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I have read (but I cannot now find where) that Arnold & Serre had a public debate on the value of Bourbaki. Does anyone have more details, or remember or know what was said?

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I have also heard of such a debate, but rather as an ongoing debate involving many interviews, opinion pieces in journals like Bulletin de la SMF, informal discussions at a tea before or at a dinner after a seminar talk, etc. rather than a single debate in from of an audience given at one single place. But perhaps there was indeed such a formal debate, that I have not heard of. –  Joël Jan 4 at 23:37
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Given that Arnold liked very much to use the expression `criminal bourbakization', I'm sure any such debate would have been lively. –  Nikita Sidorov Jan 5 at 0:01

4 Answers 4

It appears that there was an event on Bourbaki, for a general audience, at the Institut Henri Poincaré (Paris) on march 13th, 2001, with Arnold and Serre.

The announcement says:

J-P. Serre : L'apport de Bourbaki, V. Arnold : Mathématiques et Physique. Arnold et Serre sont deux des plus grands noms de l'histoire des mathématiques. Ils ont des conceptions "épistémologiques" très différentes. Leur confrontation, exceptionnelle en public, devrait intéresser tous ceux qu'intéressent une réflexion sur les Sciences.

Translating roughly to: Serre: The contribution of Bourbaki, Arnold: Mathematics and Physics. Arnold and Serre are two of the greatest names in the history of mathematics. They have very different 'epistemological' conceptions. Their confrontation, exceptional in public, should be ineteresting for everyone interested in reflections on the sciences.

Unfortunately I do not have more information about this event at the moment. However, since the existence of an actual event was also unclear it seems to be a partial answer.

Also, Pierre Colmez mentions a debate (put under quotation marks) between Arnold and Serre on Bourbaki, recalling how the latter began his contribution. I cannot be sure if this is this event he is talking about, but it would fit very well as it was not really debate but two talks yet still somehow set up one against the other, so a debate in quotation marks.

Regarding things that were said, it is recalled (by Colmez) that Arnold began with a list of bad deeds of the "Bourbaki criminals" [criminels bourbakistes], including the inclusion of $0$ among the natural numbers, something about the meaning of A implies B [that I do not understand/cannot translate], and that children cannot calculate anymore on account of Bourbaki -- recalling an anecdote about a child who in reply to the question "what is $2+5$?", said it is $5+2$ since addition is commutative [this anecdote was the starting point there, although it appears that while this apparently happened, it was actually a joke/prank of the child, whose parents were mathematicians and who is today also a mathematician].

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What is criminal is that there isn't a recording of the event :-( –  Mariano Suárez-Alvarez Jan 5 at 0:32
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I think the A implies B thing is the usual "not A or B" vs. "here is a proof of B from A that isn't 'A and not A, therefore B'"? Not sure of this. –  David Jan 5 at 0:59
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I was that child. –  Joël Jan 5 at 2:35
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! Would you be willing to tell the story of how it happened? –  Marius Kempe Jan 5 at 3:02
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It started as a joke, that I made when I was 11, a little similar in spirit to this one ( smbc-comics.com/?id=3227#comic ), though obviously less funny, that was transmitted and deformed by adults until it reached Arnold. I had a professor in middle-school (sixth, fifth, and fourth grade in the french scholar system, roughly from 11 to 14) who taught us math in a very formal fashion, starting with set theory . Once jokingly (he was intimidating even when he was joking), instead of a serious question he asked the class "combien font 2+5?" and i answered "5+2". –  Joël Jan 5 at 5:23

Arnold's account of this debate (in Russian) is here: http://www.mccme.ru/edu/index.php?ikey=viarn_burbaki&post=25742517_1256 and here http://vivovoco.rsl.ru/VV/PAPERS/NATURE/BURBAKI.HTM

Let me translate few words from the beginning.

Serre challenged me with this motivation: "I wanted to tell about the influence of Bourbaki on mathematics. If everyone will speak the same, especially panegyrics, this will be boring. Thus I started to search who would express an opposite opinion to mine. After having leafed through the list of mathematicians of the world, I understood that this must be you."

The duel happened on 13 of March 2001 in Institute Poincare.

Then he describes Serre's speach and his own. According to Arnold's account, his talk in the debates is published as "Mathematics and Physics: a parent and a child, or sisters?" English translation: Physics-Uspekhi, 1999, 42, 12 1205-1217. Here is a free version: http://ufn.ru/en/articles/1999/12/c/

However, comparison of dates shows that this cannot be the true text of Arnold's talk in the debates. I re-read the account of the duel (cited in the beginning) and on my opinion 90% of what Arnold says there (about Serre and about other things) is just not true. Unlike this account, the paper in Physics-Uspekhi looks reasonable, when Arnold writes about math.

My impression is that I also read Serre's talk somewhere. But I cannot find any trace of it in the Internet. So all we have this biased (to say the least) account by Arnold.

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I wish to warn you that this Arnold account is pure delirium, nonsense. Perhaps this is the reason no one cared to translate it. –  Alexandre Eremenko Jan 5 at 1:37
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I disagree about Russo. Once I located the exact source of Arnold's theories about "Egyptian god Toth". This is some obscure speculative book written in Russian in the middle XIX century, which Arnold happened to read in his childhood. Serious historical research does not confirm anything of what Arnold says about Egyptians etc. On the other hand, on my opinion, Russo is a serious researcher, though he exaggerrates sometimes. –  Alexandre Eremenko Jan 5 at 2:09
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Why "hidden"? What I mean, Arnold is almost always wrong when it comes to historical details. For example in the paper in Physics-Uspekhi, he says that "Descartes did not accept Newton's ideas". Newton was 8 years old when Descartes died:-) –  Alexandre Eremenko Jan 5 at 5:31
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So, Arnold was right from the point of view of the "bourbakist logic" :) –  R W Jan 5 at 6:09
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Actually, I would be very interested in following up on this literature by professional historians disputing his accounts, whether or not it is boring. –  Todd Trimble Jan 5 at 17:54

I was there. Arnol'd is one of my big mathematical heros, but I found the whole thing really sad. It was in French, but my French is decent. Arnol'd began his part with a phrase I've heard him say before: ``In Russia it is impolite to talk ill of the dead, so I will not talk about Bourbaki'' and then he proceeded to lambast
Bourbaki in as nasty a way as you've ever read in any of his writings. I just felt like hanging my head. It was embarrassing watching him insult French Mathematics in front of 500 French men and saying things that seemed silly. It went on from there, Serre with a kind of sad understated dignity, not fighting, Arnol'd wanting a fight, hurling insults. The two barely even addressed each other. And yes, he did mention Toth, and if memory serves, he stated that Toth was probably Thales and had most likely come up with Newton's inverse square law. For me, the whole event was sad, embarrassing, and myth-crushing. Well, us mathematicians, we are all humans.

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cool story brother –  Wicht Jan 5 at 23:21
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@Wicht Please clarify what you mean. There is this: knowyourmeme.com/memes/cool-story-bro –  Todd Trimble Jan 5 at 23:50
    
I mean it in the literal sense, it is really a cool story. Wish I had been there man –  Wicht Jan 7 at 1:04
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@Wicht Thanks for responding. "Cool" isn't the word I would have used myself; it seems sad and embarrassing as Richard says, considering that Arnol'd was in many respects a great and brilliant man. –  Todd Trimble Jan 7 at 23:21

In retrospect it does not seem that Arnold's 2+5=5+2 comment was that effective particularly in view of the information provided by Joël. On the other hand, there is one aspect of this matter that did not come out sufficiently in the earlier answers. Namely, Arnold is a fabulous pedagogue. To give a quick example, his book Mathematical methods of classical mechanics has benefited thousands, if not millions, of readers. Many people share his doubts about the effectiveness of the Bourbaki method. Verbal excesses of course cannot be justified but there is a genuine issue there nonetheless.

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Yes, it is true that Arnold's book are very good pedagogically. But the aim of Bourbaki was not, primarily, to teach. The preface of the "Elements" warns the reader that the book is not intended for someone who has no previous exposition to the material. The aim was to provide a consistent redaction, from the foundations to the beginnings of modern research, of significant parts of modern mathematical knowledge. In short, Bourbaki is intended to be a reference, not a textbook, and no one I know has ever took it as a textbook. Comparing it with textbooks thus misses the points. –  Joël Feb 7 at 1:47
    
@Joël, thanks for your comments. Note that Bourbaki sometimes is used as a textbook, particularly their Lie algebra volume which has been very successful. But Arnold's point is not that one shouldn't write reference books. Rather, he argued that the ethos of the Bourbaki approach has a deleterious effect on an entire generation, by creating a standard that is emulated in teaching whether or not this may have been the original intention of the Bourbaki group. From my own teaching experience in France, the approach there does tend to be more formal than other places I have taught at. –  katz Feb 7 at 8:14
    
Yes, at times the treaty is so good that it may be used as a txt book. I learned there for example, the Euler-Maclaurin summation formula and the reason if the importance of Bernoulli number all around in math. About the "formalist" tendency in France, it is real, but it is a very complicated question to determine where it comes from, and its value. In a sense, all of French thought bears the mark of an often sterile formalism. A random look at pages in wikipedia.fr on many subjects, as compared with the English version makes this clear. It is especially clear on subject such as history... –  Joël Feb 7 at 14:23
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You will see on these pages a highly structured plan with parts, subparts, and sub-subparts (in general three of each as taught by Hegel), and very few content. I do not believe that Bourbaki can be held responsible for this. On the other hand, there are examples of formalism that I like and would be proud of if I had in any way crontibuted to them. Grothendick's work can be considered as "very formalism", and I consider it one if the greatest achievement of humanity. Russian formalism in literary criticism is one of the most interesting school of thought in this discipline. –  Joël Feb 7 at 14:31
    
@Joël, you are raising a very interesting point. You didn't mention "Cartesianism" but this term is sometimes used to characterize the French tendency to formal thought. I wonder if anybody attempted to analyze the Bourbaki phenomenon in this light. –  katz Feb 9 at 8:33

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