NEOCLASSICAL THEORIZING AND

Journal of Mathematical Sociology, 1984, Vol. 10, pp. 361-393 0022-250Xy84/1003-0361$30.00/0 © 1984 Gordon and Breach, Science Publishers, Inc. Printed in the United States of America

NEOCLASSICAL THEORIZING AND FORMALIZATION IN SOCIOLOGY THOMAS J . FARARO University of Pittsburgh

INTRODUCTION Most theoretical sociologists would agree, without necessarily approving, that contemporary theoretical discussion is characterized by a great interest in "the masters of sociological thought" (Coser, 1977) . This interest is sometimes in the spirit of debunking earlier influential interpretations or of producing a new dialogue with the past, but frequently it is constructive "neoclassical" theorizing which takes a particular synthesis of certain classic ideas and contemporary developments as its main theme. In their recent book. Turner and Beeghley (1981), for instance, do more than review the ideas of an array of classic theorists. They suggest causal models and abstract principles that are in some sense suggested by the texts. But in stating principles, they employ a general action frame of reference, approximating the later Parsonian form, thereby creating a uniformity of language across the various theorists and embedding familiar ideas in an abstract and generalizing point of view. Another important excimple of the neoclassical trend is the synthesizing work of Collins (1975) who tries to link the Marx-Weber tradition to the Durkheim-Goffman line of thinking (the former, group conflict oriented; the latter ritual interaction oriented). In general, I will use the term "neoclassical" to mean this composite orientation to a alassio theoretical tradition that is valued and to some aontemporary frame of reference which calls for the advance of generalizing theory as the main purpose of sociology.^ It is in this neoclassical spirit that I wish to initiate a three part approach to the question of the current state of the nexus between sociological theory and mathematical ideas. Namely, in Part One below I will give a brief characterization of how some central ideas to be found in the classic sociological tradition relate to mathematical ideas. In many cases, I will mention actual published work in the tradition of formal mathematical thinking, but I will defer a more extended commentary on this formal tradition until Part Two. In Part Three, I will provide a point of view as to the nature of formal theoretic methods adapted to the needs of advancing sociological theory. Hence, my remarks begin [3611/143

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with the classic theoretical tradition, move on to the modern formal and mathematical tradition, and conclude with a point of view on how formal work can help us in this neoclassical epoch of theorizing. PART ONE.

CLASSICAL IDEAS IN MATHEMATICAL FORM

Comte. Perhaps the central idea for which Comte is known, an idea which is at once a key notion of his positive philosophy and a proposed sociological law, is the famous Law of Three Stages. According to Comte (Lenzer, 1975, p. 71), This law is this: that each of our leading conceptions—each branch of our knowledge—passes successively through three different theoretical conditions: the theological, or fictitious; the metaphysical, or abstract; and the scientific, or positive.

To go beyond the dismissal of this law as only an historical description (Turner and Beeghley, 1981, p. 52) we should seek its abstract form and try to capture the intuition involved. As a first step, we see immediately that it could be construed as proposing a stochastic process starting at Si (theological state) then at each time either staying in Si or moving to S2 (metaphysical state), then in each time either staying in S2 or making a transition to S3 (positive state). Once in S3, it stays there. This captures the initial condition (Si), the forward linear movement (Si->S2->-S3) allows for indeterminacy as to when the transitions will occur (the transition rate parameters), while nevertheless predicting that eventually the "conception" enters the positive state (the absorbing state S3) and stays there. A conceptual criticism of this formalization is that the "conception" may be normative, religious, ethical or esthetic — i n short, it may not be the "sort" of conception to which the law applies. A natural device, then, is a scope condition (Cohen, 1980), so that the law is said to apply universally but only under certain abstract conditions. Here Parsons (1973) has produced an array of functional components of cultural systems that helps delineate the possible scope. Any cultural system has four components: cognitive, expressive, moral-evaluative, and constitutive or religious. Comte's law, we may say, applies only to those conceptions that are predominantly cognitive in their function, i.e., predominantly functioning in adaptation to the environment of human action including the human-made parts of that environment. There is another formal interpretation of Comte's law that is suggested by thinking of it in terms of Weber's later emphasis on rationalization as a process that envelops all aspects of culture. In this interpretation, we can make use of the suggestion made by Parsons (1937, p. 752) that Weber's theme can be transformed from an historical generalization to an analytical principle by analogy to the law of increasing entropy, namely every cultural system is governed by a "law of increasing rationality." The law is conditional and "relative," in Comte's sense. One can think of it as

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describing a "gradient dynamic" (Fararo, 1978) in which the rule of process is: the state of the system changes in the direction of increasing rationalization (in Weber's sense). This can be visualized as climbing one or another slope of a potential function with possibly numerous local maxima. Such a local maximum is a stable equilibrium state. A primitive tribe's equilibrium rationalization of its music might be a very low-altitude peak compared with modern elaborated formalizations of musical notation systems (very high peaks) but in either case, the "mechanism" is one of an immanent drive toward more rational procedure if this is "in the cards." Note that in this second interpretation a continuous variable replaces the discrete states of the first interpretation. Returning to the first interpretation, one problem is that the three states and the four components of cultural systems are not conceptually articulated. If they are articulated, Comte's law becomes some sort of statement of one aspect of functional differentiation at the cultural level. The Markov process does not seem right in this interpretation: it assumes three disjoint states exist, it does not show how the states arise out of earlier states by some differentiation process. Spencer. It seems that Spencer's main contribution was to see how widely the idea of functional differentiation, from homogeneous to heterogeneous system, arises in nature and human history. If differentiation is so significant, do we have a mathematical way to treat it? Of course, we could note the principles of size, differentiation, growth and so forth (for instance, as stated by Turner and Beeghley, 1981, p. 109), symbolize the various state variables and write down approximate equations. In this formalization we could define "differentiation" as a state variable whose meaning is simply heterogeneity (Blau, 1977) . This is the analytical mode of dealing with the problem. A structural mode asks first for a representation of the structure. (The two modes are delineated as two fundamental types of conceptual schemes by Parsons, 1937, p. 30.) Differentiation in this sense is a mode of structural transformation and is probably best conceived as a "reverse homomorphism" (Fararo, 1973, p. 558) in which the simpler form is in the past: evolutionary differentiation goes from the less complex (homomorphic image) to more complex (domain of homomorphic map). This is a generalization of an idea put forward in the kinship context by Boyd (1969). Note that in one case, the analytic mode, differentiation is an analytical element, a quantitative property of a structure, while in the structural mode, it is a transformation of structures. The reverse homomorphic transformations generate more or less differentiated structures in the analytic sense, so the two ideas are complementary. We can imagine a Markov chain in which the state of the system is a structural form. The process begins with some homogeneous form and via some specified mechanism traverses states along one or another path of reverse homomorphisms. I have suggested

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this abstract type of process in a very different substantive context, namely over-time evolution of images of stratification (to be described in Part Three below). Note that in the structural transformation interpretation of differentiation we always deal with a multiplicity of possible "next states" and thereby conceptually discriminate the outcome from the generative process. As pointed out by Lind (1983) a major pitfall in evolutionary thought involves neglect of this distinction. The theorist then postulates a sequence of "stages" —namely just that sequence in state space which happened to have been historically realized from given initial conditions — t h u s eliminating the field of contingent possibilities that actually exist (as real potentialities) when we take the process or generative point of view of showing how the future state (or its probability) arises out of the current state and exogeneous inputs. Durkheim. One mode of differentiation is suggested by the idea of division of labor. Durkheim's suggestion was that, again in an evolutionary context, societies have passed from a "mechanical" integration of similar parts (via common conscience induced by similarity) to "organic" integration of dissimilar parts (via their functional interdependence). A more modern viewpoint (see for instance Parsons, 1967) is that both types of integration characterize, to one degree or another, any social system. The analytical relations among these notions are seen in the following path diagram: common cultural elements , , . , , ^ (values, ideas, ...) \ ^

^ • -, 4.1, ^ (mechanical path)

X division of labor

integration

functional interdependence

(organic path)

As the structural form exhibits transformation in the direction of increasing heterogeneity (increased division of labor), one path decreases integration but the other increases it, so that the latter may compensate for the former. Missing from this diagram is the idea that the common cultural elements cannot be compensated for indefinitely by interdependence: a threshold idea of some sort. Note that in this interpretation, Durkheim's "two types of solidarity" are two types of causal paths. "Mechanical solidarity," for instance, means integration produced by common culture. It should be mentioned that there is a second theory in Durkheim's work on division of labor in which the increasing differentiation is generated by growth in size amid competition for scarce resources in a bounded space. This process has been specified as a system of differential equations and even empirically tested (Land, 1970). A generic idea of the biological analogy in Comte, Spencer, and Durkheim is function. Durkheim carefully

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distinguishes function and cause, but from a modern standpoint and in one sense of this concept, function is a species of causation. Namely, it is contextual: whenever a network of looped causal influences has a certain form, then (subject to certain conditions) the network represents a functional system. This idea is due to Stinchcombe (1968). For instance, suppose the causal network diagram is as follows: Magical -^ Collective activity ^ ^ anxiety "*

Environmental outcome uncertainty

The theoretical condition referred to above is that for the system of which collective anxiety is a state variable, if collective anxiety does not remain within certain bounds then the system is destabilized so as to be reconstituted in a new form or so as to so drastically change as to be a different system. Then note that magical activity is functional: it counteracts the tendency for increases in environmental uncertainty to generate higher collective anxiety that, if not compensated for by the reverse causal loop, would result in system change or dissolution. Another activity might well be in a causally analogous position in the causal network in which it would be a functional alternative to magic. But note that stability is always conditional: the quantitative parameters of the (implicit) mathematical model of the relationships must satisfy certain conditions. Theorists sometimes forget that the mere existence of feedback does not assure stability. (For details, see the appendix in Powers, 1973.) Marx. According to Stinchcombe's analysis, Marx's model is a special case of the functional causal network in which special attention is given to how the various classes of actors induced by the mode of production and by the social relations of production have differential causal force on the maintenance or change of the corresponding social structure. Similarly, Sztompka (1974) argues that Marxian dialectical systems theory is one of a family of functional models. In a more mathematical context, catastrophe models are relevant to theories of social conflict processes as is pointed out in the literature (Fararo, 1978; Tiryakian, 1980). This relevance is largely due to their explicit attention to "bifurcation" and related phenomena of unstable states. The notion of "internal contradictions" in Marxian sociology can be interpreted in terms of "rank balance theory" (Zelditch and Anderson, 1966). In the social system of late eighteenth century France, for instance, the nobility and the bourgeosie have inverse relative positions in the economic institutional sphere compared with the political institutional sphere. The instability is interpreted formally as an absence of balance, defined as consistent rank positions across a multi-dimensional stratification system. Instability implies, in this balance theory, tension and efforts to change relative positions,thus theoretically "anticipating"

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a revolutionary epoch in which the balancing process works itself out. A similar but more complex and quantitative conceptualization was given by Galtung (1966). Weber and Mead. According to Weber, interpretive understanding of social actions is a prerequisite for the causal analysis of social structures and processes. In modern form, we can put it this way: there is an actual world of events and some events are behaviors. Behaviors are treated as actions when they are analyzed relative to cultural frames of reference, according to which the behavior means one or more things to the producer of the behavior and to the situational interpreters of the behavior. For instance, a behavior "pressing down a series of keys in a booth" is the action of voting in certain circumstances in certain systems of action in which institutions exist in which the described behavior counts as an instance of voting, for both the actor (who intends to vote) and the situational interpreters (who regard and formally count the behavior as a vote). If Weber's ideas are taken in conjunction with Mead's stress on interaction in situations, involving given and emergent significant symbols, we have the symbolic interactionist foundation of sociology. Two formal renditions of this Weber-Mead foundation for sociology exist: a continuous-state negative feedback model (Heise, 1979) and a discrete-state production system model drawn from artificial intelligence research, to be described in Part Three below. Both models are cybernetic in that they formulate some process of control or guidance and thus incorporate one of the most conceptually important ideas in modern science. The common idea is to generate episode-forms of institutionalized social action and interaction. Both models are partially successful. The Heise model has the advantage of being framed in an analytical framework of real numbers (in which the semantic differential is the measurement basis), permitting deductive fertility. Given (at least) these two models it is no longer true to say that the classic idea of human society as symbolic interaction fails to be reflected in formal or mathematical model-building. Simmel. In his notion of "formal sociology," Simmel saw a mode of abstraction from the institutional particulars that are highlighted in Weber's methodology. This mode of abstraction yields two central ideas. One idea is that there are important quantitative elements and relations in society. In Simmel's own essays, the discussions of dyads and triads and of group expansion illustrate this idea. The idea has received its macrosociological expression in Peter Blau's Inequality and Heterogeneity (1977). The existence of "parameters of social structure" such as age, religion, political party, wealth, and authority, gives rise to a system of univariate and multivariate distributions. The theory interrelates these quantities around the problematic element of amount of inter-group association, obtained by counting associations between members. Blau connects this purely

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conceptual starting point to the Durkheimian problem: differentiation is treated as heterogeneity, a quantitative variable defined over distributions, and integration is treated as amount of intergroup association. In a sense, Blau is asserting a causal path from differentiation to integration that seems to bypass both the mechanical and organic paths, but this is only because association is not really a sufficient condition for integration. A formalization of this systematic theory is discussed in Part Three. The network mode of representation of social systems relates to the second key idea of Simmel's formal sociology, the "geometry" of social life. Formal work on "duality" of individuals and groups by Breiger (1974) makes a direct use of Simmel's insights, now framed in network terms. This work recently has been extended to deal with three types of nodes in multilevel structures (Fararo and Doreian, 1984) in which the idea is that the structure of a concrete whole (level 1) is given by the sharing of subparts (level 3) by the parts (level 2 ) . This is a formal representation, too, of the idea of interpenetration crucial to Parsons' work. However, in one aspect we must be concerned about a potentially misleading element in Simmel's conception of the form/content distinction. In a Kantian mode, Simmel urges us to see that living processes are always synthesized in terms of forms. But in modern science a rather important distinction exists between the form of observable phenomena and the underlying generative structure that explains such forms. In genetics, for instance, the beautiful symmetries that characterize the observable form of corn on the cob must be distinguished from its causal mechanisms, constituted by some underlying structure (of genetic parts, relations, and processes). Hence, forms of interaction can be objects to be represented on the geometric analogy, but in terms of the explanatory aim of generalizing theory the focus must be on some underlying structures that generate these observable forms. Pareto. It was in the area of what Freese (1980b) very appropriately calls "theoretical methods" that Pareto excelled by comparison with other classic theorists. Central to his approach was the idea of system and a corresponding mode of analysis of dynamics and statics. Recently, Powers and Hanneman (1983) have produced a simulation of Pareto's theory, generating the expected cycles. But Pareto is also significant in terms of his influence on Homans and Parsons. In The Human Group (1950) a very sophisticated, if easily misunderstood, systems model was employed. The misunderstanding arises from interpreting Homans' propositions as "straightforward" empirical generalizations instead of the verbal equivalent of statements about partial derivatives. Subsequently, Simon (1957) formalized at least part of this system in what counts as one of the finest papers in the history of mathematical sociology. The line from Pareto to Homans to Simon is interesting: Pareto supplied the inspiration and keynote idea of couching sociological theory in a systems

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framework but did not actually employ a mathematical system for his sociological theory; Homans attempted to realize Pareto's system idea in non-mathematical form to provide systematic organization to a variety of otherwise disconnected fragments of knowledge of small groups as social systems; Simon put Homans' propositions in the form Pareto really meant, a system of differential equations analytically studied for its equilibrium and stability. In a beautiful anticipation of later ideas, Simon also produced what amounted to a "catastrophe analysis" of the model (Fararo, 1984) . In turn, Simon's model seems to have inspired similar formalizing efforts, most notably the previously mentioned work of Land (1970), which formalizes Durkheim's causal theory of the division of labor in terms of a system of differential equations. This entire generic "theoretical method" has been explicated by mathematical sociologists: the papers by Land (1975) and Fararo (1978) together provide an introduction for social scientists. Summary. In the above paragraphs, admittedly all too briefly, I have tried to show some connections between classic contributions to sociological theory and mathematical ideas, including actual models in the literature. I now want to provide a second type of treatment of the nexus between theory and mathematics by looking at how the formalization idea common to "mathematical sociology" and "formal theory" emerged and developed in the period after World War II. Following that, I will offer an elaboration of my point of view on "formal-theoretic procedures," efforts that combine the aims of formal theory and the techniques and standards developed in the mathematical models tradition. PART TWO.

FORMALIZATION TRENDS IN SOCIOLOGY SINCE WORLD WAR II

Emergence. It is appropriate to speak of what Newell and Simon (1972) call a "Zeitgeist" in the immediate post-war period of the late 1940s and early 1950s. The key developments had their origins in intellectual advances made earlier, of course, but notable works and interrelated new disciplines were to appear in a cluster in this period, including most importantly cybernetics (Wiener, 1948) and game theory (von Neumann and Morgenstern, 1947) . Operations research emerged from wartime concerns with submarines to study more generalized phenomena, such as queues. Experimental psychology, building upon its long tradition from Wundt to Hull, gave rise to mathematical theories of behavior acquisition and change (Estes, 1950, Bush and Mosteller, 1955). The extension of formalizing techniques to hximan phenomena, using logic, mathematics, and computers, was the key idea. Optimism about the future of formalized social science was strong and the successes fueled the intellectual movement. It was an epoch in which social scientists who thought of themselves as "real" scientists accepted what is now called "the positivist model of natural science" and took this as the natural

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science model for social science. For the dominant psychological frame of reference, added to these presuppositions was a behaviorist model: human behavior was mapped under a stimulus-response formula which did not attempt to represent cognition. In sociology, Homans and Parsons were each developing what they regarded as theoretical syntheses, one purely theoretical (Parsons, 1951), the other more empirically grounded but also more scope-restricted (Homans, 1950). It eas an era in which the distinction "theory versus methodology" became institutionalized in the sociology curriculum, with Parsons at Harvard as the personification of "theory" and Lazarsfeld at Columbia the personification of methodology. "The language of social research" (Lazarsfeld and Rosenberg, 1951) was different from that of sociological theory. As a science, sociology was supposed to be hypothesis-testing, based on numerical operationalization and comparison with null hypotheses in either a survey research or experimental setting. So there was a recipe for generating knowledge. Intellectual leaders such as Lazarsfeld never took this recipe too seriously, but by the time it was watered down to textbooks, college teaching, and graduate training it became "normal science." Oddly, although now the Vienna Circle seems to blamed for the crudity of all this, there is little evidence that the bulk of sociologists even heard of or took seriously the writings of Carnap, not to mention Hempel. If they attended to this philosophy of science, in Hempel (1952) they would have found ideas for a deep self-critique of their commitments and a basis for a different orientation. Articulation of Aspirations and Standards. So it was in the environment of sociology rather than in the field itself, during this period of the 1950s, that the aspirations for mathematical social science were stated and the standards articulated. Anatol Rapoport through a series of articles and books stated what amounted to a primer in mathematical model-building and provided exemplary work. Such contributions included his important expository paper in Symposium on Sociological Theory (Gross, 1959), his papers on random and biased net theory in The Bulletin of Mathematical Biophysics, his review articles in General Systems (a yearbook) and his work in Behavioral Science , publications he helped to launch in the mid-1950s. A substantial part of his work was in game theory, culminating the 1950s with a lucid comparative study of conflict models (Rapoport, 1960). Other standard-setters in this period included Simon's previously mentioned formalization of Homans' theory, the Blumen, Kogan, and McCarthy (1955) study of mobility using Markov Chains, and, very signficantly, James Coleman's critique of Rashevsky's models (Coleman, 1954) and his development of models close to known forms of sociological data, such as sociometric and diffusion data (Coleman, Katz and Menzel, 1951). The influence of Rapoport on Coleman is quite clear (Coleman, 1960) . The idea of a mathematical model as theory in mathematical form began to take hold. Writers of

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variant interests all agreed that such models permitted the logical derivation of falsifiable claims about some class of phenomena. This differentiated mathematical models from curve-fitting and data analytic reduction methods. This theoretical approach to models included the idea of "successive approximations" articulated in sociological theory first by Comte, then by Pareto and then later stressed by Homans (1950). Models were not expected to be correct in every detail nor to cover the entire potential scope of interest in a class of phenomena. They were to be modified and generalized (in a formal sense) over time. Even though such a model might include entities and processes not presently observable, the logical connections among ideas in a conceptual network assured that the theory was testable. This permitted release from the rigidity of behaviorism. A beautiful explication of standards emergent in this period later was given by Lave and March (1975) . Truth and beauty are two general types of criteria for evaluating models, each encompassing more specific evaluative criteria. In the case of beauty, for instance, simplicity, surprise, and fertility were three ways to evaluate models.

Yet this breakthrough in theoretical model-building was peripheral to the main developments of positivist quantitative sociology in the 1950s. Although it is hard to believe at this time, mathematical sociology was like ethnomethodology two decades later—something floating into sociology upsetting the image of the field as an interplay between an elite of essay-writing major theorists and the masses of statistics-gathering researchers whose interests were to be bridged by the magic wand of operational definitions. One methodological work in this period, the book on theory and verification by Zetterberg (1955) helped some of the sociologists of this period to see the limitations of their approach to social science. Yet by persuading sociologists that theory could be axiomatic although not really mathematical, Zetterberg took a route different from the standards of axiomatics in science then being beautifully articulated by Suppes (1957) and applied by him in psychology (Suppes and Atkinson, 1960). The main point is that Zetterberg omitted the key idea of representation or model (as a nonlinguistic entity which has assumed and derived properties). Subsequent work following his lead tied formalization to the awkward and unproductive use of "sentential calculus." Formal theory, in this sense, was notational translation of verbal theories rather than a creative construction of testable idealized models based on exact concepts. Later in sociology, Fararo (1973) directly adopted Suppes' methods of axiomatization. Philosopher Mario Bunge (1973) also stated an interesting variant of the axiomatic idea in science which linked the deductive system idea to the notion of "model object." (I will discuss axiomatization in detail in Part Three.) Most recently, in sociology, Freese (1980a) has clarified the whole methodological basis of all this by distinguishing between different meanings of "formal theory." An explicit critique of Zetterberg's position is very well-stated by Wilier and Heckathorn in a paper in Freese (1980b).

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The Flowering of Examples. The 1960s was a period of immense creativity in mathematical model-building and formal theorizing. Some highlights can be seen in terms of the type of model employed, often marking notable "successes" in the movement: (1) Stochastic process models became very popular among formalists. In social psychology, Cohen (1963) gave a beautiful exposition of his successively refined Markov chain model in his Conflict and Conformity. It stands today as a study every model-building novice should read. In sociology, James Coleman's pioneering An Introduction to Mathematical Sociology (1964) appeared and excited us by showing another way that mathematics could be creatively adapted to the needs of sociological analysis. Coleman looked to the form of sociological data, with two types dominant in his mind: survey-type attitudinal data and sociometric data. The key construct was the continuous-time discrete-state Markov chain applied to an aggregate of individuals. While Cohen's model could be formulated in discrete time, corresponding to experimental trials, with a reasonable assumption that individuals would not change state between such observation points, no such assumption could be made for the series of observations in a panel-study; hence, Coleman realized, events could occur at arbitrary times and required a continuous-time formulation. Today, this idea is imbedded in the "event-history" approach developed by Hannan and Tuma (1979). Yet Coleman also included sophisticated dynamic stochastic network models rather than purely aggregational (independent processes) models. This has been stressed elsewhere in a presentation of the basic logic of his approach (Fararo, 1973, Ch. 13). (2) Stochastic structure models were a second class of models employed in this period. Most notable here was the work of Anatol Rapoport. This work was based on the earlier theory developed in The Bulletin of Mathematical Biophysics but now appeared with empirical tests in more accessible mathematical form (Rapoport and Horvath, 1961; Foster, Rapoport, and Orwant, 1963). These published studies were the inspiration for a similar effort reported in Fararo and Sunshine (1964), which also tried to set out the logic of baseline random models and biased net models. Later, baseline models were developed to an extraordinary level of reference to sociological theory by Mayhew and his co-workers. (See Mayhew (1984) for the logic of this work.) Today, this idea receives its most sophisticated mathematical representation as a mode of data analysis attuned to structural properties in the works of statisticians such as Holland, Wasserman, and Frank, through collaborations with social scientists such as Samuel Leinhardt. (See, for instance, Holland and Leinhardt, 1981). (3) Algebraic structure models were advanced considerably by Harrison White with his Anatomy of Kinship (1963), which built upon the previous work of mathematicians and itself was advanced further by other mathematicians, namely Kemeny, Snell, and Thompson (1966) . Intuitively, sociologists saw in White's type of mathematics, algebras representing

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role systems, a significant advance. Here was mathematics really in touch with a central idea of the classic sociological tradition where social structure means above all institutional role structures as contrasted with interpersonal sentiment structures studied in social psychology and sociometry. But the use of modern algebra, the perception of a restriction of application to primitive kinship systems, and the general inertia of professionals in relating themselves to new ideas combined to make the spread of these ideas more difficult than that of the stochastic approach, with its quantitative and predominantly aggregate-of-individuals approach to social phenomena. As Mullins (1973) has documented, however. White's students were busily absorbing the new ideas. As one major consequence. White's "structuralism" was to dominate innovations in mathematical sociology in the 1970s. Other notable work during this period of the 1960s included William McPhee's creative and enthusiastic Formal Theories of Mass Behavior (1963) and some influential texts: Kemeny and Snell (1962) and Bartholomew (1967). Above all, there was Types of Formalization (Berger, et al., 1962) an exposition of the logic of model building. This exposition conceptualized the methodological role of the then most theoretical use of graph theory—in balance theory—under the rubric "explicational models." These are models constructed with the aim of formalizing a central concept in a theory. The key idea was to stress the function of the model in the formalization process. Any actual use of a formalism could serve any combination of three functions: explication of a theoretical concept, representation of a particular recurrent process, and explanation of a set of empirical regularities by common derivation within one framework. More recently, Rapoport (1983) has generated a typology of models by a cross-classification by theoretic goals and mathematical means. The latter are classified as classical, stochastic, and structural; the former are in terms of three types of goals: predictive, normative, and analytic-descriptive. Hence, there are nine types of model-building efforts in this taxonomy. Most distinctive in Rapoport's view of models is the emphasis he places on insight into real world phenomena produced by the study of the properties of a model, even if the model does not generate predictions. For Rapoport even if the search for laws in social science is unlikely to ever resemble natural science in its success, the process of disciplined and detached analysis—as incorporated into a recognized intellectual role of "mathematical model-builder"— has an important role to play in any society encouraging the idea that the search for knowledge is a value in its own right. Practical uses of models may be less consequential than the way their construction and discussion functions in reasoned analysis of the nature of the social world. In short, through insight they may provide understanding although not a prediction and control capability. This subdued conception of the role and prospects for mathematical social science is similar to Wilson's essay on the limits of mathematical social science (Wilson, 1984) in conclusion although different from it in content-specifics.

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Returning to the 1960s, any even cursory overview must also mention the advance in abstract generalizing theory heralded by the publication of Sociological Theories in Progress, Volume One (1966), known informally as "STP-1." As edited by Joseph Berger, Morris Zelditch and Bo Anderson, the volume sharply discriminated between a sociological abstract level of formulation and instantiations of the abstract processes. In making this distinction, Berger and his colleagues tried to provide a way for a physics-like interplay between abstract theories, appropriate formalisms, and relevant data in their own work and, by inspiration, in the work of others. (The three components just mentioned, theory, formalism (or model), and data, form the core expository device of a readable text that appeared a decade later: Leik and Meeker, 1975.) Yet Berger's methodology was linked to experimental realizations with all that this implies about the knowledge process in sociology: micro-level rapidity versus macro-level sluggishness due to the special difficulties of macroscale research. This point is developed in more detail in a paper by Heckathorn (1984) in this issue. The late 1960s were years forming an epoch of primers in "theory construction" such as the well-received work by Stinchcombe (1968). (See Freese, 1980a for a bibliography.) This contrasts sharply with the present situation: by and large sociologists have become bored with instruction on how to be scientific theorists. Neoclassical theorists produce theory, not primers on scientific methods of theorizing and this is in the spirit of STP-1. Recent Developments. If we arbitrarily label everything done from 1970 to the present as "recent," then the recent period is a veritable explosion of work in all directions, with accompanying institutional developments. Mathematical sociology, as the development of testable sociological theories couched in an appropriate formalism, received its embodiment in this journal starting in 1971, initiated by Bernhardt Lieberman and later successively edited by Gordon Lewis, Ralph Ginsberg, and Patrick Doreian. In abstract theoretical sociology we have had the second volume of Sociological Theories in Progress (Berger, Zelditch, and Anderson, 1972). This volume contains Emerson's notable exchange network theory, an apparent advance over the earlier dyadic or triadic formulations first presented by Homans in 1961 and subsequently revised by him (Homans, 1974). Later in the decade, Blau (1977) abandoned his own exchange theoretic work in favor of an abstract macrosociological theory in hypothetico-deductive form, itself subject to formalization (as discussed in the third part of this paper). Collins (1975) massively "energized" the "neoclassical" approach in a self-conscious attempt to embrace the new microsociology of Goffman and the ethnomethodologists while focussing on the problems of general theory. This work marked a new type of synthesis, comparable to Parsons' synthesis in ambition and scope and in its use of the classic theorists. In terms of a fashionable debate of the early 1970s, it adopted "the conflict model" instead of "the consensus model" of society.

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The Harrison White "structuralist" group took off in a rapid acceleration that featured a pioneering paper on algebraic structural ideas (Lorrain and White, 1971) and a conceptual shift in mobility research (White, 1970). By 197576, White and his former students (Boorman, Breiger, and others) had worked out the logic of "blockmodels" and made it available as a routine research tool. A large variety of earlier algebraic and graph-theoretic models were conceptually unified under the notion of homomorphic reduction of a structure of social relations. The whole body of work was closely linked to the conceptual probes found in the theoretical work of S. F. Nadel (1957) but otherwise remained disconnected from mainstream sociological theory. Today, "blockmodels," thanks to computers, are routine tools of basic sociological research. (For an exposition and review, see Burt, 1980.) Coleman (1973) followed-up his book of 1964 with a contribution to a problem in sociological theory: how can we articulate the level of individual action with that of collectivities as actors? The mathematics is simple and not always convincing theoretically, but has proved fruitful for directing research concerning collective action and relates to the network paradigm (Burt, 1980). Meanwhile the "positivist" tradition of the 1950s mainstream sociology continued to become more sophisticated in what has been called the "new causal theory" paradigm (Mullins, 1973). But the center of gravity of mathematical model-building was linked to structural analysis. Leinhardt (1977) collected together some of the most significant contributions to the new paradigm. (A readable review of the whole movement is given by Berkowitz (1982).) A new journal Social Networks was formed in 1978, edited by Linton Freeman. Significantly, it carried the subtitle "A Journal of Structural Analysis." This recent period, then, can be characterized by two trends. First, there was a proliferation of technical developments, such as the previously mentioned network analysis methods and techniques and the further refinement and applcation of stochastic process models and also essentially deterministic process models, often with stochastic input terms. (For a bibliography, see S«5renson, 1978.) Secondly, there was a proliferation of new theoretical work, including not only that mentioned earlier, but some important papers such as Granovetter's analysis of the role of weak ties in social systems (Granovetter, 1973) which used mathematics sparingly but significantly to address theoretical questions. I suspect that for many readers, Granovetter's style achieves a near-to-optimal mix of mathematics and sociological theory for the present epoch. Whether discussing information diffusion based on weak ties or analyzing collective behavior (Granovetter, 1978) , he keeps the theoretical problem foremost in view without overwhelming us with technical elaboration of dubious utility. Yet mathematical ideas are indispensable to his work.

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All this takes us to a third trend, which is much wider than the scope of this paper: the rethinking of the philosophical foundations of natural science and of sociology. For sociologists, the rethinking has strong elements of inherent myth: the myth, for instance, that logical positivism is somehow realized in the work of the statistical modellers, by either the earlier survey-based recipe of the 1950s or the latest new causal or loglinear recipe. My point is that, if one means Carnap or Hempel or Braithwaite, then this tradition of the philosophy of science receives its best realization as a standard guiding research not in the data-analytic refinements that have proliferated in the recent epoch but in certain of the more theoretical developments just reviewed. One prime source of the "post-positivist" viewpoint as to the nature of science stressed by Alexander (1982) is Toulmin (1953). According to Toulmin's philosophy, discovery in a theoretical science is equivalent to the invention and successful explanatory application of what I am calling an appropriate formalism, in which one learns how to theoretically model a class of phenomena. So Alexander's post-positivist philosophy of science is no embarrassment to much of the tradition I have been reviewing. It never embraced very many of the twelve positivisms recently explicated by Halfpenny (1982) and certainly it never adopted the view of science as "fact collection" with inductive discovery of laws. While this tradition is indebted to some extent to logical positivism it is now more congenial to philosophies like Toulmin's which stress models in their creative function in science. Another such philosophy is the realist view of science found both in Bunge (1973) and in the work of Harre (see Harrd and Secord, 1973). Although Harr^ discounts the significance of formalism in science, Bunge captures this idea nicely in his more formal approach. In this way, these philosophers recover an intuition about discovery in theoretical science that was lost in the more "logicist" conceptions of theorizing put forward by the logical positivists, in which a model is merely a useful but also restrictive appendage to an abstract theory. As a personal note, I may say that in writing my book (Fararo, 1973, particularly Sections 1.3 and 4.18) I tried to provide a philosophy of science for mathematical sociology which synthesized Toulmin (1953), Suppes (1957) and Hempel (1952) with an essentially cybernetic model of the interrelation of the components of an evolving science. Thus the philosophical framework which informed the book included scientific change intrinsically, as demanded in postpositivist philosophies of science. Summary. Looking back over this whole development, I am struck by how different things are today in the epoch of routine mathematical modelling as compared with the epoch of inspired first creations. As the work has proliferated in branching processes involving the usual subgoal elaborations and transformations of means into ends, it has sometimes seemed that in place of the original ideas we now have seemingly endless technical additions. The work is too complex.

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too multi-parental in its origins in terms of goals and means, for anyone to really take hold of it as a whole. We now read selectively and produce limited contributions located somewhere in the ever-growing tree of elaborations. One can feel alienated from much of this: it is our joint and human product, but it is all too much and perhaps too soon from a theoretical standpoint. Yet I suspect that any modern discipline generates the same sense of an overwhelming massive collective outpouring whose global meaning is obscure. Yet for at least some subtrees of the whole growing tree of technical elaborations there can be cumulation relative to a theoretical system. But such cumulative advance of theoretical sociology seems difficult to achieve, although essential to our current predicament, as forcefully argued by such otherwise different theorists as Freese (1980b) and Collins (1975) . What I am saying is that the formalization trend just reviewed leaves me, on the one hand, impressed by the achievements in so many directions and yet on the other hand, distressed that the proliferation of work has left us without a sense of comparable overall advance of theoretical sociology.^ So perhaps there is still some purpose to trying to state, in one's own terms, formal theoretic procedures which, if adopted more widely, might contribute to the advance of theoretical sociology. I turn to this task in my final part. PART THREE.

FORMAL-THEORETIC PROCEDURES FOR NEOCLASSICAL THEORIZING

Multi-dimensional Rigor. Although many different kinds of efforts are needed to advance the body of generalized theory in sociology, in terms of the subject-matter of this essay, I will stress the formal-theoretic element. But I will try to do this within a broader frame of reference. In recent years "theoretical sociology" has meant a commitment to realize certain standards in theoretical work. (Let me leave aside the content problem, which I am tempted to locate as some sort of neoclassical synthesis building on the contributions of Parsons, Berger and Luckmann, and Collins.) These standards involve four interrelated components of rigor: rigor in conceptualization, rigor in explanation, rigor in metatheoretical justification, and, finally, rigor in formalization. In each case, rigor means something different but the common element is a self-conscious effort to meet a higher standard with respect to that activity. Different theorists provide different implicit weightings to these four components, but all deserve to be considered in the collective enterprise of theoretical sociology. First, we can seek rigor in conceptualization. In this context, rigor means greater coherence and consistency than in an earlier epoch. Ideally, every notion should be related to every other so as to form a self-consistent system of ideas, adequate for the interpretation of some domain of phenomena. One example of work which optimizes on this component of rigor is the analysis by Sztompka (1974) of functional systems. Sztompka follows a logic of successive

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generalization, starting from the simplest functional systems and adding components in a series of logical constructions terminating in a dialectical functional system. Like Stinchcombe (1968), Sztompka is synthesizing Marxian thought with a Durkheimian tradition of functional analysis. This work is logical without being formal; its key property is logical construction. Another example of such conceptual construction is given by the work of Bates and Harvey (1975) who attempt to recast the action foundations of social systems by a constructive logical process starting from unitacts and moving in steps through norms, roles, positions, groups, and so forth. Here no claim to theory is made, in the sense of explanation; both works are seen by their authors as necessary preludes to explanatory tasks. A second component of rigor involves explanation. Here we can try to increase the extent to which we state explicit principles and provide logical if not formal arguments leading from the principles to the phenomena to be accounted for, providing some answer to a "why" question. Quite apart from the adequacy of the explanations, an example is Homans' revised general theory of elementary processes (Homans, 1974). Another example, in a different tradition, is the conflict theory presentation by Collins (1975). In both cases, explanation is given primacy over conceptualization. "Mere" conceptual schemes are criticized as, in Homans' phrase, "vocabulary without sentences." I would prefer to say that a coherent conceptual scheme is a necessary but not sufficient condition for a satisfactory explanatory theory, but not in the sense that it temporally precedes the statement of the theory but that theories found to be satisfactory will also tend to be conceptually rigorous. (A persistent criticism of Homans' work has been that it has lacked such conceptual rigor. See Turner, 1982.) A third context of rigor involves justification in the sense of some metatheoretical grounding of theoretical choices of problems, concepts, and modes of explanation. There are many examples of this demand and its response in recent sociology, but one particularly good example is given by the collaborative work of Harr6 and Secord (1973), philosopher and social psychologist, respectively. They provide a critique of conventional experimental social psychology and through rigorous argument attempt to provide a new foundation. They employ Harrd's realist philosophy of science and also make use of the modern analytical philosophy of action, in addition to drawing upon substantive themes from Goffman, Garfinkel, and other sociologists who have urged a greater attention to the microsociology of interaction. Finally, there is a fourth component of rigor. Paradoxical as it may seem, formalization itself is a context for more rigorous formulations. Here we can ask for "model objects" (using Bunge's phrase) with defined structure and derived properties. These correspond to the set-theoretical entities stressed by Suppes (1957) . As I pointed out earlier, this is in contrast to a purely sentential formalization, the mere symbolizing of given substantive theoretical sentences without the introduction of the concept of a model.

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Formal-theoretic Procedures: A Viewpoint. I would like now to state a general point of view on formalization and to relate it to other components of rigor. First I will state the point of view in general terms and later give several examples. Conceptual rigor can be incorporated into a "formaltheoretic procedure"—here I am adopting the nice phrase "theoretic procedure" from Wilier (1984)—by employing the technique of defining exact concepts (Fararo, 1973, p. 93), by which I mean concepts defined by a set of axioms on a settheoretical structure involving objects such as sets, trees, functions, and so forth. The axioms jointly define a new type of abstract entity, bringing a new type of model into being, available for model-building purposes. A familiar example is game theory, where the intuitive notion of a game is abstracted to a set of axiomatic conditions that define the set-theoretical predicate "is a game," an exact concept. The coherence aspect is clear in the mutual involvement of all the primitive entities that enter into the one coherent abstract idea termed a game: all these entities relate to each other, forming a self-consistent abstract entity formalizing our intuitions as to the essence of a game. The theory of games is elaborated on this foundation and particular gametheoretic models presuppose it. The demand for explanatory rigor can be represented in a formal-theoretic procedure by the construction of a generating process (Fararo, 1969; Fararo, 1973, p. 202). This is a process on a state space that "grinds out" the overtime changes of state and whose axiomatic stipulation allows the derivation of some constrained mathematical generator. For instance, a Markov chain might be derived. This derivation is of first importance and must be contrasted with merely positing such a chain. In the latter case, the matrix entries need only satisfy general probability constraints, while in the former case the matrix is constrained by the sociological theory given by the axioms. For instance, there may be many zero entries, which outlaw certain logically possible transformations of state. In addition, this means that the entries in the matrix are functions of theoretical parameters appearing in the axioms. There is a contrast here with the technique of displaying the parameters of such a matrix as regression functions of external variables, a method originating with Coleman (1964) . The whole idea is to generate the phenomena of interest, to provide a mechanism accounting for its detailed form. This contrasts with two other ways of thinking about explanation: the first is writing down a few sentences that logically imply the given sentence to be explained, a sentential viewpoint that misses the generating process which provides some strong intuitive sense of an explanatory mechanism. The second idea of explanation is the path diagram terminating on some dependent variable, which is based on a Humean conception of causation as contrasted with a realist conception (see Keats and Urry, 1982). One outcome of the above formal-theoretic procedures is the proof of abstract sociological theorems within an

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axiomatized conceptual framework. Let me explicate the key terms in this statement. By abstract I mean pertaining to generic form rather than to instantiations, to particular historical or cultural conditions. The structure or process involved must be phrased in genuine sociological concepts, i.e., generic concepts of structure and process that transcend the categories descriptive of particular societies at particular times and places. (See Cohen, 1980, for an explication of this distinction.) By proof 1 mean derivation employing not only logical inference rules but mathematical theorems of all sorts that are available for use as inference rules if one makes any use of the corresponding mathematical ideas. For instance, a probabilistic conception built into one's axioms allows use of all the axioms and theorems of probability theory as inference rules. (For details, see Fararo, 1973, p. 225.) Proof, in this sense, contrasts with empirical demonstration, confirmation, or verification, however it is put, based on the coordination of the abstract ideas to definite instances of the class of phenomena within the scope of the theory. Finally, by theorem I mean not any derived proposition but one which is central to the theory in the sense that the theoretician is making a judgment that it provides a proposed solution to a major problem that has instigated the theoretical work. Therefore, I would propose using the term theorem with much less frequency than some others (for instance, Blau, 1977), for otherwise the reader is unclear how to appreciate the relative significance of an array of proved propositions. Generality of coverage, with respect to all the possibilities inherent in the conceptual scheme, seems to be one formal criterion which links to the notion of centrality to the theory. Another aspect of this viewpoint on formal-theoretic procedure has to do with theory, models, and reality. Theory describes abstract models, not reality directly. Theory refers to reality indirectly via the activity of constructing models (of the type defined by the axioms of the theory) and evaluating them. I would say that the theory exactly describes the models; the models are definite entities, just what they are and not anything else. But reality is concrete and hence always in flux and bounded only by conceptual acts. The relation of model to reality is therefore approximate and involves idealization. If models, over a long period of time, seem to have properties that provide a good approximation to the phenomena observed under the appropriate scope conditions, then one can say that there exist real structures generating such phenomena such that the models describe approximations to these structures. The concepts employed in a theory, which develops in this way under empirical control, correspond to aspects of reality. The theory idealizes and studies exact entities, models, but it is not fictional. This theme is present throughout the work of Parsons as what he calls analytical realism (Parsons, 1937, p. 730) although he did not realize the value of formal-theoretic procedures of the sort I am discussing.

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The final general point that I want to make is that formalization has a contextual variation in its relation to preexisting substantive theory. One context is characterized by the simultaneous creation of a substantive theory and its formalization. The theorist is both animated by some substantive intuition about the mechanism and thinks about the world in mathematical terms. Such a creative action is not one of translating someone else's ideas into mathematics. By a nice conjunction of interests and skills, he or she who has a mathematical view of the world, who perceives aspects of things directly in mathematical terms, is also he or she who has the substantive intuition, the speculative conjecture as to some process or structure. As discussed earlier, an excellent example is Granovetter's model of collective behavior (Granovetter, 1978) in which a whole new point of view of the process generating riots and analogous collective phenomena is presented in terms of a mathematical representation of the generating process involving thresholds. Another context of relation between formalization and substantive theory is a polar opposite of the first context: here the formalization starts from some more-or-less given "systematic theory," to use the terminology of Freese (1980a). One example might be one I cited earlier: Simon's formalization of Homans' theory of groups. This is perhaps the context that many sociologists think of when they think of the meaning of formalization. Another context of interest is one in which the aim is to arrive at a formal conceptualization, a "pretheoretical" enterprise so far as explanatory theory is concerned. There is an effort to represent the intuitive content of one or more substantive conceptual schemes as a prelude to formal theory itself. Examples. So far I have stated a general point of view on formal-theoretic procedure and characterized one main outcome of the use of these procedures as that of proving abstract sociological theorems. In addition I have mentioned a contextual variation in the nexus between formalization and substantive theory. At this point, I want to provide three examples, drawing upon my own work. The first example will be the context in which theory and formalization are jointly created. The second example will focus on formalizing part of a given systematic theory. Finally, the third example will illustrate formal conceptualization problems. My example in the first type of context deals with images of stratification (Fararo, 1973, Ch. 12). The origins of this formal theory lie in my reading of Deep South (Davis, Gardner, and Gardner, 1941) . On one page of their empirical study, the authors summarize their results in diagrams that show how the perspective on the social system varies by position in it. An abstract empirical generalization can be framed which generalizes from the relationships apparent in the total arrangement of these image diagrams: in any social system, actors occupy positions in a stratification system such that they develop "reduced" images of this system; such

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images depend on location in that nearby classes appear with finer distinctions than more remote classes. By "reduced" I mean that the image preserves order but combines certain actual classes. In approaching the theoretical problem of explaining this empirical generalization, the first step is to represent these information-rich, non-numerical entities termed images. In this case, because the focus is on the construction of a generating process, on explanation, no effort is made to provide a once-and-for-all universally appropriate representation of a stratification system. This aspect is instrumental and secondary to the explanatory problem, so the representation is selected to coordinate to the intuitive notion of how images change. So explanatory rigor has primacy over conceptual rigor in this effort. In particular, the diagrams in Deep South provide the starting point. The folk descriptions are analyzed in terms of an ordered set of attributes they imply. In the example or instance of Deep South, three attributes,_in order of importance to the actors, are; white or not (w, w ) , old family or not (f, f) and rich or not (r, r ) . The "focal actor"—the arbitrary actor whose image is to be generated over time—is in some actual class, say wfr,and this actor at some_given time differentiates this class from its nearby class wfr but conceives the uppermost class with no reference to wealth: wf, whether rich or not. For this actor, blacks are simply w, whatever their other attributes. Notice that in this instance there are eight actual classes ranging from wfr to wfr, but the image is reduced in that it preserves order but has fewer classes. Other class-dependent images differ from this one in content, of course. Notice that the actual stratification of the community is represented in terms of an ordered set of ordered sets. Then an image is an entity of the same form, as to the preservation of the order, but combining classes. To describe a generating process one imagines a state space of such images. One conjectures and axiomatically specifies a process that transforms an image into another image, possibly the same as the initial image. The basic axiom is that the focal actor seeks information about alter's position, but only enough information to define alter's relative position._ So since w is above w, the w-focal-actor never gets beyond a w image for blacks, but for w-alters, such a focal actor samples for family information and discriminates wf from_wf. If the focal actor is wfr and if alter is found to be wf, then further information is needed to decide relative position by reference to the third attribute. The information search, then, terminates when a "decision" is made: ego below alter, ego above alter, or, when all key attributes are sampled, ego and alter in the same class. It is proposed that anyone starts off in early childhood from a homogeneous image, "everyone is like me" and over successive interactions expands the image as new types of people are encountered. In this way, the generating process defines a trajectory in the space of images, starting from the focal actor's own position in the system.

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It turns out that the Markov chain derived from the axioms is absorbing, which means that any such focal actor develops a stable image. It is part of the idealization involved that any two actors in the same position develop the same stable equilibrium image, although usually by different paths. As a check on the model, the set of stable equilibrium images of all classes of actors reproduces the essential features of the full array of image diagrams in Deep South when the theory is instantiated back to this case. That is, it generates the empirical outcomes that suggested the statement of the empirical generalization prompting the explanatory effort. The theory provides one explanation of that generalization. Notice how this differs from trying to write down two or three sentences from which to deduce the statement of regularity; instead, the form of the data that were the instigation for the statement of the regularity is generated from a process on a state space employing a model object introduced in the conceptualization of the phenomenon. In this example, the empirical generalization to be explained is interpreted in theoretical and formal terms that reflect several types of intuition, or convictions. First, there is the intuition of a "dynamical system." (This phrase covers a language and a set of analytic procedures described in the applied mathematical literature (Hirsch and Smale, 1974) and also by sociologists (for instance, Fararo, 1978).) An empirical generalization which appears to be atemporal is often best understood as describing some aspect of a stable equilibrium state of a dynamical system. The notion of equilibrium belongs as a special case within the array of ideas dealing with dynamics. Namely, such states are the fixed points of a transformation process. Second, there is an algebraic intuition. The relation between the actor's image and the actual stratification order is one of homomorphic reduction, an idea mentioned earlier as due to the work of Harrison White and his various colleague and students. In their work the homomorphic reduction is accomplished by the analyst of social structural data. In the above model, it is given a realist interpretation. That is, the actors have social knowledge and such knowledge means some sort of correspondence or fidelity to something else, the system including them (and which they have culturally inherited from earlier generations). In the model the correspondence is generated as a homomorphic relation of the image to the stratification aspect of the system. The actor does not, of course, intend to map social reality under a homomorphism; rather interactions through time, which are the occasions for the transformation process to operate, yield an equilibrium situation in which the image constitutes a homomorphic reduction. All this relates to deeper problems about the nature of social reality as somehow intrinsically symbolic, a point alluded to in the original concluding discussion of the model (Fararo, 1973, p. 366). Thirdly, there is a sociological intuition. In simplest terms, it is the notion that there is no place for images of structure to come from except interaction, itself constrained

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by the pre-existing structure. But the generating process speculatively proposes a mechanism, operative in interaction, that accounts for the essential property of images. These forms of intuition do not exhaust those that are brought to bear on the explanatory task (for instance, there is also a probabilistic intuition involved in the axiomatic description of the process), but they illustrate one way a formal theorist can proceed. The mathematical model is not merely a translation of an idea that originally occurs in natural language. The phenomenon is conceptualized mathematically and the stock of exact concepts intuitively known to the theorist enters directly into the "perception" of the problem. Regarding the theorist as an explanatory problemsolver, mathematics helps to structure the problem space (Newell and Simon, 1972). Namely, it contributes to the mode of representation of the problem. I will close this example with a few additional remarks. The axioms both conceptualize and represent a space of possible states and formulate a generating process, all in abstract, generic sociological terms. So a category of abstract models is defined by the axioms. To study these models, one can define narrower classes or families of models, proving propositions about their properties. In short, if a suitably significant proposition is thereby demonstrated logically, we have the outcome noted earlier: the proof of an abstract sociological theorem. All this is the formal-theoretical set-up and analysis of a category of models, to be sharply distinguished from empirical analysis with its focus on instantiating the process to certain real circumstances, identifying the theoretic entities empirically, estimating parameters, and assessing goodness of fit of a specified model to an actual system. Of course, any such class of models and its study is only a starting point for additional theoretical ideas, via successive approximations. (Current empirical tests by Kosaka (1984) relate this model to fuzzy sets and measures.) A second example will illustrate formalization of part of a given systematic theory and also illustrate in more detail the idea of defining an exact concept via approximation within set theory. The systematic theory is that developed by Blau in his book Inequality and Heterogeneity (Blau, 1977). This theory is a deductively organized set of propositions on the relation of social structure to social integration, the latter conceived in terms of rates of associations among collectivities. The social structure is conceptualized in terms of positions as in the example above except that the attributes are both unordered and ordered. Formally, one can define an exact concept by means of what Suppes (1957) calls a set-theoretical predicate, as follows: A set-theoretic structure (C, M, F, A) is a Blau model if and only if: Axiom 1. C is a set of sets of the form: C = {Ci, C2, ..., Cj., Cl', C2', ..., Cs'} with each C^ finite and unordered (i = 1, 2, ..., r) and each C j ' ordered

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(j = 1, 2, ..., s ) . Each Ci represents a nominal parameter and each C j' represents a graduated parcimeter in Blau's terms. Axiom 2. M is a finite set. It represents the members of the social system under analysis. Axiom 3.

F is a map with domain M and range:

S = Cl X Ca X . . . X Cj- X Cl' X Cz' X ... X C^' The space S represents the possible positions into which members in M are mapped by F. Axiom 4. A is an irreflexive, symmetric relation on domain M. This represents the associations among members. Given a Blau model, we can define a frequency distribution D F ( S ) over space S, based on F: namely, for each position in S, there are a certain number of members in that position. This is a defined notion, whereas each term of (C, M, F, A) is a primitive notion. A social structural Variable is a function of some subdistribution of Dp(S). For instance, the heterogeneity of a particular nominal parameter is computed from the frequency distribution over the appropriate set. A Blau model that satisfies an additional axiom, representing the speculative theoretical element rather than the conceptual structure element is teirmed a Blau theoretical model. In Fararo (1981) the axiom involves "inbreeding bias" —homophily—since the Blau theory posits that actors tend to associate with others "of the same kind." It would appear that not all characteristics and associations satisfy the inbreeding axiom. For instance, in his review of Blau's book. Bell (1978) notes that the characteristic sex and the irreflexive symmetric association marriage fail to satisfy inbreeding: actors seek their opposites, not their own kind. Hence, in my paper describing the initial formalization (Fararo, 1981) , I described a second type of bias parameter to cover such cases. Skvoretz (1983) refined, corrected, and generalized the theory to cover both types of characteristics as well as mixed types. However, a model with characteristics and associations described in terms of the "outbreeding" bias parameter is not a Blau theoretical model since then not all characteristics are inbreeding in their salience. Hence the formal-theoretic procedure reveals the unity of conceptualization and the difference in theories within the common frame of reference, the category of Blau models. Note that one obvious limitation in Blau models is that although there are types of actors (in space S) there are no types of association (since there is only one relation A ) . Blau has abstracted from the multiplexity of relations so as to characterize actors in a complex way (with multiple attributes) but not to characterize their relations beyond saying that associations exist. In this sense, the conceptual framework is structuralist in only a limited sense. Recent work

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brings multiple relations networks into formal articulation with Blau models. See Fararo and Skvoretz (forthcoming). My last example will illustrate a third type of context of relation between formal-theoretic procedure and substantive theory. The problem is initiated by noting that institutionality is the hallmark of social reality and therefore should be the basic starting point for a formal conceptualization of social structure. Nadel (1951) writes of an institution as a standardized co-activity. Parsons (1951) defines "relational institutions" as patterns of complementary status-roles. They are the main outlines of the structure of social systems, its fundamental morphological units. Berger and Luckmann (1966) in a deep neoclassical synthesis treat institutionalization as the fundamental social reality construction process, bringing forth types of actor, types of action, and types of situations. Let us introduce the notion of a "unit-institution" as a building block (Fararo and Skvoretz, 1984). It is the most primitive institutional element that still preserves the essence of institutionality: there exists a stable design for social action such that typified actors do certain typified things in certain typified situations. For instance, the design for social action relating bus passenger to bus driver is such a unit-institution. Such an entity is a cultural object that has "multiple embodiments." For instance, all over the United States, on buses throughout it, one may find interactions organized around this design. How can we think of this in formal terms? There are three formal concepts that seem intuitively close to the substantive idea. All three emerged in the Zeitgeist which I mentioned at the start of this paper. First, there is the exact concept of a game, the idea of the game in extensive form. A structure of all legitimate plays, within the rules defining players and moves, is what corresponds to the unitinstitution idea. Secondly, there is the concept of a language as implying an infinite set of acceptable utterances in a community of speakers and such that a finite set of rules generates that set. Here the unit-institution is perhaps analogous to such an infinite set of well-formed sentences— all the possible "normal forms" of interaction under certain typified rubrics as to acts, actors, and situation. The grammar corresponds to a distributed finite system of rules internal to actors who thereby can play the roles in interactions that count as instances of the design. For instance, the bus driver has one part of the rule system, while the passenger has another part. On joint activation, they "enact" the Scime interaction language—their actions coordinate in typical ways. Thirdly, there is the cybernetic idea of a plan or program for action, particularly as conceptualized by Miller, Galanter and Pibram (1960) and Newell and Simon (1972). The former authors propose a control unit called a TOTE (testoperate-test-exit), which governs performance. Complex action is generated by a hierarchy of such units in which an operation at one level is accomplished by a TOTE unit at a

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lower level. For instance, "attending to passenger intentions as to getting off" is implemented by testing the rearview mirror for images of passengers proceeding toward the door to signal a desire to leave the bus. In the NewellSimon mode of representation, "production rules" are employed. These rules have the form: situation-conditions -> action

where the left side involves tests and the right side operations or actions. These, too, are organized in a hierarchical arrangement with subroutines for any given action as ways of doing that action. Moreover, search processes and other heuristics may be employed in any such implementation of an institutional design, allowing for the full range of cognitive generativity in action. In this third mode of representation a unit-institution generator is represented by a set-theoretic entity of the form (X, V, O, P, R) where, (1) X is a set of terminal symbols, that is symbols that correspond to possible current situational and goal conditions (e.g., passengeri, passenger2, exit door, red light, exit from b u s ) ; (2) V is a set of variable or nonterminal symbols, placeholders for a range of terminal symbols (e.g., passenger taking values such as passengeri); these are essentially the generic actor types and situation types with particular actors and conditions subsumed under them (by the actors) as instances; (3) 0 is a set of operations or actions (e.g., OPEN DOOR, ENTER, P A Y ) ; these are performed by actors when productions call for them and transform current situational conditions; (4) P is a set of production rules; (5) R is a partition of P and an ordering of each part, interpreted as a "rolegram" (for instance, one part involves a set of productions generating driver social actions and a second part generates passenger social actions). The space through which the interactive system moves is implied in these axioms: the situational state keeps changing under their actions and it is "defined" in terms of the cultural categories instantiating axioms (1) and (2). The axioms define the category of unit-institution generators. This type of representation has been discussed in detail in Axten and Fararo (1977). The above formulation corresponds, in intent, to the formal definition of a grammar (Chomsky and Miller, 1963) : to express in formal, set-theoretical terms the definition of the whole category of objects introduced in formalizing a subject-matter. The well-formed infinite set of "interaction strings" satisfies the exact concept of a language, as shown by Skvoretz and Fararo (1980). (See, also, for a related earlier contribution, Nowakowska, 1973.) The set of all "normal forms" of interaction in a bounded domain comprised of actions generated by a unit-institution generator forms a language of interactions. Hence, in an exact sense, driver and passenger or passengers (with one embodiment of

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the passenger part of the production system per actor taking that role) are "talking the same language," i.e., interacting under the same "grammar," which is the unit-institution generator. This generator is not merely an algebraic generator of a static cultural object (as in Chomsky's syntactical approach to natural language) it is a generating process; that is, it moves an interaction system over a state space over time. Hence, the idea of a unit-institution generator attempts to combine the "cultural" idea of a grammar as a generator of an infinite set of well-formed strings with the "dynamical" idea of a generating process. As such it functions very much in the spirit of Weber's action approach, as explicated by Parsons (1937) : not events alone and not cultural meanings alone, but their joint realization in action comprise the "bridge" role of the action approach between the natural science model of a world of events and the hermeneutical model of a world of meanings. In a modest way, this classic idea seems to be reflected in the exact concept of a unit-institution generator. This is a contribution to neoclassical theory, one would hope. Summary. In this part of the paper I have tried to describe a general point of view on formal theory and to relate it to sociological problems via a series of examples, each example illustrating a different type of context in terms of the relation between the formal work and substantive theory. The aim has been to indicate a set of formal-theoretic procedures—such as axiomatization within set theory, specification of generating processes, and construction of definite model-objects. The procedures can help advance theoretical sociology because although they are formal they are based on one or more of the other components of rigor that refer to concept-formation, metatheory, or explanation. In this way, they involve model-building which is theoretical in focus rather than data-analytic. CONCLUDING CAUTIONARY REMARKS The construction of formal frameworks, and of models within such frameworks, is a time-extended creative process in which many sorts of intuition and judgment enter. The process cannot be mechanized in the manner of data-analytic algorithmic models which are specified in terms of inputs to a regression or loglinear program that produces certain welldefined outputs. When this is thought of as theory construction, it is misguided. Theoretical work, even when formal, remains an art-form, perhaps a craft. There is no assemblyline for producing knowledge, although we now have a good idea about the needed components, both in terms of the classic tradition as to how social action systems are constituted, operate, and change and in terms of contemporary frameworks that attempt to go forward from some combination of key elements in that tradition. For example, the axiomatic method, as discussed in Part Three above, is not always the best way to proceed from a given problematic state of affairs; it is

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one procedure that sometimes proves fruitful. Often one finds that to set out the axiom system is pedantic, that a less rigorous approach is "better" for the purpose at hand. There is no rule-book for the activation of one or another formal procedure. A good idea is more important than anything else in science. No amount of formalism will work magic to produce a significant contribution through formalization of ideas, which from a more controlling point of view bearing on the content of what we are trying to do, are merely irrelevant. But this is just to say that the great unresolved problem of mathematical sociology is that of merging formalism with theory in such a way that the former becomes indispensable to the latter.^ NOTES 1.

I regard Talcott Parsons as the first neoclassical theorist.

2. Although the networks paradigm is not itself a form of neoclassical theorizing, the ideas and techniques should be absorbed into more rigorous neoclassical syntheses. One might wonder if "network" will be to general sociological theory as "physical space" is to physical theory: a necessary but not sufficient conceptual scheme. The needed concepts and principles are being set out (Turner and Beeghley, 1981) and theoretical syntheses governed by the neoclassical aspiration exist (Collins, 1975), but this work lacks the proper notion of theoretical model-building which would bring it into contact with network models and other formal developments. 3. In a personal communication, John Skvoretz notes that having set aside the content problem in Part Three, I end up admitting that rigor alone is not sufficient for theoretical advance. Then he asks if rigor is necessary to such advance and answers affirmatively. But, he notes, many quantitative sociologists will agree about rigor but they will be thinking of models of data, in which theory is an external thing used to interpret parameters in regression equations and the like, rather than models of processes that generate the forms of the data. Obviously, I agree with his preference for the latter type of theoretical model. I also hold the view, in agreement with Harrd and Secord (1972), that Goffman is a very rigorous analyst who employs three types of models (game, drama, ritual) very creatively: but his models are not formal. Can we imbed Goffman's extraordinary depth of insight into the theatrical and ritual aspects of face-to-face interactions in some formalism? And if we cannot, should we "settle" for a new form of model-building which has no formalism? REFERENCES Alexander, J. C. (1982) Theoretical Logic in Sociology. Volume One: Positivism, Presuppositions, and Current Controversies. Berkeley: University of California Press. Axten, N. and Fararo, T. J. (1977) The information processing representation of institutionalized social action. In P. Krishnan (ed.). Mathematical Models of Sociology. Keele, U.K.: Sociological Review Monograph No. 24.

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