FOM: Reply to Professor Segal

[Adrian Mathias]

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Further remarks on Bourbaki

A. R. D. MATHIAS

Universidad de los Andes, Santa Fe de Bogota, Colombia

A preliminary version of my essay "The Ignorance of Bourbaki" 
was read to an undergraduate mathematical society, the Quintics, in 
Cambridge on October 29th, 1986, and that version was printed in the Cambridge 
undergraduate mathematical periodical Eureka, shortly afterwards.  

A revised version appeared in Physis in 1991 and in the 
Mathematical Intelligencer in July 1992; 
the text of that version may be found at 

http://www.dpmms.cam.ac.uk/ardm 

in the files bourcomp.dvi or bourcomp.ps.  

That revised version has been reviewed by Professor Sanford L Segal of 
Rochester for the Zentralblatt. I am told that it is available on-line 
at 

http://www.emis.de:80/cgi-bin/zben/MATH/math-en.html 

the review being number 764.01009, though I have failed to access that 
Website myself. I noticed, though, during my attempts that there is a link 
from the Zentralblatt page to the Laboratoire J.-A. Dieudonne at Nice. 
Is the Zentralblatt staffed by Bourbachistes ? I think we should be told. 

Light has been shed on numerous points 
in this debate by recently published remarks by two members of Bourbaki:  
the article by Armand Borel entitled 
"Twenty-Five years with Nicolas Bourbaki, 1949--1973", 
in the Notices of the American Mathematical Society, 
March 1998, pp 373 -- 380, and  
The Continuing Silence of Bourbaki --- an interview with 
Pierre Cartier, June 18, 1997.  
in a recent issue of the Mathematical Intelligencer. 

Alas, these items have probably come to my notice too late for me to 
incorporate appropriate changes in the Hungarian translation of the 1992 
version of The Ignorance of Bourbaki that is due to appear in A Termeszet 
Vilaga, a Hungarian popular scientific monthly.  So I aim to show 
here that my first point is confirmed by the statements of Cartier, 
and that my second point is not refuted by the criticisms my article 
has received. 

I am grateful for factual help given by Professor Segal during 
the preparation of this reply to his criticism. 

PLAN OF MY ANSWER: 


I. The purpose and the two theses of my essay. 

II. Answers to fine points raised by Professor Segal. 

III. My first thesis supported by recent revelations. 

IV. Bourbaki and the possibility of intellectual terrorism. 

V. Ethical questions raised by Professor Segal. 

VI. My second thesis: Bourbaki not an adequately functioning 
individual. 


1. The purpose and the two theses of my essay. 

The first question I ask on re-reading my essay in the light of 
Professor Segal's review is this:  
where in my essay is the diatribe into which he says my 
essay degenerates ? Like Lord Clive before me, I stand astounded at my 
own moderation. I am at a loss to understand the meaning that Professor Segal 
attaches to the word "unfortunately", with which he condemns some 
of my phrases. Does he mean that a sober historian would not use such 
phrases ? That might be; but 
in this piece I was not aiming to be a sober historian; 
I was aiming to describe the present-day consequences of the past 
behaviour of an influential group. 

Perhaps I should state explicitly that my piece was not intended to be a 
historical study. It was intended to challenge the near-divine status 
of Bourbaki, which in my view has meant that Bourbaki, besides the good they 
have done, have had a harmful effect. The phrase about the tall oak in the 
shade of which nothing will grow comes to mind. 

My message is a simple one. 
No one denies that the Bourbachiste uvre is a collection of very 
wonderful books written by very wonderful people; nevertheless  
I contend that they have given generations of mathematicians a 
stunted conception of logic. So I am saying to the adventurous youth of 
today: if you wish to learn what has been going on in logic in this century 
do not go to Bourbaki, for they do not know. 

In my essay I asked two questions, which I reformulated in 
various ways, and offered tentative answers. One version of the first 
question was why the foundational understanding of Bourbaki did not 
advance as foundational studies advanced; one version of the second 
was why the Bourbachistes did not notice the inadequacy of their chosen 
set theory as a foundation for mathematics. 
Before commenting further on those, let me turn to some points of 
detail raised by Professor Segal. 


2. Answers to fine points raised by Professor Segal

The purpose of my opening paragraphs with their lists of names was 
to set the scene: were they to be deleted on the grounds that I am 
playing fast and loose with history, it would in no wise affect my 
two main points about Bourbaki. 

I think the division I had in mind 
was that all the German scholars in the first list were dead by the end 
of the First World War, whereas all those in the second list 
survived it.  Let me add the dates: 

Riemann    1826--1866            1849--1925  Klein
Frobenius  1849--1917            1862--1943  Hilbert
Dedekind   1831--1916            1885--1955  Weyl
Kummer     1810--1893            1898--1962  Artin
Kronecker  1823--1891            1882--1935  Noether
Minkowski  1864--1909            1877--1938  Landau
Cantor     1845--1918            1868--1942  Hausdorff

I agree that Klein is anomalous: he outlived Poincare but was an 
established figure before the latter's meteoric rise by which Klein 
felt threatened, and hence I think of him as in some sense of an older 
generation. Of course there really are no generations, 
the division is arbitrary, as is the choice of date --- date of birth, 
date of death, date of greatest activity, date of greatest influence. 

About the French analysts, for not mentioning whom Professor Segal 
reproves me: Montel (1876--1975), 
Frechet (1878--1973), Emilio Borel (1871--1957), 
Baire (1874--1932), and Lebesgue (1875--1941) were all in their forties at the 
end of the First World War, when Dieudonne was but 12, 
so that it is surely fair to suggest that he and 
the others arriving on the mathematical scene after that holocaust 
would see its survivors as of an older generation, 
despite the remarkable longevity with which many were blest. 

I was unaware that Herbrand was a close friend of the founders; but alas 
he had died by the time they started their crusade; 
had he not been killed on the mountain 
he might have brought them to a more balanced conception of the 
role of logic. I do not see, incidentally, why his being a friend 
means that they would necessarily have been intellectually in agreement with 
him. I have many close friends who haven't a clue about mathematical logic, 
much as I admire their achievements in other domains. 

Professor Segal says that some of my speculations contradict the 
statements in Weil's reminiscences (which were not available to me 
at the time.) To me these are differences of emphasis rather than of 
fact; if one is looking for contradictions, one can find them 
even in the statements made by members of Bourbaki 
about the names of the founders, as follows.  

Armand Borel names the founders as Henri Cartan, Chevalley, Delsarte, 
Dieudonne, Weil. 

Chevalley gives Borel's list plus Mandelbrojt and de Possel. 

Cavailles gives Chevalley's list plus Ehresmann. 

Cartier agrees with Armand Borel, and goes on to list subsequent 
generations as follows: 

second:  Schwartz, Serre, Samuel, Koszul, Dixmier, Godement, Eilenberg. 

third: Borel, Grothendieck, Bruhat, Cartier, Lang, Tate. 

fourth: a group of students of Grothendieck, who by then had left in anger. 

3. My first thesis supported by recent revelations.

Professor Segal allows that I have a point concerning the Bourbachiste  
neglect of Godel. Hurrah. Someone has admitted it. 
 Supposing that Godel is indeed mentioned 
in the 1950's version, that means that his theorem had been in print for 
more than twenty years before the Bourbachistes noticed it. If one 
counts in biological generations, that is one; and in mathematical generations 
it may be more. 

This relates to the running argument in FOM: do discoveries in logic 
 affect mathematics ? I think they do; many mathematicians go into 
extraordinary contortions in order to maintain the belief that they do not. 

Cartier in his Intelligencer interview makes numerous thoughtful 
points; for the purpose of the present article the most significant 
are these:  


  "Bourbaki never seriously considered logic. Dieudonne himself was very 
   vocal against logic."

  "Dieudonne was the scribe of Bourbaki." 

Borel in his Notices essay confirms the dominant role of 
Dieudonne: he mentions shouting matches, generally led by Dieudonne 
with his stentorian voice, and says 

  "There were two reasons for the productivity of the group: the 
   unflinching commitment of the members, and the superhuman efficiency 
   of Dieudonne."  

[A peripheral question:  Borel says that there were no majority votes, 
and that all decisions had to be unanimous. Let me ask:  
were people ever expelled ? or did they just leave 
if they felt out of place ?] 

After those statements I think that the finger points at Dieudonne. 
In his last book, The Music of Mathematics he makes the same mistake 
that he made in his position papers of fifty years previously: he went to 
his grave believing that truth and provability are identical. 

That is an intuitionist position: so I confess to a feeling of glee when 
I found the following passage in the interview with Cartier: 

  "The Bourbaki were Puritans, and Puritans are strongly opposed to 
   pictorial representations of their faith. The number of Protestants and 
   Jews in the Bourbaki group was overwhelming. And you know that the French 
   Protestants especially are very close to Jews in spirit. I have some 
   Jewish background and I was raised as a Huguenot. We are people of the 
   Bible, of the Old Testament, and many Huguenots in France are more 
   enamoured of the Old Testament than of the New Testament. We worship 
   Jaweh more than Jesus sometimes."

My reason for glee is this: I made during a lecture at Oxford in 1976 
some remarks on a possible connection between religious and mathematical 
positions; they are summarised in the text of that lecture in 
the Oxford volume edited by Gandy and Hyland. Put crudely, my equations were 
Platonism = Catholicism; Intuitionism = Protestantism; Formalism = Atheism; 
Category Theory = Dialectical Materialism. 

For saying that, I was exposed to derision from 
certain quarters, though more recently people have been kind enough to 
say they find the remarks interesting. I think the above paragraph from 
Cartier vindicates me. 

The above disclosures concerning Dieudonne confirms the quotation from 
Quine's autobiography that I circulated previously. Here it is again. 

  "A Logic Colloquium was afoot in the Ecole Normale Superieure. [...]
   Dieudonne was there, a harsh reminder of the smug and uninformed
   disdain of mathematical logic that once prevailed in the rank and
   vile, one is tempted to say, of the mathematical fraternity.  His
   ever hostile interventions were directed at no detail of the 
   discussion, which he scorned, but against the enterprise as such.
   At length one of the Frenchmen asked why he had come. He replied
   J'etais invite.' "

[I should say that I have received this week an eye-witness account of 
the seminar concerned, which suggests that Quine may have been 
over-reacting to what others present knew to be 
Dieudonne's normal behaviour.]  

To me it now seems certain that there was a bias, considered or 
otherwise, against logic;  
it may be that the bias was due solely to one extremely energetic man, 
but there is a hint in Quine's remark that Dieudonne was not the 
only opponent. 

In my essay I wondered whether these attitudes might stem from 
the influence of Hilbert or from some nationalist or chauvinist 
feeling, and Professor Segal suggests that I am thereby contradicting myself. 

Perhaps I should first state that I see a distinction
between nationalism and chauvinism. Consider, for example, Janiszewski, 
who at the end of the First World War called for a small poor country to 
make its mark in foundational studies: I see him as a Polish nationalist 
but not a chauvinist. It is one thing to say "Good things are going on 
elsewhere in the world: let us try and do as well or better." 
It is another to say "Everything that is worth knowing is known by us; 
let us ignore the activities of others". 

Cartier's interview makes it clear that Hilbert and German philosophy were 
held up as models by Weil and others: 

  "The general philosophy as developed by Kant. Bourbaki is the 
   brainchild of German philosophy. Bourbaki was founded to develop and 
   propagate German philosophical views in science. All these people ... 
   were proponents of German philosophy." 

I really do not see that there is a contradiction between wishing 
to strengthen French mathematics and saying that the Germans do it better. 
One might say that the Bourbachistes were nationalist but not chauvinist. 
Further evidence comes from
"Claude Chevalley described by his daughter", where she 
says that the Bourbaki movement was started essentially because rigour 
was lacking among French mathematicians by comparison with the Germans, 
that is, the Hilbertians. 

I am delighted that Professor Segal should note my footnote 
commenting on the dearth of logic in England; the response I have had from 
leading English academics to that has resembled the wriggling of  
tobacco companies confronted with evidence of the dangers of smoking. Perhaps 
one day someone will do something. It makes a sad contrast with the positive 
response given in Poland to Janiszewski's manifesto. 

4. Bourbaki and the question of intellectual terrorism. 

Perhaps the most moving of the comments I have received from readers 
of The Ignorance of Bourbaki is 
one that came from the holder of a (C4) chair at a leading German University, 
who told me that as a young man he had been reduced to a state of 
intellectual paralysis by reading Bourbaki and that he had had to  
retire from mathematics for six months before making a fresh start.  
It may not be an exaggeration to say that he was thrilled 
to find support in my essay for the notion that it is not 
necessary to worship at the Bourbachiste shrine in order to do 
serious mathematics. 

That it might ever have been thought so necessary can be divined from 
fleeting remarks by Miles Reid in his book 
Undergraduate Algebraic Geometry, London Mathematical 
Society Student Texts, 12, first published by the Cambridge University Press 
in 1988. I quote from the historical remarks on pages 114--117 
of the 1994 reprint, which provide independent evidence of unwholesome 
tensions within the mathematical community. 

   "Rigorous foundations for algebraic geometry 
    were laid in the 1920s and 1930s by van der Waerden, Zariski and Weil. 
    (van der Waerden's contribution is often suppressed, 
    apparently because a number of mathematicians of the immediate post-war 
    period, including some of the leading algebraic geometers, considered 
    him a Nazi collaborator.)" 

   "By around 1950, Weil's system of foundations was 
    accepted as the norm, to the extent that traditional geometers 
    (such as Hodge and Pedoe) felt compelled to base their books on it, 
    much to the detriment, I believe, of their readability." 

   "From around 1955 to 1970, algebraic geometry was 
    dominated by Paris mathematicians, first Serre then more especially 
    Grothendieck." 

   "On the other hand, the Grothendieck personality cult had serious side 
    effects: many people who had devoted a large part of their 
    lives to mastering Weil foundations suffered rejection and 
    humiliation. ... The study of category theory for its
    own sake (surely one of the most sterile of all intellectual pursuits) 
    also dates from this time."

   "I understand that some of the mathematicians now involved 
    in administering French research money are individuals who suffered 
    during this period of intellectual terrorism, and that applications for 
    CNRS research projects are in consequence regularly dressed up to 
    minimise their connection with algebraic geometry." 

Let us set against Reid's remarks a comment of Armand Borel:  

   "Of course there were some grumblings against Bourbaki's influence. 
    We had witnessed progress in, and a unification of, a big chunk of 
    mathematics, chiefly through rather sophisticated (at the time) 
    essentially algebraic methods. The most successful lecturers in 
    Paris were Cartan and Serre, who had a considerable following. The 
    mathematical climate was not favourable to mathematicians with a different 
    temperament, a different approach. This was indeed unfortunate, 
    but could hardly be held against Bourbaki members, 
    who did not force anyone to carry on research in their way."

I wonder if there is an element of complacency in 
Borel's statement that no-one was forced to carry on research in the 
Bourbaki way. That opens a theme that is difficult to discuss, but 
I believe is necessary to do so. 

Suppose it were the case that over a certain period 
in numerous universities the Bourbachistes 
seized power and pursued a policy of denying jobs to non-Bourbachistes. 
How would one obtain evidence of that ? The poor non-Bourbachistes, 
being excluded from employment which would permit them to research 
would be likely to move away from universities and find jobs in 
industry or elsewhere, and indeed to lose touch with research mathematics. 
So they would be excluded from any figures that might be produced. 
People would be saying that the Bourbachiste view is the standard one; 
what would not be said is the subtext, that that state of affairs 
has come about because the opposition has been suppressed. 
In such a case there would be a political component to what 
Graham White calls mathematical practice. 

So I should very much like to hear from anyone who believes that 
unfair pressure of the kind Armand Borel says does not exist 
has been brought to bear upon them; in whatever degree of confidence 
they would like. 

Professor Segal says that I am unhappy with the neglect of logic 
by mathematicians. I wonder whether I dare to be more specific 
or will I again be accused of using "unfortunate" terminology ?  
It is not the neglect --- surely all are free to be as ignorant as 
they choose --- to which I object 
but the interference by the high-placed ignorant with the teaching of 
logic to those who wish to learn it, and the denial, 
through the mechanism farcically known as "peer review", 
 of research funds for work in this area. 

5. Ethical questions raised by Professor Segal. 

Now let me comment on what Professor Segal calls 
"the reprehensible practice of anonymous citation."  This is indeed 
serious: I was brought up to tell the truth and shame the Devil. 
It has cost me dear. 

So what should I do when someone offers comments that I find interesting, 
but asks not to be named ? Many people do not dare openly to challenge the 
Bourbachiste hegemony for fear of losing their livelihood; so I do not 
think I should betray the identity of my correspondents. 
I myself was warned not to publish my essay when I 
first drafted it, as I was told I would be "murdered" by the 
Bourbachistes. Professor Segal's review is the first opposition to my 
thesis that I have seen in print; but to my certain knowledge  
Bourbachistes have intrigued against me covertly, to the detriment of my 
career. 

Let me give another example. A mathematical logician has confided in 
me that he obtained tenure at his University by pretending that despite 
retaining an eccentric interest in logic, he in reality subscribed to his 
Department's view that "real men don't do logic". He believes, 
and I with him, 
that had he revealed the depth of his commitment to logic he would not 
have been given tenure. I could wish that now that he has landed 
safely in the Realm of the Blessed, he would speak up for logic, but 
it appears that the habit of caution is too deeply ingrained. Still, it 
is not for me to "out" him. 

Personally I think I can do more good by respecting the wish for 
anonymity of my informants. I think that if I betrayed such confidences 
I should soon cease to receive any. For example, here is one comment that was 
sent to me, about which I have yet to do anything. 

   "I was talking to someone at high table the other
    day about metamathematics (or, at least, the fact that I was interested in
    it). He remarked that at an unspecified meeting it had been said by Sir
    Michael Atiyah that there was no interest in metamathematics in Cambridge
    and that the subject wasn't worth supporting.

   "This was the first time I had heard (second-hand) views actively AGAINST 
    metamathematics/logic. Certainly I was surprised (maybe naively) that it 
    came from Atiyah, who is oherwise a bright guy." 

6. My second thesis: Bourbaki not an adequately functioning individual.

Now we come to the part of my essay dealing with my second point against 
Bourbaki, that his chosen foundations are restrictive. 

Left brain, right brain: Professor Segal states that in an adequately 
functioning individual the two halves of the brain communicate and are 
integrated via the corpus callosum. That is exactly my point: I contend that 
Bourbaki is not an adequately functioning individual; there is a gross 
imbalance in his mathematical personality. 

This is related to my debate with Mac Lane, and though I have in 
more recent essays been able to argue my case rather better than I did 
in the essay on Bourbaki, I cannot claim to have 
succeeded in conveying to devotees of category theory the limitations they 
are putting on their conceptual universe by slavishly adopting 
that mode of thought. How does one prove to someone that he is colour-blind ? 
The victim has to be willing to notice that others have perceptions 
denied to him. 

But there are signs of these different perceptions. Cartier writes 

   "(Following the collapse of the Soviet Union) 
    the Russians have brought a different style to the West, a different way 
    of looking at the problems, a new blood". 

The group I failed to mention in my essay, centred around Baire, E. Borel, 
and Lebesgue, created a new view of analysis growing out of the insights of 
Cantor. Both Lusin and Janiszewski came from the East to sit at their feet, 
and returned home with a positive message. I wonder to what 
extent the Russian style that Cartier has noticed descends through Lusin 
from Baire, just as there is in France a similar descending chain: 
Baire, Denjoy, Choquet, Louveau. 


Two last excerpts from Cartier: 


   "Most people agree now that you do need general foundations for 
    mathematics, at least if you believe in the unity of mathematics. 
    I believe now that this unity should be organic, while Bourbaki advocated 
    a structural point of view." 

   "In accordance with Hilbert's views, set theory was 
    thought by Bourbaki to provide that badly needed general framework. If 
    you need some logical foundations, categories are a more flexible tool 
    than set theory. The point 
    is that categories offer both a general philosophical foundation --- 
    that is, the encyclopaedic or taxonomic part --- and a very efficient 
    mathematical tool to be used in mathematical situations. That set theory 
    and structures are, by contrast, more rigid can be seen by reading the 
    final chapter in Bourbaki set theory, with a monstrous endeavour to 
    formulate categories without categories." 

These two quotations will really have to be the starting point of a new essay. 
In the second one it is plain that what Cartier means by set theory is the 
feeble bunch of trivialities in the Bourbaki volume of that name; a far 
cry from what set theorists mean by set theory. On the other hand, 
Cartier may be saying in the first one what I said in What is 
Mac Lane missing ?, that unity is desirable but not uniformity.

That relates to the quotation from Dieudonne with which I ended 
my earlier essay, that we have not begun to understand the relationship 
between combinatorics and conceptual mathematics. 
As in  a classical tragedy, the Bourbachistes 
are looking for something, but do not realise that what they seek is 
already to hand: it is ironical, but pleasing, that despite Bourbaki's dead 
hand, Paris has now acquired one of the largest concentrations of logicians 
on the planet. 

Meanwhile, according to Cartier, Bourbaki is struggling with dead projects. 
What about Bourbaki instead making an attempt to understand 
developments in logic ? That would be a goal worthy of their abilities, 
 and might lead to many good things.