France's political transition from the Second Empire to the Third Republic was accompanied by a mathematical transition of which one remarkable feature is an increased interest in German research. In this period, French mathematicians not only studied German work, they absorbed aspects of its dominant values. The shift toward German-style pure mathematics is not mirrored in other aspects of cultural life, and special factors mediating these developments must be sought, the more so because of the anti-German sentiment in France following the Franco-Prussian War of 1870-1871.
In this paper, I investigate the roles of Gaston Darboux and Charles Hermite in the dissemination of German work to French audiences. This was a multifaceted effort, involving the translation and publication of both abstracts and articles, the encouragement of theses on subjects of German origin, the reform of curriculum at the Paris Faculté des sciences and elsewhere, and the cultural recognition of German mathematicians through appointments to the Académie des sciences and through the award of medals. I conclude with a brief discussion of the overall impact of these efforts on French mathematics in the last two decades of the nineteenth century.
In 1559, Philipp II forbade Spanish scholars to study abroad and decided to censor foreign books. Much has been written about the ensuing Spanish intellectual isolation. The history of mathematics in Spain shows, however, that, even under the most difficult political circumstances, some individuals and even groups of scholars were perfectly aware of what was going on in mathematics elsewhere, at least at the level of school mathematics.
From the standpoint of the development of mathematics, it is thus necessary to cast the problem of isolation in terms of the mathematical community. In this sense, the process of involvement of Spain in the international mathematical mainstream could start only when a Spanish mathematical community was large enough to result in processes of institutionalization and professionalization. The efforts of leaders such as García de Galdeano, José Echegaray, and Eduardo Torroja y Caballé by the end of the 19th century allowed the Spanish mathematical community to begin this process at the turn of the 20th century. Two institutions--the Spanish Mathematical Society and the Mathematical Laboratory and Seminar of the Council of Research--and one journal--the Revista Matemática Hispano-Americana--were key agents in this development, which ended suddenly with the outbreak of the last Spanish Civil War (1936-1939).
The journal, Acta Mathematica, was founded in Stockholm in 1882. Much of the credit for its continuing reputation as one of the leading international journals in mathematical analysis goes to its founder and first editor, Gösta Mittag-Leffler. Mittag-Leffler, who was the first professor of mathematics at the new Stockholm University and the leading mathematician in Sweden, was well placed to promote an international journal, but, more significantly, he was committed to international networking. He labored tirelessly to promote connections between mathematicians across Europe and beyond, and he himself established friendships with many of the foremost mathematicians of the day, including Henri Poincaré, Georg Cantor, and Sonya Kovalevskaya, all of whom contributed to early volumes of the journal.
In this talk, I discuss Mittag-Leffler's role in the foundation and running of Acta. In particular, the contributions of Poincaré, Cantor, and Kovalevskaya receive special attention, since their involvement with the journal emphasizes the international nature of the enterprise and illustrates the symbiotic relationship that can exist between a journal and the mathematicians whose work it serves to promote. An analysis of the contents of the first 20 volumes of Acta provides an additional perspective on the first 15 years of Mittag-Leffler's administration.
During the years from the 1880s to the outbreak of World War II, mathematicians like Felix Klein, Georg Cantor, and Gösta Mittag-Leffler stressed the need for and the importance of the international organization of the mathematical community. Efforts in this direction were successful in organizing the first international congresses and in editing the first truly international mathematics journal, Acta Mathematica, but the deep enmity between the two most mathematically advanced nations, France and Germany, was a serious impediment to the official foundation of an international organization of mathematicians. Given this situation, Giovan Battista Guccia, the founder of the Circolo matematico di Palermo, recognized that local as well as international conditions were apt to transform the Circolo from a national organization to a truly international one.
Here, I discuss the role played by the Circolo matematico di Palermo in the building of an international mathematical community before the Second World War. Through his cultural politics, Guccia tried to achieve this broader goal. Ultimately, though, growing international contradictions, the cultural trends of the Italian mathematical community, and difficulties inherent in the local (Sicilian) social and economic environment all contributed to his incomplete success.
Mathematics in the West has long been associated with teachers, schools, and a variety of institutions, including universities, academies, and journals, by means of which mathematical communities have been created and defined. In the course of the 19th century, mathematics in Europe became less parochial, less nationalistic, and by the end of the century the first truly international meetings for mathematicians were being held in Europe and North America. Only recently, however, has China come into the international picture, adopting in the process a basically Western model of teaching, publishing, and institutionalizing mathematics.
Traditionally, China has supported a large class of practitioners who have used mathematics in a variety of ways for pragmatic ends. Interests that in the West would be considered didactic or theoretical were basically foreign to the Chinese temperament for a variety of reasons. Thus, important questions about society and ideology arise in trying to account for what happened when Chinese scholars first encountered Euclid and the axiomatic method through the translation efforts of Jesuits and other missionaries in the late Ming and Early Qing dynasties.
At the time, the Chinese cannot be described as a receptive audience, but this was to change. Two centuries later, following the Opium War (1839-1842) which opened China to the West, and the Self-Strengthening movement which stimulated thoughts of reform, a series of events occurred that ultimately transformed the perceptions of Chinese intellectuals. China's defeat by France in 1885, its war with Japan and eventual defeat in 1894-1895, the Boxer Rebellion and finally, the Revolution of 1911, opened the eyes of reformers to the importance of adopting both Western practice as well as theory if China were to compete and survive in the modern world.
This point of view was eventually reflected in educational changes throughout China. The constitutional reform movement of 1898 under the Emperor Guang Xu resulted in new institutions and modernization of education, including the founding of the Capital University (later Beijing University). Although the Reform Movement of 1898 only lasted 100 days, by 1905 the central government had done away with the civil service examinations and a new Education Department (Xue Bu) had been founded. After the Revolution in September of 1912, further "Amendments to the Education System" were made, and mathematics became a compulsory subject to be studied by all students.
In the course of these dramatic changes in China, Western mathematicians began to visit China, and for the first time Chinese intellectuals encountered scholars who were not missionaries interested first in conversion and only incidentally in science. Similarly, for the first time, Chinese scholars in modest numbers began to travel and study abroad. In learning Western mathematics, they were also exposed to and quickly adopted more than just its content. As the mathematical community in China became increasingly Westernized, and correspondingly internationalized, schools, colleges, universities, mathematical societies, academies, and journals all began to fashion an indigenous research community. How this occurred will be the subject of this lecture, with a focus on the international community to which modern Chinese mathematicians were indebted and on which they largely depended in the course of forging a local, indigenous mathematical community in China in the 20th century.
The nineteenth century, in Britain as elsewhere, witnessed unprecedented growth, not only in the mathematical sciences but also in the community of practitioners who considered themselves professional mathematicians. This growing community required lines of communication and outlets for the publication of research; these needs fueled a rapid increase in the medium of specialized mathematical journals.
As the British mathematical community matured during this century, it gradually emerged from the comparatively isolated position it had previously occupied in relation to mathematics abroad. This change is reflected in a slow but steady increase in the number of mathematicians from overseas who chose to publish their work in British journals.
This talk considers the period leading up to the foundation of the London Mathematical Society in 1865, as viewed through the pages of the increasing number of mathematically related journals. A look at foreign contributions to publications such as the Cambridge Mathematical Journal, the Quarterly Journal of Pure and Applied Mathematics, and the Philosophical Transactions of the Royal Society, provides a glimpse at the introduction of the British mathematical community to the international mathematical arena.
As American mathematics consolidated and grew in the early decades of this century, the American mathematical community necessarily drew from and gave to the larger international group of mathematicians as a whole. Using the influential career of Leonard Dickson as a point of departure, I explore the international influences on American mathematics as well as the American role in an increasingly international community of mathematicians.
In 1931 and 1932, George David Birkhoff and John von Neumann published the first theorems of modern ergodic theory in the Proceedings of the National Academy of Sciences ( PNAS). Many papers followed, often in the PNAS; the new subject attracted people in many places, notably A. I. Khinchin in Moscow and students or colleagues of Birkhoff's at Harvard. Fortunately, Eberhard Hopf's book, Ergodentheorie, brought much of the work together when published in 1937 in the Ergebnisse series.
In this talk, I discuss several interesting international aspects of this story: the different traditions of Birkhoff and von Neumann; the interactions between these traditions which led to the ergodic theorems; the international influence of the American work; Hopf's moves from Berlin to Cambridge (USA) in 1930 and to Leipzig in 1936; and the relative isolation of the Moscow school.
Between 1939 and 1942, the Mexican National University established the Department of Mathematics (within the Faculty of Sciences) and the Institute of Mathematics. Only a year later, in 1943, some of the participants of the first National Congress of Mathematics decided to create their own native society. It is natural to suggest that American (due to their geographical closeness) and Spanish (due to their role as political refugees during the Spanish Civil War) mathematicians must have played critical roles in this process of professionalization. In fact, they did, but in this talk, I suggest that there were other "national" influences, endorsed by philosophical, political and idiosyncratic elements, that played a more powerful role.
The internationalization of mathematical activity seems to have developed transcending national boundaries and characterization. How could such a process have evolved, however, during periods when a sense of national political identity is most strongly exerted? The evolution of France's international role after its defeat in the Franco-Prussian War of 1870 provides an interesting case study through which to examine this question.
One of the major effects of the French defeat was a national sense of the need for recovery in the scientific fields and in higher education. To these ends, two national societies, in particular, were created: the Société mathématique de France and the Association française pour l'avancement des sciences. Both developed distinct mathematical networks, on a national as well as on an international scale. French mathematicians thus pursued international activities on a variety of mathematical stages.
In this talk, I consider, in the context of late nineteenth-century France, the variety of actors, of mathematical standards and research ethos, and of outlets for mathematical activity in an effort to shed light on the complexity of internationalization process(es) in the formation of international mathematical communitie(s). This approach also provides insight into the processes of inclusion in and exclusion from communities on both national and international levels.
From the 1780s until the 1820s, French mathematics enjoyed a remarkable dominance, with Paris by far the leading center for the subject in the world. However, partly in reaction to French achievements, other countries began to produce significant mathematicians and/or revised their institutions and curricula. Mathematics has been a much more international activity ever since.
Here, I review this change from these points of view: the diffusion and distribution of French materials, the translation of major French works into other languages, and the extent of development of mathematics from non-French roots. I conclude with some remarks on the place of mathematics in the decline of science which the French claim happened from around 1860s onwards.
In this talk, I consider some of the ways in which mathematics was deemed a language around 1900, thus affording a new view on contemporary debates about the foundations of mathematics. A wide variety of positions were taken about the correct relationship between thought, language, logic, mathematics, and the world, sometimes with an intensity that can only be explained on historical and political grounds. The rediscovery of Leibniz's work stimulated the search for well-constructed languages adequate at least for science, such as the systems of Peano and Schröder were claimed to be. I close with some reflections on Hilbert's philosophy of mathematics, deliberately taking less familiar paths and skirting some of the finer views (afforded from the ridges of the Frege and Peirce mountains) in order to open up a new vantage point. Along the way, I show that for a number of reasons it is possible to speak of a distrust of language at the time, and I argue that recognizing confusions about difficult linguistic issues helps us understand the old debates.
Scientific periodicals appear as organs of scientific institutions created during the Scientific Revolution (e.g., the Philosophical Transactions of the Royal Society of London or the Journal des savans) or as a means of communication of incipient groups of philosophes (e.g., the Acta Eruditorum or the Journal de Trevoux). In this early period, the national dimensions of the European scientific community were so small that international communication was necessarily fostered.
K. H. Hindenburg published in Leipzig, between 1795 and 1805, the first mathematical journal in history, Archiv der reine und angewandte Mathematik, whose title remainded imprinted on mathematical minds. Joseph Diez Gergonne's periodical title, Annales de mathématiques pures et appliquées, was quite similar, and it is often considered the first mathematical periodical because of its greater circulation. Two other great journals, Leopold Crelle's Journal für die reine und angewandte Mathematik and Joseph Liouville's Journal de mathématiques pures et appliquées also carried similar titles.
The Leipzig periodical was local, Gergonne's Annales mainly French. Nevertheless, mathematical periodicals enjoyed a certain international hold even at this early stage: Gergonne's journal was a product of the French Revolution, which was internationalist--like all the great revolutions. The plurality of centers and subsequent rivalry came later. The first--and many later--mathematical periodicals are thus the work of individual efforts, but those conceived of as long-term and widely ranging geographically arose out of the strongest mathematical communities throughout the 19th century.
Mathematics--actually science--was among the rare human activities that adopted the intellectual and social internationalist trends that emerged in Europe from the 1848 Revolution onwards. Most of the mathematical institutions that insured the international projection of mathematical work had developed by World War I (1914-1918). The war, however, had a negative influence on this process, as the International Mathematical Congress held in 1920 in Strasbourg showed. From then on examples of differently based isolation of mathematical communities--even if they were in the vanguard--appeared, and nationalism played its role in relationships among mathematicians, which obviously influenced the editorial policy and international scope of mathematical journals. The dialectic tension between internationalist desires and emerging nationalistic realities turned the ideals of fraternity among people--and therefore among mathematicians--into hegemony. Wars generate strange phenomena, like mirages, even in mathematics.
During the nineteenth century the scientific ambiance changed dramatically. The greatly strengthened middle class in Europe showed increasing interest in science and technology; higher education expanded; and research was elevated within the universities to assume a principal role alongside teaching. The number of scientists multiplied; national scientific societies were founded in rapidly increasing numbers; and gradually, scientific cooperation across the borders began to assume organized forms. The origins of the International Mathematical Union (IMU) are connected with these broader developments.
In mathematics, an important milestone was the first International Congress of Mathematicians, held in Zürich in 1897. The Congress immediately became a permanent institution with the important task of preventing mathematics from splitting into diverse branches. Congresses could form special commissions to implement their aims, and it was in this way that an embryo of the IMU came into existence. The idea of establishing a true union, however, did not progress in the years before the outbreak of the Great War in 1914.
The IMU was founded in the aftermath of the war in 1920 by the victorious Allies. Neutral countries were invited to join, but the defeated Central Powers--Germany, Austria, Hungary, and Bulgaria--were barred from membership. The International Congresses, formally connected with the IMU, followed the same policy.
Opposition soon began to grow against this policy of exclusion. In 1926, the parent organization, the International Research Council (the forerunner of ICSU) decided to invite the former Central Powers to become members of the Council and its Unions. However, German scientists, disregarding the recommendations of the German government, declined to join.
The Italian organizers of the 1928 Congress, supported by worldwide opinion among mathematicians, made attendance at that Congress free from political restrictions. Formally, the participation of the Germans violated the rules of the IMU, whose Secretary General protested vehemently. The Congress ignored the protest with the result that the IMU lost its grip on the Congresses. Doubts about the need and usefulness of the Union were gathering strength. In 1932, the IMU was suspended in a stormy session. Yet a committee was set up to study the possibility of re-creating it. By the time of the 1936 Congress, the political situation in the world had deteriorated. The Congress endorsed the Committee's report that, although an international organization of mathematicians would be very useful, the time was not right to found one.
Preparations for the 1940 Congress, to be held in the United States, continued into 1939 and the outbreak of World War II. Developments that led to the 1950 Congress and to a new IMU were purely apolitical.
Liouville created his Journal de mathématiques pures et appliquées in 1836 as a deliberate French counterpart to Crelle's German Journal für die reine und angewandte Mathematik. Despite its nationalistic purpose, 30% of its papers were written by non-French authors during the 39 years that Liouville served as editor. These papers helped French mathematicians get an idea of what went on in other parts of the world, in particular, in Germany. In this talk, I show, by means of a number of examples, how Liouville's own research was influenced by these foreign contributions.
In the years following the unification (1861) of Italy, the mathematical community of Bologna had made some gains relative to bringing its mathematical teaching up to international standards owing to the presence on its faculty of mathematicians like Luigi Cremona, Quirico Filopanti, Eugenio Beltrami, and Domenico Chelini. At the beginning of the 1880s, Bologna finally was able to offer a full baccalaureat course of study in mathematics thanks to the addition of excellent professors--Cesare Arzelà, Salvatore Pincherle, and Luigi Donati--to the Faculty. It was in this particular historical period of renewal and intellectual growth that Cesare Arzelà, Professor of Infinitesimal Calculus and Higher Analysis, gave a public course on Galois theory in the academic year 1886-1887, the first in Italy on this subject. The audience included the future mathematician and historian, Ettore Bortolotti (1866-1947), who took the notes of the lectures. In order to present these lectures to his fourth-year students, Arzelà consulted the most significant published texts on the subject in Europe. He selected the best books that the international community had to offer--Lejeune Dirichlet's Zahlentheorie, Joseph Alfred Serret's Cours d'algèbre supérieure, Eugen Netto's Substitutionentheorie, Camille Jordan's Traité des substitutions et des équations algébriques--and he assessed the various presentations, alterating and elaborating them in full knowledge of the neeeds of his students auditors. In so doing, he brilliantly succeeded in organizing a cogent and pedagogically sound course of lectures, gathering what he viewed as the best of the best from the international published landscape. Arzelà's course exemplifies the international exchange of mathematical ideas and underscores the fruitfulness of such exchange for developing mathematical communities.
Chinese mathematics began to filter into Japan in the seventeenth century when works such as the Chinese translation of Euclid's Elements by western missionary, Matteo Ricci, and Xu Guangqi confronted wasan, or traditional Japanese mathematics. In the eighteenth century, international influences in mathematics came from a very different sector, the Netherlands, as a result of its trade contacts with Japan.
By the nineteenth century, western style mathematics had secured a strong foothold in the Japanese educational system. The years from 1855 to 1868 saw the introduction of western mathematics in Japanese military schools, notably the Nagasaki Naval Training School and the School of the Yokosuka Shipyards, while the decade from 1868 to 1878 brought more widely spread educational reform with the adoption of western arithmetic in the Japanese school system and the appointment of David Murray as an educational consultant. These dramatic changes immediately preceded the founding in 1877 of both the University of Tokyo and the Tokyo Mathematical Society and paved the way for what may be termed the "Germanization" of the Japanese political system as well as of learning in Japan.
Following the political coup d'état of 1881, new educational institutions like the Imperial University of Tokyo (founded in 1886) employed faculty such as Fujisawa Rikitaro and Takagi Teiji who had been trained in the German tradition, and these professors introduced not only the research-level of Europe but also the German seminar for the encouragement and support of research activities. By 1911, yet another university, the Tohoku Imperial University, was founded in addition to Japan's first western style, research-level journal, the Tohoku Mathematical Journal. The democratization as well as the internationalization of Japanese mathematics during the postwar period has further been evidenced by the founding of the Mathematical Society of Japan in 1946 and by Kyoto's hosting of the twenty-first International Congress of Mathematicians in 1990.
National mathematical societies began to be formed about 110 years ago. Shortly thereafter, the first International Congress was held in Zürich in 1897. At the second in Paris in 1900, thirty-eight-year-old David Hilbert presented his famous list of 23 problems for the future. Yet these early Congresses seemed to have more the flavor of nationalistic showing off of mathematical talents than that of internationalistic cooperation and community.
At the end of World War I, the French in particular (but initially also the British, Italians, Americans, and others) were determined that the militarily defeated Central Powers had excluded themselves from the human family, and so should be excluded from all future common cultural activity. This included mathematics, and Germans, Austrians, and others were excluded from "International" Congresses until 1928. Very good mathematics continued to be done by the excluded, and some extreme nationalists among them wanted to respond to exclusion with boycott.
Paradoxically, after World War II the attitude of the victors towards the defeated in cultural matters was exactly the opposite, despite the tremendous devastation. The refugees fleeing Germany for the United States from 1933 onward no doubt played a role in this movement towards a truly international community of mathematicians. However, to what extent they did is complicated by several factors, among them, their often less-than-warm reception in the United States, the possible misevaluation of their contributions, and the differing mathematical points of view of emigrants and those already working in the United States. Similar considerations seem to apply to other countries of refuge as well. That the mathematical refugees were a fortiori "international" did not by itself create an international community of mathematicians.
Hypotheses about the causes of the differences between the first fifty and second fifty years of this century will be discussed.
Here, I discuss the principal effects of Nazi rule on international mathematical communication after 1933 relative to the work of mathematicians within Nazi Germany. Restrictions due to both political isolation and militarization (e.g., secrecy regulations) and to ideological resentments on the part of the rulers as well as of some conservative mathematicians were sharpened by economic conditions (autarky and valuta shortage). These restrictions manifested themselves above all in participation in international conferences (at Oslo in 1936 and in sessions on mechanics at Cambridge in 1938) and in publications (in the Zentralblatt affair in 1938, in the preparation of the second edition of the Enzyklopaedie, and in journals). At the same time, however, there was a tendency toward the internationalization of mathematics under German hegemony which was reaching out for influence on Southern and Eastern European mathematics and which was trying to induce the collaboration of French mathematicians under the conditions of German occupation after 1940.
In this talk, I give a preliminary and relatively brief exploratory discussion of the mathematical relations between the United States and China from 1859, when an American mathematical textbook was first translated into Chinese, to 1949, the year that diplomatic relations between the two countries ceased. I open with a discussion of the transmission of modern mathematics from the United States to China, considered largely in terms of translation. This provides insight into why the Chinese were more interested in American mathematics than in that of Europe. I then focus on those Chinese students who studied mathematics in American universities, in particular those at Harvard University, who returned to China to introduce what they had learned to a new generation of Chinese students. Both this first and second generation played a crucial role in the modernization of Chinese mathematics. Moreover, some American mathematicians--like George David Birkhoff, William Fogg Osgood, and Norbert Wiener--contributed to this transmission process directly through series of invited lectures in China. Finally, some Chinese mathematicians--like Shiing-shen Chern and Tsai-han Kiang--did important research while visiting American institutions and universities, in particular the Institute for Advanced Study. Mathematical relations between China and the United States thus took a variety of forms during the period from 1859 to 1949.