Karen Hunger Parshall

The Emergence of the American Mathematical Research Community

1: An overview of American mathematics: 1776--1876. 2: A new departmental prototype: J. J. Sylvester and the Johns Hopkins University. 3: Mathematics at Sylvesters Hopkins. 4: German mathematics and the early mathematical career of Felix Klein. 5: Americas wanderlust generation. 6: Changes on the horizon. 7: The Worlds Columbian exposition of 1893 and the Chicago Mathematical Congress. 8: Surveying mathematical landscapes: The Evanston Colloquium Lectures. 9: Meeting the challenge: The University of Chicago and the American mathematical research community. 10: Epilogue: Beyond the threshold: The American mathematical research community, 1900--1933. Bibliography

Eliakim Hastings Moore and the founding of a mathematical community in America, 1892–1902

Summary In 1892, Eliakim Hastings Moore accepted the task of building a mathematics department at the University of Chicago. Working in close conjuction with the other original department members, Oskar Bolza and Heinrich Maschke, Moore established a stimulating mathematical environment not only at the University of Chicago, but also in the Midwest region and in the United States in general. In 1893, he helped organize an international congress of mathematicians. He followed this in 1896 with the organization of the Midwest Section of the New York City-based American Mathematical Society. He became the first editor-in-chief of the Societys Transactions in 1899, and rose to the presidency of the Society in 1901.

Toward a History of Nineteenth-Century Invariant Theory

Publisher Summary Before Gauss considered binary forms in his Disquisitiones Arithmeticae , Joseph-Louis Lagrange had encountered and dealt with the problem of transformation of homogeneous polynomials by linear substitutions of the variables in his two-volume Mecanique analytique . Although hinted at by Gauss in the Disquisitiones Arithmeticae in 1801, invariant theory owed its inception to George Booles work of 1841. Cayley and Sylvester developed Booles embryonic idea into a mathematical theory during the 1850s. Their work attracted the attention of the Irish mathematician, George Salmon, as well as that of Charles Hermite and Camille Jordan in France and Francesco Brioschi in Italy. Each of these mathematicians made fundamental contributions to the developing theory, and each of them was in correspondence with Sylvester and Cayley, detailing their discoveries. Therefore, an international spirit of cooperation developed. To trace the evolution of algebra in the 20th century, investigations of the development of invariant theory in the 19th century are needed.

Mathematics unbound : the evolution of an international mathematical research community, 1800-1945

The evolution of an international mathematical research community, 1800-1945: An overview and an agenda by K. H. Parshall and A. C. Rice The end of dominance: The diffusion of French mathematics elsewhere, 1820-1870 by I. Grattan-Guinness Spanish initiatives to bring mathematics in Spain into the international mainstream by E. Ausejo and M. Hormigon International mathematical contributions to British scientific journals, 1800-1900 by S. E. Despeaux International participation in Liouvilles \textit{Journal de mathematiques pures et appliquees} by J. Lutzen The effects of war on Frances international role in mathematics, 1870-1914 by H. Gispert Charles Hermite and German mathematics in France by T. Archibald Gosta Mittag-Leffler and the foundation and administration of \textit{Acta Mathematica} by J. E. Barrow-Green An episode in the evolution of a mathematical community: The case of Cesare Arzela at Bologna by L. Martini The first international mathematical community: The \textit{Circolo matematico di Palermo} by A. Brigaglia Languages for mathematics and the language of mathematics in a world of nations by J. J. Gray The emergence of the Japanese mathematical community in the modern western style, 1855-1945 by C. Sasaki Internationalizing mathematics east and west: Individuals and institutions in the emergence of a modern mathematical community in China by J. W. Dauben Chinese-U. S. mathematical relations, 1859-1949 by Y. Xu American initiatives toward internationalization: The case of Leonard Dickson by D. D. Fenster The effects of Nazi rule on the international participation of German mathematicians: An overview and two case studies by R. Siegmund-Schultze War, refugees, and the creation of an international mathematical community by S. L. Segal The formation of the international mathematical union by O. Lehto Index.

Defining a mathematical research school: the case of algebra at the University of Chicago, 1892–1945

Abstract Historians of science have long considered the concept of the “research school” as a potent analytical construct for understanding the development of the laboratory sciences. Unfortunately, their definitions fall short in the case of mathematics. Here, a definition of “ mathematical research school” is proposed in the context of a case study of algebraic work associated with the University of Chicagos Department of Mathematics from the Universitys founding in 1892 through 1945.

Perspectives on American mathematics

A research-level community of mathematicians developed in the United States in the closing quarter of the nineteenth century. Since that time, American mathematicians have regularly paused to assess the state of their community and to reflect on its mathematical output. This paper analyzes a series of such reflections—beginning with Simon Newcomb’s thoughts on the state of the exact sciences in America in 1874 and culminating with the 1988 commentaries on the “problems of mathematics” discussed at Princeton’s bicentennial celebrations in 1946—against a backdrop of broader historical

The British development of the theory of invariants (1841–1895)

The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s, Cayley, Sylvester, and the Irish mathematician, George Salmon, formulated the basic concepts, developed the key techniques, and set the research agenda for the field. As Cayley and Sylvester continued to extend the theory off and on through the 1880s, first Salmon in 1859 and later Edwin Bailey Elliott in 1895 codified it in high-level textbooks. This paper sketches the development of nineteenth-century invariant theory in British hands against a backdrop of personal, nationalistic, and internationalistic mathematical goals.

Mathematics in National Contexts (1875–1900): An International Overview

Within the international mathematical community, the last three decades have witnessed a striking number of centennial celebrations. To name just a few, the London Mathematical Society (LMS) entered a new century in 1965 with the Societe mathematique de France (SMF) following in 1972, the American Journal of Mathematics and the Circolo matematico di Palermo (CMP) saw their centenaries in 1978 and 1984, respectively, and the American Mathematical Society (AMS) passed its century mark in 1988 preceding the Deutsche Mathematiker-Vereinigung (DMV) by two years.1 These milestones suggest, at the very least, that the mathematical endeavor developed in important ways in diverse national settings during the closing quarter of the nineteenth century.

“A New Era in the Development of Our Science”: The American Mathematical Research Community, 1920-1950

It was the end of August 1950 and some 2300 mathematicians had gathered from all over the world in Cambridge, Massachusetts for the eleventh International Congress of Mathematicians (ICM). An ICM had never before been held in the United States, and the American mathematical research community had a point to make.

Chemistry Through Invariant Theory

In his Cours de Philosophie positive, the nineteenth-century French philosopher, Auguste Comte, detailed a philosophy crucially dependent upon science and scientific thought.1 In his system, human comprehension and explanation of the natural world progressed through three stages: in the first, theological stage, divine will accounted for natural phenomena; in the next, metaphysical stage, interpretations of nature took an abstract, philosophical cast; and in the final, so-called “positive” stage, scientific truth characterized the world. According to Comte, the mind reached the positive stage by moving through a well-defined hierarchy of scientific thought. At its base lay mathematics, the most complex of the sciences in his view. Furthermore, since mathematics represented a body of scientific truths, it had already reached the stage of positive knowledge and, in fact, was historically the first science to attain this goal. Upon this base rested five sciences in ascending order of their historical achievement of the positive stage and thus, to Comte’s way of thinking, in order of their decreasing complexity and increasing generality. They were astronomy, physics, chemistry, biology, and, at the top, a new science for which Comte coined the term “sociology.” Of these, he held that all but the essentially brand-new science of sociology had reached the positive stage by the mid-nineteenth century. He thus took it as his task not only to establish this discipline in terms of scientific truths but also to achieve the synthesis of all of the positive sciences into one complete and unified positive philosophy.