Customer Reviews

The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed

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1 star:		(9)

 

 

Amir Aczel is as frustrating an author as you will find anywhere. The man is bright, no question about it. He also has impeccable taste when it comes to interesting subjects to investiagte. He has written books on various mathematical subjects: Fermat's Last Theorem, Descarte, and now Nicolas Bourbaki. Yet his followthrough and his writing habits are infuriatingly inconsistent and shows signs of discordant chaos in his reasoning. He also has a disconcerting proclivity towards doing a hack job on a subject to get paid and then moving on to better things.

The story of Bourbaki is a fascinating one, so I was eager to read this book. The miniscule size of this book should have been a red flag, Aczel's reputation, at least in my head, should have been another, but I proceeded to buy it because I am an eternal optimist and I believe that people can and will surprise me and change my preconceived notions.

It didn't work this time, nor any other times when I placed my faith in Aczel. So where to begin?

1) As the previous reviewers had stated, there are no math in this book. No explanation of what Bourbaki was up to. How do you write a book on mathematicians without writing about mathematics? I understand that one does not wish to populate the book with excesive mathematical details but the power of math is in its compact notations. He does try to explain things in general terms, but a few figures and a few lines of math would have done wonders to his narrative.
2) Not enough back ground material was covered. When Aczel is trying to explain the application of structuralism in linguistics and in psychology, he was doing some extremely fine narration of extremely dense and abstract ideas and putting them into the context of what Levi-Strauss and others are trying to do, but he was not consistent in narrating the other parts of the book, he did a lot of hand waving and hot air generating.
3) As an author writing about people, one can definitely become enamoured with certain people and grow to dislike certain others. Aczel definitely fell in love with Alexander Grothendieck's story and disliked Andre and Simohne Weil. It is irresponsible, however, for Aczel come out and say that the reason for the demise of Bourbaki is because Grothendieck left the group without explaining fully WHY category theory is a more reasonable foundation. It is equally irresponsible for the ad hominem attacks on Andre Weil's character without citing specific instances of his behavior.
4) The book reads like a very bad draft, there is no continuity to the history and the book is not built around mathematical logic nor is it based on chronological order, it is as if Aczel decided to put all these bits of stories and mathematics together haphazardously. The writing is very jagged. Reading and making sense of the story is extremely fatiguing because the author made every effort to confuse the reader. Many anecdotes are repeated for no apparent reason and they are repeated without qualifiers or additional information.
5) There does not seem to be any care taken to build a case for or against anything. The author just scattered facts and his own opinions out and it was up to the readers to figure out a logic for themselves.
6) As in his previous books, the author seem to be building toward a conclusion, a crescendo in the narrative, yet after the build up, there is no crescendo, nor a diminuendo, there is just a monotone white noise in the background.

Like I said, this is a massively wasted effort towards a very interesting subject. The only thing that I have gotten from this book is the germination of various subjects that the author mentioned in passing, so thank you Amir Aczel for your bibliography and a desultory book report.

Unlike the other reviewers, I was not interested in detailed discussions of the mathematics that the Bourbaki group wrote. I heard the author plugging his book on the radio and the topic sounded interesting. However, when I started reading the book it did not cover what he was talking about in much detail. The first half of the book is a poorly conveyed life history of the various mathematicians in the group. The book lacks cohesion and continuity. The book is not very long, but the author repeats material in later chapters that was covered in earlier chapters. I can not believe that any publisher would publish such a poorly written book.

I am a Physics instructor that emphasizes the importance of good writing even in technical fields. This is a perfect example that I often cite to my students of someone in a technical field that does not know how to convey his thoughts because he failed his English course.

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.

I picked it up at my local library without previously having looked at reviews online. What a mistake! I kept thinking it had to get better, and it didn't, really, altho as one reviewer noted, the explanation of the influence of structuralism on Levi-Strauss was useful (but irritating).

In addition to no explanation of the math (which meant I kept looking stuff up online -- Aczel's choice makes some sense. Trying to explain category theory in a short, mass-market book would be even worse than what Aczel did do), choppy jumping around from person to person and within a person's lifetime (like he was trying to invent cliffhangers) and generally not being what the cover matter would lead you to believe (somehow explaining New Math, which it only barely does in passing, or telling an entertaining story of an academic prank, which it kind of does, but isn't what the book is really about), Aczel's attempts to place Bourbaki and structuralism in a larger cultural setting are intellectually bankrupt. You just cannot show that Bourbaki and a bunch of French mathematicians somehow convinced the rest of the arts and sciences of the need for structure/rigor/whatever, and ignore the fact that this was all occurring at the same time as Nazism/fascism/Stalinism. Like that is somehow an accident or coincidence? I think not. And I suspect that Aczel didn't think this through, which, given the people he's writing about, might be ironic, but apparently he's famous for this so it can't really be considered unexpected.

This book gives the impression that the Bourbaki group alone discovered and laid bare the mathematical structures studied in modern mathematics and originated the idea of founding the edifice of mathematics upon set theory. There is no mention of all the controversies and work done on the foundations and structures of mathematics prior to Bourbaki. For all the talk about Bourbaki and their investigation and championing of mathematical structures, the book says virtually nothing about what a mathematical structure is or looks like, or how the Bourbaki approach or the nature of their conception of mathematics differed from what preceded them, except to say they were rigorous and built mathematics upon set theory.

There is some mathematical terminology in the book, but no mathematics. The content is mostly biographical, with Andre Weil and Alexandre Grothendieck getting, in the final count, the most attention. The Bourbaki group's first meeting, in December of 1934, isn't mentioned until chapter seven, eighty pages in. From page 81 to page 127 the subject is Bourbaki, then the book shifts topic and is about Roman Jakobson, Claude Levi-Strauss and the rise of structuralism, including Roland Barthes, Jaques Lacan and others. The author claims that Bourbaki's structrual view of mathematics, their focus on mathematical structures and the structure of mathematics, is a major source, along with the linguists Sausurre and Jakobson, of what became, in the literary and sociological fields, Structuralism. The author displays no critical thought in these pages, and the presentation is superficial and misleading throughout.

For the mathematically adventurous, Leo Corry's book Modern Algebra and the Rise of Mathematical Structures places Bourbaki in historical context. See especially chapter 7, Nicolas Bourbaki: Theory of Structures, which is also available online as a .pdf file.

This book concentrates on the personal lives of the mathematicians who comprised the Bourbaki, a more-or-less anonymous group of (mostly) French mathematicisans who set out in the mid 1900's to reformulate the foundatons of mathematics. While this is interesting stuff, the lack of any serious description of the mathematics makes the book less interesting to a mathematician, such as myself. And it's hard for me to imagine that a non-mathematician would care whether or not the Bourbaki existed.

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.

The story is interesting. I had heard many stories about Bourbaki and Grothendieck and it was nice to know more details about them. However, there are three points which I didn't like at all:

1) The author repeats himself too much, specially when he talks about the idea of "structure" in mathematics.

He says that these ideas were first studied in linguistics and then applied to anthropology in a mathematical context. And throughout many chapters he repeats this again and again, without giving an explanation. Finally he explains what this means in chapter ten. But by that time, you are so tired of these "structures" that you don't know if you really want to read the chapter (maybe he will just repeat himself and say nothing knew as he had done before?). Perhaps this made me want to skip chapters 12 and 13 that were about other non-mathematical topics that looked boring because of the author's style.

2) The author has very radical conclusions about mathematics and mathematicians, in particular, about the influence of Grothendieck.

He constantly talks about Alexander Grothendieck as if he were a mathematical god. True, Grothendieck influenced a great part of mathematics and his work is considered extremely important. However, it all depends on what type of mathematics you do research on. Here I am speaking as a research mathematician myself. I work in an area where the work of Grothendieck on Algebraic Geometry is never discussed. So Grothendieck's work is irrelevant to my research.

He also says that Bourbaki "is dead" and it just sounds as if that was because the group did not continue in the line of Grothendieck's research. In my opinion, Bourbaki was important in the times when mathematics was not as formal as it is now, so now there is no point in writing more books. He even mentions this point... but rapidly says that Bourbaki should have given a categorical (Grothendieck-style) form to mathematics. I must mention 3 points about this: (a) work HAS been done in this direction, just not by Bourbaki, an example is Algebraic Set Theory (London Mathematical Society Lecture Note Series) where a categorical axiomatization of set theory is given, (b) it is (psychologically) easier to work in set theory and then develop category theory (this is even an exercise in Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics), "formalize category theory within ZFC"); (c) as fas as I know there is still a "Seminaire Bourbaki" in Paris and it publishes its memories (for example Seminaire Bourbaki: Volume 2008/2009 Exposes 997-1011 (Asterisque)).

Also, as I mentioned before, Grothendieck's work (and categories in general) are not used in many areas of mathematics: some parts of logic, general topology, graph theory. So I think that there is no point in thinking that we should replace the foundations of mathematics for something more difficult to understand and that will be of no use to a part of the mathematical community.

Maybe the author is not so guilty of writing this in the book, as there are some mathematicians that used to think in this same way. For example, I remember I was once visiting the library and found this papers where Mathias and MacLane fight over what is the best way to formalize mathematics. MacLane is obviously in the categorical side, and expresses many opinions that I found in this book. However, this is just one way to look at mathematics, and the other side (my side, by the way) is explained by Mathias. Maybe the author was only informed by people of the categorical side?

However, I must say that the way the author talks about this is as if "Grothendieck is God, so anybody who doesn't think in the same way as him, is wrong". I just didn't like the author's style.

3) A specific quote:

"Bourbaki had the chance, through the work of Grothendieck and his students, to refound modern mathematics on the theory of categories..... For mathematics remained based on a flawed system, set theory, rather than something that would have been much more appropriate." page 205

This is just wrong. Maybe these "flaws" he is talking about are Russel's paradox, which proved that set theory should not be taken in such an innocent way. However, this was fixed by the work of Zermelo, Skolem and Fraenkel, who wrote the axiomatics for ZF set theory (before Bourbaki, in fact) that is used nowadays. Of course, we are not able to prove the CONSISTENCY of ZF (by Gödel's second incompleteness theorem), but that doesn't give the author the right to say that set theory is "flawed".

Saying that this is a good reason to try to make category theory a foundation, is not a valid argument. Set Theory is a very active area of research in mathematics today. Making it dissapear is not a realistic idea. If ZF was in fact, flawed, then set theorists will surely find a way to sidestep these flaws (as they have done in the past) and build a new theory of sets.

Also, "more appropriate" depends on your point of view. For me, the "appropriate" foundation of mathematics is set theory, because it is what I like and what I do. It is just terrible that there are mathematicians who think that only the part of mathematics they do is important and all other things should be put in the trash can.

This book reads like a bunch of 7th grade book reports about various individuals and topics strung together in roughly chronological order. Its character studies are shallow, the writing is bland, and its exposition of key ideas are embarrassingly shallow. I got it for $2 to try out my new Kindle. The good news is that the Kindle is fantastic. What an awful book though.

As tempting as it may seem, do not get roped into reading this book. Admittedly, I purchased this book based solely on the blurb written on the back and my interest in mathematics. However, the story, as presented by Aczel, is not as intriguing as it obviously could be. In all honesty, reading the Wikipedia article on Nicolas Bourbaki is a much more worth-while experience.

The writing style is at once dull and long-winded. Ideas are repeated over and over again and far too much time is spent on topics only cursorily related to Bourbaki and its mathematics. When any detail is presented at all it is done so without exposition, without any real attempt to educate the reader.

The book literally reads like a poorly written high-school paper.