Bourbaki

In 1934 A. Weil (28 years old) and H. Cartan(30 year sold) were Maîtres de Conférences (the equivalent of assistant professors) at the University of Strasbourg. One main duty was, of course, the teaching of differential and integral calculus. The standard text was the Traité d'Analyse of E. Goursat, which they found wanting in many ways. Cartan was frequently bugging Weil with questions on how to present this material, so that at some point, to get it over with once and for all, Weil suggested they write themselves a new Traité d'Analyse. This suggestion was spread around, and soon a group of about ten mathematicians began to meet regularly to plan this treatise. It was soon decided that the work would be collective, without any acknowledgment of individual contributions. In summer 1935 the pen name Nicolas Bourbaki was chosen.

The membership varied over the years; some people in the first group dropped out quickly, others were added, and later there was a regular process of additions and retirements. I do not intend to give a detailed account. At this point let me simply mention that the true "founding fathers", those who shaped Bourbaki and gave it much of their time and thoughts until they retired, are:

Henri Cartan

Claude Chevalley

Jean Delsarte

Jean Dieudonné

André Weil

born respectively in 1904, 1909, 1903, 1906, 1906 – all former students at the École Normale Supérieure in Paris.⁵

A first question to settle was how to handle references to background material. Most existing books were found unsatisfactory. Even B. van der Waerden's Moderne Algebra, which had made a deep impression, did not seem well suited to their needs (besides being in German). Moreover, they wanted to adopt a more precise, rigorous style of exposition than had been traditionally used in France, so they decided to start from scratch and, after many discussions, divided this basic material into six "books", each consisting possibly of several volumes, namely:

1. Set Theory

2. Algebra

3. Topology

4. Functions of One Real Variable

5. Topological Vector Spaces

6. Integration

These books were to be linearly ordered: references at a given spot could only be to the previous text in the same book or to an earlier book (in the given ordering). The title "Éléments de Mathématique" was chosen in 1938. It is worth noting that they chose "Mathématique" rather than the much more usual "Mathématiques". The absence of the "s" was of course quite intentional, one way for Bourbaki to signal its belief in the unity of mathematics.

The first volumes to appear were the Fascicle of Results on Set Theory (1939) and then, in the forties, Topology and three volumes of Algebra.

5. Nota de Borel They all contributed in an essential way. For Cartan, Chevalley, Dieudonné, and Weil I could witness it at firsthand, but not for Delsarte, who was not really active anymore when I came on board. But his importance has been repeatedly stressed to me by Weil in conversations. See also [14] and comments by Cartan, Dieudonné, Schwartz in [3, pp.81–83]. In particular, he played an essential role in transforming into a coherent group, and maintaining it so, a collection of strong, some quite temperamental, individuals. Besides, otrviously, Book IV, Functions of one real variable, owes much to him. Some other early members, notably Szolem Mandelbrojt and René de Possel, also contributed substantially to the work of the group in its initial stages.

En el segundo semestre de 1968 como estudiante de matemáticas en la Universidad Nacional en Bogotá, fui alumno del profesor Jairo Charris en el curso Álgebra Multilineal con el texto de Bourbaki. Años después, cuando enseñe el curso de Cálculo en varias variables con el libro de Lima y descubrí su libro sobre Variedades me interesé seriamente por el cálculo tensorial. Chepe (Jose F. Escobar q.e.p.d.), el destacado geómetra analista exalumno de la U.Valle me recomendó el libro de Eisenhart y aprecié su poder en la geometría diferencial. Pero sólo el enfoque de Bourbaki puede liberar a la geometría multidimensional de la tiranía de las coordenadas.

En el segundo semestre de 1968 como estudiante de matemáticas fui alumno del profesor Oto Raúl Ruiz quien enseñaba el curso de Topología con el texto de Pervins. Yo leí simultáneamente con los amigos el primer volumen (los 4 primeros capítulos) del libro de Topología de Bourbaki. Puede ser que la lectura simultánea del texto (asequible) y del Libro (difícil) haya hecho posible reconocer en el libro de Bourbaki el libro perfecto. Distingo muy claramente el libro del texto.