Mathematics and the Sciences

The principal thrust of this essay is to describe the current state of interaction between mathematics and the sciences and to relate the trends to the historical development of mathematics as an intellectual discipline and of the sciences as they have developed since the seventeenth century. This story is interesting in the context of the history of present-day mathematics because it represents a shift in the preconceptions and stereotypes of both mathematicians and scientists since World War II. The notion that significant mathematical and scientific advances are closely interwoven is not particularly new. The opposing notion (associated, with whatever degree of justice, with the name of Bourbaki) was never as fashionable, at least among working mathematicians, as in the two decades immediately after World War II. The situation has changed significantly during the past decade and had begun turning even earlier. It turned not only among mathematicians, but even more significantly in such sciences as physics. The frontier of mathematical advance was seen again to be in forceful interaction with the basic problems and needs of scientific advance. This is an essay on significant trends in mathematical practice. The relations between the history and philosophy of mathematics as usually conceived and mathematical practice have often been very ambiguous. In part, this has resulted from the efforts of some historians and philosophers to impose a framework of preconceptions upon mathematical practice that had little to do with the latter. In part, however, it resulted from the diversity of mathematical practice, to lags in its perception, and to the complexity of viewpoints embedded in that practice. Let me preface this account with two statements by great American mathematicians of an earlier period who put the case in a sharp form.