Proof
A biography about a made-up mathematician whose influence went beyond numbers.

Reviewed by Charles Seife
Sunday, October 15, 2006

THE ARTIST AND THE MATHEMATICIAN

The Story of Nicolas Bourbaki, the Genius

Mathematician Who Never Existed

By Amir D. Aczel

Thunder's Mouth. 239 pp. $23.95

Nicolas Bourbaki did not exist. He was dreamt up by a playful clique of French math professors in the mid-1930s who used the Bourbaki pseudonym to tear mathematics down to its foundations. That collective tried to root out the imprecision that festered underneath the proofs of the day and replace it with more rigorous underpinnings. In so doing, the nonexistent mathematician produced more important and more original work than most real-life scholars. Alas, The Artist and the Mathematician , the new "biography" of Bourbaki by math writer Amir Aczel, is as derivative -- dangerously so -- as its subject was creative.

This is a shame, because the story of Bourbaki could be rich fodder for a talented writer. From the start, however, Aczel runs into a problem. Though Bourbaki's fictional status is given away in the title of the book, Aczel coyly treats Bourbaki as a real individual for the first few chapters. Perhaps this was because it allows him to assert that Bourbaki "was the greatest mathematician of the twentieth century." Other mathematicians have a stronger claim, at least based upon the profundity of their works. However, to Aczel, there's much more to Bourbaki than proofs and papers. Bourbaki, argues Aczel, was a product of the zeitgeist that gave birth to Cubism and Dada; Bourbaki, in turn, inspired revolutions within anthropology and literature.

In Paris at the turn of the century, painters Pablo Picasso and Georges Braque developed cubism, a movement that, as Aczel describes it, "concentrated on the relationships among the elements of a subject, as space and time related to the speed of light in relativity theory, creating a new art form." The Bourbaki movement, flush with the same spirit, studied the relationships between abstract mathematical objects -- the deep underlying structure of math. This, says Aczel, breathed life into a growing movement ("structuralism") in which anthropologists, writers, psychologists and others embraced abstraction in hopes of getting at the girders of their own fields.

Unfortunately, when Aczel strays away from purely mathematical subjects, he seems uncomfortable and unsure of himself. He sticks closely to the writings of others. For example, his passage on Georges Braque closely parallels a biography on the Guggenheim Museum's Web site.

From Aczel's book: "Then he left for Paris to study under a master decorator, and received his craftsman certificate in 1901. From 1902 to 1904, Braque worked as a painter at the Académie Humbert in Paris, and by 1906, his work, which had been associated with impressionism, was no longer in that tradition and he adopted the Fauve style. Braque spent the summer of 1906 in Antwerp with Othon Friesz, and when he returned to Paris at the end of the summer, he exhibited his Fauve paintings at the Salon des Indépendants."

From the Guggenheim: "He left for Paris to study under a master decorator to receive his craftsman certificate in 1901. From 1902 to 1904, he painted at the Académie Humbert in Paris, where he met Marie Laurencin and Francis Picabia. By 1906, Braque's work was no longer Impressionist but Fauve in style; after spending that summer in Antwerp with Othon Friesz, he showed his Fauve work the following year in the Salon des Indépendants in Paris."

Aczel again falls on the Guggenheim's language to explain the convergence of Picasso and Braque's styles: "By 1911 their styles, up to that time decidedly different, became extremely similar. In 1912 the two artists started to incorporate collage elements into their paintings and to experiment with the papier collé (pasted paper) technique. Their collaboration continued until World War I broke out in 1914."

Here's what the museum's Web site says: "By 1911, their styles were extremely similar. In 1912, they started to incorporate collage elements into their paintings and to experiment with the papier collé (pasted paper) technique. Their artistic collaboration lasted until 1914."

Aczel is not an arts writer; he made his name writing about Fermat's last theorem and other mathy subjects. However, his treatment of mathematical concepts in this book is as slipshod as his treatment of artistic ones. With little explanation, he plunks down terms like "Stokes integral," "non-commutative algebras," "Lie groups" and "Riemann-Roch theorem." A reader without a PhD in mathematics will be baffled by Aczel's prose. In all probability, so would an intellectual property lawyer. Even in passages dealing with mathematicians, Aczel seems more than a little inspired by other writers. His mini-biography of mathematician Armand Borel parallels an obituary that ran in a mathematics journal in 2004:

"The young mathematician became acquainted with key members of the group: Henri Cartan, Jean Dieudonné, and Laurent Schwartz. He also met some of the younger members of Bourbaki, including Roger Godement, Pierre Samuel, Jacques Dixmier, and Jean-Pierre Serre."

Compare that to a passage from the Notices of the American Mathematical Society: "There he quickly got acquainted with senior members of the Bourbaki group -- namely Henri Cartan, Jean Dieudonné, Laurent Schwartz -- and with the younger members -- notably Roger Godement, Pierre Samuel, Jacques Dixmier, and most importantly Jean-Pierre Serre."

If two authors writing independently about these same seven mathematicians were to list them randomly in a passage, the likelihood that the two passages would list the names in the same order is less than 1 percent, even taking into account the grouping into older and younger mathematicians.

The Artist and the Mathematician is lousy with mini-biographies -- dry, uninspired, shallow profiles that sound nearly identical. These are where Aczel cleaves most closely to his written sources. At one point, he launches into rapid-fire mini-bios of four of the founding members of Bourbaki: Henri Cartan, Claude Chevalley, Jean Delsarte and Jean Dieudonné. Very similar biographies of the four appeared (again, in the same order) on the website of Pour La Science, a French science magazine. Throughout, Aczel's prose mirrors the original. In none of these cases are there quotation marks or footnotes that would indicate the provenance of these passages.

The fact that he sticks so closely to his written sources is a sign that he doesn't really understand the subject he's writing about; this leads to an uncertain and halting style. Rather than reveling in the (often bizarre) details about the rambunctious Bourbaki crowd, Aczel shies away as if he's terrified of the story. For example, he states blandly that as Bourbaki's influence waned, there were "disagreements and power struggles within the group." What were those disagreements about? Aczel doesn't go into detail. But we know that these disagreements, whatever they are, "eventually led to a great resentment by some members of the group toward others." The group breaks up before our eyes, and we have almost no clue why. Perhaps Aczel doesn't really know himself.

It's a shame that Aczel crafted such a derivative, superficial and tedious work out of such an interesting subject. After all, he's the obvious person to write the story. Nicolas Bourbaki was trying to pass off many people's writing as the work of a single individual. It seems that so, too, is Aczel. ·

Charles Seife, a journalism professor at New York University, is the author of "Decoding the Universe: How the New Science of Information Is Explaining Everything in the Cosmos, From Our Brains to Black Holes."