Who’s responsible for the new math?

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 well before him: Bourbaki.

Nicolas Bourbaki is without a doubt the most tireless mathematical theorist of the twentieth century. In 1939 he began producing a series of treatises under the umbrella title Eléments de mathématique. The project is an attempt to rebuild the house of mathematics from the ground up, as though Descartes were the architect and Napoleon the engineer. The edifice that has taken shape over decades is a formalistic wonder, a vast deductive structure built on an austere axiomatic foundation.

Rigor and abstraction reign. Take Bourbaki’s characterization of the numeral one. After nearly two hundred pages of preliminaries, this number is finally defined in terms of 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. Deduction from the first principles–for which the French have a particular fondness–is stressed at the expense of the more English reliance on geometrical interpretation and model building.

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 axioms that would teach them mathematics Bourbaki-style? Thus was born a terrible thing: new math.

In the 1960s new math took Bourbaki into into high schools and grade schools, even into kindergartens, where a game called WFF and Proof (WFF stands for "well-formed formula") was introduced to get preschoolers to be more rigorous in their deductive inferences. I myself was a victim of new math. The nuns who taught me in elementary school had been brainwashed into believing that arithmetic was not about numbers but about 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 member is two." It was like being on Jeopardy, where the contestants have to phrase their answers in the form of a question.

New math left American parents doubly befuddled: First, they could not understand the mathematics; second, they could not understand why their children were unable to do the simple things–recite the multiplication table, figure out percentages–that they had learned to do in school. Given the Cold War atmosphere of the time, it was inevitable that some would suspect that new math was a Red conspiracy to ruin American education.

Actually, it was a conspiracy of sorts. For M. Bourbaki was, and continues to be, a figment. The real authors of Eléments de mathématique are a cabal of French mathematicians. They seem to have borrowed their collective nom de plume from a French military officer of Greek ancestry, one Charles-Denis-Sauter Bourbaki. General Bourbaki distinguished himself in the Crimean War, declined an offer of the throne of Greece in 1862, suffered a reversal of fortune in the Franco-Prussian War, and later tried to shoot himself while imprisoned in Switzerland. He missed.

One of the seven founding members of Bourbaki was the incomparable André Weil, who died this summer in Princeton at the age of ninety-two. The brother of the religious mystic Simone Weil, André Weil was long considered the world’s greatest living mathematician. Like many great mathematicians, he was capable of indolence. When a student of his once needed a particular result for his thesis, Weil told him that he was sure the result was true and could be safely used but that he was too lazy to write out a proof. So the student employed the result and credited it to "Nicolas Bourbaki of the Royal Academy of Poldavia."

The membership of Bourbaki is thought to vary from ten to twenty mathematicians. With a few conspicuous exceptions–like the late Samuel Eilenberg of Columbia University–the conspirators have always been French. Retirement from this société anonyme is compulsory at the age of fifty, in order to keep the project from getting sclerotic. Yet its allure has waned within the profession. The feeling is that, as Freeman Dyson once put it, Bourbaki’s obsession with generality and structure makes the approach insensible to "things of accidental beauty," things that "have a quality of strangeness, of unexpectedness."

Meanwhile, volumes of Eléments de mathématique continue to appear, albeit less frequently than in the past. And there are those who persist in believing that Bourbaki is a flesh-and-blood theorist–if a reclusive one. An invitation issued to M. Bourbaki to attend an English conference in 1968 elicited the excuse that his "well-known timidity and modesty prevented him from speaking in public."

As for the skeptics, while they may make bold to assail Bourbaki formalism, they should think twice before trying to cast doubt on the existence of the polycephalic author himself. When a paragraph in Encyclopaedia Britannica, written by one Ralph P. Boas, described Bourbaki as a group, the editors 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.


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