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/MATH/
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.
I. 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 oeuvre 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.
II. 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), Fre'chet (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 ro^le 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.
Cavaille\s 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.
III. My first thesis supported by recent revelations.
Professor Segal allows that I have a point concerning the Bourbachiste
neglect of Go"del. Hurrah. Someone has admitted it. Supposing that Go"del
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 ro^le 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 Supe'rieure. [...]
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.
IV. 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.
V. 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."
VI. 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.