangement of the subject, but the clarification so achieved is confined to the one subject, and often seems like quibbling.

A good deal of the new mathematics consists of introducing such axiomatizations of elementary parts of mathematics at an early stage of the curriculum, in the hope of facilitating understanding. Bourbaki’s axiomatization is in a different spirit. His axioms are for parts of mathematics with the widest possible scope, which he calls structures. A mathematical structure consists, in principle, of a set of objects of unspecified nature, and of certain relationships among them. For example, the structure called a group consists of a set of elements such that any two can be combined to give a third. The way in which this is done must be subject to suitable axioms. The structure called an order consists of a set of elements with a relationship between any two of them, corresponding abstractly to the statement (for numbers) that one is greater than the other.

Having studied a structure, one may add axioms to make it more special (finite group or commutative group, for example). One can combine two structures, assuming that the objects considered satisfy the axioms of both (obtaining, for example, the theory of ordered groups). By proceeding in this way, one obtains more and more complicated structures, and often more and more interesting mathematics. Bourbaki, then, organizes mathematics as an arrangement of structures, the more complex growing out of the simpler.

There are great advantages in dealing with mathematics in this way. A theorem, once proved for an abstract structure, is immediately applicable to any realization of the structure, that is, to any mathematical system that satisfies the axioms. Thus, for example, a theorem about abstract groups will yield results (which superficially may look quite different) about groups of numbers, groups of matrices, or groups of permutations. Again, once it is recognized that the theory of measure and the theory of probability are realizations of a common set of axioms, all results in either theory can be reinterpreted in the other. Historically, in fact, these two theories were developed independently of each other for many years before their equivalence was recognized. Bourbaki tries to make each part of mathematics as general as possible in order to obtain the widest possible domain of applicability. His detractors object that he loses contact with the actual content of the subject, so that students who have studied only his approach are likely to know only general theorems without specific instances. Of course, the choice of an axiom system is never arbitrary. Bourbaki’s collaborators are well aware of the concrete theories they are generalizing, and select their axioms accordingly.

Bourbaki has been influential for a number of reasons. For one thing, he gave the first systematic account of some topics that previously had been available only in scattered articles. His orderly and very general approach, his insistence on precision of terminology and of argument, his advocacy of the axiomatic method, all had a strong appeal to pure mathematicians, who in any case were proceeding in the same direction. Since mathematicians had to learn Bourbaki’s terminology in order to read his work, that terminology has become widely known and has changed much of the vocabulary of research. The effect of the work in the development of mathematics has been fully commensurate with the great effort that has gone into it.

BIBLIOGRAPHY

I. Original Works. Works by Bourbaki include “The Architecture of Mathematics,” in American MathematicalMonthly, 57 (1950), 221–232; and Éléments de mathématique, many numbers in the series Actualités Scientifiques et Industrielles (Paris, 1939-).

II. Secondary Literature. André Delachet, “L’école Bourbaki,” in L’analyse mathématique (Paris, 1949), pp. 113–116; Paul R. Halmos, “Nicolas Bourbaki,” in Scientific American, 196 , no. 5 (May 1957), 88–99.

R. P. Boas, Jr.