The Organization of the Mathematics Community

Overview

Along with the establishment of international fellowships, institutes and prizes such as the Fields Medal, both the formal and the informal formation of an international community of scholars of mathematics proved a powerful shaping force to the course of twentieth-century mathematical thought and research. In essence, the rise of an international mathematics community in the early part of the twentieth century was, with particular regard to scholarly communication, analogous to the Internet revolution at the end of the century. In both cases new communications media revolutionized mathematics teaching, publishing, and institutions.

Background

During the nineteenth century mathematical methods and applications became increasingly useful to science, engineering, and economics. Mathematicians also developed mathematical logic and abstract definitions, complex relations, and theorems of pure mathematics. Although there were subtle divisions of mathematics at the beginning of the nineteenth century, by the end of the century there were full and formal divisions of pure and applied mathematics. Moreover, within these divisions, mathematicians took on increasingly specialized roles that resulted in a rapid compartmentalization and specialization of mathematics. This often left mathematicians disconnected from their scholarly colleagues.

During the early decades of the twentieth century, the specialization in mathematics accelerated. Mathematicians also increasingly worked in the shadows of tremendous advancements in the characterization of physical law. What was once pure mathematical theory found new expression and emphasis in the advent of relativity and quantum theories. In stark contrast to the insulated and isolated traditions of prior centuries, in such an intellectually tumultuous age, increased communication among scholars was a prerequisite for serious work.

Compared with the nineteenth century, the mathematics community of the early twentieth century seemed relatively cohesive. This cohesion resulted from nineteenth-century efforts to establish an international community of mathematicians and to increase communication between mathematical scholars. There were, for example, efforts to organizing an international meeting of mathematicians during the 1893 Chicago's World Fair. Finding the exchange of information and ideas invigorating, scholars moved to establish the International Congress of Mathematicians.

Beginning with an 1897 meeting in Switzerland, the International Congress of Mathematicians resolved to meet every four years, one of their prime objectives being to prevent the complete isolation of mathematics' diverging branches. Another goal of the early international congresses was to facilitate international communication and cooperation, although they often devolved to national competitiveness regarding the display of mathematical prowess.

The process for selecting the site of the quadrennial congress eventually came under the control of the International Mathematical Union (IMU). The site-selection process eventually became similar in many regards to the selection process for the Olympic Games. IMU members evaluate bids by potential cites to host the congress, and many political and economic factors can influence the ultimate decision regarding site selection. These matters are more than logistical details: decisions concerning where to host a conference can profoundly influence the content and tone of the proceedings. Because the general intent of the congress is to facilitate communication and exchange between mathematical scholars, the location must be accessible both physically and politically for potential participants.

Even more controversial are decisions regarding the content of congressional proceedings. Selected members of the IMU gather to screen potential participants and issue invitations to speakers for the lectures to be delivered at the congress. This also can profoundly influence the mathematics community. Nowhere was this more important than in the proceedings of the 1900 International Congress of Mathematicians in Paris. At the Paris Congress German mathematician David Hilbert (1862-1943) presented 23 famous problems that spurred mathematics research for scholars around the world. Hilbert's articulation of the major problems facing mathematicians profoundly influenced the course of twentieth-century mathematics. Indeed, because of the broad scope of many of Hilbert's problems, some problems may continue to provides challenges to mathematicians for centuries to come. During the later half of the twentieth century it became traditional to award the Fields medals and the Nevanlinna Prize (for information science) on the first day of each Congress.

Organizing the International Congress of Mathematicians was only one of many steps in the formation of an international mathematics community. Prior to the outbreak of World War II, various local and national organizations helped provide an international forum for scholars. The publication of the journal Acta Mathematica, for example, also marked a major step in linking international scholars.

Despite the success and popularity of the International Congress of Mathematics, the organization of the mathematics community was not instantaneous. Progress was made in fitful spurts and often, especially during times of political turbulence and discord, various mathematicians were called upon to renew efforts to establish permanent international ties. During the first half of the twentieth century, however, continuing and bitter feuds between France and Germany not only resulted in two World Wars but also spilled over into the academic world, carving deep divides between these two countries, both of which had outstanding mathematical traditions.

War played an important part in shaping the character of the international mathematics community. Unlike the emergent political institutions painted on the post-World War I political map, most of the international institutions dedicated to the advancement of mathematics were already in place and functioning—at least to a limited degree—early in the twentieth century. The goal of these societies, especially the International Mathematical Congress, was often to simply weather the storms of war.

War-heightened nationalism crept into what otherwise should have been purely academic decisions and affected both the submissions and published content of both mathematical societies and journals. Following World War I, the International Mathematical Union formally bound only the former Allied powers and most of the neutral nations. The vanquished countries, including Germany and Austria, were excluded from membership. As a result of these exclusions research work done in Germany was often ignored or shunned. To the dismay of many scholars, the International Congress also excluded scholars from former other Central Power nations as well.

Most scholars bristled against such exclusions, however, and the isolations were short-lived. By 1926 scholars of all nations were once again welcomed into the IMU. Nationalism was not, however, exclusive to the victorious powers, and many German mathematicians declined the invitation. In an effort to heal wounds, the 1928 conference, held in Italy, adopted a policy that attendance at the International Congress must be free of political and ethnic restrictions. A dispute between the organizers of the Italian conference and the governing powers of the IMU led to a loss of control by the IMU over the organization of the congress. In 1932 the IMU essentially ceased to function, and attempts to revive the union were thwarted by the gathering clouds of the Second World War.

In the aftermath of World War II, scholars found themselves caught in the crossfire of a very different Cold War. Early Cold War tensions prevented a complete reformation of the IMU (along less political lines) until mid-century.

Impact

One benefit of the growth of an international community of mathematicians during the twentieth century was the proliferation of peer-reviewed journals both of a general and specialized nature. In addition, beginning in the later portion of the nineteenth century there were increasing numbers of foreign-based contributions to institutional publications.

A growing international community of mathematical scholars also provided additional outlets for the dissemination of work that, prior to that time, was critiqued only at the more provincial level of the university or academy. The rise of the international community also tended to shunt the dissemination of work among selected scholars—often along nationalistic lines.

After the two World Wars, the world entered an era of superpower military industrialization, in which exhausted nations such as France could no longer maintain a dominant international position. Accordingly, these nations increasingly relied on scientific, mathematical, and cultural advancement as a source of pride. The Société Mathématique de France and the Association Française played in important role in providing a forum for furthering contacts between the rich French mathematical culture and the international mathematical community.

The repressive policies of Nazi Germany had two immediate effects on the international mathematics community. The first was a practical reisolation of Germany and German mathematicians. Few Germans participated in the 1936 Oslo International Congress. The second was a mass exodus of mathematicians from Germany in the 1930s and 1940s, an exodus and dissemination of talent to all parts of the Western world—especially to the United States and Canada. This migration provided the basis for improving postwar international communication.

Progress in formulating a truly international community of mathematicians was also not limited to the West. During the twentieth century the IMU and the International Congress forged links between Western and Eastern nations and, following World War II, Japan and China sent increasing numbers of mathematicians for training in the West.

Especially in a modern world where international dialogue is taken for granted as a byproduct of Internet development, it is important to note the very different conditions that existed at the start of the twentieth century, when mathematicians labored in relative isolation or—at best—with other members of their academic institutions. The rise of the international community provided new forums for communication that further revolutionized mathematics teaching, publishing, and research.

Work to establish an award for mathematical achievement culminated with the institution of the International Medal for Outstanding Discoveries in Mathematics, more commonly known as the Fields Medal. More than just an international research award, the Fields Medal was also designed to promote promising academic talent. For mathematicians, the medals eventually became the equivalent of a Nobel Prize. The awarding of the medals, and the spotlight of publicity brought to the honored mathematicians and topics, profoundly influenced the course of mathematics research during the twentieth century.

That such a general trend toward international cooperation in mathematics could occur in spite of strident nationalistic fervor and the distractions of war was testimony both to the higher ideals of scholarship and to the practical benefits of scholarly discourse.

K. LEE LERNER