The Flowering of Applied Mathematics in America

Abstract

Mathematicians are notoriously bad historians; they describe development of an idea as it should logically have unfolded rather than as it actually did, by fits and starts, often false starts, and buffeted by forces outside of mathematics. In this sketchy account I will describe twists and turns as well as thrusts of in America. Applied is alive and well in America today. Looking at 18 lectures [presented at Centennial Meeting of American Mathematical Society] chosen to describe frontiers of research, we see that topics include physiological modeling, fluid flow and combustion, computer science, and formation of atoms within framework of statistical mechanics. We also see that subject of one lecture and starting point of several others are physical theories; conclusions reached are of interest to physicists and mathematicians alike. This was not always so. For a few decades, late 1930s through early 1950s, predominant view in American mathematical circles was same as Bourbaki's: is an autonomous abstract subject, with no need of any input from real world, with its own criteria of depth and beauty, and with an internal compass for guiding further growth. Applications come later by accident; mathematical ideas filter down to sciences and engineering. Most of creators of modern mathematics-certainly Gauss, Riemann, Poincare, Hilbert, Hadamard, Birkhoff, Weyl, Wiener, and von Neumann-would have regarded this view as utterly wrongheaded. Today we can safely say that tide of purity has turned; most mathematicians are keenly aware that does not trickle down to applications, but that and sciences, mainly but by no means only physics, are equal partners, feeding ideas, concepts, problems, and solutions to each other. Whereas in not so distant past a mathematician asserting applied is bad mathematics or the best is pure mathematics could count on a measure of assent and applause, today a person making such statements would be regarded as ignorant. How did this change come about? Several plausible reasons can be discerned. But first a bit of selective history. World War II, a watershed for our social institutions, concepts and thinking, permanently changed status of in America. That is not to say that there was no worthwhile in America before 1945; after all, already in nineteenth century, Gibbs' contributions to statistical mechanics as well as to vector analysis and Fourier series, and Hill's studies of Hill's equation, had put America on map. The leading American analysts in 1920s and 1930s were Birkhoff, renowned worldwide for his work in dynamics, and Wiener, a pioneer in study of physical processes driven by chance influences such