Mathematics is the human activity of constructing axiomatic definitions of. abstract patterns among unspecified or arbitrary elements and studying the

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1 Published in Dictionnaire de la Pensee Sociologique [Dictionary of Sociological Thought (ed. by M. Borlandi, R. Boudon, M. Cherkaoui & B. Valade). Presses Universitaires de France, Paris Mathematics and Sociology: Evaluation Mathematics is the human activity of constructing axiomatic definitions of abstract patterns among unspecified or arbitrary elements and studying the properties of such patterns by deductive elaboration, using principles of logic. Any such abstract pattern, arising in such a context, may be said to define a class of mathematical objects, e.g., "Markov chains," "semi-groups," "vector spaces," and the like. If T is the axiomatic theory that defines a class M of mathematical objects, then any entity in M is said to be a T-model. Such models play a central role in the sciences. As a science, sociology includes the use of such mathematical models. For most sociologists, however, this connection between mathematics and sociology is confined to problems of data analysis, employing statistical models. In other words, the mathematical theory (T) is the theory of statistics, and the T- models deal with such things as linear regression and statistical significance tests. Nevertheless, this is far from the whole story about the linkage between mathematics and sociology. After World War Two, as part of a more general zeitgeist involving the deepening and broadening of the interpenetration of mathematics and the social and behavioral sciences, some sociologists began to employ mathematical models in contexts different from traditional data analysis. Their point of view was a common one in the newly developing field of "mathematical social science." The idea was to create more rigorous scientific theories than had hitherto existed in the social and behavioral sciences. Traditionally, for instance,

2 sociological theories were strong in intuitive content, but weak from a formal point of view. Assumptions and definitions were not clearly stipulated and distinguished from factual descriptions and inferences. In particular, there was rarely a formal deduction of a conclusion from specified premises. The new and preferred style was encapsulated in the phrase "constructing a mathematical model." This means making specified assumptions about some mathematical objects and providing an empirical interpretation for the ideas. It also means deducing properties of the model and comparing these with relevant empirical data. "Mathematical sociology" was part of this general movement in the social and behavioral sciences. Sociologists who contributed most to the development of mathematical sociology were very much influenced by these wider developments. Some of the latter will be briefly described because of their particular importance in this respect. (Some other examples may be found in the compendium edited by Lazarsfeld and Henry 1966). Influential Early Developments Starting in the late 1940s, the mathematical biologist Anatol Rapoport developed a probabilistic approach to the characterization of large networks. Starting from a baseline of a "random net," then introducing "bias parameters," Rapoport logically derived formulas connecting parameters such as density of contacts to important global network features, especially connectivity (Rapoport 1957). In another early "social networks" development, mathematician Frank Harary and social psychologist Dorwin Cartwright collaborated in a discrete 2

3 mathematical approach to social networks, featuring the theory of graphs -- large parts of which were being created by Harary and his collaborators as they worked on social science problems. Starting from a representation of positive and negative sentiment relations among persons, Harary and Cartwright went on to prove the important and non-obvious Structure Theorem (Cartwright and Harary 1956). The theorem says that if a structure of interrelated positive and negative ties is balanced -- illustrated by the psychological consistency of "my friend's enemy is my enemy" -- then it consists of two substructures, with positive ties within and negative ties between them. (There is a special case where one of the two substructures is empty.) In these two developments we have mathematical models bearing upon the analysis of structure. Other early influential developments pertained to process. In the analysis of processes, two types of mathematical models are relevant: deterministic and stochastic. A key example of the former was produced by Herbert Simon (1952), a mathematical formalization of a social systems theory. Mechanisms are described and embedded in a system of differential equations. The system is then studied in its abstract form, leading to theorems about the dynamics and the implied equilibrium states. The stochastic approach was strongly developed in mathematical learning theory (Bush and Mosteller 1955). The general probabilistic approach came to be known as stimulus sampling theory, in which the human being is viewed as sampling stimulus elements and connecting these to responses as a function of reinforcement contingencies. 3

4 Research Programs and Mathematics What these sorts of developments meant for some sociologists in the late 1950s and 1960s was the existence of a new and promising intellectual environment. At least three key research programs with lasting impact in the late 20 th century were initiated in this period and serve to illustrate the growing nexus between mathematics and sociology around the idea of constructing mathematical models. James S. Coleman came to sociology from an engineering background, studying with Paul Lazarsfeld at Columbia University in the 1950s. As an engineer, he thought about social processes in terms of differential equations, as had Simon and others. But how could one connect differential equations to the data of sociology? That was Coleman's question. He noted that surveys reported results in the forms of proportions. Yet the proportion of people believing or doing something at a given time had to be correctly interpreted. First, it was not necessarily a stable proportion, since it could change. So such proportions should be conceptualized as states of a probabilistic dynamic system, with a flow of probabilities over time that might indeed have some equilibrium state. Second, although each person held a belief or voted a certain way, the process by which these individual orientations came about was socially mediated. That is, we should understand the process by which the probability state changed over time as a network process in which individuals influence each other to change orientations. 4

5 The results of these sorts of considerations were embodied in Coleman's Introduction to Mathematical Sociology (1964). The publication of this book marks the legitimation of mathematical sociology as a distinctive and important part of sociology. Coleman's innovation was to show how processes in social networks could be analyzed in such a way as to come to grips with relevant sociological data, allowing empirical identification of abstractions, estimation of parameters, and calculations of goodness of fit between model and data. Coleman's interest in purposive action as the foundation for understanding social processes culminated in a major work on rational choice theory in sociology, including the use of the mathematics of general equilibrium theory (Coleman 1990). While Coleman's work demonstrated how mathematical models of social processes could be constructed, Harrison White's 1963 monograph The Anatomy of Kinship illustrated how modern mathematical ideas could be applied to the analysis of social structure. The immediate background was the structuralist tradition in social science, especially the analysis of kinship structures by Levi- Strauss. Some of the latter's ideas had led to formalization by mathematicians with interests in the application of finite mathematics to social science, especially Kemeny, Snell, and Thompson (1957). White, a holder of a Ph.D. in theoretical physics and hence well-trained in mathematical methods, continued this formal development. White set out a set of axioms describing a certain type of prescribed marriage system and formally analyzed this class of systems using the formal methods of group theory. Hence, the analysis of social structure was 5

6 directly linked to "the new mathematics," i.e., discrete mathematics featuring abstract algebraic concepts and methods. White went on to become a central figure in mathematical sociology and especially in the newly emerging field of social network analysis. By the 1990s, the social network paradigm had a conceptual core strongly linked to modern abstract algebra (Pattison 1993) as well as to a wide variety of other mathematical tools (Wasserman and Faust 1994). In the 1950s, the Harvard department of sociology was a major center of developments in sociological theory and in small group research. Three graduate students at Harvard at that time went on to initiate intertwined careers of fruitful collaboration, establishing an experimental laboratory and graduate program at Stanford University: Joseph Berger, Morris Zelditch Jr. and Bernard P. Cohen. In particular, Berger initiated a highly influential research program in which the central idea was the use of the theoretical construct "expectation state" to construct theoretical models to explain interpersonal processes. Over time, much of the theoretical work became linked to mathematical model building. The generations of mathematical model-builders that followed Coleman, White and Berger, among others, drew upon their work in a variety of ways. For instance, Fararo and Skvoretz (1986) constructed a mathematical approach called "E-state structuralism" that synthesized components of the social network paradigm with core components of expectation-states theory. 6

7 Major sociological theorists in the classic tradition had not made any important connections between theory and mathematics. The new developments that began to create such links called for work that would elucidate the nature of the efforts. How did using mathematics advance theory? An early attempt to provide such elucidation was Types of Formalization in Small Group Research (1962) authored by members of the Stanford group (along with mathematician J. Laurie Snell). In this book, the authors formulated a typology of models in terms of the goal of the model-builder. One goal is to formalize an important concept in a theory, as in the Cartwright-Harary graph-theoretic formalization of the concept of structural balance. A second goal is to formally represent a recurrent process, as in Coleman's process model building. Finally, a third goal is the formalization of a theory of a broad class of phenomena, illustrated by stimulus sampling theory. These types of formalization are very visible, for instance, in the field now called "group processes," which includes a variety of long-term research programs. In many of them, mathematical model building is a strong component feature, including expectation-states theory, affect control theory, and exchange network theory, among others (see (Berger and Zelditch 1993). Although most of these programs focus on social interaction, mathematical model building in sociology has not been limited to microsociology. For example, the representation of processes of social mobility through Coleman-type stochastic 7

8 processes and related mathematical ideas has been extensive (Bartholomew: 1982). Institutionalization of Mathematical Sociology The institutionalization of a field is indicated by the appearance of such entities as textbooks, bibliographic surveys, journals, and graduate programs. Mathematical sociology textbooks cover a variety of models, usually explaining the required mathematical background before discussing important work in the literature (Fararo 1973, Leik and Meeker 1975). Sørenson and Sørenson (1975) provide an extensive survey and bibliography of the developments in the early decades of the field. The Journal of Mathematical Sociology (started in 1971) has been open to papers covering a broad spectrum of topics employing a variety of types of mathematics, especially through frequent special issues. Three specialized publication outlets emerged for contributors to the three families of research programs originated out of the works reviewed above: Rationality and Society, Social Networks, and Advances in Group Processes (an annual publication). One could say that these are a kind of routinization of the intellectual charisma of Coleman, White and Berger, respectively. The journal Quality and Quantity has been one of the main publication outlets for European mathematical sociology. In addition, and this is important as an indicator of the penetration of mathematical model building into sociological research, the major comprehensive journals in sociology, especially The American Journal of Sociology and The American Sociological Review, regularly publish articles featuring mathematical formulations. 8

9 Thus, mathematical model building has become a recognized and widely employed method in sociological research. The special field of mathematical sociology had emerged in 1960s and was associated with notable landmark works such as those cited earlier. Yet, there have been some disappointments. Graduate programs in the field are difficult to sustain because few students who enter graduate school have the requisite formal skills and mathematical training. Given the very diversity and depth of mathematical ideas now employed in sociology, it is difficult to find a mechanism for training students who do not already have the needed skills. The graduate program situation is one symptom of a deeper problem. Namely, the mathematical approach has not penetrated deeply into the general theoretical frameworks of sociology. There is a great deal of mathematical work that is highly theoretical, but such work is difficult for most sociologists to assimilate in a field that is not already mathematical at its core. Norms that are invoked in mathematical work in science -- such as idealization in constructing models, simplicity of framing assumptions, fertility of deductive consequences -- are ignored or ill-understood despite efforts to communicate these standards to the field (see, for instance, Lave and March 1975; Fararo 1984). As in the case of many social and cultural phenomena, the result is that the socialization process has an inherent pattern-maintaining aspect. In this sense, mathematical model building has become common in much sociological research and formaltheoretical analysis is prominent in some research areas, but there is a 9

10 continuing problem of absorbing the spirit and content of mathematical model building into general sociological theory. See also: Axiomatics. Models/Model-Building. Bibliography Bartholomew, David J Stochastic Models for Social Processes. 3 rd ed. Wiley. Berger, Joseph and Morris Zelditch, Jr Editors. Theoretical Research Programs: Studies in the Growth of Theory. Stanford University Press. Berger, Joseph, Bernard P. Cohen, J. Laurie Snell, and Morris Zelditch, Jr Types of Formalization in Small Group Research. Houghton-Mifflin. Bush, Robert R. and Frederick Mosteller Stochastic Models of Learning. Wiley. Cartwright, Dorwin and Frank Harary "Structural Balance: A Generalization of Heider's Theory." Psychological Review 63: Coleman, James S An Introduction to Mathematical Sociology. Free Press Foundations of Social Theory. Harvard University Press. Fararo, Thomas J Mathematical Sociology. Wiley Editor. Mathematical Ideas and Sociological Theory. Gordon and Breach. and John Skvoretz "E-state Structuralism: A Theoretical Method." American Sociological Review 51: Kemeny, John G., J. Laurie Snell, and G. L. Thompson Introduction to Finite Mathematics. Prentice-Hall. 10

11 Lave, Charles and James March An Introduction to Models in the Social Sciences. Harper and Row. Lazarsfeld, Paul F. and Neil W. Henry Editors. Readings in Mathematical Social Science. MIT Press. Leik, Robert K. and Barbara F. Meeker Mathematical Sociology. Prentice-Hall. Pattison, Philippa Algebraic Models for Social Networks. Cambridge University Press. Rapoport, Anatol "Contributions to the Theory of Random and Biased Nets." Bulletin of Mathematical Biophysics 19: Simon, Herbert A "A Formal Theory of Interaction in Social Groups." American Sociological Review 17: Sørenson, Aage and Annemette Sørenson "Mathematical Sociology: A Trend Report and a Bibliography." Current Sociology XXIII:No. 3. Mouton. Wasserman, Stanley and Katherine Faust Social Network Analysis: Methods and Applications. Cambridge University Press. White, Harrison C An Anatomy of Kinship. Prentice-Hall. Thomas J. Fararo University of Pittsburgh 11

Theory and Formalization: Some Reflections on Experience*

Theory and Formalization: Some Reflections on Experience* Joseph Berger Stanford University I describe in this paper some of my efforts in developing formal theories of social processes. These include