Mathematics and Social Theory

A WIDE variety of symbolic devices are employed in sociology for the representation of facts and ideas. These range all the way from descriptive devices such as maps, pictures, and charts to more conceptual devices like words, statistics, and mathematics. Of all these, one of the less exploited and yet most promising types of symbolism is that of mathematics. Particularly in the field of social theory, where emphasis is upon the relationships among data rather than upon the data themselves, does mathematics offer us a useful and exact language. The truth of this becomes evident upon consideration of the nature of theory itself. Essentially any theory is a system of abstract statements. Each one of these statements describes a general mode of relationship between the terms which comprise the statement. Each of these terms, in turn, refers to a general class of data, i.e., to a universal as distinct from a particular. It follows that all of the statements comprising the theory are themselves generalizations. That is to say, they specify certain common attributes, certain regularities in the phenomenon described. Theory is not primarily concerned with unique relationships; rather it is concerned with invariant relationships. Consider now the nature of mathematics. It too is characterized by abstractness. Its statements, like those of theory, specify general modes of relationship between the component terms of the system. Again, these terms are by their very nature universals rather than particulars. But here a new feature appears, one which endows mathematics with certain properties that are not characteristic of theory in general. The terms which form the elements of a mathematical system are variables. As such these terms have the special property of denoting any one of a class of elements each of which may possess a different value. The statements which are composed of variables-i.e., of terms having this property-are known as propositional functions, or formulas. By itself, of course, a formula or propositional function has no empirical reference whatever. But when specific values are put in the place of the variables which comprise the formula, a proposition is expressed which can be empirically tested. A set of variables thus related to one another by a formula is the most general kind of statement that can possibly be made about a phenomenon. A system of such formulas has a flexibility and a rigor which perhaps no other theoretical system can equal. By means of it a complexity and subtlety of relationships can be indicated between facts and ideas which wholly escape a verbal symbolism. Mathematics, it may be suggested, is the most elegant language in which a theory can be expressed. In the development of social theory there have been numerous attempts to employ a mathematical symbolism, some of them being of remarkable antiquity. As long ago as the ninth century the Arab mathematician Al-Khowarizmi evolved an algebra of property inheritance by which he endeavored to show the logic of Arab property laws.' Spinoza, as is well known, used a geometrical language in his Ethics. Early in the seventeen hundreds Ceva outlined the logical requirements for a mathematical theory of value.2 At about the same time Weigel sought to formulate a universal theory of moral quantities. 3 During the last half century there have appeared numerous sociological systems which in one way or another have employed mathematical symbolism, among them being the works of Haret, Barcelo, and Pareto,4 and most recently the contributions of Menger, Dodd, Chapple, Rashevsky, and Von Neumann and Morgenstern.6 A study of these mathematical theories of society