Mathematical Sociology

by

Guillermina Jasso

• LAST REVIEWED: 23 October 2018
• LAST MODIFIED: 28 November 2016
• DOI: 10.1093/obo/9780199756384-0174

Introduction

Mathematical sociology is sociology expressed in the language of mathematics. It has no special subject matter or special domain, for all of sociology is its domain and all human behavioral and social phenomena are its subject matter. What is distinctive about mathematical sociology is its language, its vocabulary. While articles on particular topical domains expressed in ordinary language have sentences as their main elements—with a subject, a verb, perhaps an object, perhaps embellished with adjectives and adverbs—articles on particular topical domains expressed mathematically have equations as their main elements—a term to the left of the equals sign, to the right a term or terms linked by plus and minus signs, perhaps embellished with subscripts and superscripts. In the same way that sentences are combined into paragraphs, equations are combined into multi-equation models. And in the same way that nouns and verbs are modified by adjectives and adverbs, the terms in equations are modified by transformations and parameters. The task of mathematical sociology is mathematical statement of the terms and relations in all of sociology—from the foundational ideas of the discipline to the starting ideas for its subfields to the predictions and possibilities for all topical domains. Importantly, the task is not embraced for its own sake, though it would be easy to do so based on notions of parsimony, precision, and beauty. Rather, the task is embraced because mathematics is the tool par excellence for advancing knowledge. Two of the ways that mathematics shows its power for advancing knowledge pertain to sociological theory, that is to the very foundations of sociology. First, mathematics is a power tool for deriving testable predictions, including novel predictions, from the foundational postulates in the discipline and the starting ideas in its subfields. Second, mathematics is a power tool for theoretical unification, helping the discipline to reach the goal of understanding more and more by less and less.

Further Introductory Remarks

As noted, two of the ways that mathematics advances sociological knowledge pertain to theory. First, mathematics is a power tool for deriving predictions from the foundational postulates in the discipline and the starting ideas in its subfields. While pure words can produce “instantiation” in abundance, only mathematics enables the “marvellous deductive unfolding” (Popper 1963, p. 221) of a theory, yielding the “derivations far afield from its original domain,” that “permit an increasingly broad and diversified basis for testing the theory” (Danto 1967, pp. 299–300). Mathematics makes it possible to generate a large number of predictions, including novel predictions, from a small set of postulates (Jasso 1988), thus making it possible for sociology to achieve the objective of theory—to “deduce the largest number of empirical findings from the smallest number of general propositions” (Homans 1967, p. 27). Second, mathematics is a power tool for theoretical unification. The goal of scientific work is to understand more and more by less and less, and in this effort theoretical unification plays a key part (Fararo 1989). While pure words can suggest similarities and commonalities, mathematics can render unambiguous a range of exact equivalences. The fact that mathematical sociology has no topical domain of its own has implications for an article such as this. The task of a bibliography in mathematical sociology is not to trace the development of knowledge about a topical domain. Rather, the task is to provide a flavor for the way mathematical sociologists work. Accordingly, this article traces the use of mathematics in sociology, highlights basic tools, and notes the partnerships formed with mathematicians, statisticians, and engineers. Clogg 1992 offers the fascinating and still timely observation that sociological questions have spawned advances in statistical methodology; as time passes, the same may be said about advances in mathematics. What is the future of mathematical sociology? At this stage in the development of sociology, mathematical sociology is on the upswing, the discipline now awakening from the dark night of math anxiety. But it remains the case that contributions to specific topical domains will appear in both a topical article and a mathematical sociology article. One can imagine a period of ferment and growth, inclusive of intriguing new puzzles and paradoxes, culminating in all the topical domains achieving a mathematical core. At such time, mathematical sociology as a “specialty,” or “niche,” will disappear. It will linger in the history books and, of course, in methods handbooks and manuals. However, its work will be done.

• Clogg, Clifford C. 1992. The impact of sociological methodology on statistical methodology. Statistical Science 7.2: 183–196.

  Early observation that sociological questions have spawned advances in statistical methodology.

• Danto, Arthur C. 1967. Philosophy of science, problems of. In Encyclopedia of philosophy. Vol. 6. Edited by Paul Edwards, 296–300. New York: Macmillan.

  Cogent discussion of how theoretical predictions far afield from the postulates permit “an increasingly broad and diversified basis” for theory testing.

• Fararo, Thomas J. 1989. The spirit of unification in sociological theory. Sociological Theory 7.2: 175–190.

  DOI: 10.2307/201894

  Insightful discussion of the key part played by theoretical unification in scientific work.

• Homans, George Caspar. 1967. The nature of social science. New York: Harcourt, Brace, and World.

  Incisive statement of the goal in theory development—to deduce a maximum of predictions from a minimum of postulates.

• Jasso, Guillermina. 1988. Principles of theoretical analysis. Sociological Theory 6.1: 1–20.

  DOI: 10.2307/201910

  Discussion and illustration of deductive theory using the language of mathematics, including twenty-five predictions of justice theory.

• Popper, Karl R. 1963. Conjectures and refutations: The growth of scientific knowledge. New York: Basic Books.

  Masterly discussion of the part played by deductive theory in the growth of scientific knowledge.