Bourbaki, Nicolas (1939)

Mike Conlay

written 6/11/95
presented 6/15/95
Nicolas Bourbaki was not an actual person. Rather his name is a pseudonym for a society of young twentieth century french mathematicians. This group of mathematicians stemed from a group of young graduates of the Ecole Normal Superieure in Paris. These men sought to jointly write a textbook on analysis geared toward French university students. This book was intended to replace the classic annalysis textbook used in France that had been writen by Goursat. They wished for it to live up to the quality of Goursat's work yet modernize it so that it incorporated mathmatical advancements that had occured in the twentieth century. Thus began the mathematical society that collaborated under the pseudonym Bourbaki.

The founding members of the Bourbaki movement, or membres fondateurs, were Henri Cartan, Claude Chevalley (also writen as Chevallier), Jean Dieudonne, Jean Deslarte, and Andre Weil. Andre Weil was a universalist, proficient in almost every field of mathematics. He contributed substantially on various topics such as harmonic analysis, differential geometry, algebraic number theory, functions of several complex variables, and many other mathmatical subjects. He was one of the prime catalysts of the Bourbaki movement and was so involved in the movement that he named his daughter Nicolette after Bourbaki's first name. Henri Cartan was the son of Elie Cartan, a master of differential geometry and a protege of Poincare. He studied in the field of analytic functions of several complex variables, among others. He made significant contributions in almost every major field of algebraic topology. Chevalley did not cover as many fields as his aforementioned cohorts, yet he worked deeply in the area of algebraic geometry and algebraic number theory. Dieudonne was sometimes referred to as the chief scribe of Bourbaki's works and it is his personality that is greatly emmited from Bourbakis work. He worked in many diverse fields from a vigorous and unambiguous attitude. His personality is reflected in a paper he published called "On an Error of M. Bourbaki": He critiqued a paper that he had originally had a hand in writing! Deslartes is the one founding member who has been hardly well known. Over the years other bright young French, and sometimes international, mathematicians have contributed to Bourbaki's works.

Bourbaki, or rather the men behind the name, undertook the task of producing a fully axiomatized presentation of mathematics in entirety. Many of the mathematicians of the time were convinced that such a construction of mathematics was feasible but had never undertaken the laborious task. This massive adventure began with the publishing of Bourbaki's publishing of the "Elements" in 1939. This volominous series of books sought to construct mathematics in a way that unified all of its various and remote fields into one connected saga. Bourbaki felt was concerned with the effects that had been produced from the rapid development of twentieth century mathematics. Various fields had exploded onto the scene and there was no apparant relation between most of them. Thus by 1968 thirty-three volumes of Elements were produced. Bourbaki wrote from a rigid abstrcact approach and didn't stray from the arguments at hand. Thus the contributions of previous figures from mathematical history were not payed tribute until 1960 when Bourbaki published Elements d'historique des mathematiques. Although this was not a truly great documentation of math history, it did explain the work of previous mathematicians and give credit where credit is due.

Though they were brilliant, most of the Bourbaki contributors were not universal mathematicians and they knew that such an enterprise as theirs was beyond individual capabilities. Thus they sought the best method to produce their work collectively. They couldn't simply divide out topics to the ones most proficent in the specific areas because then they would simply produce a typical mathematical encyclopedia, which is furthest from what they wanted. In order to produce a synthesis of the various mathematical fields, they sacrificed their pride and forsake studying in their respective mathematical disciplines, started fresh, and for some years diligently studied fundamental concepts of all branches of mathematics. Finally they began meeting together as the Bourbaki-congres three times a year for a week or two at a time in various pleasent vacation spots. They worked about eight hours a day and spent the rest of the day reflecting on their endeavors. The congresses were a time to make plans for forthcoming volumes of the Elements and to prepare each volume. One member would agree to write the first draft for a certain topic and latter send mimeographed copies of the completed draft to each member for discussion at the next congress. At the next congress, after lively discussions, a joint outline is produced of the same work and is passed on to another member to rewrite a new work on the same topic again. This would happen a number of times and sometimes the topic, after much labor, would be abandoned and may even be brought up again at later congreses. The numerous revisions made sure that the work was of the uptmost quality and also made it impossible to detect the presence of any individual contributions to the work. Bourbaki became his own person. Also, the selfless founding members made sure Bourbaki would never grow old, rather his works would always reflect the spirit and energy of the original young contributors. They did this by establishing that all Bourbaki members must retire before age 50.

The format of Elements is very abstract and strictly logical. Thus it may appear boring or above the capabilities of less experienced mathematicians. Bourbaki once wrote that the Elements were "...directed especially to those who have a good knowledge of at least the contents of the first year or two of a university mathematics course." This, however, refered to those who attended the Ecole Normale Superieue in Paris, one of the finest mathematical institutions in France where such notables as Lagrange had taught. Furthermore, the problems in "Excercise", which accompanied the Elements, are too difficult for any undergraduate. But through this style, Bourbaki's Elements presented an intrinsically complex unified and simplified account of the products of mathematics, even the most recent ones. The Elements were left open to constant revision to accomodate for the ever developing world of mathematics.

J. Dauben "The History of Mathematics from Antiquity to the Present", Garland Publishing, New York 1985
J. Fang "Bourbaki: Towards a Philosophy of Modern Mathematics", Paideia Press, Hauppauge, New York 1970

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