Olivier, that last paragraph paints a wonderfully specific portrait of the work of a unique individual; thank you for taking the time to write it, for the benefit of those of us whose knowledge of AG's work, like mine, is mostly second-hand. Was Grothendieck aware of the veneration with which he was viewed by so many, and if so, how did he feel about it? I can imagine it making him uncomfortable, given his devotion to the dream of an impersonal, perfect mathematics. Jim Propp On Monday, November 17, 2014, Olivier Gerard <olivier.gerard@gmail.com> wrote:
The main cause of trouble in all this is that we are all guilty of propagating bad journalism methods and angles
- Who is the best mathematician of the century, the three most important theorems of all time ? - Who will win ? Who is right ? Bourbaki-style abstract nonsense with their species-loving crypto allies who never take an empty set for granted or down-to-earth no-nonsense experimental mathematicians who know how to count the leaves on a tree when they see one ?
I suggest that keen amateurs and professional of mathematics would be better off leaving this kind of sensationalism to people magazines.
Simon is right that A. Grothendieck mathematical writings was very much influenced by the Bourbaki style. He was in early direct contact with several important members such as Henri Cartan, Jean Dieudonné, Laurent Schwartz and joined the group later for a few years.
Another related point is that Jean Dieudonné, who had a strong role in the final editorial stages of many Bourbaki volumes decided to become a kind of assistant and co-author of the younger Grothendieck for the hundreds of pages of the E.G.A. volumes (= Elements of Algebraic Geometry).
But both Grothendieck and Dieudonné had their own special voices, interest and projects and diverged on many points. Other important members of the Bourbaki group such as André Weil, Ehresmann, ... did not write their own books in the manner of the Bourbaki volumes. It would not have made sense anyway. But they used many of its conventions and notations.
The Bourbaki style was a collective, the results of relentless criticism between members of the intermediate or alternate redactions of sections and chapters. The goals have changed over the course of the group's life, but the starting point was to write down what the members saw as the core of mathematics, allowing others to elaborate on say, functional analysis.
It is also really wrong to use Bourbaki as a scapegoat for some of the mathematical viewpoints one has had no chance to really practice and appreciate. I would like to write about the mutual interest and strong relevance of all this for combinatorics among other things, but this will be another time.
Bourbaki's treaty by itself is not hegemonic on mathematics. I except Bourbaki historical notes, often written by Dieudonné, that I find very partial. But some of the most vocal members such as André Weil and Jean Dieudonné made very strong, sometimes provocative, statements and were very influential in the professional world. Their personal tastes were as biased and as subjective as anyone but their position made that these were not easily dismissed as tastes and opinions but took a programmatic turn and motivated frightful epigons. One can appreciate Bourbaki treatises but have widely different opinions from the well-known members.
Most things that Simon alludes to are not Bourbaki's. Most of the times Bourbaki's treaty is collecting, ordering and systematizing important past research, proposing new names and notations of which many have become standard now, starting with the most basic (such as set theory or rings) but most ideas come from people such as Frege, Russel, Peano, Zermelo, Hilbert, Hasse, Noether, Artin, Van der Waerden, Borel, Banach, von Neumann, etc.
Those ones are the authors and the inspirators. Simon was amused by some parts of Book I about set theory I guess, of perhaps the "fascicule" preprint. I wonder what he would think of Russel and Whitehead's Principia Mathematica and many other works by logicians, including to this day. But there are fascinating things going on on this front. Not very readable for the uninitiated, I agree.
Bourbaki books are very close to an encyclopedic dictionary with no special interest in being original but the most general possible, even if this requires tremendous work, knowledge and ingenuity. One can read them with the same kind of pleasure you read the entries for important words in a large dictionary such as the O.E.D., not for the spelling and the main definition but for all the meanings you would not have listed yourself even if you could grasp them easily enough when reading widely. It expands the mind to consider uses of langage you have not been exposed to. You are free to steal them or reject them. You are free to invent something else as a result.
The fact that Simon (and countless others) attributes to Bourbaki what has been the outcome of heated debates and a fantastic renovation of mathematics in the 1890s to the 1940s by many great logicians and mathematicians is a bit ironic. They just wanted to be an easy and homogenous reference for all this. They strived for unity in mathematics. You may think it is a bit naïve.
Grothendieck's mathematical style has evolved a lot and is quite in contrast with his litterary or epistolary style. In mathematics, it was a mixture of large-scale engineering, akin to building tunnels and bridges across oceans instead of rivers, and search for harmony, generality and security in a way most people would find insecure : by removing special cases, historical examples and concepts.
What is common between his mathematics and his other writings is the energy, the will to encompass, to settle as much of the Universe as possible, a longing for the absolute, a rejection for compromise, a certain view of civilisation which finds no human being worthy of being called civilized and wants to start again and again.
Olivier Gérard _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun