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

YCor
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Marius Kempe
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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 04 '14 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 05 '14 at 00:01

4 Answers4

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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.

Marius Kempe
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Richard Montgomery
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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.

Alexandre Eremenko
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  • Thanks for the interesting links! Unfortunately I cannot read Russian but it seems Google translate handles the text alright so that I can still get the idea. –  Jan 05 '14 at 01:23
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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 05 '14 at 01:37
  • There is a recent book on the sort of historical theories that Arnold espouses (similar in spirit but different in detail), 'The Forgotten Revolution' by L. Russo. – Marius Kempe Jan 05 '14 at 01:57
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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 05 '14 at 02:09
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    That's very interesting. I don't think we disagree, but I was vague; all I meant was that Russo, like Arnold, argues for hidden influences from antiquity on modern science. – Marius Kempe Jan 05 '14 at 03:01
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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 05 '14 at 05:31
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    So, Arnold was right from the point of view of the "bourbakist logic" :) – R W Jan 05 '14 at 06:09
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    In his account Arnold ascribes to his opponents, alive and dead, views which are clearly insane. – sds Jan 05 '14 at 08:10
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    Is there a systematic (written) account of Arnold's misrepresentations in history? – Igor Belegradek Jan 05 '14 at 14:00
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    @AlexandreEremenko I find this thread very interesting. "What I mean, Arnold is almost always wrong when it comes to historical details." Does this also apply to his well-known book Huygens and Barrow, Newton and Hooke? What do you think of his scholarship there? – Todd Trimble Jan 05 '14 at 14:01
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    @Todd Trimble: No, I don't think so. This applies only to his later writings, especially those where he mentions "Egyptian God Toth", Greek mathematics, etc. About Newton and his contemporaries, most what he writes is OK, though he sometimes exaggerrates to make his thought clearer. – Alexandre Eremenko Jan 05 '14 at 15:51
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    @Igor Belgradek: No, I don't think so. What for? There is a large literature written by professional historians who dispute what he says in "Huygens, Barrow...", but unlike Arnold's book, these papers are boring. The things he wrote about Egyptians, nobody takes seriously, I guess. – Alexandre Eremenko Jan 05 '14 at 16:02
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    @AlexandreEremenko, so "there is a large literature written by professional historians who dispute what he says in "Huygens, Barrow..."", but you disagree with the historians and think Arnold is basically accurate in that book. Interesting... – Igor Belegradek Jan 05 '14 at 16:09
  • I think MO rules discourage such long discussions in the comments... – Alexandre Eremenko Jan 05 '14 at 16:16
  • There is no discussion. I merely wish to clarify what you said: it seems you dispute not only Arnold's "later writings" but also "a large literature ... by professional historians". Could you confirm whether that's you meant to say? – Igor Belegradek Jan 05 '14 at 16:35
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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 05 '14 at 17:54
  • I would be interested too – Joël Jan 05 '14 at 19:47
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    Igor, Todd and Joel, here are some examples: R. Weinstock, Inverse-square orbits in Newton..., Arch. Hist. Exact Sci. 55 (2000) 137-162. B. Pourciau, Integrability of ovals...,Arch. Hist. Exact Sci. 55 (2001) 479--499. – Alexandre Eremenko Jan 06 '14 at 01:51
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    @Igor Belgradek: I confirm. Arnold's historical statements are frequently inaccurate/exaggerated. His stories about Egypt and relations of Egyptian and Greek science are fantasies, contradicting to what is known about early history of math. (See, for example books by O. Neugebauer and van der Waerden. – Alexandre Eremenko Jan 06 '14 at 01:56
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    I glanced over the papers Arch. Hist. Exact Sci. and it seems the disagreements of their authors with Arnold are mathematical, not historical: in brief Arnold thinks the certain Newton's proofs are correct, while the authors claim otherwise. The issue is that Newton's statements are not rigorous by modern standards, and the discussion revolves around whether they can be made rigorous. – Igor Belegradek Jan 06 '14 at 02:39
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    Yes. Most factual, historical detail in his book about Barrow,... etc. seem correct. And I tend to agree with his interpretation of Newton's proofs. In this particular discussion I am on the side of Arnold, not historians. – Alexandre Eremenko Jun 30 '17 at 19:39
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    There is a book of Arnold, "What's mathematics" (2 Russian editions, 2002, 2008) where unlike in his other similar writings he gives some references on his sources. In particular he names the Russian 19 century book from which he learned about Toth. And repeats that the ancients knew about elliptic orbits at the time of Numa Pompilius. – Alexandre Eremenko Jun 30 '17 at 19:45
  • What he says about Descartes, is that true? "Не следует экспериментально проверять исходные положения наших теорий: это просто произвольные аксиомы, и их отношение к реальности отношения к науке не имеет. Столь же бессмысленно сравнивать с реальностью и окончательные выводы: вряд ли они согласуются с ней лучше исходных аксиом." ("There is no need to check the postulates experimantally: they are just axioms, their relation with reality has nothing in common with science. Similarly there is no sense in comparison the conclusions with reality.") Did Descartes write this? – Sergei Akbarov Oct 11 '17 at 20:53
  • And about Manin: "Чем математика действительно полезна, так это своим огромным вкладом в решение основной проблемы современного постиндустриального человечества. Проблема же эта состоит вовсе не в том, чтобы, как думают некоторые, ускорять прогресс человечества, а напротив, в том, чтобы этот прогресс всемерно тормозить." ("The use of mathematics is that it gives a solution of the main problem of the post-industrial society. This problem is not to accelerate progress, but on the contrary to brake it.") Can these be true Manin's words? – Sergei Akbarov Oct 11 '17 at 20:54
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    @SergeiAkbarov We can never prove that Descartes did not write something, but the quote seems a misunderstanding of Descartes' A Discourse on Method, which is devoted to, in retrospect, metaphysics and ontology. In Part IV, he wrote that I supposed that all the objects (presentations) that had ever entered into my mind when awake, had in them no more truth than the illusions of my dreams and then, he observed that I think, therefore I am (COGITO ERGO SUM). In a nutshell, he saw everything experimentally presents might be distorted ... –  Dec 19 '19 at 13:05
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    @SergeiAkbarov [continued] that is to say, he tried to establish "absolute truth". Quoting him: In the next place, I attentively examined what I was and as I observed that I could suppose that I had no body, and that there was no world nor any place in which I might be; but that I could not therefore suppose that I was not; and that, on the contrary, from the very circumstance that I thought to doubt of the truth of other things, it most clearly and certainly followed that I was;... If you are interested, I suggest you find a Russian translation: I believe that as non-native speaker ... –  Dec 19 '19 at 13:08
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    @SergeiAkbarov [continued] It is quite difficult to understand these philosophical thoughts. It is even difficult in mother tongues. I read it several years ago in mother tongue, and only understood very little bit. My impression is that this is quite different from Arnold's interpretation. It is not depriving formalism from reality, but see reality distorted and try to base our knowledge on the absolute. He also tried to deduce transcendental facts, such as the existence of God. Philosophy treats these, and I guess that Descartes is also a mathematician therefore attracts Arnold's attention. –  Dec 19 '19 at 13:13
  • @Yai0Phah yes, I have the same feeling. And, actually, I must say that this negligence is typical in Russia. Thank you! – Sergei Akbarov Dec 22 '19 at 13:39
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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].

j.c.
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    What is criminal is that there isn't a recording of the event :-( – Mariano Suárez-Álvarez Jan 05 '14 at 00:32
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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 05 '14 at 00:59
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    That $2+5$ anecdote never gets old. (Said that, the canonical version is $2+3$, I believe.) – Nikita Sidorov Jan 05 '14 at 01:07
  • @quid: Thanks. The information in your answer helped me to locate the sources I gave. – Alexandre Eremenko Jan 05 '14 at 01:40
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    The comments about the "2+5" also appear in Arnold's essay on teaching Mathematics , accessible at http://www.math.fsu.edu/~wxm/Arnold.htm – Piyush Grover Jan 05 '14 at 02:02
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    I was that child. – Joël Jan 05 '14 at 02:35
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    ! Would you be willing to tell the story of how it happened? – Marius Kempe Jan 05 '14 at 03:02
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    It started as a joke, that I made when I was 11, a little similar in spirit to this one ( http://www.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 05 '14 at 05:23
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    @Joël that's great! I always liked that story, in a way different from the intended one, now it is still better. –  Jan 05 '14 at 13:52
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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.

Mikhail Katz
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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 07 '14 at 01:47
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    @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. – Mikhail Katz Feb 07 '14 at 08:14
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    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 of 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 07 '14 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 07 '14 at 14:31
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    @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. – Mikhail Katz Feb 09 '14 at 08:33