In the late 1930’s, an obscure Russian mathematician by the name of Nicolas Bourbaki began publishing. Over the years, these books would go on to become a treatise Éléments de mathématique (Elements of Mathematics). Several research papers accompanied these volumes. Bourbaki’s aim was to reinvigorate rigorous mathematics and he succeeded in doing so. Bourbaki’s influence on today’s mathematics is enormous. Small things which people take for granted, like the symbol $latex \O$ for an empty set, are Bourbaki’s initiations.
Mathematicians agree that Bourbaki was one of the most prolific and influential mathematicians of the twentieth century. There is just one catch – Nicolas Bourbaki did not exist!
Almost a generation of mathematicians was killed in the First World War. The rising French mathematicians of the École normale supérieure of the 1930’s, then just twenty-somethings, had no one to look up to. They thought the standard textbooks of the time were inadequate for research. So they decided to take-up the initiative. Nicolas Bourbaki was the nom de plume (pseudonym) under which they published their works.
Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, René de Possel and André Weil were the founding members of Bourbaki. They would meet after regular intervals at the Café Capoulade for discussions. There was no leader. Everything they would publish had to have a unanimous agreement. They were young, passionate and full of ideas. André Weil wrote:
“[We maintained] in our discussions a carefully disorganized character. In a meeting of the group, there has never been a president. Anyone speaks who wants to and everyone has the right to interrupt him … The anarchic character of these discussions has been maintained throughout the existence of the group … A good organization would have no doubt required that everyone be assigned a topic or a chapter, but the idea to do this never occurred to us … What is to be learned concretely from that experience is that any effort at organization would have ended up with a treatise like any other …”
The first Bourbaki meeting took place in July 1935. If someone did not like the draft of a new chapter, it was common to throw the papers out. Their passionate, heated discussions looked only for rigorous, formal proof. There was no place for things like ‘intuition’! For Bourbaki, abstraction and generality were necessary for perfection. If someone witnessed this gathering for the first time, Jean Dieudonné wrote, they would:
“… always come out with the impression that it is a gathering of madmen. They could not imagine how these people, shouting — sometimes three or four at the same time — could ever come up with something intelligent. It is perhaps a mystery but everything calms down in the end.”
Bourbaki believed in the abstract mathematics and was vehement in denying to handle anything related to mathematical physics, such as combinatorics or mathematical logic. They disliked figures and denied to work on any applied mathematical field such as applied partial differential equations. Influential mathematicians such as Benoit Mandelbrot had to leave France because his ideas were based solely on pictures. They did not prefer intuition either. Bourbaki’s work is devoid of figures, intuition or applications. But their work looked for generality and form which believed
“Structures are the weapons of the mathematician.”
They also created a story of Bourbaki’s origins. The story says that the Russian mathematician Bourbaki had met these young mathematicians in Paris.
“Nicolas Bourbaki, who became somewhat of a misanthropist following his misfortunes, refuses to see anyone except the collaborators he has chosen for himself. This gave rise to the legend that he is merely a pseudonym, but anyone who has met him knows the strength and vitality of his extraordinary personality, to which even his collaborators are prone to ascribe somewhat mysterious powers.”
Although I find their adherence to sticking to the purest of pure mathematics annoying and frankly elitist, there is no doubt that Bourbaki raised importance of ‘the formal proof’. Rigor, abstraction and generality gained their status in the world of mathematics because of Bourbaki’s seminal work. There may be varying opinions on how Bourbaki chose to work, but there is no contradiction when saying that Bourbaki changed the way mathematics is done.
image source : here
Some good references on Bourbaki:
Bourbaki: A Secret Society of Mathematicians
Audio:
YouTube: BBC – Nicolas Bourbaki – A brief history of Mathematics
Book Review:
Bourbaki, A Secret Society of Mathematicians and The Artist and the Mathematician
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Nice article and awesome image.
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