Bakerian Lecture, 1975: Global geometry

Abstract

It must be many years since a pure mathematician was asked to address this society on such an occasion. Certainly none appears in the records going back to 1940. This is of course no reflection on the impartiality of the Council of the Royal Society, but simply an acknowledgement of the wide gap that separates pure mathematicians from other scientists and of the serious difficulties of communicating across that gap. Fortunately we have our intermediaries - the applied mathematicians - who extract from the body of mathematical knowledge the most useful parts and bring them to bear on recognizable scientific problems in a wide variety of fields, all the way from the traditional areas of physical science right through to the biological and social sciences. Many distinguished applied mathe­maticians have indeed spoken to the society on problems such as aerodynamic noise or stellar evolution, which are heavily dependent on mathematical analysis, but which can be explained in physical terms readily understood by a wide audience. On such occasions the mathematical techniques involved will quite properly have been relegated to decent obscurity. As a result scientists at large probably have only the vaguest ideas about mathematical research per se . They will understand mathematical work related to their particular field, frequently, I may add, better than the mathematicians themselves, but they must find it hard to visualize mathematics in the abstract. It is therefore perhaps worthwhile for a pure mathematician to attempt to explain how we view our subject and what motivates our research in the absence of any particular scientific interpretation or application. If I might summarize the situation diagrammatically, consider mathematics as some kind of giant computer with a large number of terminals on its periphery, representing fields of application. A practising scientist is like the terminal user. He is primarily interested in the output and will know something about what the computer can do for him, but he is not involved in what goes on inside the heart of the computer. In the early days of computers, users and designers were frequently the same people, but with their rapid growth and sophistication this is now the exception rather than the rule. Similarly it is the increasing sophistication of mathematics which has led to the large gap between ‘users’ and ‘designers’.