Some Social Characteristics of Mathematicians and Their Work

The effectiveness of science has often been seen to lie in its ability to solve the problems posed by its subject matter. Scientific progress is displayed by inverted pyramids of ever more concise and comprehensive solutions: Newton, Lagrange, Einstein. For some scholars, problems are the essence of science.2 But problems do not exist by themselves. Like all intellectual products, they are social artifacts within the complex communities whose members create them. In this essay, I will examine the relationships between some characteristics of the contemporary world of mathematics and the actions of mathematicians attempting to solve a long-standing problem. To do this, I will indicate some social structures of the contemporary community of mathematicians, and then look at them as they interact with the historically given problem. Finally, I will point to some consequences this situation has for the men and their discipline. Together they will illustrate the kinds of variability that exist in what to outsiders appears to be a unified scientific activity. Before proceeding, some comments are necessary concerning the nature of problem solving, the scope of this study, and the methods of research used. The expression "problem solving" can have a wide variety of meanings. In the everyday world, it refers to anything from a problem in an arithmetic book to the problem of achieving happiness. What constitutes the problem and its solution in each of these situations differs. Arithmetic creates a simple task which is accomplished when a number placed upon