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

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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
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