The Artist and the Mathematician, Amir D. Aczel. New York: Thunder’s Mouth Press, 2006.

Summary: The story of the Bourbaki, named after the greatest mathematician who never existed, who led a revolution in the emergence of the “new math,” introducing a new rigor into the field.

When I was in middle school, we were introduced to “the new math.” One of the things I was always curious about was why the first thing we did was learn about sets. I was reminded of this when I read this book, which explained why sets were foundational to the approach.

This is the story of Nicolas Bourbaki, who convened a group of mostly French mathematicians around him, creating a tremendously productive group that in its day revolutionized the practice and teaching of math. Aczel introduces us to the key figures in this group–Andre Weil (brother of philosopher Simone Weil, not to be confused, as I did, with Andrew Wiles, who solved Fermat’s Last Theorem), Laurent Swartz, Henri Cartan, Claude Chevally, Jean Delsarte and Jean Dieudonne. We are also introduced to Alexandre Grothendieck, perhaps the most brilliant and also eccentric of them.

The most striking thing we learn is that the group formed around a mathematical joke upon which Weil built. Nicolas Bourbaki never existed except as a made up identity that reflected the collective effort of this group to rehabilitate and revolutionize mathematics in France that had fallen into the backwaters of German mathematics and science. These mathematicians met regularly and forged a consensus on how math would be practiced and taught in France that resulted in the prolific production of texts, revolutionized not only math education throughout the world, but touched a variety of other disciplines. Their approach was founded on set theory. They emphasized math in the abstract, focusing on mathematical proofs and rigor.

They were trying to articulate the structure of mathematics and this led to interesting interactions with pioneering anthropologist Claude Levi-Strauss, child psychologist Jean Piaget, linguistic theorists, and even writers including Italo Calvino. Aczel traces how structuralism for a time replaced existentialism in philosophy until the turn to the post-modern.

During the war Weil fled to America and stayed there, and gradually, his influence in Bourbaki waned. In the early 1950’s Alexandre Grothendieck joined for a time. His brilliance both stimulated the work of the Bourbaki and led to his departure as he recognized the weakness of set theory as a basis for Bourbaki, trying and failing to convince them of the idea of categories. Grothendieck differed from the Bourbaki, preferring to work alone.

The parting spelled a turning point for both. While Bourbaki continued to have a spreading influence for a time, it was more on the basis of past work. Grothendieck went on to do innovative work for a time, and directing students into significant problems. He held a position at the IHES, a French version of the Institute for Advance Study. Then he became more engaged in political and environmental causes, and when his efforts failed in these areas, he retreated to the Pyrenees, where his whereabouts remained unknown. After this work was published, he died in 2014 in Saint-Girons, Ariège.

The title of this work is a bit of a puzzle. Apart from a chapter on cubism, Braque, and Picasso, and its connections to antecedents to the Bourbaki, this is not a book about artists, unless this is a contrasting reference to Grothendieck and Weil, which was opaque to this reader. I found the organization of the book a bit labyrinthine. Nevertheless, it was an intriguing account of a movement in mathematics I’d never heard of. It was fascinating to see how productive this group was for a period and yet how significant the human factors were in the ultimate fate of Bourbaki.