Nicholas Bourbaki was an influential 20th-century mathematician who published a series of texts begnning in the 1930s, trying to rigourously describe the major core areas of mathematics based on the foundations of set theory. Many of his books became standard references in their fields. Among the Bourbaki contributions that have become familiar to modern math students are the use of the slashed zero to represent an empty set, and the terms injective, surjective, and bijective. The ill-fated New Math movement in education, which I discussed back in podcast 145 (http://mathmutation.blogspot.com/2011/12/145-why-johnny-couldnt-add.html), was also largely inspired by Bourbaki ideas. But if you try to find out about the life of Bourbaki himself, you will be in for a bit of a surprise: Nicholas Bourbaki did not actually exist.

Actually, to be more precise, Bourbaki was not an individual, but a secretive society of mathematicians established by Andre Weil, Henri Cartan, and other young mathematicians in Paris in the 1930s. They originally got together because they believed there was a serious gap in the available mathematics texts of the time, a problem magnified by the fact that a generation of potential leaders in the field had been wiped out by World War I. They hoped to re-establish the rigorous foundations of mathematics, while at the same time to provide a standard series of reference works. While secret societies tend to sound sinister in concept, their secrecy actually stemmed from rather noble motives: they wanted to ensure that any works they produced would be judged on the basis of their content, not on factors related to personal egos. So they created the ficticious persona of Nicholas Bourbaki, member of the Royal Academy of Poldavia, and agreed that he would be credited as the author of all books they produced.

The members of the Bourbaki group enjoyed having fun with the semi-secret nature of their small club. To anyone who asked, they would claim that Bourbaki was a real person of their acquaintance, and they even printed up a set of mathematical-pun-laden wedding invitations from Bourbaki's daughter, to show to anyone who doubted his authenticity. According to the invitation, "The trivial isomorphism (aka the sacrament of matrimony) will be given to them by P. Adic, of the Diophantine Order, at the Principal Cohomology of the Universal Variety, the 3 Cartember, year VI, at the usual hour.". You wouldn't think that would fool too many people-- but this prank backfired horribly during World War II, when Andre Weil (who had fled to Finland) was arrested on suspicion of spying. This apparently encoded letter from a strange foreign contact was considered a key piece of evidence. Weil was sentenced to death, though at the last minute a friend with government contacts managed to intervene and get him pardoned.

You would think they would be done with practical jokes after that, but the Bourbaki group tried to continue the whimsical spirit, perhaps a welcome break from the incredbly serious work they were attempting. In the late 1940s, American mathematician Ralph Boas was contacted by the Encyclopedia Britannica to assist with an article on modern mathematics, and since he knew Weil, he mentioned in his article that Bourbaki was actually a pseudonym for a collaboration rather than an actual person. Boas and the Encylopedia then received a letter, claiming to have been written by Bourbaki from an ashram in the Himalayas, asking "You miserable worm, how dare you say that I do not exist?" A series of letters went back and forth on the topic, though the encyclopedia editors were eventually convinced of the truth. But then Bourbaki members began to spread a new rumor, that Ralph Boas did not actually exist, and was actually a collective pseudonym for a group of American mathematicians!

The Bourbaki group lasted for several generations, producing a series of comprehensive texts on the foundations of modern mathematics. While they made many lasting contributions, they were also criticized for focusing too narrowly on foundations, omitting areas relevant to widely applicable topics such as logic and mathematical physics. They also had a strange aversion to pictures and illustrations, which meant that although some of their books were very useful as references, they ironically were not very good as textbooks. The group gradually declined after the 1970s, with their last major text, "Spectral Theory", being published in 1983. The decline may have been hastened by a long legal battle with their publishing company over royalties and translation rights: Pierre Cartier, one of the members during the final productive period, described the result as "both parties lost and the lawyer got rich."

In the show notes, you can find a link to a 1997 interview with Cartier, who shares some fascinating thoughts on the rise and fall of the Bourbaki group. Aside from the legal issues, one of the other major reasons he gave for Bourbaki's decline was that they had successfully achieved their objectives. As Cartier describes it, "In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. Now we are again at the beginning of a new revolution."

And this has been your math mutation for today.

References:

- Bourbaki at Wikipedia: http://en.wikipedia.org/wiki/Nicolas_Bourbaki
- Bourbaki at PlanetMath: http://planetmath.org/encyclopedia/NicolasBourbaki.html
- Cartier interview: http://ega-math.narod.ru/Bbaki/Cartier.htm
- Article on Bourbaki-Boas clash: http://www.springerlink.com/content/x15h030150620688/
- Article on Bourbaki wedding: http://www.neverendingbooks.org/index.php/when-was-the-bourbaki-wedding.html
- Article on Weil's arrest: http://www.neverendingbooks.org/index.php/the-wedding-invitation-that-nearly-killed-andre-weil.html

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