Karen V. H. Parshall" A Parisian Café and Ten Proto-Bourbaki ...

cal, even though some of the Committee participants
may have had leftist leanings. With ihe exception of
some discussion of and intervention in local academic
politics, the proto-Bourbakis shied away from the political
arena. In their case, then, the café served as a
public place for mathematical practice, while it stopped
short of providing a metaphor for fuller public and
political involvement.
How could the Committee reconcile these seemingly
disparate goals? In particular, how could a textbook on
analysis be used equally by mathematics students, professional
scientsts, and the man in the street (however
studious he might be)? The proto-Bourbaki meetings
found their participants groping to reconcile these different
objectives.
A partial solution to their dilemma lay in their decision,
at the first meeting, to "start from scratch." They
would open with a preliminary set of abstract and elementary
notions which would appear in the first sections
of the treatise. They quickly decided, however, to
restrict this "abstract package" to a minimum and,
during the whole semester, very little was done to produce
even that. In fact, the minimal "abstract package"
apparently ceased to be an immediate preoccupation.
Instead, two levels of questions came to the fore: globally,
which topics should go into the heatise; and
locally, for each topic considered, what material
should be covered and from what point of view?
The Committee concentrated on inventories and
outlines for its eventual analysis treatise, in no particular
order of presentation. Notions went undefined;
theorems went unstated and, of course, unproved.
Each meeting, thus, resembled a brainstorming session,
with many suggestione-but few final resultsbursting
forth. While the Committee did not concern
itself so much with an overall plan, it did discuss some
26 Translation by the author. There is a pun here on the words "chercheurc"
: "researchers" or "seekers" and "houveuts" : "finders."
of the topics to be included in the treatise.zT Of the 10
meetings, 5 were devoted mostly to differential equations,
integral equations, and partial differential equations.
These topics constituted the bulk of the material
of the old Cours d'analyse. Integration theory, analytic
functions, and a little algebra were also discussed during
that semester.
The Committee parceled out the various topics to
separate subcommissions ând mandated each to
skeich out its topic. Most subcommissions consisted of
Mathematical Shoptalk:
three people, no more than two of whom were sup-
Discussions and Sketches
posed to be specialists in the field. Subcommissions
At first, the Committee's aim seemed quite clear: to were created for the following topics: algebra, analytic
change the teaching of mathematics at the university functions, integration theory, differential equations,
level by writing a treatise on analysis. Soon, however, integral equations, etstence theorems (for differential
another purpose emerged:
equations), partial differential equations, differentials
and differential forms, topology, calculus of variations,
We must wr:ite a treatise which will be useful to all: to special functions, geometry, Fourier series and Fourier
rcsearcherc (bona fide or not), "finders," aspirants to posts
integrals,
in public
and representation
education, physicists,
of
and all technicians.
functions. There was
As a
criterion, we can say that we should (without thought of no special subcommission for set theory (which was
monetary gain) be able to recommend this treatise, or at considered part of algebra), and the subcommission for
least its most important sections, to any self-taught stu- topology was formed only belatedly.2E Altogether the
dent, presumably of average intelligence. . . . Mostly, we subcommissions had to concentrate on a lot of "hard
must provide users with a collection of tools, which
classical
should
analysis."2e But,
be as powerful and universal as possible.
of course, they had to rework
Usefulness
and convenience should be our guiciing principles." these traditional topics into a modern idiom.
The queries inspired by analytic functions especially
revealed the Committee's puzzlement over the task of
merging the tradltional and the modern. A classical
topic of ihe Cours d'analyse, function theory had been
sirongly affected by the ideas of René Baire, Émile
Borel, and Henri Lebesgue, especially in the area of
functions of real variables. The works of Lars Ahlfors,
Ludwig Bieberbach, Constantin Carathéodory, Nicolas
Lusin, and Rolf Nevanlinna further changed the field.
The Committee had its own specialists at hand: Henri
Cartan, Dieudonné, Mandeibrojt; to some extent,
Leray and de Possel worked with analytic functions as
well. When Mandelbrojt voiced the opinion that the
treatise should not overemphasize entire functions, his
colleagues asked whether it should not include such
important matters âs Picard's theorem, conformal representation,
elliptic functions, Abelian functions, infinite
products, etc. A variety of suggestions followed
and, in the resulting confusion, no decision was made.
Analytic functions reappeared on the agenda at another
meeting when the participants put together their
27
this aspect of the work of the fust semester is mentioned in Weil,
Souoefiirs d' apprentissage, pp. 109:110.
'?3
The list of topics comes from the report on the eighth meeting.
Because Alexandroff and Hopfs book on topology only appeared in
1935, the Committee did not even hav€ this reference available for
consideration of topological topics. See Paul S. Alexandroff and
Heinz Hopf, Topologie, Vol. l, Berlin: Springer-Verlag (1935).
'-
Benoit Mandelbrot criricized Bourbali---among other thing+for
hâving neglected what he termed "hard dassical analysis" in its
Ê.\émmX. See Benoit Mandelbrot, Chaos, Bourbaki, and Poincaré,
Mathematical lntelligencer, Vol. 11(1989), no. 3, 10-12.
THE MÀTHEMATcAL INTELUcENcER vol-. It No. 1,1993 33