Wednesday, February 11, 2009

Wednesday Math, Vol. 60: Bourbaki

World War I, known in its day as The Great War, was like nothing that came before it. Europe had seen war many times in its past, and with Napoleon nearly the entire continent had been embroiled at about the same time, but technological advancements from the time of Napoleon's defeat through the beginning of the 20th Century turned the battlefields of the 1914-1918 war into an unprecedented slaughter. We laugh now about the name The War To End All Wars, but consider that World War II only began because a madman ruled a major industrial power, and another megalomaniac stood against him as he tried to conquer the East with its major oil reserves. If humanity had a lick of sense, there would never have been anything to match the Great War, let alone surpass it.

The loss of so many lives was felt deeply in many European countries, as an entire generation was sacrificed in the trenches. In France after 1918, there was a gap in the mathematical profession. Mathematicians were either too young or too old, and an age group that should have kept the connection in the discipline was gone. The too young set took it upon themselves to be the repository of mathematics in France, and decided to write the definitive texts in as many sub-disciplines as possible. It was a very talented group who first got together in 1935 and planned to write these texts, and instead of naming each book for the person who would be the main contributor to that field, all the books would be published under the pseudonym Nicolas Bourbaki.


Here is a picture from 1951 of the Bourbaki group meeting at some pleasant retreat. From left to right, we have Jacques Dixmier, Jean Dieudonné taking notes, Pierre Samuel lighting a cigarette in a holder, André Weil wearing shorts, Jean Delsarte leaning back in a chair and Laurent Schwartz under the parasol, also busy scribbling away. Obviously, they are no longer young men in this picture, but the old lions of French mathematics, still ruling the roost they decided to claim some sixteen years earlier. This is not the complete list of mathematicians who worked in Bourbaki group. To give one example, the picture does not include Henri Cartan, who passed away just last year.


Bourbaki is famous still today for the rigor of definition they decided to impose on the field and the consistency of terminolgy and symbols. For example, the ideas of functions being onto, one to one or both onto and one to one became replaced with the more official sounding terms surjective, injective and bijective. One symbol first popularized by Bourbaki, using a circle with a slash in it to represent the empty set, is now the worldwide standard symbol, as are many others first employed by the group working together.



Another idea Bourbaki is given credit for inventing is the "dangerous bend" sign placed in the text. People who have a hard time in math may feel a little sense of fraternity with the eggheads and poindexters who are really good at math when we admit that there are some concepts in mathematics that are hard to grasp, even for us. This sign on a page in Bourbaki meant that the ideas here were difficult, and it would make sense to read slowly or even re-read a passage to make sure things were clear, because like all math, it would be referenced again later. The "dangerous bend" sign made it into the texts of other mathematicians, even the non-French, including Donald Knuth, the Wisconsin born professor emeritus of computer science at Stanford who took it upon himself to write a Bourbaki like, multi-volume text for the computer science field, The Art of Computer Programming.

Yay, the French! Do not worry, though. To keep my American citizenship, my next math post will revert to the more common American practice of making fun of the French.

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