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

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

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


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The <strong>Café</strong><br />

<strong>Karen</strong> V. H. Parshall"<br />

A <strong>Parisian</strong> <strong>Café</strong> <strong>and</strong> <strong>Ten</strong> <strong>Proto</strong>-<strong>Bourbaki</strong><br />

Meetings (L934-L935)<br />

Liliane Beaulieu<br />

Capoulade came from Auvergne. Like many of his fellow<br />

countrymen, he migrated to Paris where he became<br />

a café owner. From the nineteenth centLrry, bou<br />

gnafsl abounded in Paris, where modest neighborhood<br />

cafés often doubled as coal stores. ln the interwar period,<br />

the café business flourished, despite the fickleness<br />

of the French economy. Some Auaergnafs opened<br />

or bought gr<strong>and</strong>es brasseries, in Montparnasse or Saint-<br />

ments of Montparnasse or Saint-Germain.' Its clientele<br />

also included students, professors, publishers, <strong>and</strong><br />

people who worked in the Latin Quarter where the<br />

Sorbonne, the École Normale Supérieure, the Ëcole<br />

Polytechnique. the Ecole des Mines, many scientific<br />

laboratories, <strong>and</strong> several renowned lycées were concentrated.<br />

At Capoulade, a group of mathematicians met<br />

inforrnally but regularly during the academic year<br />

3<br />

See Herbert R. Lottmann, La Ril',egar.fts, Points, Paris: Seuil (1981).<br />

Cermain-des-Prés, places such as La Coupole, Le<br />

Dôme, La Brasserie Lipp, Le <strong>Café</strong> de Flore, <strong>and</strong> Les<br />

Deux Masots. These became world-famous institutions<br />

when politicians, artists, actors, writers, <strong>and</strong> scientists<br />

from all over France, Europe, <strong>and</strong> the United<br />

States poured into Paris. They tasted Ia bohème or partook<br />

of the intense intellectual life which the city<br />

seemed to spawn. With Georges Braque, Pablo Picasso,<br />

André Breton, <strong>and</strong> Paul Éluard, among many others,<br />

Liliane Beaulieu<br />

setting the trends in arts <strong>and</strong> letters, <strong>Parisian</strong> cafés served<br />

as the breeding grounds of their followers.<br />

Less visible, but not less crowded, were the cafés,<br />

tabacs, <strong>and</strong>. brasseries of the Latin Quarter, establishments<br />

Iike Balzar, Capoulade. Làcipière, <strong>and</strong> Mahieu.2<br />

Capoulade's café was located at 63 boulevard Saint-<br />

Liliane Beaulieu wrote her dissertation on the work done by<br />

Michel, at the corner of Soufflot, near the Panthéon<br />

<strong>Bourbaki</strong> between lq34 <strong>and</strong> 1944. After receivins hir<br />

<strong>and</strong> the Luxembourg Gardens. It was the regular Ph.D. from lhe Lniversité de Montreal, .he .pent i vear<br />

ha,Jnt of littersti who preferred to meet in its basement <strong>and</strong> a hall as a postdoctoral lellow at the University of<br />

room rather than frequent the glamorous establish- California, Berkeley. She is currently writing a book on<br />

<strong>Bourbaki</strong>'s first 20 years. Her other rcsearch projects include<br />

a studv of the controversies that surrounded the<br />

'<br />

The people from Auvergne, as well as their shops, were commonly foundation o? the M athemqtical Reaie:aos <strong>and</strong> an analysis of<br />

called àolgnafs.<br />

shifts in the conceptual settings of linear algebra <strong>and</strong> its<br />

2<br />

When they were not câlled "café de . . - ," Pa sian establishments teaching in North American universities. She counts sing-<br />

were often refeûed to by their owner's patronym.<br />

ing, cross-country skiing, gardening, <strong>and</strong> cooking among<br />

* Column Editoi's address: Departments of Mâthematics <strong>and</strong> His- her hobbies.<br />

tory, University of Virginia, Charlottesville, VA 22903 USA.<br />

THE MATHEM^TIC^L I^'TEI-I-ICENC! VOL 15, NO I O 1993 SprinAe.-V€rlas New York 27

I<br />

1934-35. The loose circle gradually formed a real working<br />

team <strong>and</strong> later became famous for its rnanyvolumed,<br />

perpetually revise d treatise entltled ÊIéments<br />

de Mathêmatique,a which has been published under the<br />

pseudonym "Nicolas <strong>Bourbaki</strong>" from 1939 to the present.<br />

The First Meeting <strong>and</strong> Its Context .<br />

December 70, 19U. At the sacred hour of noon, Capoulade<br />

is filling up. Henri Cartan, Claude Chevalley,<br />

fean Delsarte, Jean Dieudonné, René de Possel, <strong>and</strong><br />

André Weil meet----over a lunch of cabbage soup <strong>and</strong><br />

grilled meats served with endùtes braisées or Wmmes<br />

" The singular was fâvored over the usual plulal. BouÈaki chose this<br />

title around 1938. The pseudonlan was chosen in the summer of<br />

1935.<br />

soufflées-to exchange views on a new proiect.s Weil<br />

opens the discussion with a general but firm statement<br />

which maps out the perspective. The goal of the venture<br />

will be "to define for 25 years the syllabus for the<br />

certificate in differential <strong>and</strong> integral calculus by writing,<br />

collectively, a treatise on analysis. Of course, this<br />

treatise will be as modern as possible."6 All are ardently<br />

in favor. The principle of collective writing<br />

makes good sense. Because analysis touches on such<br />

di{ferent specialties, the work of many people will allow<br />

better <strong>and</strong> more thorough coverage. Delsarte is<br />

particularly insistent on this issue, but he also realizes<br />

that by sharing the writing the group will leave no<br />

" The reports on the meetings, which were used as a prirnary soutce<br />

for this account, are analyzed <strong>and</strong> described extensively by Liliane<br />

Beaulier l<strong>Bourbaki</strong>. Une histoie du gtoupe de mathélruticiefis fronçik et<br />

(k ses trqLvux, 1934-1944, 2 vols. (Ph.D. dissertation, Université de<br />

MontÉal, 1989)1.<br />

6<br />

Translation by the âuthor.<br />

"<strong>Café</strong>, grill-room, A. Capoulade," 63 boulevard Saint-Michel, comer of Soufflot, Paris (1934). Until recentlv, this café was<br />

a m€eting Point for the artiets, students, professors, publishers,_<strong>and</strong> employees who haunted the neighborÉood where the<br />

Sorbonne, Ecole Normale Supérieure, École Polytechnique, École des l\iines, scientific laboratories, <strong>and</strong> h1cées wete<br />

co,ncenttated, Inside the café, trdo rooms accommodaled pahons for meals <strong>and</strong> drinks. The basement room wasïften used<br />

for meetings. In the unrestrained atmosphere oI the Latin Quarter <strong>and</strong> free from the constraints of the university, it served<br />

the needs oI the proto-<strong>Bourbaki</strong>s. Copyright Agence Roger-Viollet, photo Boyet.<br />

28 rrc verruueTlcel TNTELUGENCER vol. rs. No. r. ree3

At the sarne address now st<strong>and</strong>s a fast-food restaurant in the "QuiclC' chain, The population of the Latin Quarter, especially<br />

,ycéestudents, still constitutes part of the clientele, but the style of the restaurant does not lend itself to lengthy <strong>and</strong> intense<br />

meetings. Good old "côte" wine is no longer served, <strong>and</strong> the basement room now houses arrays of huge freezers. Like a<br />

reminder of the int€rwar years, th€ poster on the right in the foreground advertises an exhibition of work; by André Breton<br />

ar.d his cafés-habit tés, surrealist companions. Photo by Liliane Beaulieu, June 1991.<br />

traces of individual authorship <strong>and</strong> mav therebv safeguaïd<br />

against future claims to intelectual prcpêrq.7<br />

In the ensuing discussion, all but one participant<br />

agree that the treâtise should be geared to teaching<br />

rather than reference. Someone mentions that it<br />

should cover about 1000 pages. Delsarte urges that the<br />

treatise should appear quickly, within 6 months, so as<br />

to assure surprise. Consequently, they agiee to set a<br />

definitive table of contents by the summer of 1935,<br />

when a general meeting will convene to parcel out the<br />

various writing tasks.<br />

At the time of the fust meeting, "differential <strong>and</strong><br />

integral calculus" <strong>and</strong> "analysis" were nearly synonymous<br />

in French mathematics instruction. The courses<br />

offered to students who registered for the licence in<br />

mathematics changed very little in the twenties <strong>and</strong><br />

the thirties. In Paris, thev chose from: differential <strong>and</strong><br />

7 These thoughts of Delsarte were not lecorded in the report on the<br />

fust meeting. Most likely they were expre$ed at some other time.<br />

They were recounted by his colleagues, Cartan <strong>and</strong> Weil. See Henri<br />

Cartan, SurJean Delsarte, in "Hommâge à Jeàn D;lsarte," Nicftifutsll<br />

Bunka 25 (1970\, 27-&.<br />

integral calculus, advanced geometry, râtional mechanics<br />

(or analytic mechanics <strong>and</strong> celestial mechanics),<br />

application of analysis to geometry, group theory<br />

<strong>and</strong> calculus of variations (or function theory <strong>and</strong> the<br />

theory of transformations), probability <strong>and</strong> mathematical<br />

physics, <strong>and</strong> general physics.8 Another course,<br />

mathématiques générales, was a prerequisite for physics<br />

<strong>and</strong> mathematics students alike. Each course <strong>and</strong> its<br />

examinations constituted a "certificate" <strong>and</strong> three such<br />

certificates were required for the /icence degree.e A provrnclal<br />

faculté des sciences oflered. fewer choices. Everywhere,<br />

though, "calculus" was the core of the mathematics<br />

curriculum, <strong>and</strong> so it was a traditional Dractice<br />

for professors to write a textbook on analysis.<br />

-<br />

The French Cours d'analyse belonged to a particular<br />

textbook genre. Some of these, like Édouard Goursafs<br />

Cours d'analyse mathématiqu3 <strong>and</strong> Jacques Hadamard's<br />

Cours d'analyse professé à I'EcoIe Polytechnique, sternrned<br />

3<br />

Course announcements, Faculté des sciences de Paris, 1920-1939,<br />

Archives nationales, Box 61AJ161.<br />

e<br />

The licence was then roughly equivalent to the upp€r levels of underqraduate<br />

instruction in the befter American univergities.<br />

THE MATHEMATTCAL tNrËLucENcER vol. 1s, No. 1, r99o 29

F<br />

essentially ftom <strong>and</strong> were written for the dassroom.lo<br />

Others, like Camille Jordan's Cours d'analyse de l'Ê,cole<br />

Polytechnique (despite its title) <strong>and</strong> Émile Picard's Tralté<br />

d'analyse,,were more elaborate, true general analysis<br />

treatises. " These books integrated relatively recent results<br />

together with some of their authors' own mathematical<br />

work.<br />

The <strong>Parisian</strong> professorial lectures dominated French<br />

mathematics instruction, <strong>and</strong> some consensus had de-<br />

Many physicists who use methods of integration <strong>and</strong> operations<br />

on series obtain exact numerical tesults but are<br />

convinced meanwhile that they are committing some<br />

mathematical heresies. This stems from the fact that, in<br />

most analysis treatises, fundamental theorems, such as<br />

methods of calculation, existence theorems, etc., are introduced<br />

with a rather impressive wealth of precautions.<br />

These theorems contain an overabundance ofÏl'potheses.<br />

ln many cases,,it wilJ be necessary for us to reconsider<br />

such theorems."<br />

veloped among university teachers ihroughout the<br />

country as to which textbooks to adopt. Thus, not<br />

atypically, Goursa/s book had a long shelflife, guiding<br />

French mathematicians in the "calculus" even in the<br />

1950s. Although a system as centralized as the French<br />

Unioercité network ostensibly fostered neither change<br />

nor originality it did, however, support them in actual<br />

practice. Instructors, <strong>and</strong> especially professors, were<br />

free to teâch their subiect as they wanted. So, in conbast<br />

to the ossified, albeit remarkable teachings of immortal<br />

masters, the instructors <strong>and</strong> professois in the<br />

different facultés des sciences often tâught from their<br />

own notes. Provincial university teachers mav have<br />

enjoyed even greater freedom in this regard than their<br />

more established <strong>Parisian</strong> counterparts. The proto-<br />

<strong>Bourbaki</strong>s intended to make full use of that freedom.<br />

Judging by the contents of Goursafs book, a French<br />

course on differential <strong>and</strong> integral calculus typically<br />

drew from among the following topics: differentiaton,<br />

integration (definite <strong>and</strong> indefinite integrals, multiple<br />

integrals), series approximations of functions, geometry<br />

(envelopes, curves, surfaces), functions of a complex<br />

variable, analytic functions (Cauchy's theory, holomorphic<br />

functions, analytic continuation, functons<br />

of several variables), solution of differential equations<br />

(etstence theorems, linear <strong>and</strong> nonlinear differential<br />

equations), partial differential equations (first-order,<br />

Monge-Ampère, linear, elliptic, harmonic), integral<br />

equations (solution by approximations, Fredholm's<br />

theorems, applications), <strong>and</strong> introduction to the calculus<br />

of variations. The group assembled at Capoulade<br />

did not object to these topics in <strong>and</strong> of themselves. Its<br />

main qualm about Goursat's book <strong>and</strong> other current<br />

Cours d'analyse was that they misled their readers, especially<br />

physicists, on the nature of mathematical<br />

ngor.<br />

As the Committee sees it, special conditions do not<br />

necessarily yield more rigorous results. In the Coars<br />

d'analyse, theorems tended to appear several times in<br />

the text, each time with an added set of hvpothes€s.<br />

Thus, Cauchy's theorem was followed by boursat's<br />

version of it, <strong>and</strong> Stokes's theorem received similar<br />

beatment. The group at Capoulade wants to avoid this<br />

sort of repetition <strong>and</strong> to present material in a more<br />

general <strong>and</strong> modern setting.l3<br />

Indeed, they want their h€atise to be "as modem as<br />

possible"-with emphasis on the word "modem"-in<br />

contrast to a type of knowledge which they deem outmoded.<br />

But what does "as modern as possible" really<br />

mean? Where should they start? What topics should<br />

they include? These questions occupy them hom soup<br />

to coffee. Weil states that no topic should be eliminated<br />

a priori, whereas Cartan wonders whether it is appropriate<br />

to include algebra in an analysis treatise. As far<br />

as level is concemed, Cartan suggests that the equivalent<br />

of mathématiques générales be assumed- His colleagues<br />

protest: "We should start from saatch!"<br />

Soon after, they argue about which topic should<br />

come first: functions of real or complex variables?<br />

Whereas a majority favors treating the real before the<br />

complex case, all promptly agree with Weil <strong>and</strong> Chevalley<br />

that it would be best to introduce algebraic aspects<br />

of the theory of complex functions fi$t. Delsarte<br />

points out that, more generally, the treatis€ should<br />

stârt with an abstract <strong>and</strong> axiomatic exposition of some<br />

general but essential notions such as field, operation,<br />

set, <strong>and</strong> group, as in van der Waerden's boôk.Ia The<br />

group agrees <strong>and</strong> informally calls this introductory section<br />

the "abstract package lpaquet abstraitl." They further<br />

concur that it should be kept to a minimum, however,<br />

because notions can always be introduced later,<br />

as the needs of exposition dem<strong>and</strong>. Next, they argue<br />

10<br />

Édouard Goursat, Co urs d'ûfiollse mathématiqxe, 3 vols., Paris: Gauthier-Vilars<br />

(1902-1914 (a fifth edirion of the third volume appeared<br />

in 1956); A fj|sf cou6e it nqlhemûlical anallsis {Earle Rayrnond Heddck,<br />

trans.),3 vols., Boston-New York: Ginn & Co. (190+1914 (a<br />

Dover edition of this tnnslation appeared between 1959 <strong>and</strong> 1964);<br />

<strong>and</strong> Jacques Hadamâr d,, Cours d'analyse prolessé à I'Êale Pofutechniqte,<br />

Paris: Hermânn (1927). See also Hadamard's preface wherc he points<br />

out whi(h extra material he deemed useful io add.<br />

rr<br />

Camille Jordan, Cours d'arulyse de l'É,cole Polytechnique, 3 vols.,<br />

Paris: Gauthier-Villars (1882-1884 (a third edition oI volume three<br />

appeared in 1915); <strong>and</strong> Émile Picard, Traité d'analuse,3 vols., Parisl<br />

Cauthier-Villa$ (1891-1896) (a rhird edition of rh; rhird volume appeared<br />

in 1928).<br />

30 rne vetttpuerrcat- INTELLIcENCER vol. 15. No. r.1993<br />

12 Translation by the author. These were repoltedly Weil's own<br />

words.<br />

13<br />

Related in André Wetl, Oeuores scientifiqueÊ---Collected Popefi, Vol.<br />

1, New York Spdnger-Veilâg (1980), 563j a^d Souoenirs d'aryrentissage,<br />

Vita mathematica, Basel-Boston Berlin: Birkhàuser (191), 103_<br />

104. The latter work appeared in English translaiion as The Apprenticeship<br />

of a Mathenaticiar, Boston: Bitkhâuser (1992). A discussion of<br />

Bou:rbaki's treatment of Stokes's theorem appears in Liliâne Beau-<br />

lieu, Proofs in expository writing: Some eximples from <strong>Bourbaki</strong>,s<br />

early worlr, lntcrchange 2l (19901, 3145, on pp. 36-38.<br />

'"<br />

Bârtel L. van der Waerden, Moderne Algebra, 2 vols., Berlin:<br />

Springer-Verlag (1930-1931).

over the starting point: sets or operations <strong>and</strong> fields?<br />

They close the meeting unresolved. Weil convenes another<br />

meeting for rnid-]anuary, same time, same place.<br />

Each participant is told to bring along a list of iopics<br />

which he thinks should appear in the treatise.<br />

As it set out to write its modern analogue of the<br />

Cours d'analyse, the Committee already knew where it<br />

wanted to publish its work-with Hermann in its collection<br />

entitled "Actualités scientifiques et industrielles."<br />

Gauthier-Villars, the official acâdemic publisher<br />

The "Committee on Analysis" <strong>and</strong><br />

Its Participants<br />

of mathematics at the time, was out of lhe quàsdon for<br />

them. Some of the most conservative French mathematcians<br />

dominated its editorial board, iust as they<br />

controlled every other institutional body in the field. In<br />

contrast, Hermann stood somewhat on the frinee. It<br />

was a small, independent firm directed by an e-nterprising<br />

<strong>and</strong> eccentric Latin Quarter character, Enrique<br />

Freymann, who was always willing to venture into<br />

new proiects regardless of how financially unsound<br />

they might appear.<br />

Respectable mathematical texts, such as Éte Cartan's<br />

Lryons sur les inoariants intégraux <strong>and</strong> Hadamard, s<br />

Cours d'analyse had already beèn publshed by Hermann.rs<br />

Fuithermore, sevéral members of the Committee<br />

had recently had fust-h<strong>and</strong> experience with the<br />

publisher <strong>and</strong> its "Actualités." Soon àfter Lhe early dernise<br />

of their friend <strong>and</strong> colleague. Jacques Herbi<strong>and</strong>,<br />

Chevalley <strong>and</strong> Weil decided to publish a memorial volume<br />

of papers. They brought their proiect to Freymànn,<br />

<strong>and</strong> th,e articles soon appeared as a series in the<br />

"Actualités."'6 As a collection, the ,,Actualités,,<br />

The inJormal circle first called itself the<br />

had<br />

originated in 1929 <strong>and</strong> published monographs in the<br />

form of rather homely, paper-bound booklets. It encompassed<br />

a variety of series on science, some of<br />

which pertained to mathematics. Each series was<br />

headed by a dhector, who had full reign over content,<br />

qualty, <strong>and</strong> quantity of publication. For instance, Élie<br />

Cartan led a series on geometry, Maurice Fréchet<br />

headed one on general analysis, <strong>and</strong> Hadamard edited<br />

another on mathematical analysis <strong>and</strong> its applicâtions.<br />

A given series had limitations neither on duration nor<br />

on number of publicatons. This flexibility welcomed<br />

innovations <strong>and</strong> encouraged autonomy. Freymann<br />

<strong>and</strong> his "Actualités" thus provided the Capoulade<br />

group with the full editorial freedom it sought.l7<br />

,,Committee on<br />

Analysis" or "Committee for the Analysis Treatise.,,<br />

Delsarte apparently decided simply to wdte the title<br />

"Traité d'analyse" at the top of the reports on the<br />

meetings, although René de Possel once insisted that it<br />

should be changed to "Treatise on Mathematics.,, No<br />

one else supported this motion, <strong>and</strong> the title-page of<br />

the minutes remained the same for some time. -<br />

The notion of membership in the group was not<br />

clearly defined either. At the second meeting, the<br />

Committee decided to limit its numbers to the fàllowing<br />

nine pârticipants: Cartan, Chevalley, Delsarte,<br />

Dieudonné, Paul Dubreil, Jean Leray, Szolem M<strong>and</strong>elbrojt,18<br />

de Possel, <strong>and</strong> Weil. Dubré attended orùy a<br />

couple of meetings, <strong>and</strong> in May, Jean Coulomb, a<br />

physicist with â strong mathematical background, replaced<br />

him. Leray officially stayed on until the summer.<br />

Although he did not attend very regularly, he<br />

wrote out some of the more elaborate plans which occupied<br />

ihe Committee for several meetings. Later,<br />

when too few members showed up to discuss differentiàl<br />

equations, a quorum became essential. It was<br />

only in the summer of 1935 that the Committee<br />

adopted the pseudonym "<strong>Bourbaki</strong>.,, The participants<br />

who were present at that general meeting declared<br />

thernselves "official members" <strong>and</strong> in so doing finally<br />

defined membership.le<br />

From time to time, the members informallv invited<br />

others to join the discussions as guests o, âdrriro.".<br />

Thus, Emil Ariin attended a meeting of the Committee<br />

as a guest while visiting Paris Érom Hamburg in February<br />

1935. His presence did not seem to influlnce the<br />

meeting, however: a five-line-long outline for ,,set theory"<br />

went undiscussed, <strong>and</strong> measure <strong>and</strong> inteqration<br />

was the topic of conversation. To provide thé extra<br />

manpower needed to tackle di{fereniial equations, the<br />

Committee invited the physicist, Yves Roiard,2o as an<br />

advisor to inform them on what would be especiallv<br />

useful to physicists. Although Rocard's suggestions<br />

were labeled "the physicists' desiderata,,, they seemed<br />

mostly related to vibration <strong>and</strong> stability problerns, <strong>and</strong><br />

they did not convey a general picture ôf what physicists<br />

at the time might have needed as mathematical<br />

13<br />

Szolem M<strong>and</strong>elbrojt is an uncle of the mathemahcian Benoit M<strong>and</strong>elbrot<br />

of ftactals fame.<br />

D<br />

Charles Ehresmann was asked to join the group in place of Leray.<br />

As a point of nomenclature, I use the term ,,ploto-<strong>Bourbaki</strong>,, to<br />

designàte a pàrticipant in the early Committee. A ,<strong>Bourbaki</strong>,, 15<br />

Élie Cartan, Leçons sur les h1L\;iriants intéÙraut, pads: Hernann<br />

(1922). lor the relerence to Hadamard's texf, see footnote 10.<br />

16<br />

Claude Chevalley <strong>and</strong> André Weil (eds.), Êxwsés matb/natiql)es<br />

publiës à 1û mémoirc de I. Herbr<strong>and</strong>, Paris: Hermann (t934-lca5). This<br />

was an intemational project involving French a5 wejl às foreign conlributors.<br />

Among the French contributors, Henri Cafiàn, Delsarte.<br />

is<br />

Dieudonné,<br />

a<br />

Dubreil, <strong>and</strong> Weil were also among the proto- member of the group Bou.baki <strong>and</strong> is the official designation coined<br />

<strong>Bourbaki</strong>s. The papers presented in this sedes are listea in Be;uiieu, b), the gloup itself. A "<strong>Bourbaki</strong>st" is a follower ot B-ourbaki.<br />

Boufuaki. Unc Histoirc, Vol.2, pp.66a7.<br />

20<br />

17<br />

Rocard was in the same class as Delsalte <strong>and</strong> Weil at the École<br />

See Beaulieu, <strong>Bourbaki</strong>. tJne Histoire, Vol. 1, pp. 13g-140 <strong>and</strong> 142- Normale Supérieure <strong>and</strong> is known for his work on the A-bomb <strong>and</strong><br />

148. On En-rique Frefmann ând his publishing house, consult Weil, in the theory of vibrations. He is the fâther of Michel Rocard,<br />

Souoenirs<br />

a<br />

d'aryreùis,age, pp. 102-109 (footnote 13).<br />

French socialist politician <strong>and</strong> recent Prime Minister of Fnnce.<br />


l'<br />

tools. Although Rocard's suggestions were tabled until men formed an elite in a highly hierarchical systemi<br />

some future rneeting, some of them ultimately found a they were the designated aspirants to the leadership of<br />

p_lace in one of the projects on differential equations. French mathematics.<br />

Elie Cartan was also called in----at the tenth meeting- Paris, of course, was the best place for them to meet.<br />

to help out the Comrnittee on integral equations, <strong>and</strong> They frequently went to the iapital city to keep in<br />

although no formal decision was reached at that time, touch with its scientific life, to visit fellow mathemati-<br />

like Rocard's, some of his advice was also later heeded. cians, <strong>and</strong> to frequent bookstores <strong>and</strong> libraries. Most of<br />

Born between 1899 <strong>and</strong> 1909, the regular participants them also commuted twice a month, on Mondavs, to<br />

were graduates of the École Normale Supeileurefwith take pârt in a Sèminaire de mathématiques, which Àet at<br />

the exception of M<strong>and</strong>elbrojt who received rnost of the recently built Institut Henri-Poincaré. They usually<br />

education in Pol<strong>and</strong> <strong>and</strong> came to Paris for his doctor- held their Committee meetings in the hours bêfore the<br />

ate. They shared a corunon background, but did not semlnar.<br />

know or use the same mathernatical tools, <strong>and</strong> they did Gaston lulia, professor at the Paris faculté des sciences,<br />

not even work in the same specialties. Many of them officially convened this seminar which was primarily<br />

had traveled abroad after their doctorate or while do- organized <strong>and</strong> animated by a h<strong>and</strong>ful of devbtees. Àt<br />

ing their doctoral research, as fellows of some granting the beginning of each year, the organizers would<br />

agency, the most generous of which was the Rockefel- choose the theory which was going to be the topic of<br />

ler Foundation. Although their hosts were not always the year, <strong>and</strong> draft a list of possible talks <strong>and</strong> speakers<br />

real mentors to them, those fellows who spent re- (usually recruited from their own ranks). The<br />

search time in Denmark, Germany, Hungary, Italy,<br />

Sweden, Switzerl<strong>and</strong>, <strong>and</strong> the United States became<br />

acquainted with areas of research, methods, <strong>and</strong> resources<br />

which were not common in their native<br />

France. Partly as a result of their contacts with foreign<br />

mathematicians in France or in other countries, some<br />

participants were more familiar with set-theoretical,<br />

axiomatic, algebraic, or topological methods. Others<br />

remained closer in their work to function-theoretical<br />

questions <strong>and</strong> analytic methods on which the reputation<br />

of French mathematics had been built. Their ëommittee<br />

discussions reflected this heterogeneity.2l<br />

As mathematical researchers, the proto-<strong>Bourbaki</strong>s<br />

had each already published several papers.22 Furthermore,<br />

they were the recipients of fellowships, prizes,<br />

<strong>and</strong> other distinctions. In fact, most of these men received<br />

several of the prizes of the Académie des Sciences<br />

de Paris early in their careers.23 As professors,<br />

they had also followed the then current pâttern of taking<br />

a position in a French prov'rncial t'nculté des sciences<br />

before seeking a call to Paris. In 1934-1935, Henri Cartan<br />

<strong>and</strong> André Weil taught in Strasbourg, Delsarte <strong>and</strong><br />

Dubreil were in Nancy, Dieudonné had a position in<br />

Rennes, <strong>and</strong> M<strong>and</strong>elbroit, de Possel <strong>and</strong> Coulomb in<br />

Clermont-Ferr<strong>and</strong>. Chevalley <strong>and</strong> Leray were grantees<br />

of the Caisse nationale that yéar.2a From their pôvincial<br />

vantage points, they had the freedom to experiment<br />

with new ideas <strong>and</strong> fresh approaches which might<br />

have been less accepted in Paris. Nevertheless, these<br />

,Julia<br />

seminar," as it was called, studied the following topics<br />

between 1933 <strong>and</strong> 1939: group theory <strong>and</strong> algebras,<br />

Hilbert spaces, topology, the works of Élie Cartan, algebraic<br />

functions, <strong>and</strong> calculus of variations. Its obiective<br />

was neither to serve as a teaching seminar nor to<br />

inform the audience on current liteàrure. Rather, it<br />

offered, to its speakers as well as to its audience, an<br />

opportunity to work through large parts of recently<br />

developed theories <strong>and</strong> methods. It forced the speakers<br />

to synthesize in{ormation which was otherwise<br />

scattered in the literature. Since the seminar had virtually<br />

no official institutional ties, it provided a free<br />

forum for criticism. There, the members of the Committee<br />

eventuâlly assimilated, together, the approaches<br />

which they later tried to integrate into their<br />

collective expository writng.25<br />

The Committee had neither money nor a set administrative<br />

structure. Delsarte acted as "secretary" or<br />

"manager." He wrote up minutes <strong>and</strong> sent around<br />

reminders, but he had no particular powers. A de facto<br />

hierarchy developed quickly, however, based on a<br />

mixed measure of mathematical excellence <strong>and</strong> exoertise,<br />

intellectual sophistication, strength of voice, ànd<br />

determination. Indeed, the meetings of the Committee,<br />

like the later gatherings of <strong>Bourbaki</strong>, reportedly<br />

took place amid great noise <strong>and</strong> confusion. This should<br />

come as no surprise, given that a Latin Quarter café<br />

served as their laboratory. Still, the ambience of the<br />

Latin Quarter of the mid-30s-with its strikes, demonstrations,<br />

<strong>and</strong> political street fights-did not totally<br />

dominate the group's meetings. As a mathematical<br />

2r<br />

On trips <strong>and</strong> traveling fellowships, see Beaulieu, <strong>Bourbaki</strong>. Ilne team, it largely managed to remain staunchly apoliti-<br />

Histoile, YoL 1, pp. 69-105.<br />

2<br />

On average, the proto-<strong>Bourbaki</strong> had published 23 papers, with a<br />

minimum of 4 <strong>and</strong> a mayjmuin of 75, according to a rough count.<br />

æ<br />

On prizes <strong>and</strong> other distinctions, see Beatljiu, Boutbaù. LIne Histoirc,<br />

Vol. 1, rp. 87-114.<br />

'?a<br />

On teaching posts <strong>and</strong> provincial French frrultés iles scimces, see<br />

<strong>Bourbaki</strong>. Une Histore, Vol. 1, pp.<br />

5<br />

The texts of the talks were mimeogûphed. A near complete set is<br />

deposited at the libÉry of the Institut Henri-Poincaré in Paris. For<br />

the list, see <strong>Bourbaki</strong>. Une Histoire, Yol. 2, pp, 63*65. On the Julia<br />

Seminar, see Borl,rh. Ufie Histoire, Vol. 1, pp. 133_132.<br />

'n4122.<br />

32 TrIE MATHEMATICAL tNrELucENcER vo|-. 15. No. 1.1993

cal, even though some of the Committee participants<br />

may have had leftist leanings. With ihe exception of<br />

some discussion of <strong>and</strong> intervention in local academic<br />

politics, the proto-<strong>Bourbaki</strong>s shied away from the political<br />

arena. In their case, then, the café served as a<br />

public place for mathematical practice, while it stopped<br />

short of providing a metaphor for fuller public <strong>and</strong><br />

political involvement.<br />

How could the Committee reconcile these seemingly<br />

disparate goals? In particular, how could a textbook on<br />

analysis be used equally by mathematics students, professional<br />

scientsts, <strong>and</strong> the man in the street (however<br />

studious he might be)? The proto-<strong>Bourbaki</strong> meetings<br />

found their participants groping to reconcile these different<br />

objectives.<br />

A partial solution to their dilemma lay in their decision,<br />

at the first meeting, to "start from scratch." They<br />

would open with a preliminary set of abstract <strong>and</strong> elementary<br />

notions which would appear in the first sections<br />

of the treatise. They quickly decided, however, to<br />

restrict this "abstract package" to a minimum <strong>and</strong>,<br />

during the whole semester, very little was done to produce<br />

even that. In fact, the minimal "abstract package"<br />

apparently ceased to be an immediate preoccupation.<br />

Instead, two levels of questions came to the fore: globally,<br />

which topics should go into the heatise; <strong>and</strong><br />

locally, for each topic considered, what material<br />

should be covered <strong>and</strong> from what point of view?<br />

The Committee concentrated on inventories <strong>and</strong><br />

outlines for its eventual analysis treatise, in no particular<br />

order of presentation. Notions went undefined;<br />

theorems went unstated <strong>and</strong>, of course, unproved.<br />

Each meeting, thus, resembled a brainstorming session,<br />

with many suggestione-but few final resultsbursting<br />

forth. While the Committee did not concern<br />

itself so much with an overall plan, it did discuss some<br />

26 Translation by the author. There is a pun here on the words "chercheurc"<br />

: "researchers" or "seekers" <strong>and</strong> "houveuts" : "finders."<br />

of the topics to be included in the treatise.zT Of the 10<br />

meetings, 5 were devoted mostly to differential equations,<br />

integral equations, <strong>and</strong> partial differential equations.<br />

These topics constituted the bulk of the material<br />

of the old Cours d'analyse. Integration theory, analytic<br />

functions, <strong>and</strong> a little algebra were also discussed during<br />

that semester.<br />

The Committee parceled out the various topics to<br />

separate subcommissions ând m<strong>and</strong>ated each to<br />

skeich out its topic. Most subcommissions consisted of<br />

Mathematical Shoptalk:<br />

three people, no more than two of whom were sup-<br />

Discussions <strong>and</strong> Sketches<br />

posed to be specialists in the field. Subcommissions<br />

At first, the Committee's aim seemed quite clear: to were created for the following topics: algebra, analytic<br />

change the teaching of mathematics at the university functions, integration theory, differential equations,<br />

level by writing a treatise on analysis. Soon, however, integral equations, etstence theorems (for differential<br />

another purpose emerged:<br />

equations), partial differential equations, differentials<br />

<strong>and</strong> differential forms, topology, calculus of variations,<br />

We must wr:ite a treatise which will be useful to all: to special functions, geometry, Fourier series <strong>and</strong> Fourier<br />

rcsearcherc (bona fide or not), "finders," aspirants to posts<br />

integrals,<br />

in public<br />

<strong>and</strong> representation<br />

education, physicists,<br />

of<br />

<strong>and</strong> all technicians.<br />

functions. There was<br />

As a<br />

criterion, we can say that we should (without thought of no special subcommission for set theory (which was<br />

monetary gain) be able to recommend this treatise, or at considered part of algebra), <strong>and</strong> the subcommission for<br />

least its most important sections, to any self-taught stu- topology was formed only belatedly.2E Altogether the<br />

dent, presumably of average intelligence. . . . Mostly, we subcommissions had to concentrate on a lot of "hard<br />

must provide users with a collection of tools, which<br />

classical<br />

should<br />

analysis."2e But,<br />

be as powerful <strong>and</strong> universal as possible.<br />

of course, they had to rework<br />

Usefulness<br />

<strong>and</strong> convenience should be our guiciing principles." these traditional topics into a modern idiom.<br />

The queries inspired by analytic functions especially<br />

revealed the Committee's puzzlement over the task of<br />

merging the tradltional <strong>and</strong> the modern. A classical<br />

topic of ihe Cours d'analyse, function theory had been<br />

sirongly affected by the ideas of René Baire, Émile<br />

Borel, <strong>and</strong> Henri Lebesgue, especially in the area of<br />

functions of real variables. The works of Lars Ahlfors,<br />

Ludwig Bieberbach, Constantin Carathéodory, Nicolas<br />

Lusin, <strong>and</strong> Rolf Nevanlinna further changed the field.<br />

The Committee had its own specialists at h<strong>and</strong>: Henri<br />

Cartan, Dieudonné, M<strong>and</strong>eibrojt; to some extent,<br />

Leray <strong>and</strong> de Possel worked with analytic functions as<br />

well. When M<strong>and</strong>elbrojt voiced the opinion that the<br />

treatise should not overemphasize entire functions, his<br />

colleagues asked whether it should not include such<br />

important matters âs Picard's theorem, conformal representation,<br />

elliptic functions, Abelian functions, infinite<br />

products, etc. A variety of suggestions followed<br />

<strong>and</strong>, in the resulting confusion, no decision was made.<br />

Analytic functions reappeared on the agenda at another<br />

meeting when the participants put together their<br />

27<br />

this aspect of the work of the fust semester is mentioned in Weil,<br />

Souoefiirs d' apprentissage, pp. 109:110.<br />

'?3<br />

The list of topics comes from the report on the eighth meeting.<br />

Because Alex<strong>and</strong>roff <strong>and</strong> Hopfs book on topology only appeared in<br />

1935, the Committee did not even hav€ this reference available for<br />

consideration of topological topics. See Paul S. Alex<strong>and</strong>roff <strong>and</strong><br />

Heinz Hopf, Topologie, Vol. l, Berlin: Springer-Verlag (1935).<br />

'-<br />

Benoit M<strong>and</strong>elbrot criricized Bourbali---among other thing+for<br />

hâving neglected what he termed "hard dassical analysis" in its<br />

Ê.\émmX. See Benoit M<strong>and</strong>elbrot, Chaos, <strong>Bourbaki</strong>, <strong>and</strong> Poincaré,<br />

Mathematical lntelligencer, Vol. 11(1989), no. 3, 10-12.<br />

THE MÀTHEMATcAL INTELUcENcER vol-. It No. 1,1993 33

î'<br />

written outlines on the subiect. From these, the Committee<br />

drafted a proposal which attempted to place the<br />

material in an algebraic <strong>and</strong> topological setting. It<br />

started with the geometdc representation of cornplex<br />

numbers <strong>and</strong> stressed that these form a field. It next<br />

proposed to move to the topology of open <strong>and</strong> dosed<br />

surfaces. This sketch also featured some of the more<br />

"usual" material on analytic functions: Jordan's theorem,<br />

series, convergence, differentiation, integration,<br />

Cauchy's theorem <strong>and</strong> Cauchy's integral, Taylor's <strong>and</strong><br />

Laurent's theorems for singular points of uniform<br />

functions, conformal representation, entire functions,<br />

Weierstrass's theorem, Mittag-Leffler's theorem, analytic<br />

continuation, etc. It was suggested that the general<br />

notion of analytic function be highlighted <strong>and</strong> that<br />

two sections be devoted to algebraic <strong>and</strong> automorphic<br />

functions, elliptic functions, <strong>and</strong> the theta function<br />

(the latter, most likely, with number theory in mind).<br />

The meeting also considered whether it should introduce<br />

analytic functions of many complex variables. It<br />

delegated the decision making on these matters to its<br />

subcommission.<br />

The Committee's initial work on integration theory<br />

looked more promising. At least, one point was certain<br />

at the out6et: Integration would be done from Lebesgue's<br />

point of view, which had already undergone<br />

different levels of qeneralization <strong>and</strong> extension since<br />

1901 . 30 kinds of measures. At any rate, the Comrnittee had<br />

already decided to restrict its exposition on integration<br />

theory in the treatise.<br />

This sketch of contents resulted from discussions between<br />

Chevalley <strong>and</strong> de Possel, who advocated a thorough<br />

exposition on measure, <strong>and</strong> Delsarte, Dieudonné,<br />

<strong>and</strong> Dubreil, who thought that a less elaborate<br />

presentation of measure <strong>and</strong> integration might be<br />

more appropriate for the treatise. Most probably, WeiI<br />

<strong>and</strong> de Possel opposed each other also: Whereas de<br />

Possel wanted to do measure <strong>and</strong> integration on arbitrâry<br />

sets orùy, Weil wanted to involve vector spaces<br />

<strong>and</strong> topological groups. The subcommission on integration<br />

had to r.eionsid"t these different opinions.32<br />

Delsarte <strong>and</strong> Leray were the main protagonists in<br />

the area of differential equations. Delsarte first suggested<br />

subdividing the study of differential equations<br />

into three sections: existence theorems, eigenvalue<br />

problems (crucial in physics), <strong>and</strong> the study of local<br />

<strong>and</strong> global properties of solutions. The Committee<br />

agreed with his choices, at least in principle. Then it<br />

examined a draft by Leray on existence theorems. Contrary<br />

to most other plans, this draft was not merely a<br />

Iist of items. It was more like a brief introduction to ar<br />

abstract theory for systems of n equations in n unknowns.<br />

Leray introduced concepts from topology <strong>and</strong><br />

Although some French Cozrs d'analyse did nen-<br />

functional analysis which he had been using in his<br />

own work, especially the notions of the differential of<br />

tion the Lebesgue integral (usually in passing), they a function of n variables (as a linear functional) <strong>and</strong> of<br />

did not give an exposition of Lebesgue's theory. The the topological degree of a continuous transformation.<br />

older texts used Cauchy's approach, <strong>and</strong> the more re- His approach set the study of differential equations<br />

cent ones inhoduced the Riemann integral. All con- squarely in line with works by Riesz, Banach, <strong>and</strong><br />

centrated on developing the techniques of integrâtion Hahn on normed spaces. Leray's draft included the<br />

<strong>and</strong> their applications. By choosing Lebesgue integra- statements of fundamental theorems of global existion,<br />

the proto-<strong>Bourbaki</strong>s were more in line with texts tence <strong>and</strong> of local existence (with uniqueness) of a so-<br />

such as those by Constantin Carathéodory, Charles de lution. The Committee thought that, although it was<br />

La Vallée Poussin, <strong>and</strong> Stanislaw Saks."<br />

all very interestinç, Leny's project involved topologi-<br />

The Committee resolved that rneasure <strong>and</strong> integracal notions which were too specialized for the treatise.<br />

tion should not be seDarated in the presentation. It Perhaps for this reason, no final decision on differen-<br />

drafted a list of potentiàl topics which itarted with the tial equations was reached at this point in their delib-<br />

notion of measure <strong>and</strong> proceeded to the integral eranons.<br />

viewed as a linear functional, stressing the equivalence At the end of its eighth meeting, the Committee it-<br />

of the two concepts. Next followed particular types of self drafted a provisional <strong>and</strong> rather jumbled outline<br />

measures <strong>and</strong> integrals on topological spaces, Radon for differential equations. This plan comprised general<br />

measures, <strong>and</strong> Haar measures on topological groups. efstence theorems, global existence for reai differen-<br />

Although order of presentation was not stressed, it tial equations over any domain where the conditions<br />

appeared that the latter were meant not to be the pil- for local existence hold, a classical theory of general<br />

lars of the theorv, but were introduced rather as special linear equations, systems of z first-order linear equations<br />

in z unknowns, <strong>and</strong> second-order linear equations<br />

with constant coefficients. Among the applications<br />

to physics, some of Rocard's old suggestions re-<br />

s<br />

For an historical account oI Lebesgue's ideâs, see Thomas Hawki^s,<br />

Izbesgue's Theory of lftte9ratiofl: lts Oigi\s afld Danloryent, Nevr<br />

Yorkr Chelsea (1975).<br />

31<br />

See Constantin Carathéodory, Vorlesungen iiber reelle Funktiotutt,<br />

Leipzig-Berlin: Teubner (1918); Charles de La Vallée Poussin, lnfégrûles<br />

de Lebesgue. Fonctiont il'ensemble. Classes de Baire,lsl ed., Paris:<br />

Gauthier-Villa$ (1916); 2nd ed., 1936; <strong>and</strong> Stard'slaw Saks, Théorie de<br />

I'intéyale, Mo ogmlie matematyczne, 1st ed., Vol. 2, warsaw: Zsubwencji<br />

funduezu kultury narodowe (1933); 2nd ed., 1937.<br />

34 rm uerr+uercel rNrELucENcER vot-. 15, No. 1,1s93<br />

32<br />

These differences in opinion are discussed in Beâulieu, Bo!rù!,ti.<br />

Une Histoîe, !ol. 1, pp. 178-188. Chevalley <strong>and</strong> de Possel put forth<br />

their ideas in "Un théorème sur les fonctions d'ensemble complète.<br />

ment additives," Cot?rp. tutd. Acad. Sci. Palis 197 (1933), 88H87. See<br />

also René de Possel, "Notion générale de mesure et d'intégrale," 5émiwie<br />

de msthénaliques llA (1934), mimeographed.

surfaced, but the conceptual setting of Leray's project<br />

had disappeared. The Committee had adopted the<br />

functional approach earler in its work, yet it did not<br />

seem to envisage all the implications of this choice.<br />

With the intention of covering a lot of ground, it delegated<br />

work on differential equations to two subcommissions,<br />

one responsible for existence theorems <strong>and</strong><br />

the other for the rest.<br />

The Committee's work on integral equations proceeded<br />

differently. An initial discussion, involving<br />

Cartan, Delsarte, Deudonné, <strong>and</strong> Weil, emphasized<br />

three approaches to the subject: bounded operators on<br />

Hilbert spaces, Fredholm's point of view, <strong>and</strong> the more<br />

recent line developed by Riesz <strong>and</strong> Leray (using<br />

nomed vector spaces). In the absence of Leray, the<br />

Committee temporarily opted for bounded operators<br />

on Hilbert spaces, which they thought was a complete<br />

<strong>and</strong> beautiful theory. It hesiiated over the third approach<br />

because it was less familiar to most of those<br />

present <strong>and</strong> postponed its decisions until Leray could<br />

be reached. Paradoxically, the Committee was willing<br />

to favor bounded operators, even though quantum<br />

mechanics required unbounded operators. Thus, despite<br />

expressed intentions to serve physicists, the<br />

Committee was not always guided by a close eye on<br />

what was being done in that field. Other considerations<br />

sometimes prevailed. 33<br />

Two meetings làter, Leray came to the rescue. He<br />

found Fredholm's approach rather useless for the<br />

group's purposes, although Delsarte disagreed. The<br />

two also discussed how much should be done on<br />

bounded operators. Leray actually suggested two<br />

ideas for integral equations. The first one introduced<br />

nonsymmetric integral equations as a special case of<br />

equations of the form )c + g(x) : 0, where x is an<br />

element of a Banach space <strong>and</strong> I is a completely continuous<br />

operator. The second one viewed symmetric<br />

integral equations as special cases of Hermitian operators<br />

in a Hilbert space. The Committee adopted Leray's<br />

approach without further ado, <strong>and</strong> so the third<br />

point of view ultimately won the day.<br />

For partial differential equations, Delsarte drafted a<br />

brief outline which emphasized two aspects: local<br />

problems (where the notion of characteristic would be<br />

fundamental) <strong>and</strong> limit ptoblems (related to inte$al<br />

equations). After some discussion, the Committee decided<br />

that the study of a single fust-order partial differential<br />

equation with z unknowns represented the<br />

bare minimum of what needed to be covered. But what<br />

else should be included? Élie Cartan, who attended<br />

that meeting, suggested instead that they restrict their<br />

s On bounded linear operators in Hilbert space, see Jean Delsarte,<br />

"L'axiomatique des opérateuls linéaires dans l'espace de F{ilbert:<br />

opélateurs borrrés," Sêminaire de fiothérû.atiques UC (1934), mimeo-<br />

$aphed; <strong>and</strong> Jean Leray <strong>and</strong> J. Schauder, "Topologie et équations<br />

fonctionnelles," Comp. Rend. Acad. Sci. Pais 197 (7933), 115f1,<br />

study to systems of linear partial differential equations<br />

with one unknown function. He also stressed that both<br />

the "classical" point of view <strong>and</strong> the Pfaffian equations<br />

should be introduced. These points were well taken,<br />

but the group still did not commit itself on partial differential<br />

equations .<br />

The Committee did not intend to treat set theorv.<br />

algebra, topology, or even integration for their own<br />

sakes. As it stood, then, the treatise would include<br />

only enough algebra to deal with systems of equations,<br />

some integration theory to support analytic functions,<br />

but mostly differential equations, integral equations,<br />

<strong>and</strong> partial differential equations. The Cours d'analyse<br />

exerted the main pull on the topical choices of the<br />

proto-<strong>Bourbaki</strong>s. Wanting to cast this material in a<br />

modern setting, the Committee sought a workable<br />

ground between approaches which otherwise appeared<br />

either too general or too specialized. The Capoulade<br />

group made only minimal progress toward<br />

this goal, however, <strong>and</strong> its aim to provide mathematicians<br />

<strong>and</strong> physicists with the tools of their trade stll<br />

seemed far from reach. Also, the initially expressed<br />

concem for the man in the street <strong>and</strong> the average selftaught<br />

student lost its edge in the course of on-going<br />

debates.<br />

Epilogue<br />

In choosing a café as the setting for its meetings, the<br />

proto-<strong>Bourbaki</strong>s were not particuJarly original. Indeed,<br />

it was usual among politicians, artists, or scientists to<br />

meet in <strong>Parisian</strong> cafés. It was also common for mathematicians,<br />

as well as writers, to enjoy working at a<br />

favorite bistro table. Capoulade's café was a famil.iar<br />

rendezvous which provided freedom from strict university<br />

institutons while being conveniently close to<br />

all the amenities of the neighborhood. This mixture of<br />

Latin Quarter parochiality <strong>and</strong> a taste for autonomy<br />

typifies the proto-<strong>Bourbaki</strong> attitude. The narrative of<br />

biweekly gatherings, however, shows disunity in the<br />

party, hesitation in ideology, <strong>and</strong> uncertainty in purpose.<br />

Indeed, this was a time of ill-defined goals <strong>and</strong><br />

unfinished business. The Committee orùy investigated<br />

possibilities, <strong>and</strong>, despite hea*y arguments, it hesitated<br />

to strive either to make things work or to establish<br />

truths. It willfully postponed decisions to the general<br />

meeting of the summer of 1935, when a definitive<br />

overall plan of the treatise was expected to emerge.<br />

That meeting, which took place in Besse-en-<br />

Ch<strong>and</strong>esse, started another phase in the history of<br />

<strong>Bourbaki</strong>. Through negotiations <strong>and</strong> selections, opportunities<br />

were foreclosed, but new options were foreseen.<br />

DEartement de Mathématiques et d'Informatique<br />

Untuersité du Québec à Montftal<br />

C. P. 8888, Succursale "A"<br />

Montrtul, Québec H3C 3P8<br />

Cannda<br />


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