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