The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed - Amir D. Aczel

fluff & fuss, but where's the math?

This book is disappointing on a number of levels. I'll mention just a few. First, it is peppered with overstated superlatives. Every mathematician seems to be extremely important and every theorem is extremely important and every text is extremely important and we are rarely shown what is so important about anything. After awhile, the sensationalism looses its impact.

In stressing the importance of Nicholas Bourbaki, the book often ignores the contributions of others. So while Bourbaki contributed to the notion of building mathematics on the foundation of set theory, this misses the previous work of others such as Giuseppe Peano. Also, others deserve some credit for the level of precision mathematics now enjoys--David Hilbert and Alfred Tarski, to name just two.

At times it seems poorly edited. For example on page 101 we find "if we look at the set of numbers 123, the various possible orders form a group." Now, if you already know some group theory you can figure out what he meant to say. But the newcomer is more likely to say, "set of numbers???? I see only one number and it is one hundred and twenty three." As another example, we are given an illustration on page 104. We see a collection of spheres and a collection of arrows. It is supposed to illustrate how a topological space can be associated with a vector space. How the picture is meant to illustrate anything is puzzling.

One page 117 we are told "Algebraic geometry is an area in which the geometry of numbers is studied." In fact, algebraic geometry is a wide field connected to many branches of math including number theory and topology. It uses mostly commutative algebras to attack problems in geometry. So when I find a mistake like this, it calls into question remarks made in the rest of the text on topics that I'm unfamiliar with. I have to wonder if the author knows what he's talking about.

The book tends to repeat stories. It reminds me of visiting a nursing home where a resident with a memory problem keeps telling the same stories over and over. So for example, on three occasions in the book we're told that wedding invitations were printed for Bourbaki's imaginary daughter. And the fact is indirectly referenced at a fourth point. The first time it was interesting. On pages 69 and 119 we're told that Henri Poincare was called "the last universalist" and in both places we're told why. One page 125 we read, "As we shall see, the modern idea of structure originated in linguistics..." But we already saw this on page 102. My guess is the author didn't have enough to fill a book, so rather than doing a good job of explaining some of the mathematical ideas, he fluffed it out by repeating things.

Indeed, many interesting ideas are presented in the book but nothing seems well developed. So one last example, from page 197. Here we're told, "And then, of course, there are the great paradoxes in set theory, which make the discipline full of theoretical holes." What are these paradoxes? How do they poke holes in set theory? If Aczel has answers, he doesn't explain them. I only know of Russel's Paradox and the Cantor Paradox. The first has been taken care of and the second isn't really a paradox in the classical sense, but only used in a reductio ad absurdum argument.

Having said all these horrible things I'll acknowledge I did read the book to the end. It has some colorful characters and anecdotes that were new. I just wish the story was better told.

Bell Curve Appraisal

The first third to half of the book read as an encyclopedia article i.e. Nicolas's father was born in 1893. He worked in a grocery store for five years. His mother etc., etc.,... and so it went for many others mentioned in the book. Puleze!

Eventually, Aczel gets to the point where the Bourbaki group is frustrated by conventional mathematics teaching techniques and present the idea "that relationships and structure were the key elements of mathematics." From there they put together syllabii for many areas of mathematics. We learn of Modern Mathematics rise and fall. All very interesting.

In the last part of the book, Aczel proffers that mathematical structures are inherent in the human brain and so are transferable to many other disciplines e.g. linguistics, psychiatry, economics and others.

I would have preferred an entire book devoted to the middle section. The last of the book was of moderate interest.

If you want to waste your money....

I am very sorry to say but this is a very bad book.
For people who are familiar with the Bourbaki group it adds nothing to what is easily available in journals, biographies, interviews and so on.
For those who are not but may be interested I would strongly recommend "Bourbaki: A Secret Society of Mathematicians ", by Maurice Mashaal (Author), Anna Pierrehumbert (Translator).
Actually I should have known better. The book on Fermat's last theorem by the same author is also of very poor quality.
For some years now I have been waiting by an announced book on Bourbaki by Liliane Beaulieu, someone whose work is of great quality. I wonder if the project was abandoned.

For the dissatisfied, an alternate book on Bourbaki

I am not going to pass judgement on Aczel's book, though I have flipped through it in the bookstore and I can see why people are dissatisfied. I put in a star rating in line with the average of what's above, only because the Amazon review form forces me to specify something. Some previous commentators indicated a desire to learn more about Bourbaki, but found that Aczel's book didn't do the trick for them. For the benefit of such people, I heartily suggest taking a look at the following, recently published in translation by AMS, and available through Amazon:

Bourbaki: A Secret Society of Mathematicians

I will admit a small family connection with this book, but I offer this only in the spirit of providing information that may be of use to somebody. I've read this book a few times, and have enjoyed it thoroughly and I think I've retained enough objectivity to say that others are likely to enjoy it as well.

The Origin of the Beauty - New Math à la Bourbaki

Before reading this book, I was ignorant of Nicolas Bourbaki, although I did my New Math education in France in mid 70s, personally experienced the abstractness of New Math syllabus based on Axioms and Structures. I love the Bourbaki's New Math with its rigour and structure, but I hate its extreme abstractness against our normal human thinking process from special cases to generalisation.

The demise of Bourbaki does not change the abstract culture in New Math - unfortunately still an obstacle for math lovers coming from traditional math background.

This book should go a bit deeper in math demonstration on some interesting topics like Levi-Strauss' Anthropology using Group Theory and Piaget's linguistic math structure, etc.

Overall, I think Aczel had done a good job on the history side of New Math, although not enough to satisfy bigger appetite of mathematician readers who want more solid math illustrations.