Bourbaki Lives! His Archives Are Open

N. Bourbaki was not a real person. But the many French mathematicians who have collaborated under a name they picked out of the air have produced more than 40 books and 30 revisions since the 1930s. These anonymous, spirited mathematicians have produced a "fruitful source of concepts and references," said Jean-Paul Pier (University of Luxembourg). Pier's anecdotal talk, called "Bourbaki, an Epiphenomenon in the History of Mathematics," which he gave at the MAA's Carriage House on the evening of April 3, was a witty, insightful, and informative look at the movement's origins, impact, and meaning.

Motivated by David Hilbert's work, the Bourbaki—founding members included Henri Cartan, Claude Chevalley, Jean Dieudonné, Jean Delsarte, and Andre Weil—undertook a major task in 1935: producing a fully axiomatized presentation of mathematics. The adventure to synthesize mathematics, which began with the publishing of Éléments de Mathématique in 1939, is still active today. Bourbaki has become one of the most striking and influential movements in the history of 20th-century mathematics, unique in conception and duration. Present-day members maintain anonymity in a movement that Pier called a "magnificent French garden."

Thirty-five years after Hilbert's fully axiomatized treatise on elementary geometry, the Bourbaki group launched the axiomatized presentation of large mathematical domains, "in their utmost useful generality," according to Pier. This garden aimed to construct mathematics in a way that unifies various and remote fields of mathematics while harkening back to the ideas of Euclid. It had to enhance the growth of mathematics while moving from the simple to the complex and from the general to the specific. Moreover, the mathematical work had to be "useful." Bourbaki had written that Elements was "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." Unfortunately, this referred to students at the Ecole Normale Superieure in Paris, one of the finest mathematical institutions in France where such notables as Joseph Louis Lagrange had taught.

Most of the Bourbaki members were under 40 years of age. In order to work together and produce a synthesis of the various mathematical fields, the strong-willed members were sometimes forced to sacrifice their pride. While some members lived abroad, no matter how and where they worked, their mathematics reflected the "axiomatic method," stressed the idea that "intuition is a major form of discoveries," and that their output offered "multipurpose knowledge," said Pier. To ensure that Bourbaki would never grow old, members had to retire by middle age. Dieudonné, said Pier, was the "sergeant major" of the founders.

By the mid-1960s--the high point of the life of Bourbaki--when 33 volumes of Elements had been published, the contributions of earlier figures from mathematical history were acknowledged. Elements has been left open to revision to accommodate the ever-growing and developing world of mathematics.

Now, thousands of pages of the archives of Bourbaki, up through 1953, have become available for study and examination. The archives will soon become available online, accompanied by comments and explanations. There is much to be gleaned from the work of Bourbaki, said Pier.

One note of interest: founding member and "Dean of Mathematics" Henri Cartan is still alive at 104!—H. Waldman