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–Andrew Weil (who later solved Fermat’s Last Theorem and brother of philosopher Simone Weil), 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. ( )

Interesting subject, irritating writing style, no math. ( )

C'était une lecture entendue. La lecture d'Impératif catégorique de Jacques Roubaud m'a replongé dans une époque où l'appréhension structurelle des mathématiques était ma vie, une période où Bourbaki était encore au centre des développements de cette science.

Nicolas Bourbaki, membre éminent de l'Académie des sciences de Poldévie, n'a, en effet, jamais existé. Il est le nom emprunté, sur la base d'un canular, par un groupe de mathématiciens français regroupés autour d'Henri Cartan, Claude Chevalley, Jean Dieudonné et André Weil (frère de la philosophe Simone Weil). Cette association des amis de Nicolas Bourbaki, à laquelle se sont joints plusieurs autres mathématiciens influents, a été le phare de l'organisation rigoureuse des mathématiques pendant le XXe siècle. La publication, sous le nom de Nicolas Bourbaki, des différents volumes de l'œuvre collective que constitue les Éléments de mathématique a marqué l'évolution, l'organisation, le développement ainsi que l'enseignement des mathématiques jusqu'au début des années soixante-dix et même au-delà.

Comme mathématicien, j'étais bien conscient de cette prépondérance bourbakiste dans le développement structurel des mathématiques du XXe siècle. Ce que je connaissais moins et que cette lecture m'a fait découvrir, c'est la place que Bourbaki a occupé dans la naissance du structuralisme.

Je ne savais pas que le structuralisme trouvait son origine en linguistique, que c'est Saussure et Jakobson (dans les années trente) qui avaient délimité les principes fondateurs de la linguistique structurale, que cette linguistique structurale s'exprime par la prééminence de la structure dans l'étude d'une langue [nos linguistes me corrigeront, j'espère]. J'ai appris que l'utilisation que Lévi-Strauss a fait du concept en anthropologie résidait en partie sur un lien qu'il avait établi avec André Weil (de Bourbaki) et que c'est l'application de la théorie des groupes à un problème proposé par Lévi-Strauss qui initie le mouvement structuraliste à l'extérieur de la sphère linguistique.

Finalement, cet ouvrage m'aura permis de constater les liens multiples entre un groupe de mathématiciens, la notion de structures en mathématiques et la ramification de l'influence structuraliste dans des domaines aussi divers que la psychologie avec Piaget, la psychiatrie et la psychanalyse avec Lacan, l'économie avec Leontief, ... et la littérature avec l'Oulipo, l'Ouvroir de littérature potentielle (eh oui). La boucle est donc bouclée.

[http://rivesderives.blogspot.ca/2009/08/nicolas-bourbaki-histoire-dun-genie-des.html] ( )

I was really looking forward to reading this book -- I'd read other books by Aczel and enjoyed them, I wanted to know more about the fictional mathematician Nicolas Bourbaki, and the title intrigued me. What could be more interesting to a math geek than a 'biography' of a fictional mathematician who shaped 20th century mathematics?

Unfortunately, I was very disappointed in this book for many reasons. Here are several of the reasons:

1) There are sections that seem totally unedited. The same ideas are repeated over and over, as if the author was trying to perfect the sentence structure. No new material is added -- just variations on the basic sentences.

2) I still have no clear idea what Bourbaki did for the math world. The author, for example, asserts that Bourbaki was responsible for the New Math ideas which influenced teaching of math in the US, but that is, honestly, as much detail as the author gives.

3) There are several sections which deal with structuralism in linguistics, psychology, anthropology, and even economics. Fine, but these sections were not linked to the topic of Bourbaki.

4) While the math that Bourbaki did change is fairly complex and abstract, the author made no attempt to explain mathematically the changes Bourbaki wrought.

5) The author was fascinated by the life and mathematics of Alexandre Grothendieck. In some ways, he was the primary focus of the book, and perhaps, the author should have written a biography of him and not Bourbaki.

This book frustrated me and left me sorely disappointed. ( )

Nicolas Bourbaki fue el nombre que un grupo de matemáticos usó para reescribir los libros de texto de su tiempo y para desarrollar una serie de innovadoras propuestas teóricas que tuvieron repercusiones importantes en los campos de la sociología, el arte y la filosofía. Este libro nos presenta esta historia singular y vincula algunos de sus protagonistas: Pablo Picasso, Claude Lévi-Strauss y Simone Weil. ( )