Toward a Calculus for Social Science

The growth of science has been tied to the development of tools and methods which help the scientist sharpen her/his ability to use the evidence of the senses. Mathematics has been one of the most important of these tools, not as a means for extracting further evidence from the senses but as a method for comparing and contrasting the various bits of evidence. Mathematics fills the need for a method of reckoning—a calculus—which is precise (i.e., rigorous), which leads to the simplification of complex phenomena, and which exposes underlying relationships among the phenomena being considered. But besides rigor, the growth of science also depends on relevance and validity. Relevance is the ability of a method to lay bare the relationships that underlie apparently diverse phenomena. Relevance is what distinguishes, for example, alchemy from chemistry. Validity is the concurrence of rigor and relevance. It is what distinguishes science fiction from science.

The development of the physical sciences brought with it new forms of mathematics: Newton’s calculus was developed to express the relationships he was studying in his scientific research; quantum mechanics was developed to express the forces of sub-atomic physics. These techniques, defining mathematical “operations” on generalized “terms,” are not, however, somehow completely distinct from other means of expressing the relationships among various phenomena. They can be “translated” into another form of mathematics, Boolean algebra. This particular mathematics is very easily transformed into another “calculus,” formal logic, which is in turn a translation of everyday language. In other words, mathematics can be seen as a language, a very restrictive and specialized language, but a language nonetheless.

The social sciences, for the most part, have used common language (or, at their worst, an uncommon language of jargon) to expose the underlying relationships among the phenomena with which they are concerned. When the social sciences have used mathematics, they have tended to use mathematical techniques developed in the physical sciences. Thus, parametric statistics (correlation, analysis of variance, and other variants on the general linear model) are commonly used in sociological and psychological research. These statistical tools were developed in the physical sciences to explore the behavior of grains of sand on the beach and other similar phenomena. The tool assumes that the elements under consideration are randomly distributed in space and that each element behaves autonomously. Unfortunately, people and societies tend to behave in a patterned rather than random fashion, and they tend to have reciprocal influence on each other’s behavior. While parametric statistics may be appropriate for analyzing some situations in the social sciences, they are inappropriate for many more situations.

In some respects, nonparametric statistics (also developed in the physical sciences—particularly the life sciences) offer a better tool for social science research. While nonparametric statistics still assume that each element behaves autonomously, at least the techniques make no assumption (or make more flexible assumptions) about the distribution of elements. A poisson distribution, for example, assumes that a single occurrence is more likely than a double occurrence which is more likely than a triple occurrence, and so on: this could be a very useful model for many social science research activities. Further, nonparametric statistics can be used to examine the relationship between elements that are not infinitely divisible: nominal categories (like male/female, Black/white) and ordinal categories (like preference scales or voting choices) are not suited to parametric statistical analysis.

The problem with statistical analysis—whether parametric or nonparametric—is the requirement that elements behave autonomously. This assumption is necessary if one is to use the mathematics of probability to explain the underlying relationships. But this emphasis on the “independent” behavior of elements has led to the slighting of the reciprocal nature of personal and social behavior—and it has led to a “social physics” rather than a science of society and behavior.

There is a third mathematical technique used in the social sciences, particularly economics, called linear programming. This technique was taken from astronomy and engineering where it was developed precisely to take into account the mutual interaction of phenomena. The problem with linear programming, however, is that it requires that the elements under consideration be infinitely divisible and that they be very accurately measured. With few exceptions, precise quantitative measurement is not available to the social sciences.

Where might we look for new tools that could be used to develop a “calculus” for the social sciences? There are two areas of research—one in mathematics, the other in philosophy—that hold particular promise for a science of society and behavior. Just as physics turned to pure mathematics—to the study of non-Euclidean geometries—to find a tool which could be used to explore Einstein’s theories of relativity, so the social sciences could do well to turn to topology. Topology is the mathematical study of relationships. It makes no assumptions about quantity and is intended to describe distributions (rather than to assume them). The application of topology is already being worked out in techniques known jointly as “graph theory.”

In philosophy, a twentieth-century European school called “phenomenology” has been focusing on the study of “meaning”—when we say that something is a “cup,” what exactly do we mean? What is “cup-ness”? Phenomenology arrives at its conclusions through a process called “the reduction”: describe the object, then take away each element of the description in turn, seeing if the essence of the object remains. When all of the inessential elements have been reduced away, what remains will be only the essential meaning. This technique has already begun to be applied in the social sciences, particularly in ethnomethodology and the sociology of knowledge.

The point is not that methods derived from the physical sciences should be drummed from the corps of social science methods. Social physics clearly has its place in the social sciences. But it should not be mistaken for an adequate study of the science of society and behavior.

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