The Promotion of Mathematical Research

Overview

The makers of mathematics and their modes of practice were transformed during the nineteenth century. New mathematical ideas were publicized in mathematical journals. Professional mathematicians began to meet together to share their research in local societies, which grew into national organizations. At the same time, these mathematicians created opportunities to complete their research as university professors and to train mathematics professionals. Toward the end of the century, an international congress was established. Indeed, by 1900 a growing community of specialized mathematicians cooperated around the world.

Background

Before 1800, mathematicians were generally isolated in their own nations. Mathematicians supported themselves through patronage from nobles or the state, or with their own personal wealth; even Isaac Newton (1642-1727) served as Master of the Mint in London. They generally exchanged ideas sporadically, in private letters or by publishing books containing their life's work. Although it remained possible for one person to master all existing mathematical knowledge, the body of mathematics was increasing. Still, amateurs and individuals outside institutions continued to dominate the exploration of mathematics since mathematics teachers were usually too busy to do research or to learn the new results on their own.

Around the turn of the nineteenth century, however, mathematicians increasingly gained awareness of mathematical improvements in other nations. One of the most striking examples of this new awareness was in Great Britain. Because of the bitter priority controversy between Newton and Gottfried Wilhelm Leibniz (1646-1716) over the invention of the calculus, British mathematicians were especially disconnected from mathematical developments on the European continent during the eighteenth century. By the 1810s, though, professors such as John Playfair (1748-1819) and Robert Woodhouse noted the superiority of Continental analysis. Then, three students at Cambridge University, Charles Babbage (1792-1871), John Herschel (1792-1871), and George Peacock (1791-1858), founded an "Analytical Society" in 1813 to urge the university to use Leibniz's differential notation, rather than Newton's fluxions, on the annual tripos examination. They also argued for a translation of a calculus textbook by Silvestre François Lacroix, which was printed in 1816. The achievements of the society were limited overall, but the group served as a harbinger of a new, modern attitude toward mathematics in Great Britain.

Communication between mathematicians was further facilitated by the publication of journals. While earlier periodicals such as the English Ladies' Diary popularized mathematics and problem-solving, and encyclopedias informed a larger public of relatively recent results, French mathematicians started to directly report their research in publications such as Journal de l'École Polytechnique. In 1810 Joseph Diaz Gergonne (1771-1859) established the first private journal wholly devoted to mathematics—Annales de Mathématiques Pures et Appliquées. Although Gergonne's original intent was to provide material for teachers, he mainly published new results. By 1832, when the journal ceased publication, Gergonne had publicized areas of mathematics often neglected in France and influenced the direction of future research.

After the middle of the century, mathematicians who sought places where they could discuss mathematical journals and research formed societies devoted specifically to mathematics. These societies eventually grew beyond local boundaries. The first to flourish, founded in 1865, was the London Mathematical Society, originating as a college club at University College. Because the group was supported by Augustus De Morgan (1806-1871), one of Britain's leading mathematicians, other major British mathematicians soon joined the society. The group began publishing original papers immediately, with the first by James Joseph Sylvester (1814-1897). By the end of 1866, the society was national in membership and activity.

Mathematical meetings turned fully international in scope with the congress held in conjunction with the 1893 World's Fair in Chicago. The New York Mathematical Society (renamed the American Mathematical Society the following year) issued invitations to prominent European mathematicians. Felix Klein (1849-1925), a German mathematician additionally involved in educational reforms and the application of mathematical ideas, accepted. From August 21-26, 1893, he gave talks surveying all of mathematics. Afterwards, Klein stayed an additional two weeks at Northwestern University for a colloquium during which he presented 12 lectures exposing his audience—from Austria, France, Germany, Italy, Russia, Switzerland, and the United States—to current areas of mathematical research. He also met personally with the participants. The New York Mathematical Society later issued these lectures in book form. The congress inspired further meetings both in the United States and as a series of international mathematics congresses held every four years, beginning in 1897 in Zurich.

Impact

These developments both influenced and were influenced by changes in the practice of mathematics. For example, the technical standards of mathematics were elevated as mathematicians such as Augustin-Louis Cauchy (1789-1857) strove for greater rigor in proof. Furthermore, as mathematicians expanded their areas of study, mathematics as a whole became a more specialized endeavor. It was no longer possible for one person to master all mathematical knowledge. New fields ranged from the mathematical theory of heat studied by Joseph Fourier (1768-1830), to research in elliptical functions by Carl Gustav Jacobi (1804-1851), to the invariants studied by Arthur Cayley (1821-1895) and Sylvester, all mathematicians who reported their results to mathematical societies and in journals.

Mathematicians who studied these specialized subfields needed certain qualifications, most notably a doctoral degree. Traditional higher education policies emphasizing Euclid's Elements, the value of mathematics in training the mind to reason properly, and memorization were increasingly considered unsatisfactory. Rather, military academies often led the way early in the century by providing specialized and technical training. Then, colleges and universities in Europe and the United States gradually focused on advanced mathematics. German institutions established the model for the modern graduate school, where students met in seminars with professors and were awarded doctorates in mathematics departments upon completing dissertations on original research.

These trained professionals were then recruited back to universities to conduct research. University leaders had recognized the prestige mathematical research could bring to their institutions. For example, after Daniel Coit Gilman (1831-1908) established Johns Hopkins University in Baltimore as primarily a graduate school, he hired Sylvester in 1876 to lead the mathematics department. Sylvester relished the opportunity to escape teaching introductory mathematics courses. He carried on his research into the properties of invariants, among other interests, and founded the American Journal of Mathematics in 1878, securing contributions from prominent European mathematicians. In turn, with the university's new mission to both transmit and generate mathematical ideas, professors such as Sylvester trained advanced students who earned doctorates and generally embarked on careers as mathematical researchers themselves.

The success of Gergonne's Annales opened the way for additional mathematical journals. For instance, the Journal für die reine und angewandte Mathematik, founded by August Leopold Crelle (1780-1855) in 1826, took the European lead as a periodical of advanced content after Annales folded. "Crelle's journal" accepted contributions from nations outside the German lands and promoted travel and research by mathematicians. Meanwhile, by mid-century, French mathematicians published their work in more than 30 journals, magazines, and newspapers, including the Journal de Mathématiques Pures et Appliquées, founded by Joseph Liouville (1809-1882) in 1836. In the last quarter of the century, Italian journals promoted the dissemination of mathematical research and bridged the gulf between French and German mathematicians caused by the Franco-Prussian War, while Acta Mathematica, based in Scandinavia, successfully drew international contributions on the largest scale for that time. These periodicals also started to include material on the history of mathematics. The major journals in the United States were Sylvester's American Journal of Mathematics and Annals of Mathematics, founded in 1884.

Similarly, the London Mathematical Society's example in facilitating communication between mathematicians, publicizing international developments, and holding talks given by and to mathematics researchers was followed by mathematical societies established around the world in the last quarter of the nineteenth century. These organizations included the Moscow Mathematical Society, founded in 1867, the Société Mathématique de France, founded in 1872, the Edinburgh Mathematical Society, founded in 1883, and the Circolo Matematico di Palermo, founded in 1884. These societies often founded journals to publicize research results or published the proceedings of their meetings. Sometimes they presented awards to outstanding mathematicians.

In addition, new centers of mathematics rose in importance. While mathematicians in Paris dominated mathematical achievement into the 1830s, German mathematicians were then generally most productive for at least the next third of the century. London and Italy were also home to accomplished mathematicians. The United States as a whole rapidly gained prominence late in the nineteenth century, with the success of the 1893 congress bringing specific attention to the University of Chicago as an educational institution offering advanced mathematical training comparable to that available in Europe.

Mathematicians came to view mathematics as an enterprise dependent upon international cooperation. National societies brought in distinguished mathematicians by electing foreign members. For instance, just two years after the London Mathematical Society was founded, French geometer Michel Chasles (1793-1880) was named to an honorary position. In the last decade of the nineteenth century, in addition to the congresses that followed the 1893 meeting, the International Mathematical Union was founded. Although the upheavals of the two world wars would disrupt its activities in the twentieth century, the Union would foster communication and interaction between mathematicians on an unprecedented level. It then became global as mathematicians in such areas as eastern Asia and South America professionalized and communicated with other mathematicians.

None of these developments could have taken place without the new technologies of the nineteenth century. Improvements in transportation, such as the railroad, made it easier for mathematicians to visit top researchers. Shorter travel times also meant books and journals reached distant mathematicians soon after they were printed, while changes in printing materials and techniques enabled mathematical research to be published more quickly and cheaply.

In summary, the promotion of mathematical research in the nineteenth century coincided with the evolution of a new definition for "mathematician." A professional mathematician had earned a doctoral degree, produced research, and shared the results in talks and papers with other mathematicians, a community of scholars with a sense of identity. Nineteenth-century mathematicians promoted professionalization by establishing journals and forming mathematical societies. On the other hand, the dissemination of research made it possible for more people to engage in advanced mathematics and for mathematical productivity and diversity to increase even more dramatically. As these various factors acted upon each other, the result was an attitude among mathematicians that centered upon the promotion of mathematical research.

AMY ACKERBERG-HASTINGS