Nicolas Bourbaki, whose mathematical publications began to appear in the late 1930s and continued to be published through most of the twentieth century, was a direct product as well as a major force behind an important revolution that took place in the early decades of the twentieth century that completely changed Western culture.

Pure mathematics, the area of Bourbaki's work, seems on the surface to be an abstract field of human study with no direct connection with the real world. In reality, however, it is closely intertwined with the general culture that surrounds it. Major developments in mathematics have often followed important trends in popular culture; developments in mathematics have acted as harbingers of change in the surrounding human culture.

The seeds of change, the beginnings of the revolution that swept the Western world in the early decades of the twentieth century (both in mathematics and in other areas) were sown late in the previous century. This is the story both of Bourbaki and the world that created him in that time. It is the story of an elaborate intellectual joke: because Bourbaki, one of the foremost mathematicians of his day, never existed.

"Lay readers interested in mathematical history will learn a lot they didn't know from Aczel's latest book, which focuses on a group of French mathematicians who in the 1930s decided to publish their collective work under an imaginary name. But readers may also get the feeling that this able math and science popularizer is running out of suitable topics. It's not that the contributions of the Bourbaki school weren't important — their rigorous approach to proofs and emphasis on set theory provided the basis for what became known as the New Math — it's just that this curious story isn't as inherently dramatic as, say, that of Andrew Wiles's solving Fermat's Last Theorem. Aczel surveys with his usual panache the careers of some major members of the group, like the eccentric Alexandre Grothendieck, who in 1991 became a hermit in the Pyrenees, but Aczel is less convincing when he draws simplistic parallels between advances in mathematics and modern art. While always readable, this diffuse narrative (including chapters on Bourbaki's influence on anthropology and linguistics) strains to pull its disparate parts into a satisfactory whole." Publishers Weekly (Copyright Reed Business Information, Inc.)

"While a competing collection (Reviews, July 31) found the majority of its articles in mainstream publications like the New Yorker, guest editor Greene (The Elegant Universe) sticks to the fundamentals in the seventh volume of Houghton's science anthology. In line with his belief that scientific literacy is increasingly vital to full participation in contemporary culture, Greene draws heavily from the scientific press — six selections come from Scientific American alone. These articles lay out the facts about topics like lupus and the nature of mass with admirable clarity, but can fall short of the excitement level in other pieces that have a more personal touch. John Hockenberry, for example, shows how blogging technology has radically changed the way U.S. soldiers fighting in Iraq communicate with friends, family and even total strangers, while Mark Dowie thoughtfully considers how environmentalist zeal threatens to disrupt indigenous communities. Other writers focus on the compelling stories of individual scientists, from Drake Bennett's profile of 'the godfather of Ecstasy' to Oliver Sacks's memories of his lively correspondence with Francis Crick, the co-discoverer of DNA." Publishers Weekly (Copyright Reed Business Information, Inc.)

"Lay readers interested in mathematical history will learn a lot they didn't know from Aczel's latest book, which focuses on a group of French mathematicians who in the 1930s decided to publish their collective work under an imaginary name. But readers may also get the feeling that this able math and science popularizer is running out of suitable topics. It's not that the contributions of the Bourbaki school weren't important — their rigorous approach to proofs and emphasis on set theory provided the basis for what became known as the New Math — it's just that this curious story isn't as inherently dramatic as, say, that of Andrew Wiles's solving Fermat's Last Theorem. Aczel surveys with his usual panache the careers of some major members of the group, like the eccentric Alexandre Grothendieck, who in 1991 became a hermit in the Pyrenees, but Aczel is less convincing when he draws simplistic parallels between advances in mathematics and modern art. While always readable, this diffuse narrative (including chapters on Bourbaki's influence on anthropology and linguistics) strains to pull its disparate parts into a satisfactory whole." Publishers Weekly (Copyright Reed Business Information, Inc.)

"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... Washington Post Book Review (read the entire Washington Post review) 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 Academie 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 Independants.'

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 Academie 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 Independants 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 colle (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 colle (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 Dieudonne, 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 Dieudonne, 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 Dieudonne. 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.'" Reviewed by Charles Seife, Washington Post Book World (Copyright 2006 Washington Post Book World Service/Washington Post Writers Group)
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"A fascinating topic." Kirkus Reviews

This is the story both of Nicolas Bourbaki — whose mathematical publications began to appear in the late 1930s and continued to be published through most of the 20th century — and the world that created him in that time. It is the story of an elaborate intellectual joke, because Bourbaki, one of the foremost mathematicians of his day, never existed.