In the Footsteps of Bourbaki

In an article titled “The Joy of Sets” in the October 1998 issue of Lingua Franca, Jim Holt began “Why is it that French theory so often ends up having a baneful effect on American pedagogy? I am thinking not of Derrida, but of another figure, one whose influence reached these shores long before him: Nicolas Bourbaki.”

“In 1939 he began producing a series of treatises under the umbrella title Elements de mathematique. The project is an attempt to rebuild the house of mathematics from the ground up.”

“Rigor and abstraction reign. Take Bourbaki’s characterization of the numeral one. After nearly two hundred pages of preliminaries, the number is finally defined in a forbidding concatenation of formal symbols — which, a footnote adds, is only an abbreviation. The unabbreviated form of the definition, the reader is informed, would require many tens of thousands of symbols.”

“Some mathematicians have been hostile to Bourbaki’s philosophy. By neglecting physical intuition and problem solving, they felt, it divorced mathematics from the real world, making the subject into a kind of logical theology.”

“Yet by the late 1950s Bourbaki’s prestige had become irresistible for American educators. Why, they wondered, should children waste precious years solving concrete problems with numbers when they could be imbibing abstract axoims that would teach them mathematics Bourbaki style? Thus was born a terrible thing: new math.”

“In the 1960s new math took Bourbaki into high schools and grade schools, even into kindergartens…” Teachers were taught that arithmetic was not about numbers, but more abstract entities called sets. “Thus if you were asked to solve the equation x + 3 = 5 you didn’t dare say ‘two.’ You said ‘the set whose number is two.’ It was like being on Jeopardy, where contestants have to phrase their answers in the form of a question.”

Parents were dismayed because they couldn’t understand the new math, and their children performed poorly in math. Parents would have been even more dismayed if they had learned that the great mathematical theorist Nicolas Bourbaki didn’t exist. He was an invention of a secret group of French mathematicians who have produced volume after volume of Elements de mathematique.

“When a paragraph in Encyclopedia Britannica, written by one Ralph P. Boas, described Bourbaki as a group, the editor at once received a letter of protest signed by an injured ‘N. Bourbaki.’ Soon it was being put about that Ralph P. Boas did not exist but was merely a pseudonym for a shady group of American mathematicians.”

This summary of Jim Holt’s article is composed of extracts from beginning to end, but the entire article is much more inclusive.