Nicolas Bourbaki is the
collective pseudonym under which a group
of (mainly French)
20th-century mathematicians wrote a
series of books presenting an exposition of modern advanced
mathematics, beginning in 1935.
With the goal of founding all of mathematics on
set theory, the group strove for
rigour and generality. Their work led to the
discovery of several concepts and terminologies still
discussed.
While
Nicolas Bourbaki is an invented personage, the Bourbaki
group is officially known as the Association des
collaborateurs de Nicolas Bourbaki (Association of
Collaborators of Nicolas Bourbaki), which has an office at the
École Normale Supérieure in Paris.
Books by Bourbaki
Aiming at a completely self-contained treatment of the core areas
of modern mathematics based on set theory, the group produced
Elements of Mathematics (
Éléments de mathématique) series,
which contain the following volumes (with the original French
titles in parentheses):
The book
Variétés différentielles et analytiques was a
fascicule de résultats, that is, a summary of results, on
the theory of
manifolds, rather than a
worked-out exposition. A final volume IX on
spectral theory (
Théories
spectrales) from 1983 marked the presumed end of the
publishing project; but a further commutative algebra
fascicle was produced in 1998.
While several of Bourbaki's books have become standard references
in their fields, some have felt that the austere presentation makes
them unsuitable as textbooks. The books' influence may have been at
its strongest when few other graduate-level texts in current pure
mathematics were available, between 1950 and 1960.
Notations introduced by Bourbaki include: the symbol \varnothing
for the
empty set and a
dangerous bend symbol, and
the terms
injective,
surjective, and
bijective.
It is frequently claimed that the use of the
blackboard bold letters for the various sets
of
numbers was first introduced by the group.
There are several reasons to doubt this claim.
Influence on mathematics in general
The emphasis on
rigour may be seen as a
reaction to the work of
Henri
Poincaré, who stressed the importance of free-flowing
mathematical intuition, at a cost of completeness in presentation.
The impact of Bourbaki's work initially was great on many active
research mathematicians world-wide.
It provoked some hostility, too, mostly on the side of
classical analysts; they approved of
rigour but not of high abstraction. Around 1950, also, some parts
of
geometry were still not fully axiomatic
— in less prominent developments, one way or another, these were
brought into line with the new foundational standards, or quietly
dropped. This undoubtedly led to a gulf with the way
theoretical physics is practiced.
Bourbaki's direct influence has decreased over time. This is partly
because certain concepts which are now important, such as the
machinery of
category theory, are
not covered in the treatise. The completely uniform and essentially
linear referential structure of the books became difficult to apply
to areas closer to current research than the already mature ones
treated in the published books, and thus publishing activity
diminished significantly from the 1970s. It also mattered that
while especially
algebraic
structures can be naturally defined in Bourbaki's terms, there
are areas where the Bourbaki approach was less straightforward to
apply.
On the other hand, the approach and rigour advocated by Bourbaki
have permeated the current mathematical practices to such extent
that the task undertaken was completed. This is particularly true
for the less applied parts of mathematics.
The
Bourbaki seminar series founded
in post-WWII Paris continues. It is an important source of
survey articles, written in a prescribed,
careful style. The idea is that the presentation should be on the
level of absolute specialists, but for an audience which is
not specialized in the particular field.
The group
Accounts of the early days vary, but original documents have now
come to light.
The founding members were all connected to
the Ecole Normale
Supérieure in Paris and included
Henri Cartan, Claude Chevalley, Jean Coulomb, Jean
Delsarte, Jean Dieudonné,
Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil. There was a preliminary
meeting, towards the end of 1934.
Jean
Leray and
Paul Dubreil were present
at the preliminary meeting but dropped out before the group
actually formed. Other notable participants in later days were
Laurent Schwartz,
Jean-Pierre Serre,
Alexander Grothendieck,
Samuel Eilenberg,
Serge Lang and
Roger
Godement.
The original goal of the group had been to compile an improved
mathematical analysis text; it
was soon decided that a more comprehensive treatment of all of
mathematics was necessary. There was no official status of
membership, and at the time the group was quite secretive and also
fond of supplying disinformation. Regular meetings were scheduled,
during which the whole group would discuss vigorously every
proposed line of every book. Members had to resign by age 50.
The atmosphere in the group can be illustrated by an anecdote told
by Laurent Schwartz. Dieudonné regularly and spectacularly
threatened to resign unless topics were treated in their logical
order, and after a while others played on this for a joke.
Godement's wife wanted to see Dieudonné announcing his resignation,
and so on one occasion while she was there Schwartz deliberately
brought up again the question of permuting the order in which
measure theory and
topological vector spaces were to
be handled, to precipitate a guaranteed crisis.
The name "Bourbaki" refers to a French general
Charles Denis Bourbaki; it was
adopted by the group as a reference to a student anecdote about a
hoax mathematical lecture, and also possibly to a statue. It was
certainly a reference to
Greek
mathematics, Bourbaki being of Greek extraction. It is a valid
reading to take the name as implying a transplantation of the
tradition of
Euclid to a France of the 1930s,
with soured expectations.
Appraisal of the Bourbaki perspective
The
underlying drive, in Weil and Chevalley at least, was the perceived
need for French mathematics to absorb the best ideas of the
Göttingen school, particularly Hilbert and the modern algebra school of
Emmy Noether, Artin and van
der Waerden. It is fairly clear that the Bourbaki point
of view, while
encyclopedic, was never intended as
neutral. Quite the opposite: it was more a question of
trying to make a consistent whole out of some enthusiasms, for
example for Hilbert's legacy, with emphasis on formalism and
axiomatics. But always through a transforming process of reception
and selection — their ability to sustain this collective, critical
approach has been described as "something unusual".
The following is a list of some of the criticisms commonly made of
the Bourbaki approach:
Furthermore, Bourbaki make only limited use of pictures in their
presentation. In general, Bourbaki has been criticized for reducing
geometry as a whole to
abstract algebra and
soft analysis.
Dieudonné as speaker for Bourbaki
Public discussion of, and justification for, Bourbaki's thoughts
has in general been through
Jean
Dieudonné (who initially was the 'scribe' of the group) writing
under his own name. In a survey of
le choix bourbachique
written in 1977, he did not shy away from a hierarchical
development of the 'important' mathematics of the time.
He also wrote extensively under his own name: nine volumes on
analysis, perhaps in belated
fulfillment of the original project or pretext; and also on other
topics mostly connected with
algebraic geometry. While Dieudonné could
reasonably speak on Bourbaki's encyclopedic tendency, and tradition
(after innumerable frank
tais-toi, Dieudonné! ("Hush,
Dieudonné!") remarks at the meetings), it may be doubted whether
all others agreed with him about mathematical writing and research.
In particular Serre has often criticised the way the Bourbaki works
were written , and has championed in France greater attention to
problem-solving, within
number theory
especially, not an area treated in the main Bourbaki texts.
Dieudonné stated the view that most workers in mathematics were
doing ground-clearing work, in order that a future
Riemann could find the way ahead
intuitively open. He pointed to the way the axiomatic method can be
used as a tool for problem-solving, for example by
Alexander Grothendieck. Others found
him too close to Grothendieck to be an unbiased observer. Comments
in
Pal Turán's 1970 speech on the
award of a
Fields Medal to
Alan Baker about theory-building
and problem-solving were a reply from the traditionalist camp at
the next opportunity , Grothendieck having received the previous
Fields Medal
in absentia in 1966.
Bourbaki's influence on mathematics education
In the longer term, the manifesto of Bourbaki has had a definite
and deep influence. In secondary education the
new math movement corresponded to teachers
influenced by Bourbaki. In France the change was secured by the
Lichnerowicz Commission.
The influence on graduate education in pure mathematics is perhaps
most noticeable in the treatment now current of
Lie groups and
Lie
algebras. Dieudonné at one point said 'one can do nothing
serious without them', for which he was reproached; but the change
in Lie theory to its everyday usage owes much to the type of
exposition Bourbaki championed. Beforehand
Jacques Hadamard despaired of ever getting
a clear idea of it.
See also
Notes
- Confronted by the task of appraising a book by N. Bourbaki,
this reviewer feels as if he were required to climb the Nordwand of
the Eiger. The presentation is austere and monolithic. The route is
beset by scores of definitions, many of them apparently
unmotivated. Always there are hordes of exercises to be worked
through painfully. One must be prepared to make constant
cross-references to the author's many other works. [1]
- ...by 1958 when the original six books were completed, the
first few of these books were already almost 20 years out of
date. [2]
- (1) the symbols do not appear in Bourbaki publications (rather,
ordinary bold is used) at or near the era when they began to be
used elsewhere, for instance, in typewritten lecture notes from
Princeton University (achieved in some cases by overstriking R or C
with I), and (an apparent first) typeset in Gunning and Rossi's
textbook on several complex variables; (2) Jean-Pierre
Serre, a member of the Bourbaki group, has publicly inveighed
against the use of "blackboard bold" anywhere other than on a
blackboard.
- Bourbaki came to terms with Poincaré only after a long
struggle. When I joined the group in the fifties it was not the
fashion to value Poincaré at all. He was old-fashioned. Pierre
Cartier interviewed by Marjorie Senechall. [3]
- Ian Stewart: Mathematicians knew how to decode Bourbakist
messages, but the rest of the world didn't. This led to unfortunate
misunderstandings, and by the end of the sixties, mathematics and
physics departments were no longer on speaking terms.
- Borel (1998)
- Chevalley in Guedj (1985)
- The minutes are in the Bourbaki archives — for a full
description of the initial meeting consult Liliane Beaulieu in the
Mathematical
Intelligencer.
- This resulted in a complete change of personnel by 1958; see
Robert Mainard paper cited below. However, the Aubin paper cited
below quotes the historian Liliane Beaulieu as never having found
written affirmation of this rule.
- Charles Denis Bourbaki fought in the
Crimean War and
Franco-Prussian War, refer to A. Weil:
The Apprenticeship of a Mathematician, Birkhäuser Verlag 1992, pp
93-122.
- It is said that Weil's wife Evelyne supplied Nicolas.
( Mentioned by McCleary (PDF). This is more or less
confirmed by Robert Mainard((PDF), a long article in French, which
gives numerous further details: why N?, and the prank lecture of
Raoul Husson in a false beard that gave rise to Bourbaki's
theorem). They married in 1937, she having previously been
with de Possel; who then unsurprisingly left the group.
- Hector C. Sabelli, Louis H. Kauffman, BIOS (2005), p.
423.
- Pierre Cartier, a Bourbaki
member 1955–1983, comments explicitly on several of these points (
The Continuing Silence of Bourbaki, article from
the Mathematical Intelligencer): ...essentially no
analysis beyond the foundations: nothing about partial differential
equations, nothing about probability. There is also nothing about
combinatorics, nothing about algebraic topology, nothing about
concrete geometry. And Bourbaki never seriously considered logic.
Dieudonné himself was very vocal against logic. Anything connected
with mathematical physics is totally absent from Bourbaki's
text.
- This is one of the reasons for diminishing influence: Le
développement des mathématiques dites appliquées, de la statistique
et des probabilités, des théories liées à l'informatique a diminué
l'influence de Bourbaki[4]
- Tim Gowers
discusses at length the distinction between mathematicians who
regard their central aim as being to solve problems, and those who
are more concerned with building and understanding theories in
his The Two Cultures of Mathematics
(PDF).
- Lennart Carleson spoke of this in an
interview ( Infomat August 2006 (PDF)): ...that book
[from 1968] was written mostly as a way to encourage the teachers
to stay with established values. That was during the Bourbaki and
New Math period and
mathematics was really going to pieces, I think. The teachers were
very worried and they had very little backing.
- Heinz König: The traditional abstract measure theory which
emerged from the achievements of Borel and Lebesgue in the first
two decades of the 20th century is burdened with its total
limitation to sequential procedures and its neglect of regularity.
The alternative theory due to Bourbaki which arose in the middle of
the century was able to relieve these burdens, but produced new
ones. In particular its fundamental turn to inner regularity, based
on the profound role of compactness, was done with the
inappropriate weapons from the outer arsenal, which subsequently
enforced that unfortunate construction named the essential one. All
this produced serious obstacles against a unified theory of measure
and integration, for example for the notion of signed measures, the
formation of products and for the representation theorems of
Daniell-Stone and Riesz types.[5]
- Discussed by the set theorist Adrian Mathias ( The Ignorance of Bourbaki (PDF)). See also
Mashaal (2006), p.120, "Lack of interest in foundations".
- Pierre Cartier, in the article cited above, is quoted as later
saying The Bourbaki were Puritans, and Puritans are strongly
opposed to pictorial representations of truths of their
faith.
- In the French context it has been said that geometry was in
effect exiled from secondary teaching: Pour ce qui est des
années 1960, l’effet de la réforme dite des mathématiques modernes
sur l’enseignement de la géométrie est bien connu : si Dieudonné,
comme Bourlet finalement, lance "A bas Euclide", le résultat n’est
pas l’élaboration d’une géométrie plus expérimentale, plus
intuitive. C’est l’effacement de la géométrie derrière l’algèbre
linéaire et la quasi-disparition de l’enseignement de la géométrie
élémentaire au collège et au lycée pour une dizaine
d’années.—"As for the 1960s, the effect of this reform of
modern mathematics on the teaching of geometry is well-known: if
Dieudonné, like Bourlet finally, says "push Euclid back," the
result is not the development of a geometry that is more
experimental, more intuitive. It's the erasure of geometry behind
linear algebra, and the quasi-disappearance of the teaching of
elementary geometry in high school, for ten years."[6]
- On the Work of Alan Baker
- Mashaal (2006) Ch.10: New Math in the Classroom
References
External links