How We Got Where We Are: An International Overview of Mathematics in National Contexts (1875-1900) Karen Hunger Parshall

n the last thirty years or so the mathemat- At the very least, these milestones suggest that ical community internationally has ob- the mathematical endeavor developed in served a remarkable number of centenni- important ways in diverse national settings dur- als. To name just a few, the Moscow ing the closing quarter of the nineteenth century, Mathematical Society entered a new century but … how? How did we get where we are? How inI 1964, with the London Mathematical Society did we become an international mathematical (LMS) following one year later; the Mathemati- community, working together across political sche Annalen turned one hundred in 1968 four boundaries on common problems using shared years before the Société Mathématique de France techniques? (SMF); the American Journal of Mathematics and Prior to the nineteenth century, scientists (as the Circolo Matematico di Palermo (CMP) reached opposed to the modern notion of “specialists”), their century marks in 1978 and 1984, respec- formed communities centered around institu- tively; and the American Mathematical Society tions like the Royal Society of London or the (AMS) celebrated its centenary in 1988 just two Académie des Sciences in Paris. These institu- years before the Deutsche Mathematiker-Ver- tions, together with the courts of various mon- einigung (DMV). 1 archs and the salons of wealthy patrons, en- couraged research, sponsored general journals Karen Parshall is associate professor of mathematics like the Philosophical Transactions, and sup- and history at the University of Virginia. ported scientific communications. The univer- A version of this paper was delivered at the Interna- sities, on the other hand, taught an essentially tional Congress of Mathematicians in Zürich in 1994. medieval “liberal arts” curriculum within the An abbreviated text of that talk has just appeared in context of faculties of philosophy, law, and med- the Congress Proceedings published by Birkhäuser Ver- icine. Euclid’s mathematics formed a key part of lag. I thank Birkhäuser Verlag for permission to pub- this curriculum, with the calculus added on at lish the version presented here. some universities in the eighteenth century, but the emphasis was on imparting a body of knowl- 1For publications honoring the centennials of the LMS, edge, not on imparting a body of knowledge in AMS, and DMV, see [6], [8], and [11], respectively. Hélène order to further knowledge, not on imparting the Gispert detailed the history of the SMF from 1872 to “latest” knowledge in the field, and not on ac- 1914 in her book [12], while Aldo Brigaglia and Guido tively training others to make original research Masotto chronicled the early history of the CMP in [5]. contributions. These latter ideals, so character-

MARCH 1996 NOTICES OF THE AMS 287 istic of the university today, only became part importance of pure research over the utilitarian of its educational mission during the nineteenth concerns perceived as dominant within the post- century, first in Germany 2 and, by the closing Revolutionary educational system in France. This quarter of the century, in many other countries emphasis on research accompanied and com- as well. Educational reform—whether spurred plemented a strong insistence on academic free- partially by political unification in the case of dom which developed into the ideals of Lehr- und Italy, by the loss of the Franco-Prussian War in Lernfreiheit, that is, the freedom to teach and to the case of France, by the philanthropy of wealthy learn without political or religious interference. industrialists in the case of the United States— Such educational reforms aimed not only to sup- is a kind of international com- port the faculty’s search for new knowledge but mon denominator relative to also to train independent-minded, creative, and the formation of mathematical original thinkers within an atmosphere of dis- communities in individual na- interested, scholarly pursuit. Von tional contexts between 1875 Although the full philosophical range of von and 1900. What other attrib- Humboldt’s educational vision was never realized, Humboldt’s utes transcended national the latter, more immediate aims became char- vision of boundaries? An answer to this acteristic of the Prussian system. Beginning with at least hints at “how we got philosophy and soon extending to the natural sci- higher where we are”. ences, mathematics, and other developing disci- plines, teaching and research defined the uni- education… The Mathematical Scene in versity professor’s mission. As Gert Schubring has Nineteenth-Century argued, “[t]he transition from laying the main em- stressed the Germany phasis on teaching, which compelled the teach- The opening decade of the nine- ers to seek additional part-time teaching posts, importance of teenth century was one of great to a dual activity in which teaching comprised pure research political reorganization in the only the lesser part of the remunerated activity German states owing to the ef- marks the decisive step towards professional- over fects of the Napoleonic Wars. In ization” [23, p. 123]. In the case of mathematics, Prussia, for example, the years moreover, this research ethic ultimately brought utilitarian from 1806 and the Prussian de- with it increasing specialization in the field, as feat at the Battle of Jena to 1810 mathematicians and mathematicians-to-be concerns… and the founding of the Uni- tended to focus their studies in an effort to make versity of Berlin witnessed a se- their own personal contributions. At the same ries of fundamental political, time, the emphasis on disinterested—as opposed socioeconomic, and educational to more applications-oriented—research resulted reforms. The latter, spearheaded by Wilhelm in the evolution of a fundamentally purist ap- von Humboldt, the elder brother of the cele- proach to the discipline.4 Perhaps nowhere were brated naturalist and traveler Alexander von these interrelated aspects of the development of Humboldt, came quickly to dominate not just the mathematics in Germany more in evidence early educational system in Prussia but those in the on than at the University of Berlin. other predominantly Protestant German states Under the influence of Dirichlet beginning in as well. 3 Von Humboldt’s vision of higher edu- the 1830s, the University rather swiftly estab- cation, which reflected the influences of both ide- lished itself as the dominant force in mathemat- alist philosophy and neohumanism, stressed the ics in the German-speaking world.5 This domi-

2As Gert Schubring has pointed out [24], there was, in 4The processes underlying these developments as well as fact, no one German model. Observers from other coun- the interrelations between pure and applied mathematics tries actually focused on the Prussian model, as defined are, however, much more complicated than these state- not only by the policies of the Prussian Ministry of Edu- ments might suggest. See, for example, [23] and [24]. cation but also by the specific programs in place at uni- versities like Berlin and, after 1866 when it became part 5Just as American would-be mathematicians traveled of Prussia, Göttingen. After the German unification, this abroad in the near absence of high-level training at the model increasingly influenced the very different edu- end of the nineteenth century, Dirichlet had journeyed cational environment which had developed in the pre- to Paris in the 1820s for his mathematical education. dominantly Catholic southern German states. From 1800 to the 1830s at least, France was Europe’s 3This panoply of issues has also been thoroughly ex- acknowledged leader in mathematical research. For amined. See, for example, the references provided in [18, more on the unparalleled strength of the French math- pp. 24–26] and [23]. Wilhelm von Humboldt was both ematical community in the opening decades of the an influential statesman within the Prussian government nineteenth century, see [14]. France, however, lost its and a philologist, aestheticist, and philosopher. place to Germany after midcentury. See below.

288 NOTICES OF THE AMS VOLUME 43, NUMBER 3 nance only intensified during the last half of the as an advocate for the field nineteenth century owing to the triumvirate of in his efforts to stimulate Kummer, Weierstrass, and Kronecker. Their Berlin further the German math- advocated a highly formal and rigorous approach ematical community as a to mathematics which produced stunning results, whole. These institutions— particularly in analysis, algebra, and number the- the graduate seminar, the ory [3]. By the century’s closing quarter, however, specialized journal, the spe- the University of Berlin’s position at the head of cialized society—together the mathematical hierarchy in Germany came with the twin values of teach- under increasing challenge from the formerly ing and research largely de- Hanoverian university in Göttingen. fined the profession and, in Opened in 1737, Göttingen University, too, subtler ways, the discipline had quickly embraced the ideals of Lehr- und of mathematics as it had de- Lernfreiheit and had developed strong research- veloped in Germany by the oriented programs in, first, philology and some- end of the nineteenth cen- what later the natural sciences. In mathematics, tury. These same institutions the near-legendary Gauss is often perceived by and values informed the the twentieth-century mathematician as repre- emergence of mathematical senting the beginning of a long, distinguished, research communities in a and continuous research tradition at Göttingen number of other countries Photograph courtesy of the London Mathematical Society which included the names of Dirichlet, Riemann, as well and thereby served to Clebsch, Klein, Hilbert, and Weyl, among others. build a common foundation Enrico Betti Yet the establishment and maintenance of a true for the subsequent inter- research tradition hinges not only on gifted, re- nationalization of the field. search-minded professors ready to impart their knowledge at a research level but also on a corps The Reverberations in Mathematics of of talented students willing and able to put those Educational Reform lessons to use. A man of eighteenth-century sen- It is not accidental that this brief sketch of the sibilities, the austere Gauss had no real desire context of German mathematical developments to train future researchers, and, consequently, in the nineteenth century opened with a men- few students benefitted directly from his guid- tion of educational reform. Changes in higher ed- ance. Dirichlet, Riemann, and Clebsch, although ucation and in its overall objectives naturally committed to the new nineteenth-century ideals spurred changes at the level of the individual dis- of teaching and research, each had careers at Göt- ciplines. Educational reform also tended to af- tingen cut short and so were able to influence fect mathematics all the more directly, since few students. It was Felix Klein, who, after as- one of the key features distinguishing the math- suming the Göttingen chair in mathematics in ematical endeavor of the nineteenth century 1886, established that university as a rival stan- from that of the preceding hundred-year period dard-bearer of mathematics in Germany [21; and was its venue, namely, the university as opposed 18, pp. 151–154]. to an academy of sciences, a royal court, or else- As a professor, Klein brilliantly employed the where [23, p. 111]. Its effects were not always seminar, an institution which had evolved in the positive relative to the development of research- German context as the principal vehicle for the level mathematics in a given national setting active training of young researchers, to animate though, as a comparison of the situations in a thriving mathematical community in Göttingen Italy and France with those in Spain and England [18, pp. 189–234, 239–254].6 Moreover, as a underscores. mathematical activist, he lobbied energetically Prior to the diplomatic recognition in 1861 of and successfully for mathematics with the offi- a unified Italy, several of the Italian states had cials of the Prussian Ministry of Education, edited supported schools with mathematicians whose the Mathematische Annalen, supported and par- research attracted the attention and earned the ticipated in the activities of the Deutsche Math- respect of those whom they viewed as standard- ematiker-Vereinigung [22], and generally served bearers in the field in France, Germany, and Great Britain. In the 1850s, for example, Enrico Betti pursued researches in Pisa on questions in 6 Obviously, Berlin and Göttingen did not support the only both Galois theory and the theory of substitu- active graduate programs in mathematics in late nine- tions which favorably impressed Charles Her- teenth-century Germany. Paul Gordan and Max Noether at Erlangen, Sophus Lie at Leipzig, and others through- mite, while Francesco Brioschi did work in Pavia out Germany contributed to the overall profile of higher on the theory of determinants and, more gen- education in mathematics. For more on the Leipzig erally, on the theory of forms which met with ap- seminar in particular, consult [2]. proval from James Joseph Sylvester. These two

MARCH 1996 NOTICES OF THE AMS 289 Photograph courtesy of the London Mathematical Society and the Istituto Nazionale de Alta Mathematica 290 Francesco Brioschi publication ofthe samename. lation ofCrelle’sGermanjournal andLiouville’sFrench to the e Fisiche 7 Salvatore Pincherle,andFederigo Enriques[4,pp. later colleagueUlisseDiniaswellVitoVolterra, at Pisaincluded,amongothers,hisstudentand The groupofmathematiciansinfluencedbyBetti them toinfluencedevelopmentsinpoliticsgen posts withinthenewgovernmentwhichallowed mathematicians, butmostnotablyBrioschi,held ory andinthegeometryof cially ofRiemann’sideasincomplexfunctionthe school’s directorthroughhispromotionespe ing hisalmostthirty-year-longtenureasthe Betti animatedanactivecircleofresearchersdur the modelofParis’sÉcoleNormaleSupérieure, an institutionfoundedin1808byNapoleonon and research[4,pp.275–276]. who couldmeetthechallengesofbothteaching itself providednewjobopportunitiesforthose braic geometry—atthesametimethatschool well asbyCremona’scontinuingworkinalge sorati’s workinRiemannianfunctiontheoryas within thisacademicsetting—asevidencedbyCa fostered anatmosphereconducivetoresearch dent LuigiCremonafrom1867to1873.Brioschi only Casoratiandhimselfbutalsohisformerstu training engineersandbroughttoitsfacultynot Tecnico Superioreinitseducationalmissionof 1863, heguidedMilan’snewlyfoundedIstituto form atalllevels.Inparticular,beginningin cision-making processregardingeducationalre years, Brioschiexercisedhisinfluenceinthede Ministry’s ExecutiveCouncilforsomethirty tion from1861to1862andasamemberofthat eral secretaryoftheMinistryPublicInstruc erally andineducationspecifically.Asthegen Tortolini foundedthe Similarly, atPisa’sScuolaNormaleSuperiore, Annali diMatematicaPuraedApplicata in 1850.In1858thejournalchanged itsname unification, anumberof Tortolini’s the publicationsinBarnaba cases suggest,andalookat 272]. research atahighlevel[4,p. tration inItalyonalgebraic witnessed acertainconcen that the1850shadalready Berlin, andParis. ematical centersofGöttingen, 1858 pilgrimagetothemath cians returnedfromtheir after thesethreemathemati Brioschi, andFeliceCasorati circles surroundingBetti, further inthemathematical search wasonlystrengthened tance anddesirabilityofre Annali diScienzeMatematiche Following Italy’spolitical 7 A senseoftheimpor n -dimensional space. N Annali OTICES OF confirms, THE in emu AMS ------8 other thanEnglish presentedherearemyown. pass usbeforetoolong”[12,p.19]. I thinkthatifcontinues,theItalianswillsur Germans getthebetterofusthereaselsewhere. to mendour[systemof]highereducation.The had alreadycausetoremarkthat“weneed Prussian Warin1870–1871,GastonDarboux ened fromcomplacencybyitslossoftheFranco- noticed elsewhereinEurope.InaFranceawak ing quarterofthenineteenthcentury. faculty levelthroughoutmuchofItalybytheclos others—already fairlyfirmlyentrenchedatthe thanks totheeffortsofBrioschi,Betti,and large extent,buttheidealstheyreflectedwere— Michele Coppino,rolledbackthesereformstoa research, andgraduatetraining.Hissuccessor, the Germanmodelwithitsemphasisonteaching, Italy’s institutionsofhighereducationcloserto introduced reformsaimedexplicitlyatbringing minister ofeducation,RuggieroBonghi,officially specific individuals,itwasnotuntil1875thatthe taking placeatcertainschoolsundertheswayof 279–280]. Althoughchangessuchasthesewere cipally intheprovinces), looseningoftheold p. 303].Indeed,thecreation ofnewjobs(prin could out-performtheGerman institutions”[25, French setting,“the sciencefaculties ing variousaspectsoftheGermanmodelto The reformersbelievedthatbyconsciouslyadapt tering acommitmenttoteaching tem ofeducation,therebyallowingforandfos ties betweenthe program; andthird,weakenedthetraditional enhancement oftheoverallgraduatetraining port ofmorejuniorscholarsandthe sition of cluding mathematics,aswellthesalariedpo chairs inthevariousscientificdisciplines, enroll inthe courage agreaternumberofstrongstudentsto tuted ascholarshipprograminordertoen between 1876and1900,theFrench,first,insti the Germanmodel[25,pp.302–303]. politicians andeducationalreformersfoundin gions, neededanewfocus,whichtheFrench the this objective,theseotherinstitutions,principally tutions ofhighereducation.Inordertorealize with previousregimesbutalternateinsti not withthe part, byfosteringanintellectualéliteassociated strengthen theirpoliticalposition,atleastin newly formedThirdRepublicsoughtto adequately fortimesofcrisis,leadersthe perhaps notgrandenoughtopreparetheFrench plication thattheso-called spurred largelybythemilitarydefeatanditsim All translationsofquotationsoriginally inlanguages These developmentsinItalydidnotgoun In aseriesofmajorreformswhichtookplace facultés maître deconférence in eachofFrance’sadministrativere grandes écoles facultés facultés ; second,establishednew and thesecondarysys V as hadbeenthecase grandes écoles OLUME for boththesup and 43, N 8 research. UMBER In fact, were 3 ------disciplinary boundaries through the creation of creased number of professorships, the advent chairs in various subdisciplines (as exemplified of fellowships based purely on merit, and the end by Camille Jordan’s chair, not in mathematics but of the religious tests for faculty members and in higher algebra per se), and the adoption of re- students. They also resulted in the strengthen- search productivity as a criterion for determin- ing of the university structure from the resources ing salary, all contributed to the rise of a more of the associated colleges, with an eye specifi- specialized, research-oriented mathematical pro- cally toward improving training in the sciences. fession in France on a par by Despite such changes, the fun- 1900 with that in place in Ger- damental goal of the English many [12, pp. 59–63]. As Hélène system remained the liberal ed- Gispert has characterized it in ucation of gentleman- (and, be- her study of the Société Math- ginning in the 1870s, lady-) ématique de France, France, scholars, and the pedagogical which had entered the Dark In Milan, emphasis still lay largely on the Ages relative to mathematics passing of set examinations. during the decades from 1830 to Brioschi Throughout the nineteenth cen- 1870, experienced an emergent tury, the English educational period between 1870 and 1900 fostered an system continued to focus pri- that resulted in a new golden marily on the diffusion and not age after the turn of the twen- atmosphere the advancement of knowledge, tieth century [12, p. 113].9 That and, in mathematics as in the new golden age owed, in large conducive to other sciences, this tended to measure, to the stimulus pro- research militate against the definition of vided by educational reform. a mathematical profession in Still, such reform did not nec- within the terms of teaching and research. essarily have a positive impact As the examples of Italy, relative to the development of academic France, Spain, and England il- mathematics at the research lustrate, widespread educa- level. In 1857 Spain adopted a setting tional reforms in the last quar- centralized educational system ter of the nineteenth century modeled on the one put in place affected the development of in France under Napoleon mathematics in countries around the turn of the nine- throughout Europe (and the teenth century. Under Madrid’s United States could be cited firm control, further change here as an example as well). The came only slowly. In mathematics, that control creation of new academic chairs and institu- translated into the dominance in the advanced tions, the addition of new grades of instructors, curriculum of the projective geometry which the direct emulation of the German ideals link- Karl von Staudt had developed around midcen- ing teaching, research, and the production of fu- tury and which Madrid’s Eduardo Torroja y Ca- ture researchers—these aspects of reform com- ballé embraced beginning in the 1870s. Although plemented one another in those countries where Torroja did advocate doing mathematics in his mathematics at the research level came to de- courses at Madrid, he clung doggedly to an area fine the professional standard. The absence of which, over the closing decades of the nine- one or more of them, however, tended to thwart teenth century, grew increasingly distant from, that sort of development. for instance, the more purist Riemannian fron- tiers of geometrical research [13, p. 1508]. In so The Production of Future Researchers doing, Torroja and his adherents in Madrid ob- In turn-of-the-century France, Émile Picard sum- structed the efforts of others in Spain, like Gar- marized well the key role educational reforms cía de Galdeano [15, pp. 112–114], to encourage had played in the professionalization of high- the sort of mathematics being done elsewhere level mathematics. “Beyond their mission of in Europe and particularly in Germany, France, making the sciences known and understood,” he and Italy [1, pp. 162–163]. wrote, “the institutions of higher education … In England, on the other hand, educational re- have another [mission], just as noble as all the forms in 1858, 1871, and 1877 brought an in- others, that of advancing science and of contin- 9Although the circumstances surrounding their “emer- ually initiating new generations of researchers gent periods” were certainly quite different, France to the methods of invention and of discovery” and the United States were influenced by many of the [12, p. 60]. As he clearly stressed, a sense of the same external factors in this same thirty-year period. importance of the training of future researchers Compare [12] and [18]. represented one crucial byproduct of these Ger-

MARCH 1996 NOTICES OF THE AMS 291 man-inspired reforms. Thus, educators and math- and Berlin. In addition to the regular lecture ematicians in other countries who looked to- courses they offered in the established areas of ward Germany and France for their inspiration late nineteenth-century mathematics—invariant and guidance in the final quarter of the nine- theory, the theory of substitutions, elliptic func- teenth century increasingly conceived of this tion theory, among others—the Chicago math- “noble mission” as an integral part of their en- ematicians also incorporated the seminar into deavor. The United States and Russia provide their overall pedagogical approach. As especially just two of a number of examples of this sort Bolza and Maschke knew from firsthand expe- of influence. rience, the seminar served as a fertile seedbed The years between 1875 and for the germination of new mathematical ideas 1900 represented a period of along more specialized lines. The Chicagoans fur- growth and financial prosperity ther augmented this learning device with what in the United States which had they called the “Mathematical Club”, a series of E. H. Moore, important repercussions in biweekly meetings throughout the academic year higher education. As great for- in which speakers, both faculty and students, with Bolza tunes were made on the rail- presented expositions of the recently published roads, the telegraphs, and in- results of other mathematicians or of their own and Maschke, dustrial expansion in general, evolving ideas. The atmosphere fostered by this individuals like Johns Hopkins faculty and through these means produced in implemented and John D. Rockefeller en- short order a number of first-rate mathemati- dowed universities through cians, notably Leonard E. Dickson, Oswald Ve- a training their private philanthropy. The blen, Robert L. Moore, and George D. Birkhoff (see program in presidents of these new [18, pp. 372–393] for a brief overview of some schools, well aware of the edu- of the research these mathematicians did within mathematics cational scenes abroad and es- this mathematical environment). pecially in Germany, France, and Although the broader cultural and political cir- at Chicago Great Britain, crafted their new cumstances in Moscow could perhaps not have institutional philosophies in- been more different than those in the Chicago which rivaled formed by the examples of of the late nineteenth century, Moscow Univer- those foreign systems. In par- sity, like the University of Chicago, supported a that of many ticular, many of them adopted mathematics program under an activist influ- the production of research and enced by contemporaneous mathematical de- of their of future researchers as explicit velopments in both Germany and France. Niko- German missions for their faculties and lai Vasil’evich Bugaev journeyed to Berlin in schools [18, pp. 261–294]. 1863, where he studied under Kummer and competitors. At the University of Chicago, Weierstrass, and then continued on to Paris to for example, a university fi- hear the lectures of Liouville, Chasles, Serret, and nanced by Rockefeller and others. After some two and a half years abroad, opened in 1892, a strong em- Bugaev returned to Moscow, where he took a pro- phasis was placed from the very fessorship in 1867, the year after earning his doc- beginning on securing a talented torate there for work on numerical identities in- research faculty. In the words of the university’s volving the exponential function ex. From 1867 first president, William Rainey Harper, this until his death in 1903, he continued his work faculty would seek “not to stock the student’s in number theory, striving in particular to de- mind with knowledge of what has already been velop general methods in a subject given to accomplished in a given field, but rather so clever solutions of often limited applicability to train him that he himself may be able to [16, pp. 198–200]. push out along new lines of investigation” [18, Bugaev, however, did more at Moscow than p. 278]. In mathematics, Harper succeeded in his own mathematical research. He influenced a bringing together three men who embodied these corps of colleagues and students through his same ideals. broader conception of mathematics. For Bugaev, The American Eliakim Hastings Moore and mathematics involved communication, which he the two Germans, Oskar Bolza and Heinrich fostered through his vigorous support of the Maschke, implemented a training program in Moscow Mathematical Society and, after 1866, of mathematics at Chicago which rivaled that of its journal, Matematicheskii Sbornik. It also hinged many of their German competitors [18, p. 367]. on its university setting, which he worked to This comes as no surprise in light of the facts strengthen and enhance at Moscow through his that Bolza and Maschke had learned their trade efforts first as secretary and then as dean of its from Felix Klein and that Moore had spent a faculty of physics and mathematics. Most year abroad studying mathematics in Göttingen importantly, it depended on training students

292 NOTICES OF THE AMS VOLUME 43, NUMBER 3 capable of contributing to its further develop- 1865, served as a model for ment. To the latter end, Bugaev taught a wide mathematical organizers range of courses—in, for example, number the- throughout Europe and in ory, the theory of elliptic functions, the calculus the United States. Not only of variations, and the theory of analytic func- did it bring together math- tions—which aimed to introduce his students to ematicians in and around these subjects at the research level. He also fos- London and eventually tered and contributed to a philosophical atmos- throughout England for the phere in which mathematics was interpreted es- presentation and discus- sentially as a theory of functions and where the sion of mathematical re- theory of discontinuous functions played a key sults, but it also published role. This conception not only proved conducive from the outset the Pro- to the acceptance of Georg Cantor’s novel set-the- ceedings of the LMS for the oretic ideas but also served as the foundation of further dissemination of the Moscow school of function theory, spear- original research [20; and 6, headed in the early decades of the twentieth cen- pp. 577–581]. tury by Bugaev’s student, D. F. Egorov [7], and per- Looking across the petuated by Egorov’s disciple, N. N. Luzin. This Channel, the Society’s first school, which also included such influential twen- foreign member, Michel tieth-century mathematicians as P. S. Aleksan- Chasles, called for his own drov, A. Ya. Khinchin, and D. E. Menshov, made countrymen to follow the seminal contributions to the advancement of mea- British example. In his 1870 sure theory and the general theory of functions report on the progress of Photograph courtesy of the Department Mathematics, University Chicago of a real variable [19]. mathematics in France, Eliakim Hastings Moore The cases of both Moscow University and the Chasles strongly advocated University of Chicago drive home the obvious the formation of a society point that the success of the mathematical en- specifically for math- deavor in a given national context depends cru- ematicians, in contradis- cially on the process of training talented students tinction to one in which in areas rich in interesting open questions. At its mathematicians repre- core, mathematics undeniably involves proving sented only one of the sci- theorems, and these students not only learned entific constituencies. Such how to carry out that creative process success- an organization would fully but also embraced the belief that they serve to focus mathemati- should pass on their insights to a subsequent cians on their discipline per generation. As they had been trained, so should se, its technical develop- they train—this philosophy came to character- ment as well as its broader ize the mathematical mission internationally in structural needs. He also the latter quarter of the nineteenth century. underscored the impor- Moreover, in concert with the other factors ex- tance of liberal member- amined here, it encouraged the formation of ship policies which would self-sustaining mathematical communities, that permit all mathematicians is, interacting groups of people linked by com- to join and participate. In mon interests. his view, the élitist and ex- clusionary membership The Establishment of Lines of policies of institutions like Communication the Académie des Sciences Photograph courtesy of the London Mathematical Society The formation of a community, however, also did little to promote the Michel Chasles turns upon the ability of its members to com- overall French mathemati- municate effectively. The time period cal endeavor. Moreover, the 1875–1900—one in which telegraphy, railroad extant journals—the Comptes Rendus of the systems, steamships, and the printed word Académie, the Annales Scientifiques of the École linked nations internally and with each other— Normale Supérieure, and even Liouville’s Jour- witnessed the widespread creation of at least two nal des Mathématiques Pures et Appliquées—all sorts of communications vehicles dependent on limited access to publication for reasons inde- this new level of mobility: the mathematical so- pendent of mathematical quality. A French math- ciety and the specialized mathematical journal. ematical society could provide, in a sense, a free Although the Moscow Mathematical Society and independent outlet for the publication of its predated it, the London Mathematical Society, members’ work [12, pp. 14–17]. The institutional which first met under that name in January of void which Chasles sensed in French math-

MARCH 1996 NOTICES OF THE AMS 293 ematics was ultimately filled in 1872 following dred members [18, pp. 267–268]. As with the SMF the formation of the Société Mathématique de in France, the AMS came to define the primary France and its Bulletin. locus of professional mathematical activity at the Similarly, the example of the English math- research level in the United States by 1900. ematicians, at least to some extent, informed ini- As this brief discussion of the establishment tiatives taken in Palermo and in New York City of national mathematical societies highlights, for the promotion of research-level mathemat- mathematicians during the closing quarter of the ics. Giovan Battista Guccia, a student of both nineteenth century recognized the importance Brioschi and Cremona, estab- of communication both in person and in print lished the Circolo Matematico for the advancement of their discipline. Their or- di Palermo in 1884, inspired by ganization of societies further reflected their the examples of both the growing sense of mathematics as a profession, The time French Association for the Ad- while their creation of new publication outlets vancement of Science and the reinforced the standards adopted for that pro- period LMS. His objectives for the new fession.10 By the late nineteenth century, to be society, however, were in some a mathematician meant the same thing in- 1875–1900… sense more outward-looking ternationally: namely, to produce and to share than those of either of the so- the results of original research with like-minded witnessed cieties which served as his members of an extended community of math- model. Guccia sought to create ematical scholars both at home and abroad.11 the…creation an organization which united of two sorts of mathematicians internationally An International Overview: Some Common at the same time that it spurred Denominators advanced work in Italy by draw- communication: This brief comparative inquiry now suggests a ing it into the wider interna- number of the parameters crucial to the devel- tional arena [5, pp. 51–75]. He the opment of the mathematical endeavor during the worked to achieve these goals last quarter of the nineteenth century. First and by actively soliciting foreign mathematical foremost, the establishment internationally of a members as well as by pub- mathematical profession during this time period society lishing, beginning in 1885, the largely—although not exclusively—hinged on Circolo’s Rendiconti. Guccia rec- changes in higher education in the various national and the ognized the distinct advantages settings. Although these changes came about of an interna- often through very different sequences of events— specialized tional as opposed to a strictly mathematical national posture for the overall political, financial, philosophical, pedagogical— vitality of mathematics at the and so for very different reasons in different na- journal. research level. tional contexts, they nevertheless tended to Although the vision of his provide increasingly conducive settings within contemporaries in New York higher education for the study and pursuit of re- City was initially not as far- search-level mathematics. In particular, the adop- reaching as Guccia’s, the tion of nationally tailored versions of the twin Ger- founders of the New York Mathematical Society man principles of Lehr- und Lernfreiheit in various also had the general furtherance of mathemat- countries brought with it a redefinition of the ics as their primary goal when they met for the role of the professor of mathematics, which in- first time in 1888. In six years, this group, which creasingly encompassed the dual activites of had started out as little more than a mathematics teaching and the production of original research. club for Columbia College with a total member- Education at a graduate level thus developed in ship of only six, had begun publishing its Bul- order to train students capable of realizing these letin, had assumed a nationwide purpose in two goals, and this training took place in the lec- changing its name to the American Mathemati- ture hall as well as within the context of another cal Society, and had grown to well over two hun- keystone borrowed from the German educational system, the seminar. 10Here, we could clearly also cite many examples of jour- As the discovery of new mathematical results nals founded during this time period which were in- came to set the standard of entry into the evolv- dependent of the mathematical societies formed: the ing profession, the discipline defined itself in in- Mathematische Annalen founded by Alfred Clebsch in Germany in 1868, Gaston Darboux’s Bulletin des Sci- 11For a quantitative sense of the depth of the Ameri- ences Mathématiques started in 1870, and the Amer- can mathematical research community, see, for ex- ican Journal of Mathematics begun by James Joseph ample, [9] and [10]. [12] provides an analysis of the Sylvester in the United States in 1878, to name just three broader French mathematical constituency, and [17] of the earlier periodicals. gives some indication of the situation in Spain.

294 NOTICES OF THE AMS VOLUME 43, NUMBER 3 creasingly specialized terms. Universities split References their chairs of mathematics and physics or of mathematics and astronomy and even created [1] Elena Ausejo and Ana MillÁn, The Span- chairs in specific mathematical areas such as ish Mathematical Society and its periodicals geometry and higher algebra. This specialization in the first third of the 20th century, in Mes- resulted both in the sharpening definition of sengers of Mathematics: European Math- subdisciplines within mathematics and in an in- ematical Journals (1800-1946) (Elena Ausejo crease in the number of positions available in the and Mariano Hormignón, eds.), Siglo XXI de field. This latter aspect of the evolution of a España Editore, Madrid, 1993, pp. 159–187. profession was also influenced by the estab- [2] Herbert Beckert and Horst Schumann lishment of new grades of instructors under the (eds.), 100 Jahre Mathematisches Seminar professor (Dozenten, maîtres de conférences, der Karl-Marx-Universität Leipzig, VEB assistant and associate professors, etc.). As Deutscher Verlag der Wissenschaften, Berlin, individuals sought out this graduate training, as 1981. they assumed these new positions, as they [3] Kurt-R. Biermann, Die Mathematik und ihre adopted these values of teaching and research, Dozenten an der Berliner Universität, they banded together in national or broadly 1810–1933, Akademie-Verlag, Berlin, 1988. [4] Umberto Bottazzini, Il diciannovesimo se- based mathematical societies and shared their colo in Italia, in Mathematica: Un Profilo new research through specialized journals tar- Storico by Dirk Struik, Il Mulino, Bologna, geted at an appreciative and understanding au- 1981, pp. 249–312. dience. The individual nationalization of math- [5] Aldo Brigaglia and Guido Masotto, Il Cir- ematics was thus well under way by the end of colo matematico di Palermo, Edizioni Dedalo, the nineteenth century; and since the model em- Bari, 1982. ulated was largely the same in the various na- [6] E. F. Collingwood, A century of the London tional contexts, this implies that the internatio- Mathematical Society, Journal of the Lon- nalization of the field was likewise in process. don Mathematical Society 41 (1966), Perhaps no one piece of evidence supports this 577–594. latter conclusion better than the fact that Zürich [7] Sergei Demidov, N. V. Bougaiev et la créa- hosted the first International Congress of Math- tion de l’école de Moscou de la théorie des ematicians in 1897. fonctions d’une variable réelle, in Mathe- An international perspective on the develop- mata: Festschrift für Helmuth Gericke ment of mathematics over the period from (Menso Folkerts and Uta Lindgren, eds.), roughly 1875 to 1900 thus uncovers a number Franz Steiner Verlag Wiesbaden GMBH, of factors common to particular national settings Stuttgart, 1985, pp. 651–673. which strictly nationally oriented studies tend [8] Peter Duren et al. (eds.), A century of math- perhaps to obscure. It also provides at least a ematics in America, History of Mathematics, sense of the complexity of the process of the in- vols. 1–3, American Mathematical Society, ternationalization of mathematics. Mathemati- Providence, RI, 1988–1989. cians today tend to take the international nature [9] Della Dumbaugh Fenster and Karen of the field for granted, but it is really an aspect Hunger Parshall, A profile of the American of the discipline that has come about primarily mathematical research community: in the twentieth century as a result of dynamic 1891–1906, in The History of Modern Math- changes, especially during the closing quarter of ematics (Eberhard Knobloch and David E. Rowe, eds.), vol. 3, Academic Press, Boston, the nineteenth century. 1994, pp. 179–227. [10] ———, Women in the American mathematical research community: 1891-1906, in The His- tory of Modern Mathematics (Eberhard Knobloch and David E. Rowe, eds.), vol. 3, Academic Press, Boston, 1994, pp. 229–261. [11] Gerd Fischer, Friedrich Hirzebruch, Win- fried Scharlau, and Willi TÖrnig (eds.), Ein Jahrhundert Mathematik 1890–1990: Festschrift zum Jubiläum der DMV, Doku- mente zur Geschichte der Mathematik, vol. 6, Friedr. Vieweg & Sohn, Braunsch- weig/Wiesbaden, 1990. [12] HÉlÈne Gispert, La France mathématique: La Société mathématique de France

MARCH 1996 NOTICES OF THE AMS 295 (1872–1914), Société française d’histoire des tory of Mathematics, vol. 8, American Math- sciences et des techniques & Société math- ematical Society, Providence, RI, and London ématique de France, Paris, 1991. Mathematical Society, London, 1994. [13] Ivor Grattan-Guinness (ed.), Companion [19] Esther Phillips, Nicolai Nicolaevich Luzin encyclopedia of the history and philosophy and the Moscow School of the Theory of of the mathematical sciences, 2 vols., Rout- Functions, Historia Mathematica 5 (1978), ledge, London and New York, 1994, 275–305. 2:1427–1539 on higher education and in- [20] Adrian C. Rice, Robin Wilson, and J. Helen stitutions. Gardner, From student club to national so- [14] ———, Convolutions in French mathematics, ciety: The London Mathematical Society 1800–1840: From the calculus and mechanics 1865–1867, Historia Mathematica 22 (1995), to mathematical analysis and mathemati- to appear. cal physics, 3 vols., Birkhäuser Verlag, Basel, [21] David E. Rowe, Klein, Hilbert, and the Göt- 1990. tingen mathematical tradition, Science in [15] Mariano HormigÓn, García de Galdeano Germany: The Intersection of Institutional and El Progreso Matematico, in Messengers and Intellectual Issues (Kathryn M. Olesko, of Mathematics: European Mathematical ed.), Osiris 5 (1989), 189–213. Journals (1800-1946) (Elena Ausejo and Mar- [22] Norbert Schappacher, Fachverband—In- iano Hormignón, eds.), Siglo XXI de España stitut—Staat, in [11], pp. 1–82. Editores, Madrid, 1993, pp. 95–115. [23] Gert Schubring, The conception of pure [16] A. N. Kolmogorov and A. P. Yushkevich, mathematics as an instrument in the pro- Mathematics of the 19th century: Math- fessionalization of mathematics, in Social ematical logic, algebra, number theory, prob- History of Nineteenth Century Mathematics ability theory (H. Grant, Abe Schenitzer et al., (Herbert Mehrtens, Henk Bos, and Ivo Schnei- trans.), Birkhäuser Verlag, Basel, 1992. der, eds.), Birkhäuser, Boston, 1981, pp. [17] Eduardo Ortiz, El rol de las revistas 111–134. matemáticas intermedias en el estable- [24] ———, Pure and applied mathematics in di- cimiento de contactos entre las comunidades vergent institutional settings in Germany: de Francia y España hacia fines del siglo The role and impact of Felix Klein, in The His- XIX, in Contra los Titanes de la Rutina/Con- tory of Modern Mathematics (David E. Rowe tre les Titans de la Routine (Santiago Garma, and John McCleary, eds.), 2 vols., Academic Dominique Flament, and Victor Navarro, Press, Boston, 1989, 2:171–220. eds.), Comunidad de Madrid/C.S.I.C., Madrid, [25] Terry Shinn, The French science faculty sys- 1994, pp. 367–382. tem, 1808–1914: Institutional changes and [18] Karen Hunger Parshall and David E. research potential in mathematics and the Rowe, The emergence of the American math- physical sciences, Historical Studies in the ematical research community (1876-1900): Physical Sciences 10 (1979), 271–332. J. J. Sylvester, Felix Klein, E. H. Moore, His-

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