Mathematics educators have become increasingly interested in recent years in the usefulness of the subject. This interest has taken many forms, such as, for example, the preparation of source books of applied problems, emphasis on applications at international congresses on mathematics education, Oxford seminars on industrial problems, internships in industry for students and faculty, and a desire to understand what mathematical scientists outside of education actually do and how they should be prepared for it. This paper contains the views of one industrial mathematician on this last topic.
The first question we need to examine is what mathematicians actually do in industry. The obvious answer, that is to solve specific mathematical problems bothering other people, is only part of the truth, in fact a relatively small part. One of the mathematician's activities is indeed to solve mathematical problems precisely formulated in mathematical terms. For example, someone may want to know the length of the shortest tree connecting n points in a unit square, either the expected length if these points are placed at random or the maximum possible length no matter how they are placed. Someone else may give you a 3-term recursion relation for a family of polynomials and ask you for the measure with respect to which these polynomials are orthogonal. Or they might just ask you to sum the following series. Is it wise to plunge in and immediately try to solve the problem? Maybe. The mathematician learns after a number of experiences to question why the visitor is interested in the particular problem. It turns out that it is by no means true that other people always ask the right mathematical .question. This does not imply malevolence on their part. They have, after all, tried to solve the problem, but their mathematical knowledge may not have been sufficient to lead them to the best formulation.
Besides mathematical problems precisely formulated in mathematical terms, the mathematician will often be asked to investigate precisely formulated situations in some other field. What purely resistive circuits can you make by printed circuit techniques? Why does a strand of spaghetti slurped sufficiently rapidly come up and hit your nose? Which parts of this composition were probably written by Mozart, and which by some graduate students? What is the best way to lay out a running track? What is the best strategy in playing blackjack against the house rules in Las Vegas, and can you win in the long run? These questions are all well formulated in the field of application, but the process of