The Artist and the Mathematician by Amir D Aczel (PDF)

Description

Nicolas Bourbaki, whose mathematical publications began to appear in the late 1930s and continued to be published through most of the twentieth century, was a direct product as well as a major force behind an important revolution that took place in the early decades of the twentieth century that completely changed Western culture. Pure mathematics, the area of Bourbaki’s work, seems on the surface to be an abstract field of human study with no direct connection with the real world. In reality, however, it is closely intertwined with the general culture that surrounds it. Major developments in mathematics have often followed important trends in popular culture; developments in mathematics have acted as harbingers of change in the surrounding human culture. The seeds of change, the beginnings of the revolution that swept the Western world in the early decades of the twentieth century — both in mathematics and in other areas — were sown late in the previous century. This is the story both of Bourbaki and the world that created him in that time. It is the story of an elaborate intellectual joke — because Bourbaki, one of the foremost mathematicians of his day — never existed.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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

⭐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

⭐, “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

⭐).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 205This 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.

⭐The idea of a collective individual named Nicolas Bourbakiseems to be a communism ( small C) of mathematics.The idea came from Andrew Wiles originally it seems as way to produce needed new mathematics texts as a joint venture of elite French mathematicians.In many case the result tends to generalizations, axiomatics approaches and texts that are hard to read and understand, if superior to the Frenchtexts that existed before 1933. The world war scattered the peopleinvolved and delayed publications until the 1950’s of the most representative texts. The collective seems to have declined after the hey days of the ’60’s as a result of Bourbaki liberal political activity.Alexandre Grothendieck seems to have been associated also with the decline as he wanted to rewrite the basic foundation from set theory to category theory.There is no doubt of the influence of this era of mathematics,but the “structualism” philosophy the text reports is lessimportant maybe that the author gives credit?I liked this book better than

⭐as is is clearly just biography and history of mathematicsand pretty much leaves the mathematics alone.Radical claims about Grothendieck’s extreme genius seem to be beliedby the lack of concrete everyday results: that he thought 57 was primeseems to show he was a generalizer/ abstractor but not someone who left a lot of concrete examples like his dessins l’infant which the author never mentions ( but where I first ran into Grothendieck in mathematics).The great French mathematician Benoit Mandelbrot who had a history of a Jew in France very much parallel to Grothendieck is not mentioned,but his uncle is twice. Rene Thom gets one mention near the end.Yet it is chaos theory and fractals that brought down the structualist approach to mathematics?Mathematics is moving on from Nicolas Bourbaki,but it appears he has left us some classic mathematical textsthat are very hard to find in English translations?

⭐The book covers a lot of ground, and that can make the storyline feel disjointed at times compounded by not always clear timeline jumps. The core of the story is however very interesting and the concept of structuralism and it’s great influence across several areas of human thought is made clearly understood. Overall a very worthwhile read despite the minor narrative issues mentioned.

⭐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.

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