Some aspects of mathematics in social sciences

The ultimate aim of this paper is to examine the compromises which can be made in order to reach a point where measurement and computation are feasible in some of the quantitative aspects of the social sciences. The basic mathematical theories upon which the social sciences depend include: 1. Linear attrition theory including Feller's abstract formulation of parabolic equations. 2. Utility theory as a branch of the calculus of variations, described, for example, by G. E. Evans. 3. Combination of utility theories and competition in many play games, including Bellman's theory of dynamic programming. Basic empirical aspects of the social sciences include: 1. The acquisition of information about probability distributions, such as distribution of demand. 2. The testing of hypotheses related to reality, such as a hypothesis of rational behavior as formulated and tested by J. Marschak. 3. The intelligent formation of a priori Bayes probabilities and their correction as demanded by L. J. Savage in his The Foundations of Statistics and as carried out by W. A. Smith. 4. Observations of value of various competitive situations, based on simulation, experience, or other valid means of practical realization of quantities introduced theoretically in dynamic programming studies. Practical considerations require action after limited observation of existing conditions, and limitations of computational facilities may prohibit complete numerical analysis of situations whose underlying parameters are believed to have been accurately evaluated. This leads to an almost epistemological question as to what can be learned through aggregation of data related to seemingly similar situations and to difficult mathematical questions of reducing the computational extent of problems. Some means of examining these questions will be treated, but the paper will not attempt to present a unified theory covering any substantial set of these problems.