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

The Café
Karen V. H. Parshall"
A Parisian Café and Ten Proto-Bourbaki
Meetings (L934-L935)
Liliane Beaulieu
Capoulade came from Auvergne. Like many of his fellow
countrymen, he migrated to Paris where he became
a café owner. From the nineteenth centLrry, bou
gnafsl abounded in Paris, where modest neighborhood
cafés often doubled as coal stores. ln the interwar period,
the café business flourished, despite the fickleness
of the French economy. Some Auaergnafs opened
or bought grandes brasseries, in Montparnasse or Saint-
ments of Montparnasse or Saint-Germain.' Its clientele
also included students, professors, publishers, and
people who worked in the Latin Quarter where the
Sorbonne, the École Normale Supérieure, the Ëcole
Polytechnique. the Ecole des Mines, many scientific
laboratories, and several renowned lycées were concentrated.
At Capoulade, a group of mathematicians met
inforrnally but regularly during the academic year
3
See Herbert R. Lottmann, La Ril',egar.fts, Points, Paris: Seuil (1981).
Cermain-des-Prés, places such as La Coupole, Le
Dôme, La Brasserie Lipp, Le Café de Flore, and Les
Deux Masots. These became world-famous institutions
when politicians, artists, actors, writers, and scientists
from all over France, Europe, and the United
States poured into Paris. They tasted Ia bohème or partook
of the intense intellectual life which the city
seemed to spawn. With Georges Braque, Pablo Picasso,
André Breton, and Paul Éluard, among many others,
Liliane Beaulieu
setting the trends in arts and letters, Parisian cafés served
as the breeding grounds of their followers.
Less visible, but not less crowded, were the cafés,
tabacs, and. brasseries of the Latin Quarter, establishments
Iike Balzar, Capoulade. Làcipière, and Mahieu.2
Capoulade's café was located at 63 boulevard Saint-
Liliane Beaulieu wrote her dissertation on the work done by
Michel, at the corner of Soufflot, near the Panthéon
Bourbaki between lq34 and 1944. After receivins hir
and the Luxembourg Gardens. It was the regular Ph.D. from lhe Lniversité de Montreal, .he .pent i vear
ha,Jnt of littersti who preferred to meet in its basement and a hall as a postdoctoral lellow at the University of
room rather than frequent the glamorous establish- California, Berkeley. She is currently writing a book on
Bourbaki's first 20 years. Her other rcsearch projects include
a studv of the controversies that surrounded the
'
The people from Auvergne, as well as their shops, were commonly foundation o? the M athemqtical Reaie:aos and an analysis of
called àolgnafs.
shifts in the conceptual settings of linear algebra and its
2
When they were not câlled "café de . . - ," Pa sian establishments teaching in North American universities. She counts sing-
were often refeûed to by their owner's patronym.
ing, cross-country skiing, gardening, and cooking among
* Column Editoi's address: Departments of Mâthematics and His- her hobbies.
tory, University of Virginia, Charlottesville, VA 22903 USA.
THE MATHEM^TIC^L I^'TEI-I-ICENC! VOL 15, NO I O 1993 SprinAe.-V€rlas New York 27

I
1934-35. The loose circle gradually formed a real working
team and later became famous for its rnanyvolumed,
perpetually revise d treatise entltled ÊIéments
de Mathêmatique,a which has been published under the
pseudonym "Nicolas Bourbaki" from 1939 to the present.
The First Meeting and Its Context .
December 70, 19U. At the sacred hour of noon, Capoulade
is filling up. Henri Cartan, Claude Chevalley,
fean Delsarte, Jean Dieudonné, René de Possel, and
André Weil meet----over a lunch of cabbage soup and
grilled meats served with endùtes braisées or Wmmes
" The singular was fâvored over the usual plulal. BouÈaki chose this
title around 1938. The pseudonlan was chosen in the summer of
1935.
soufflées-to exchange views on a new proiect.s Weil
opens the discussion with a general but firm statement
which maps out the perspective. The goal of the venture
will be "to define for 25 years the syllabus for the
certificate in differential and integral calculus by writing,
collectively, a treatise on analysis. Of course, this
treatise will be as modern as possible."6 All are ardently
in favor. The principle of collective writing
makes good sense. Because analysis touches on such
di{ferent specialties, the work of many people will allow
better and more thorough coverage. Delsarte is
particularly insistent on this issue, but he also realizes
that by sharing the writing the group will leave no
" The reports on the meetings, which were used as a prirnary soutce
for this account, are analyzed and described extensively by Liliane
Beaulier lBourbaki. Une histoie du gtoupe de mathélruticiefis fronçik et
(k ses trqLvux, 1934-1944, 2 vols. (Ph.D. dissertation, Université de
MontÉal, 1989)1.
6
Translation by the âuthor.
"Café, grill-room, A. Capoulade," 63 boulevard Saint-Michel, comer of Soufflot, Paris (1934). Until recentlv, this café was
a m€eting Point for the artiets, students, professors, publishers,_and employees who haunted the neighborÉood where the
Sorbonne, Ecole Normale Supérieure, École Polytechnique, École des l\iines, scientific laboratories, and h1cées wete
co,ncenttated, Inside the café, trdo rooms accommodaled pahons for meals and drinks. The basement room wasïften used
for meetings. In the unrestrained atmosphere oI the Latin Quarter and free from the constraints of the university, it served
the needs oI the proto-Bourbakis. Copyright Agence Roger-Viollet, photo Boyet.
28 rrc verruueTlcel TNTELUGENCER vol. rs. No. r. ree3

At the sarne address now stands a fast-food restaurant in the "QuiclC' chain, The population of the Latin Quarter, especially
,ycéestudents, still constitutes part of the clientele, but the style of the restaurant does not lend itself to lengthy and intense
meetings. Good old "côte" wine is no longer served, and the basement room now houses arrays of huge freezers. Like a
reminder of the int€rwar years, th€ poster on the right in the foreground advertises an exhibition of work; by André Breton
ar.d his cafés-habit tés, surrealist companions. Photo by Liliane Beaulieu, June 1991.
traces of individual authorship and mav therebv safeguaïd
against future claims to intelectual prcpêrq.7
In the ensuing discussion, all but one participant
agree that the treâtise should be geared to teaching
rather than reference. Someone mentions that it
should cover about 1000 pages. Delsarte urges that the
treatise should appear quickly, within 6 months, so as
to assure surprise. Consequently, they agiee to set a
definitive table of contents by the summer of 1935,
when a general meeting will convene to parcel out the
various writing tasks.
At the time of the fust meeting, "differential and
integral calculus" and "analysis" were nearly synonymous
in French mathematics instruction. The courses
offered to students who registered for the licence in
mathematics changed very little in the twenties and
the thirties. In Paris, thev chose from: differential and
7 These thoughts of Delsarte were not lecorded in the report on the
fust meeting. Most likely they were expre$ed at some other time.
They were recounted by his colleagues, Cartan and Weil. See Henri
Cartan, SurJean Delsarte, in "Hommâge à Jeàn D;lsarte," Nicftifutsll
Bunka 25 (1970\, 27-&.
integral calculus, advanced geometry, râtional mechanics
(or analytic mechanics and celestial mechanics),
application of analysis to geometry, group theory
and calculus of variations (or function theory and the
theory of transformations), probability and mathematical
physics, and general physics.8 Another course,
mathématiques générales, was a prerequisite for physics
and mathematics students alike. Each course and its
examinations constituted a "certificate" and three such
certificates were required for the /icence degree.e A provrnclal
faculté des sciences oflered. fewer choices. Everywhere,
though, "calculus" was the core of the mathematics
curriculum, and so it was a traditional Dractice
for professors to write a textbook on analysis.
-
The French Cours d'analyse belonged to a particular
textbook genre. Some of these, like Édouard Goursafs
Cours d'analyse mathématiqu3 and Jacques Hadamard's
Cours d'analyse professé à I'EcoIe Polytechnique, sternrned
3
Course announcements, Faculté des sciences de Paris, 1920-1939,
Archives nationales, Box 61AJ161.
e
The licence was then roughly equivalent to the upp€r levels of underqraduate
instruction in the befter American univergities.
THE MATHEMATTCAL tNrËLucENcER vol. 1s, No. 1, r99o 29

F
essentially ftom and were written for the dassroom.lo
Others, like Camille Jordan's Cours d'analyse de l'Ê,cole
Polytechnique (despite its title) and Émile Picard's Tralté
d'analyse,,were more elaborate, true general analysis
treatises. " These books integrated relatively recent results
together with some of their authors' own mathematical
work.
The Parisian professorial lectures dominated French
mathematics instruction, and some consensus had de-
Many physicists who use methods of integration and operations
on series obtain exact numerical tesults but are
convinced meanwhile that they are committing some
mathematical heresies. This stems from the fact that, in
most analysis treatises, fundamental theorems, such as
methods of calculation, existence theorems, etc., are introduced
with a rather impressive wealth of precautions.
These theorems contain an overabundance ofÏl'potheses.
ln many cases,,it wilJ be necessary for us to reconsider
such theorems."
veloped among university teachers ihroughout the
country as to which textbooks to adopt. Thus, not
atypically, Goursa/s book had a long shelflife, guiding
French mathematicians in the "calculus" even in the
1950s. Although a system as centralized as the French
Unioercité network ostensibly fostered neither change
nor originality it did, however, support them in actual
practice. Instructors, and especially professors, were
free to teâch their subiect as they wanted. So, in conbast
to the ossified, albeit remarkable teachings of immortal
masters, the instructors and professois in the
different facultés des sciences often tâught from their
own notes. Provincial university teachers mav have
enjoyed even greater freedom in this regard than their
more established Parisian counterparts. The proto-
Bourbakis intended to make full use of that freedom.
Judging by the contents of Goursafs book, a French
course on differential and integral calculus typically
drew from among the following topics: differentiaton,
integration (definite and indefinite integrals, multiple
integrals), series approximations of functions, geometry
(envelopes, curves, surfaces), functions of a complex
variable, analytic functions (Cauchy's theory, holomorphic
functions, analytic continuation, functons
of several variables), solution of differential equations
(etstence theorems, linear and nonlinear differential
equations), partial differential equations (first-order,
Monge-Ampère, linear, elliptic, harmonic), integral
equations (solution by approximations, Fredholm's
theorems, applications), and introduction to the calculus
of variations. The group assembled at Capoulade
did not object to these topics in and of themselves. Its
main qualm about Goursat's book and other current
Cours d'analyse was that they misled their readers, especially
physicists, on the nature of mathematical
ngor.
As the Committee sees it, special conditions do not
necessarily yield more rigorous results. In the Coars
d'analyse, theorems tended to appear several times in
the text, each time with an added set of hvpothes€s.
Thus, Cauchy's theorem was followed by boursat's
version of it, and Stokes's theorem received similar
beatment. The group at Capoulade wants to avoid this
sort of repetition and to present material in a more
general and modern setting.l3
Indeed, they want their h€atise to be "as modem as
possible"-with emphasis on the word "modem"-in
contrast to a type of knowledge which they deem outmoded.
But what does "as modern as possible" really
mean? Where should they start? What topics should
they include? These questions occupy them hom soup
to coffee. Weil states that no topic should be eliminated
a priori, whereas Cartan wonders whether it is appropriate
to include algebra in an analysis treatise. As far
as level is concemed, Cartan suggests that the equivalent
of mathématiques générales be assumed- His colleagues
protest: "We should start from saatch!"
Soon after, they argue about which topic should
come first: functions of real or complex variables?
Whereas a majority favors treating the real before the
complex case, all promptly agree with Weil and Chevalley
that it would be best to introduce algebraic aspects
of the theory of complex functions fi$t. Delsarte
points out that, more generally, the treatis€ should
stârt with an abstract and axiomatic exposition of some
general but essential notions such as field, operation,
set, and group, as in van der Waerden's boôk.Ia The
group agrees and informally calls this introductory section
the "abstract package lpaquet abstraitl." They further
concur that it should be kept to a minimum, however,
because notions can always be introduced later,
as the needs of exposition demand. Next, they argue
10
Édouard Goursat, Co urs d'ûfiollse mathématiqxe, 3 vols., Paris: Gauthier-Vilars
(1902-1914 (a fifth edirion of the third volume appeared
in 1956); A fj|sf cou6e it nqlhemûlical anallsis {Earle Rayrnond Heddck,
trans.),3 vols., Boston-New York: Ginn & Co. (190+1914 (a
Dover edition of this tnnslation appeared between 1959 and 1964);
and Jacques Hadamâr d,, Cours d'analyse prolessé à I'Êale Pofutechniqte,
Paris: Hermânn (1927). See also Hadamard's preface wherc he points
out whi(h extra material he deemed useful io add.
rr
Camille Jordan, Cours d'arulyse de l'É,cole Polytechnique, 3 vols.,
Paris: Gauthier-Villars (1882-1884 (a third edition oI volume three
appeared in 1915); and Émile Picard, Traité d'analuse,3 vols., Parisl
Cauthier-Villa$ (1891-1896) (a rhird edition of rh; rhird volume appeared
in 1928).
30 rne vetttpuerrcat- INTELLIcENCER vol. 15. No. r.1993
12 Translation by the author. These were repoltedly Weil's own
words.
13
Related in André Wetl, Oeuores scientifiqueÊ---Collected Popefi, Vol.
1, New York Spdnger-Veilâg (1980), 563j a^d Souoenirs d'aryrentissage,
Vita mathematica, Basel-Boston Berlin: Birkhàuser (191), 103_
104. The latter work appeared in English translaiion as The Apprenticeship
of a Mathenaticiar, Boston: Bitkhâuser (1992). A discussion of
Bou:rbaki's treatment of Stokes's theorem appears in Liliâne Beau-
lieu, Proofs in expository writing: Some eximples from Bourbaki,s
early worlr, lntcrchange 2l (19901, 3145, on pp. 36-38.
'"
Bârtel L. van der Waerden, Moderne Algebra, 2 vols., Berlin:
Springer-Verlag (1930-1931).

over the starting point: sets or operations and fields?
They close the meeting unresolved. Weil convenes another
meeting for rnid-]anuary, same time, same place.
Each participant is told to bring along a list of iopics
which he thinks should appear in the treatise.
As it set out to write its modern analogue of the
Cours d'analyse, the Committee already knew where it
wanted to publish its work-with Hermann in its collection
entitled "Actualités scientifiques et industrielles."
Gauthier-Villars, the official acâdemic publisher
The "Committee on Analysis" and
Its Participants
of mathematics at the time, was out of lhe quàsdon for
them. Some of the most conservative French mathematcians
dominated its editorial board, iust as they
controlled every other institutional body in the field. In
contrast, Hermann stood somewhat on the frinee. It
was a small, independent firm directed by an e-nterprising
and eccentric Latin Quarter character, Enrique
Freymann, who was always willing to venture into
new proiects regardless of how financially unsound
they might appear.
Respectable mathematical texts, such as Éte Cartan's
Lryons sur les inoariants intégraux and Hadamard, s
Cours d'analyse had already beèn publshed by Hermann.rs
Fuithermore, sevéral members of the Committee
had recently had fust-hand experience with the
publisher and its "Actualités." Soon àfter Lhe early dernise
of their friend and colleague. Jacques Herbiand,
Chevalley and Weil decided to publish a memorial volume
of papers. They brought their proiect to Freymànn,
and th,e articles soon appeared as a series in the
"Actualités."'6 As a collection, the ,,Actualités,,
The inJormal circle first called itself the
had
originated in 1929 and published monographs in the
form of rather homely, paper-bound booklets. It encompassed
a variety of series on science, some of
which pertained to mathematics. Each series was
headed by a dhector, who had full reign over content,
qualty, and quantity of publication. For instance, Élie
Cartan led a series on geometry, Maurice Fréchet
headed one on general analysis, and Hadamard edited
another on mathematical analysis and its applicâtions.
A given series had limitations neither on duration nor
on number of publicatons. This flexibility welcomed
innovations and encouraged autonomy. Freymann
and his "Actualités" thus provided the Capoulade
group with the full editorial freedom it sought.l7
,,Committee on
Analysis" or "Committee for the Analysis Treatise.,,
Delsarte apparently decided simply to wdte the title
"Traité d'analyse" at the top of the reports on the
meetings, although René de Possel once insisted that it
should be changed to "Treatise on Mathematics.,, No
one else supported this motion, and the title-page of
the minutes remained the same for some time. -
The notion of membership in the group was not
clearly defined either. At the second meeting, the
Committee decided to limit its numbers to the fàllowing
nine pârticipants: Cartan, Chevalley, Delsarte,
Dieudonné, Paul Dubreil, Jean Leray, Szolem Mandelbrojt,18
de Possel, and Weil. Dubré attended orùy a
couple of meetings, and in May, Jean Coulomb, a
physicist with â strong mathematical background, replaced
him. Leray officially stayed on until the summer.
Although he did not attend very regularly, he
wrote out some of the more elaborate plans which occupied
ihe Committee for several meetings. Later,
when too few members showed up to discuss differentiàl
equations, a quorum became essential. It was
only in the summer of 1935 that the Committee
adopted the pseudonym "Bourbaki.,, The participants
who were present at that general meeting declared
thernselves "official members" and in so doing finally
defined membership.le
From time to time, the members informallv invited
others to join the discussions as guests o, âdrriro.".
Thus, Emil Ariin attended a meeting of the Committee
as a guest while visiting Paris Érom Hamburg in February
1935. His presence did not seem to influlnce the
meeting, however: a five-line-long outline for ,,set theory"
went undiscussed, and measure and inteqration
was the topic of conversation. To provide thé extra
manpower needed to tackle di{fereniial equations, the
Committee invited the physicist, Yves Roiard,2o as an
advisor to inform them on what would be especiallv
useful to physicists. Although Rocard's suggestions
were labeled "the physicists' desiderata,,, they seemed
mostly related to vibration and stability problerns, and
they did not convey a general picture ôf what physicists
at the time might have needed as mathematical
13
Szolem Mandelbrojt is an uncle of the mathemahcian Benoit Mandelbrot
of ftactals fame.
D
Charles Ehresmann was asked to join the group in place of Leray.
As a point of nomenclature, I use the term ,,ploto-Bourbaki,, to
designàte a pàrticipant in the early Committee. A ,Bourbaki,, 15
Élie Cartan, Leçons sur les h1L\;iriants intéÙraut, pads: Hernann
(1922). lor the relerence to Hadamard's texf, see footnote 10.
16
Claude Chevalley and André Weil (eds.), Êxwsés matb/natiql)es
publiës à 1û mémoirc de I. Herbrand, Paris: Hermann (t934-lca5). This
was an intemational project involving French a5 wejl às foreign conlributors.
Among the French contributors, Henri Cafiàn, Delsarte.
is
Dieudonné,
a
Dubreil, and Weil were also among the proto- member of the group Bou.baki and is the official designation coined
Bourbakis. The papers presented in this sedes are listea in Be;uiieu, b), the gloup itself. A "Bourbakist" is a follower ot B-ourbaki.
Boufuaki. Unc Histoirc, Vol.2, pp.66a7.
20
17
Rocard was in the same class as Delsalte and Weil at the École
See Beaulieu, Bourbaki. tJne Histoire, Vol. 1, pp. 13g-140 and 142- Normale Supérieure and is known for his work on the A-bomb and
148. On En-rique Frefmann ând his publishing house, consult Weil, in the theory of vibrations. He is the fâther of Michel Rocard,
Souoenirs
a
d'aryreùis,age, pp. 102-109 (footnote 13).
French socialist politician and recent Prime Minister of Fnnce.
THE MÀITIEMÀTICAL INTELLIGENCER VOL. 15, NO. T, 1993 31

l'
tools. Although Rocard's suggestions were tabled until men formed an elite in a highly hierarchical systemi
some future rneeting, some of them ultimately found a they were the designated aspirants to the leadership of
p_lace in one of the projects on differential equations. French mathematics.
Elie Cartan was also called in----at the tenth meeting- Paris, of course, was the best place for them to meet.
to help out the Comrnittee on integral equations, and They frequently went to the iapital city to keep in
although no formal decision was reached at that time, touch with its scientific life, to visit fellow mathemati-
like Rocard's, some of his advice was also later heeded. cians, and to frequent bookstores and libraries. Most of
Born between 1899 and 1909, the regular participants them also commuted twice a month, on Mondavs, to
were graduates of the École Normale Supeileurefwith take pârt in a Sèminaire de mathématiques, which Àet at
the exception of Mandelbrojt who received rnost of the recently built Institut Henri-Poincaré. They usually
education in Poland and came to Paris for his doctor- held their Committee meetings in the hours bêfore the
ate. They shared a corunon background, but did not semlnar.
know or use the same mathernatical tools, and they did Gaston lulia, professor at the Paris faculté des sciences,
not even work in the same specialties. Many of them officially convened this seminar which was primarily
had traveled abroad after their doctorate or while do- organized and animated by a handful of devbtees. Àt
ing their doctoral research, as fellows of some granting the beginning of each year, the organizers would
agency, the most generous of which was the Rockefel- choose the theory which was going to be the topic of
ler Foundation. Although their hosts were not always the year, and draft a list of possible talks and speakers
real mentors to them, those fellows who spent re- (usually recruited from their own ranks). The
search time in Denmark, Germany, Hungary, Italy,
Sweden, Switzerland, and the United States became
acquainted with areas of research, methods, and resources
which were not common in their native
France. Partly as a result of their contacts with foreign
mathematicians in France or in other countries, some
participants were more familiar with set-theoretical,
axiomatic, algebraic, or topological methods. Others
remained closer in their work to function-theoretical
questions and analytic methods on which the reputation
of French mathematics had been built. Their ëommittee
discussions reflected this heterogeneity.2l
As mathematical researchers, the proto-Bourbakis
had each already published several papers.22 Furthermore,
they were the recipients of fellowships, prizes,
and other distinctions. In fact, most of these men received
several of the prizes of the Académie des Sciences
de Paris early in their careers.23 As professors,
they had also followed the then current pâttern of taking
a position in a French prov'rncial t'nculté des sciences
before seeking a call to Paris. In 1934-1935, Henri Cartan
and André Weil taught in Strasbourg, Delsarte and
Dubreil were in Nancy, Dieudonné had a position in
Rennes, and Mandelbroit, de Possel and Coulomb in
Clermont-Ferrand. Chevalley and Leray were grantees
of the Caisse nationale that yéar.2a From their pôvincial
vantage points, they had the freedom to experiment
with new ideas and fresh approaches which might
have been less accepted in Paris. Nevertheless, these
,Julia
seminar," as it was called, studied the following topics
between 1933 and 1939: group theory and algebras,
Hilbert spaces, topology, the works of Élie Cartan, algebraic
functions, and calculus of variations. Its obiective
was neither to serve as a teaching seminar nor to
inform the audience on current liteàrure. Rather, it
offered, to its speakers as well as to its audience, an
opportunity to work through large parts of recently
developed theories and methods. It forced the speakers
to synthesize in{ormation which was otherwise
scattered in the literature. Since the seminar had virtually
no official institutional ties, it provided a free
forum for criticism. There, the members of the Committee
eventuâlly assimilated, together, the approaches
which they later tried to integrate into their
collective expository writng.25
The Committee had neither money nor a set administrative
structure. Delsarte acted as "secretary" or
"manager." He wrote up minutes and sent around
reminders, but he had no particular powers. A de facto
hierarchy developed quickly, however, based on a
mixed measure of mathematical excellence and exoertise,
intellectual sophistication, strength of voice, ànd
determination. Indeed, the meetings of the Committee,
like the later gatherings of Bourbaki, reportedly
took place amid great noise and confusion. This should
come as no surprise, given that a Latin Quarter café
served as their laboratory. Still, the ambience of the
Latin Quarter of the mid-30s-with its strikes, demonstrations,
and political street fights-did not totally
dominate the group's meetings. As a mathematical
2r
On trips and traveling fellowships, see Beaulieu, Bourbaki. Ilne team, it largely managed to remain staunchly apoliti-
Histoile, YoL 1, pp. 69-105.
2
On average, the proto-Bourbaki had published 23 papers, with a
minimum of 4 and a mayjmuin of 75, according to a rough count.
æ
On prizes and other distinctions, see Beatljiu, Boutbaù. LIne Histoirc,
Vol. 1, rp. 87-114.
'?a
On teaching posts and provincial French frrultés iles scimces, see
Bourbaki. Une Histore, Vol. 1, pp.
5
The texts of the talks were mimeogûphed. A near complete set is
deposited at the libÉry of the Institut Henri-Poincaré in Paris. For
the list, see Bourbaki. Une Histoire, Yol. 2, pp, 63*65. On the Julia
Seminar, see Borl,rh. Ufie Histoire, Vol. 1, pp. 133_132.
'n4122.
32 TrIE MATHEMATICAL tNrELucENcER vo|-. 15. No. 1.1993

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

î'
written outlines on the subiect. From these, the Committee
drafted a proposal which attempted to place the
material in an algebraic and topological setting. It
started with the geometdc representation of cornplex
numbers and stressed that these form a field. It next
proposed to move to the topology of open and dosed
surfaces. This sketch also featured some of the more
"usual" material on analytic functions: Jordan's theorem,
series, convergence, differentiation, integration,
Cauchy's theorem and Cauchy's integral, Taylor's and
Laurent's theorems for singular points of uniform
functions, conformal representation, entire functions,
Weierstrass's theorem, Mittag-Leffler's theorem, analytic
continuation, etc. It was suggested that the general
notion of analytic function be highlighted and that
two sections be devoted to algebraic and automorphic
functions, elliptic functions, and the theta function
(the latter, most likely, with number theory in mind).
The meeting also considered whether it should introduce
analytic functions of many complex variables. It
delegated the decision making on these matters to its
subcommission.
The Committee's initial work on integration theory
looked more promising. At least, one point was certain
at the out6et: Integration would be done from Lebesgue's
point of view, which had already undergone
different levels of qeneralization and extension since
1901 . 30 kinds of measures. At any rate, the Comrnittee had
already decided to restrict its exposition on integration
theory in the treatise.
This sketch of contents resulted from discussions between
Chevalley and de Possel, who advocated a thorough
exposition on measure, and Delsarte, Dieudonné,
and Dubreil, who thought that a less elaborate
presentation of measure and integration might be
more appropriate for the treatise. Most probably, WeiI
and de Possel opposed each other also: Whereas de
Possel wanted to do measure and integration on arbitrâry
sets orùy, Weil wanted to involve vector spaces
and topological groups. The subcommission on integration
had to r.eionsid"t these different opinions.32
Delsarte and Leray were the main protagonists in
the area of differential equations. Delsarte first suggested
subdividing the study of differential equations
into three sections: existence theorems, eigenvalue
problems (crucial in physics), and the study of local
and global properties of solutions. The Committee
agreed with his choices, at least in principle. Then it
examined a draft by Leray on existence theorems. Contrary
to most other plans, this draft was not merely a
Iist of items. It was more like a brief introduction to ar
abstract theory for systems of n equations in n unknowns.
Leray introduced concepts from topology and
Although some French Cozrs d'analyse did nen-
functional analysis which he had been using in his
own work, especially the notions of the differential of
tion the Lebesgue integral (usually in passing), they a function of n variables (as a linear functional) and of
did not give an exposition of Lebesgue's theory. The the topological degree of a continuous transformation.
older texts used Cauchy's approach, and the more re- His approach set the study of differential equations
cent ones inhoduced the Riemann integral. All con- squarely in line with works by Riesz, Banach, and
centrated on developing the techniques of integrâtion Hahn on normed spaces. Leray's draft included the
and their applications. By choosing Lebesgue integra- statements of fundamental theorems of global existion,
the proto-Bourbakis were more in line with texts tence and of local existence (with uniqueness) of a so-
such as those by Constantin Carathéodory, Charles de lution. The Committee thought that, although it was
La Vallée Poussin, and Stanislaw Saks."
all very interestinç, Leny's project involved topologi-
The Committee resolved that rneasure and integracal notions which were too specialized for the treatise.
tion should not be seDarated in the presentation. It Perhaps for this reason, no final decision on differen-
drafted a list of potentiàl topics which itarted with the tial equations was reached at this point in their delib-
notion of measure and proceeded to the integral eranons.
viewed as a linear functional, stressing the equivalence At the end of its eighth meeting, the Committee it-
of the two concepts. Next followed particular types of self drafted a provisional and rather jumbled outline
measures and integrals on topological spaces, Radon for differential equations. This plan comprised general
measures, and Haar measures on topological groups. efstence theorems, global existence for reai differen-
Although order of presentation was not stressed, it tial equations over any domain where the conditions
appeared that the latter were meant not to be the pil- for local existence hold, a classical theory of general
lars of the theorv, but were introduced rather as special linear equations, systems of z first-order linear equations
in z unknowns, and second-order linear equations
with constant coefficients. Among the applications
to physics, some of Rocard's old suggestions re-
s
For an historical account oI Lebesgue's ideâs, see Thomas Hawki^s,
Izbesgue's Theory of lftte9ratiofl: lts Oigi\s afld Danloryent, Nevr
Yorkr Chelsea (1975).
31
See Constantin Carathéodory, Vorlesungen iiber reelle Funktiotutt,
Leipzig-Berlin: Teubner (1918); Charles de La Vallée Poussin, lnfégrûles
de Lebesgue. Fonctiont il'ensemble. Classes de Baire,lsl ed., Paris:
Gauthier-Villa$ (1916); 2nd ed., 1936; and Stard'slaw Saks, Théorie de
I'intéyale, Mo ogmlie matematyczne, 1st ed., Vol. 2, warsaw: Zsubwencji
funduezu kultury narodowe (1933); 2nd ed., 1937.
34 rm uerr+uercel rNrELucENcER vot-. 15, No. 1,1s93
32
These differences in opinion are discussed in Beâulieu, Bo!rù!,ti.
Une Histoîe, !ol. 1, pp. 178-188. Chevalley and de Possel put forth
their ideas in "Un théorème sur les fonctions d'ensemble complète.
ment additives," Cot?rp. tutd. Acad. Sci. Palis 197 (1933), 88H87. See
also René de Possel, "Notion générale de mesure et d'intégrale," 5émiwie
de msthénaliques llA (1934), mimeographed.

surfaced, but the conceptual setting of Leray's project
had disappeared. The Committee had adopted the
functional approach earler in its work, yet it did not
seem to envisage all the implications of this choice.
With the intention of covering a lot of ground, it delegated
work on differential equations to two subcommissions,
one responsible for existence theorems and
the other for the rest.
The Committee's work on integral equations proceeded
differently. An initial discussion, involving
Cartan, Delsarte, Deudonné, and Weil, emphasized
three approaches to the subject: bounded operators on
Hilbert spaces, Fredholm's point of view, and the more
recent line developed by Riesz and Leray (using
nomed vector spaces). In the absence of Leray, the
Committee temporarily opted for bounded operators
on Hilbert spaces, which they thought was a complete
and beautiful theory. It hesiiated over the third approach
because it was less familiar to most of those
present and postponed its decisions until Leray could
be reached. Paradoxically, the Committee was willing
to favor bounded operators, even though quantum
mechanics required unbounded operators. Thus, despite
expressed intentions to serve physicists, the
Committee was not always guided by a close eye on
what was being done in that field. Other considerations
sometimes prevailed. 33
Two meetings làter, Leray came to the rescue. He
found Fredholm's approach rather useless for the
group's purposes, although Delsarte disagreed. The
two also discussed how much should be done on
bounded operators. Leray actually suggested two
ideas for integral equations. The first one introduced
nonsymmetric integral equations as a special case of
equations of the form )c + g(x) : 0, where x is an
element of a Banach space and I is a completely continuous
operator. The second one viewed symmetric
integral equations as special cases of Hermitian operators
in a Hilbert space. The Committee adopted Leray's
approach without further ado, and so the third
point of view ultimately won the day.
For partial differential equations, Delsarte drafted a
brief outline which emphasized two aspects: local
problems (where the notion of characteristic would be
fundamental) and limit ptoblems (related to inte$al
equations). After some discussion, the Committee decided
that the study of a single fust-order partial differential
equation with z unknowns represented the
bare minimum of what needed to be covered. But what
else should be included? Élie Cartan, who attended
that meeting, suggested instead that they restrict their
s On bounded linear operators in Hilbert space, see Jean Delsarte,
"L'axiomatique des opérateuls linéaires dans l'espace de F{ilbert:
opélateurs borrrés," Sêminaire de fiothérû.atiques UC (1934), mimeo-
$aphed; and Jean Leray and J. Schauder, "Topologie et équations
fonctionnelles," Comp. Rend. Acad. Sci. Pais 197 (7933), 115f1,
study to systems of linear partial differential equations
with one unknown function. He also stressed that both
the "classical" point of view and the Pfaffian equations
should be introduced. These points were well taken,
but the group still did not commit itself on partial differential
equations .
The Committee did not intend to treat set theorv.
algebra, topology, or even integration for their own
sakes. As it stood, then, the treatise would include
only enough algebra to deal with systems of equations,
some integration theory to support analytic functions,
but mostly differential equations, integral equations,
and partial differential equations. The Cours d'analyse
exerted the main pull on the topical choices of the
proto-Bourbakis. Wanting to cast this material in a
modern setting, the Committee sought a workable
ground between approaches which otherwise appeared
either too general or too specialized. The Capoulade
group made only minimal progress toward
this goal, however, and its aim to provide mathematicians
and physicists with the tools of their trade stll
seemed far from reach. Also, the initially expressed
concem for the man in the street and the average selftaught
student lost its edge in the course of on-going
debates.
Epilogue
In choosing a café as the setting for its meetings, the
proto-Bourbakis were not particuJarly original. Indeed,
it was usual among politicians, artists, or scientists to
meet in Parisian cafés. It was also common for mathematicians,
as well as writers, to enjoy working at a
favorite bistro table. Capoulade's café was a famil.iar
rendezvous which provided freedom from strict university
institutons while being conveniently close to
all the amenities of the neighborhood. This mixture of
Latin Quarter parochiality and a taste for autonomy
typifies the proto-Bourbaki attitude. The narrative of
biweekly gatherings, however, shows disunity in the
party, hesitation in ideology, and uncertainty in purpose.
Indeed, this was a time of ill-defined goals and
unfinished business. The Committee orùy investigated
possibilities, and, despite hea*y arguments, it hesitated
to strive either to make things work or to establish
truths. It willfully postponed decisions to the general
meeting of the summer of 1935, when a definitive
overall plan of the treatise was expected to emerge.
That meeting, which took place in Besse-en-
Chandesse, started another phase in the history of
Bourbaki. Through negotiations and selections, opportunities
were foreclosed, but new options were foreseen.
DEartement de Mathématiques et d'Informatique
Untuersité du Québec à Montftal
C. P. 8888, Succursale "A"
Montrtul, Québec H3C 3P8
Cannda
lHE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1gq3 35