Bourbaki: A Secret Society of Mathematicians

by Maurice Mashaal

The name Bourbaki is known to every mathematician. Many also know something of the origins of Bourbaki, yet few know the full story. In 1935, a small group of young mathematicians in France decided to write a fundamental treatise on analysis to replace the standard texts of the time. They ended up writing the most influential and sweeping mathematical treatise of the twentieth century, Les élements de mathématique. Maurice Mashaal lifts the veil from this secret society, showing us how heated debates, schoolboy humor, and the devotion and hard work of the members produced the ten books that took them over sixty years to write. The book has many first-hand accounts of the origins of Bourbaki, their meetings, their seminars, and the members themselves. He also discusses the lasting influence that Bourbaki has had on mathematics, through both the Élements and the Seminaires. The book is illustrated with numerous remarkable photographs. Readership Students, mathematicians, and historians interested in the group of mathematicians known as Bourbaki.

Genres: Mathematics, Science, Biography

260 pages, Paperback

First published March 1, 2002

Reviews

[Paolo Gianlucca]

To a mathematician who asked him
"Can I ask you a stupid question?"
André Weil replied,
"You just did..."

[Mike · Rating 3 out of 5]

Bourbaki began in order to write a book on Analysis. (pg 4) They were particularly unsatisfied with Goursat's Analysis book that was the standard book used at the time. (pg 6) People like V.I. Arnold claimed Goursat's book is still a standard for analysis today. Having only glanced at Bourbaki and Goursat's texts, I can tell you neither are suitable for and introduction and Bourbaki is far more difficult. The book was decent, I have no interest to read it again. It does try and explain math concepts and I think it is a waste. It is a history book, if you try and explain the math you will only confuse the reader, as textbooks do already. The Bourbaki groups main contributions to mathematical research are in algebraic geometry, algebraic topology, and lie groups. (Pg 100)

[Curtis Penner · Rating 4 out of 5]

This is a nice book on the history of the Bourkaki Group, its politics (internal), and its relationship to the history of mathematics. I found it a good read to learn about this influential group of mathematics (mostly from the French Schools) who wanted the improve and make more rigid their fields of mathematics.

I nice addition, are portraits of a large number of mathematician. It is nice to see them as people and not just have a abstract name connected to slice of mathematics.

[Bruno Oliveira · Rating 4 out of 5]

Uma excelente biografia da matemático fictício que além de cobrir de forma detalhada a parte histórica, não deixa de fora suas ideias matemáticas explicando de forma acessível e simples ao leigo o conteúdo dos Elements des Mathematiques.

[Tue Le · Rating 4 out of 5]

A fascinating book on the history of Nicolas Bourbaki, the pseudonym for team of mathematicians whose original goal was to write a textbook in analysis to replace the one by Édouard Goursat. But their project soon expanded to encompass much of the known mathematics of their time, since they insisted on presenting everything in a rigorous axiomatic basis. Ultimately, they, as a group, proved no ground-breaking theorems and devised any new penetrating techniques, though individual members were notable mathematicians in their own right. In particular, Laurent Schwartz received the Fields Medal in 1950 for creating a rigorous theory of distributions (generalized functions). But their way of presenting mathematics continues to be influential. Certain mathematical symbols, like the one denoting an empty set, were their inventions.

How the Bourbaki group approached mathematics proved to be controversial. Proponents thought it was going to revolutionize higher education by teaching students newer mathematics rather than that from the youth of elderly professors ("Daddy's function theory"). Opponents decried its overemphasis on abstraction and detachment not just from intuition but also branches of mathematics Bourbaki did not care about (such as probability theory) and the needs of the sciences and engineering. While Bourbaki never intended, at least not explicitly, to reform secondary education, their style was adopted for the New Math movement, which began in France and spread to many other countries around the world. Jean Dieudonné exclaimation "Down with Euclid!" was taken to be his criticism of the mathematics taught in secondary schools, intuitive and useful topics such as Euclidean geometry (the "theory of triangles"). This was regrettable and led to reforms that was opposed by many experts in STEM, including not just educational psychologists but also many mathematicians, including (former) members of the Bourbaki group, like Schwartz.

Given how much mathematics was grown, nay, ballooned, since the mid-twentieth century, it is doubtful that anyone or any group, Bourbaki included, would be able to present mathematics in a single set of textbooks or references in a unifying manner. Teams like Bourbaki work best as a small group rather than a large committee. As a result, the amount of mathematics the entire team knows and is interested can only be a small fraction of what is available. Even Bourbaki neglected topics like probability and statistics, or all of applied mathematics, let alone anything of interest to the sciences and engineering.

The book explains everything in some detail, and with some background information to contextualize the discussion. Maurice Mashaal did a fine job researching what he was writing about. Anne Pierrehumbert is a competent translator of the original French into (American) English. I noticed a number of errors, however. For example, the malapropism "In principal, ..." should be changed to "In principle, ..." The word verb "to perilize" does not exist in the English language; we say "to imperil" instead. I deducted one star because there were more of these errors than I would like to see in a book.

[Fernando Pestana da Costa · Rating 4 out of 5]

An history of the group of French mathematicians, created in 1935, that had such a tremendous influence in mathematics and its teaching in the second half of the twentieth century. A very interesting book (although with a rather shameful proofreading, both on the Portuguese and on the mathematical levels) with a lot o interesting and curious informations and anecdotes about the group and its members (e.g.: the symbol ∅ for the empty set is an André Weil's, a Bourbaki member, invention of 1937.) Apart from the unfortunate exaggerations of the late 1960's and 1970's, namely the disastrous consequences of the pedagogical experiments of "Modern Mathematics", the original idea and much of the work of Bourbaki had important and everlasting effects in the way mathematics is presented and published. An interesting book about this episode of the history of last century's mathematical and intellectual lives.

[Sudip Paul · Rating 2 out of 5]

The book should have focused more on the mathematics of Bourbaki than on the interpersonal aspects. I believe a book by a mathematician would have been better.

[Dan Saattrup · Rating 2 out of 5]

As a mathematics student, I'm clearly not in the target group for this book. I read it to learn about the Bourbaki group, how it was formed and what its purpose was. This _was_ in fact explained, but at something like a third of the entire book. The rest was a very vague explanation of mathematical areas, the foundational crisis of mathematics and the "New Math" movement - which didn't correspond at all to what I expected the book to be.

Furthermore, there's a decent amount of redundancy in the book (e.g. groups are defined 2-3 times), which interrupts the reading flow. Overall, it was a decent book, but not one I would recommend other mathematically inclined people to read, unfortunately.