Customer Reviews

Bourbaki: A Secret Society of Mathematicians

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This is a very solid overview of the Bourbaki school, though obviously nothing like a full-fledged biography or monograph. The treatment is slightly more sophisticated than what you might find in a series of Scientific American articles (e.g., the author is confident that readers won't be scared off by an occasional integral sign or 2x2 matrix). I read the French edition, so I can't comment on the translation, though I was abashed to learn that the French adjectival form is not "bourbakien," as I might have guessed, but "bourbachique".

One of the main virtues of the book is that it's frank enough to include many thoughtful criticisms of the Bourbaki style and content. For example, although the Bourbaki were dedicated to following an axiomatic method, they ignored Gödel's incompleteness theorem, aside from an occasional dismissive reference. (That theorem shows that if you start from a system of axioms, you can't deduce all "mathematical truths" from them -- i.e., you can run across some statements that are consistent with the axioms but that cannot be deduced from them). The Bourbaki also ignored category theory, even though one of its inventors (Samuel Eilenberg) was a member of the group for a while. Today category theory is the dominant framework for describing the fundamental structures of mathematics. The Bourbaki also disdained so-called applied mathematics, including probability theory and dynamics, for its lack of "purity," even though it has yielded much mathematical fruit in the past 50 years. (Indeed it represents much of the lasting glory of French mathematics, e.g. the work of Fourier, Legendre, Lebesgue, Henri Poincaré and even Jacques Hadamard, whose seminars were a role model for the Bourbakis' and who supervised the PhDs of two of the group's founding members.) In short, the Bourbaki seem to have ignored or disdained rather lots of stuff. The bourbachique closed-mindedness ultimately contributed to the obsolescence of their approach.

The book's candor is also a bit of a flaw. By the end of the book the members of the group come across as the dogmatic, elitist clique their contemporaries accused them of being. They do seem to have been a livelier bunch in person than what I'd expected from their impersonal, austere, diagram-less presentation of mathematics. But while having a sense of humor was a prerequisite for being invited to join, the examples of their humor are for the most part sophomoric, and occasionally mean-spirited. Readers who already weren't Bourbaki fans might feel vindicated after reading this book; I can't vouch for what their fans might feel. Overall a quite interesting, but not uplifting, brief intellectual biography.

This is very interesting for fans of any of Bourbaki's mathematics texts. It has photos of many of the members taken at their meetings, and information about how the group operated. You'd never know it from the final product, but their original goal was to write a calculus book! Several of the founding members had just begun teaching, and they were unhappy with the standard French calculus text of the day--a multivolume work by Goursat that they considered out of date. They decided to write a little bit of background material on algebra and general topology, and somehow ended up with the books we all know and love.

The book also describes the personality quirks of the members, and has some commentary on the contents of the various texts of Bourbaki. It doesn't explain technical details like why Bourbaki chose to define integrals the way they did.

If you didn't major in math, this book probably won't interest you.

For values and beliefs for a large group of mathematicians, this is a helpful book. For Bourbakian values and beliefs have been, by and large, adopted by the academic masses, alas. (expression of a Bourbakian value and a Bourbakian belief from p.75 "basic concepts are treated as abstract entities whose nature and concrete meaning are insignificant") Still, it's a nice tourist guide, even has lots and lots of color snaps.

But the text just reads and feels odd. It might just be due to some weaknesses in the text's translation from the French. All the same, sometimes I have no idea what is being written about or why.

For example, on page 41 I read "the fundamental theorem of algebra, which states that any polynomial with real coefficients has at least one complex root (in other words, there is a complex numer x that makes the expression equal to zero)" WHAT?!?!?!?!?!?! Not only useless, it's nonsense. The basic thought on complex numbers and polynomials with real-valued coefficients is the following. If any polynomial equation with real coefficients has a complex number z as a root, then the complex conjugate of z is also a root of the polynomial equation.