Figure It Out!

Theoretical Questions About Visual Theorizing

Given this current state of the conversation, it is not necessary to review here general arguments about the nature and value of visual thinking. Placing those in the background, we may identify some open questions about graphical social theorizing in particular.

Some questions concern experimentation with novel graphical techniques and empirical study of theory visualization. For instance, questions remain about the visual grammar and vocabulary most appropriate for tackling theoretical problems in sociology. And we lack a comprehensive catalog of the visuals that sociological theorists have actually deployed throughout the history of the discipline. Such a catalog would provide a basis for taxonomies of theory visuals; histories of how visual forms have changed, waxed, and waned; studies of the life course of particular diagrams as they are reinterpreted by various authors; critical examination of successful and unsuccessful visuals; and baseline knowledge from which to build novel techniques. With collaborators, I have compiled the rudiments of this type of catalog, and we are currently studying it. I sometimes draw on it here as a source of illustrative examples.

Nevertheless, more basic theoretical problems remain open as well. These involve questions concerning the contributions of visuals to the theoretical process itself. In the following sections, I develop four interlocking lines of argument: (1) that visuals may aid theorizing not only with empirics (conjunctively) but also independently (disjunctively), (2) that the value of diagrams comes not only in the problems or topics they allow us to represent but in the inferential procedures or theoretical methodologies they help to consolidate and advance, (3) that not only does creative diagrammatic reasoning in sociological theory span the entire reasoning process from abduction to deduction to induction but that also abduction with diagrams may be cultivated for its own sake, and (4) that metaphorical associations are often woven into the very fabric of theory visuals, and these associations in turn encode basic assumptions informing them. These arguments work together to articulate multiple interconnected processes through which visualization can contribute to the creative intellectual-material practice of sociological theory and point the way toward a more robust integration of visualization into the sociological theorist’s repertoire.

Logical Versus Scientific Diagrams

But the issue runs deeper and involves appreciating the radical nature of Peirce’s own innovations in logic. The sharp distinction between logical and scientific diagrams—upon which the recommendation that sociological theory should be modeled on the latter but not the former—is not in keeping with Peirce’s understanding of logic. Indeed, the mere fact that Peirce used logical diagrams and thought visually is not what makes him distinctive in the history of logic. Even Aristotle apparently used diagrams as pedagogical heuristics for helping students to remember his sentential syllogistic forms (Greaves 2002). Peirce’s diagrams were far more radical than his predecessors’, however, since they drew out the radical implication of his pragmatism and semiotics: that logic and other apparently purely theoretical enterprises are experimental and observational practices like any other science. For Peirce, even the most abstract reasoning involves the same underlying processes as its empirical counterpart, namely, observation, experimentation, and manipulation (Grieves 2001:172). The implications of this pragmatist theory of theory apply to sociological theory as well—it means that theorizing in the disjunctive as well as the conjunctive mode involves the characteristic operations of creative scientific thinking.

Diagrams play a crucial role in Peirce’s arguments questioning the basic assumptions that had historically separated pure (sentential) logic from empirical observation. Indeed, it is in virtue of diagrams that logic can become an observational science: They provide the objects that the logician may experiment upon (Peirce 1994:362). We assign meaning to physical signs and undertake an “experimental methodology of manipulating signs and directly observing the consequences of this manipulation” (Greaves 2002:172). We learn about logic by learning about the implications of a well-constructed diagrammatic system, in much the same way that Kepler or Feynman and others have learned about physics through the iconic representation of physical relations (Cheng and Simon 1995; Kaiser 2009).

Thus, contrasting Kepler’s empirical diagrams with Peirce’s logical diagrams fails to appreciate the implications of the radical claim driving Peirce’s logical diagramming. Even the “purest” most abstract theorizing can become an experimental empirical practice, by giving itself something to observe, namely, a diagram. Indeed, several Peirce interpreters argue that because the “purest” theoretical ideas are the least visible, the pragmatic maxim to clarify our ideas through observing their practical implications applies to them the most stringently; in such cases, it is a maxim to draw and to learn from examining the difference our drawings make to our theoretical habits and practices (Fabbrichesi 2011:117; Kent 1997; Misak 2004:22; Otte 2011; Stjernfelt 2011:397). ¹

To be fair, Swedberg correctly observes that Peirce did have an unusual and innovative notion of logic and of deduction. He viewed diagrams—of any form, scientific, logical, artistic, whatever (Stjerfeldt 2007)—as vehicles for semiotic exploration and experimentation, where we trace out multiple paths from premises to conclusions (Peirce 1994:1501; Shin 2002). ² Yet by reenforcing the sharp distinction between logical and scientific diagrams that Peirce worked so hard to blur, Swedberg draws precisely the opposite conclusion that Peirce’s logic would seem to invite: Pragmatist sociological theorists, like Peircian logicians, may adopt a more open conception of deduction by using diagrams to experimentally observe and come to grips with various implications and possibilities of their ideas, with or without data.

Used as a tool for this sort of deductive reasoning, in other words, the diagram becomes a vehicle for creatively clarifying, confronting, and even discovering what our theories commit us to. Diagrams can do this to the extent they transform our concepts into tangible and visible objects for discovering our theoretical commitments. For example, simply drawing a standard path diagram of a causal theory requires one to take a stand: Does A cause B or vice versa? Do I believe that other variables cause C or just these? Are they correlated or uncorrelated? Directed acyclic graphs (DAGs) pose even more stringent demands on how their users formulate their ideas as a causal theory (Morgan and Winship 2014). If you change the diagram, you are committed to a change in your causal theory; an altered diagram represents a different set of commitments. Thus, to draw the diagram involves “research commitments… to accept a causal structure for the sake of argument” (Griesemer 1991:173). In Peirce’s terms, diagrammatic reasoning has a normative component—a demand to define and clarify our commitments—which Peirce himself saw at work in deduction as a process of gaining greater self-control over our reasoning practices.

Diagrammatic Methods Versus Diagrammatic Topics

What are the conditions of possibility for engaging in this kind of sociological experimentation? This question moves us to the second problem noted above, namely, the extent to which theory diagrams in sociology are constituted primarily by characteristic subject matters or also by characteristic procedures or methods. Swedberg suggests we apply standard diagrammatic forms—lines, arrows, circles, and dashes—to specific but challenging sociological problems, highlighting Coleman’s use of a diagram to theorize about the micro–macro problem. The major moments of experimentation and creativity come in the effort to “capture such phenomena as social structure, social action, and social forms” (Swedberg 2016:258) and then to use the diagram to learn more about those phenomena. On this view, the theorizing diagram is defined by its ability to handle particular—though not singular—topics and problems such as micro–macro or the communication of meaning.

This is an eminently sensible recommendation. But it raises an additional theoretical question that Swedberg does not pursue, about the possibility that theorizing diagrams embody general-purpose theoretical methods, widely applicable to many topics, and that such methods may evolve through collective communication and practice. A helpful resource for contemplating this question is Wilhem Baldamus. Baldamus’ (1976) work, especially his The Structure of Sociological Inference, provides a useful, more methodologically oriented counterpoint. ³

Although Baldamus was by no means a card-carrying pragmatist, he was certainly familiar with pragmatist ideas. For instance, he explicitly points out that his analysis of methodological and theoretical discovery in sociology was “in essence an adaption of C. S. Peirce’s general method of abduction” (Baldamus 1976:40). Yet, in contrast to the pragmatist philosophers, Baldamus offers a more directly sociological template for studying sociological theory “empirically,” that is, “by examining the concrete conduct of practicing sociologists” (p. 154). Moreover, his focus on the techniques of theorizing gives concrete purchase to the general pragmatist principle that our tools themselves open up (or foreclose) various lines and goals of action. And finally, the central place he gives to error in the study of theoretical practice is an application of the pragmatist (especially Peircian) argument that the strongest “fixation of beliefs” arises through adopting procedures that constantly expose beliefs to disturbance, frustration, and surprise—for it is when we are surprised that we most powerfully experience a recalcitrant reality that does not automatically bend to our wills. For Baldamus and Peirce alike, error and surprise are the practical conditions of our ongoing contact with reality.

Thus, as a resource for understanding the practice of (visual) sociological theory, Baldamus offers a distinctive advantage over the classical pragmatists, Peirce included. He actually sought to apply pragmatist principles to sociological theory. The results of his analysis are highly instructive; they offer guidance for conceptualizing diagrammatic theorizing as a collective method, with its own procedures, sources of potential error, surprise, and adaptation.

Let us begin with the very proposition that theory has a method and then extend this proposition to theory diagrams. Baldamus points out that we are comfortable and familiar with the notion of empirical methods that can be developed relatively separately from application to any specific subject matter, and whose utility outlasts and cuts across various historical and political changes and analytical foci. By contrast, we tend to think of theory as emerging from spontaneous inspiration and as strongly dependent on its historical–social context (Baldamus 1976:85). Certainly, the number of textbooks in empirical methodology dwarfs that of theoretical methodology. And theorists rarely produce accounts of their theorizing practice that could be readily and critically taught and shared.

Baldamus argues, however, that empirical methodologies are in fact much more “of their time” than their theoretical counterparts. “It does not seem possible to connect the origin and the development of theoretical methods with specific societal changes. What is possible, however, is to do just that in respect of empirical methods” (Baldamus 1976:151). Setting aside that intriguing claim about relative durability and scope, we can appreciate Baldamus’ point about the possibility of theoretical method. For Baldamus, we can find evidence of such a method by looking past the content and topics theorists may have discussed. Instead he scrutinizes tables of contents, grammatical forms, and persistent linguistic structures (such as paradox and pleonasm) in order to discern recurrent nonverbal inferential procedures that pattern theoretical creativity.

This search for skeletal forms of theoretical inference made diagrams a natural topic. Baldamus was particularly fascinated by cross-classification diagrams. He scrutinized Parsons’ diagrams most carefully, but he treated Parsons’ as the apotheosis of a tradition that had been collectively if implicitly searching for theorizing procedures for decades. It has persisted since, constituting one of the most common diagrammatic forms in the discipline. And like empirical methods (whether quantitative or qualitative), the use of cross-classification diagrams is not restricted to a particular ideological orientation: Baldamus shows how Marx’s basic theoretical concepts grew out of the process of creating and combining dichotomies embodied in the cross-classification diagram, and some of Marx’s most prominent contemporary heirs in sociology use them routinely (Figure 2; e.g., Burawoy 2005; Mills 1959). Moreover, these formal procedures of a cross-classifying diagram are neutral with respect to their context of application. One can use them—as Habermas did—to examine religion, politics, culture, and more. Cross-classifying diagrams thus embody a theorizing methodology that spans time, space, empirical content, and ideological orientation.

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Figure 2.

These figures illustrate the general conceptual form of a cross-classifying diagram and show a relatively recent and prominent usage in Burawoy (2005) to articulate the “division of sociological labor.”

Methodological Adventures

Swedberg (2016) dismisses the sorts of diagrams used by Parsons and Habermas as “conventional” and “quite simple” uses of “a few boxes and arrows” that do not attempt to express difficult social phenomena (p. 259). Baldamus instead took a more pragmatist stance and saw in their very conventionality a sign of their widespread practical utility. He sought to uncover in their simple and unassuming graphical forms a set of procedures that could generate quite complex and surprising theoretical discoveries.

According to Baldamus, the fundamental feature of Parsons’ “methodological achievement” was twofold. First to specify a concept across various levels of abstraction, so that “as the content of the concept becomes smaller and smaller and the number of combined attributes larger and larger” (Baldamus 1976:112). High-level abstractions have large coverage but simple definitions; lower level abstractions have complex definitions and narrow coverage. Second that at any given level of abstraction, all the various dichotomies are equivalent and thus capable of being flexibly combined to generate novel syntheses.

While uncovering these implicit rules is important in itself, Baldamus’ description of the actual use of cross-classifying diagrams again shows its fundamental pragmatism in stressing that these procedures’ value come from exposing the theorist to a creative process marked by frustration, error, surprise, and discovery. Eldrige (2010) aptly describes it as an “adventure.” “It is only the negative experience of ‘bad fits’, the frustration of recalcitrant discordance in the overall system…which keeps the never-ending process going” (Baldamus 1971:67-68). While cross-classifying procedures operate in abstractions, they demand a creative relationship with such abstractions, with characteristic experiences of discovery as one learns how a set of concepts may interlock—or not (Eldridge 2010). The result is, as Mills had also insisted, “the genesis of qualitatively new concepts, ideas, frameworks, models” (Baldamus 1976:101).

While Swedberg defines sociological diagrams by their content, for Baldamus what makes cross-classifying diagrams distinctively sociological is the methodological procedure informing them: the infinite interplay of dichotomies. The procedure itself embodies a metaphysical conception of what, at bottom, social reality is, namely, a set of hierarchically ranked binaries. “The strange tendency of cross-classification toward an ever-increasing complexity of telescopically interlocked ‘boxes’ reflects an in-built structure of social reality” (Baldamus 1971:73). This notion is not unique to Baldamus. For example, another avid user of the cross-classifying diagram, Alexander (2006), has advanced it explicitly. Such a conception similarly underlies work such as Jenks (1998) that gathers the numerous dichotomies that define various sociological theories. Crucial in the present context is not so much the specific details of these various theories. Rather, more important is that this basic ontological assumption about the nature of society (as a set of hierarchical ranked binaries) is not evident in the content or problems of cross-classifying diagrams but in their formal methods. In using such a diagram one assumes—practically—a particular notion of reality. The form generates its own content. ⁴

Two Engines of Progress in Diagrammatic Methodology: Error and Theorematic Reasoning

While Baldamus certainly admired Parsons’ and others methodological achievements, he was not uncritical of the cross-classifying diagram as a theorizing method. Yet, even his criticism illustrates something important, namely, that diagrammatic theorizing conventions in sociology can themselves be occasions for a collective and cumulative project of methodological innovation, not fundamentally dissimilar from any other (at least locally) progressive scientific tradition. We can use Baldamus’ example to draw out some major principles at work in this sort of innovation.

One principle involves identifying sources of systematic error and searching for alternatives that might remedy them. Baldamus, for example, points out that cross-classifying in the Parsonian mode of equivalent dichotomies has no clear stopping rule to prevent the theorist from falling into an abyss of ever-finer distinctions with decreasing analytical payoff and contact with reality. He thus recommends experimenting with procedures for restricting the abstraction process such as asymmetric dichotomies, trichotomies, and explicitly ranked hierarchies of dichotomies. He also discussed the potential of grounding stopping rules and the human tendency to binary constructions in patterns of historical development, or in genetic-cognitive propensities and limitations. Thus, while Parsons may have carried his procedures deep into what Baldamus perceived to be a “great error,” Baldamus also observes that the very fact that Parsons’ diagrams were capable of a great error is a mark of their importance: Recognizing the possibility of being wrong is the condition of being a part of an enterprise that might be right. For sociological theory, this is no small feat.

As part of his account of theoretical method itself as an unfolding scientific practice lodged in diagrammatic procedures, Baldamus highlights how graphical innovations in cross-classifying diagrams could facilitate inferences that the traditional diagram cannot easily accommodate. These examples point toward a second principle through which diagrammatic theorizing procedures may grow: through what Peirce called theorematic reasoning. For Peirce, theorematic experiments constituted the most demanding and creative moment in reasoning. The classical example of theorematic reasoning is the Euclidean proof that the sum of the angles of a triangle is equal to two right angles. The proof requires adding auxiliary lines and extending the sides of the triangle. Theorematical reasoning adds some new elements to a diagram that permit one to reach a new conclusion. Peirce highlighted theorematic constructions especially in enabling deduction to move through paths that would otherwise be blocked.

The tradition of cross-classifying diagramming reveals a similar sort of creativity. A simple and powerful example is Habermas’ addition of a diagonal in the upper left corner of his diagrams. This addition, Baldamus (1992) writes, “ranks among his most daring ventures” (p. 106).

We may see a similarly bold innovation in Hayes (1993; Figure 3). Hayes uses a novel diagrammatic structure to elaborate the “circular” logic at work in Marx’s analyses of French class structure, in which multiple categories intersect in various ways that cannot be understood on the “polarized” logic of dichotomies underlying most of Marx’s writing. Hayes in turn uses the diagram to demonstrate the logical possibility of certain categories (e.g., “degenerate workers”) that Marx had ideological reasons to omit.

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Figure 3.

These two diagrams come from Hayes (1993) who uses them to discover the implications of the “circular,” nondichotomous logic of class in Marx’s analysis of the French class structure. Hayes describes the left diagram as follows: “categories containing the proletariat, bourgeoisie, and big bourgeoisie all lie above the line that separates being productive from being unproductive. Therefore, these three categories are productive…. The circular class structure underlying Marx’s analysis of events in France represents a departure from the polarized class conflict described in the Manifesto. The structure is no longer based on the polarization of the bourgeoisie and proletariat as the class categories into which they fall lie adjacent to each other.” Describing the right figure, Hayes writes: “The technique Marx used to analyze events through the circular class structure…consisted of shifting between different combinations of class attributes in order to group class categories in various ways as well as to step between them.” Hayes’ article illustrates both an instance of diagrammatic innovation to capture inferential procedures not possible with previous forms (the dichotomous method of the traditional cross-classifying diagram) and the utility of the diagram for working through a logic, uncovering surprising combinations, and elaborating the logic driving of a method of explanation (in this case Marx’s method in his analysis of French class structure).

In the present context, the merits of such novel procedures and additions are less important than is the appreciation of their impulse. This is the impulse to identify the strengths and limits of collective theorizing practices and suggest incremental remedies that can open up new styles of reasoning that would be otherwise blocked. Theorizing diagrams are not only efforts to capture a particular problem but encode the evolving working theoretical procedures of an intellectual community.

Baldamus’ analysis illuminates how a kind of collective creativity can advance not only the capacity of diagrams to represent and handle a particular problem or topic but also to utilize a set of theorizing procedures. In particular, it shows how various innovations in cross-classifying diagramming involve an ongoing search for conventions and shared habits that could permit conceptual exploration and discovery. In this process, practitioners figured out where and why some habits work better than others. They did so by manipulating their cross-classifying diagrams and observing the kinds of novel conceptual combinations they could produce under various structural configurations. But they did not only play with existing formal devices, they added new forms: a diagonal, a box within a box, a connecting arrow.

But this creativity only appears if we admit that theorizing diagrams are defined not only by their ability to express particular sociological subjects or theoretical problems but also through their utility as vehicles for carrying out general-purpose theoretical procedures. Such procedures may be studied and refined for their own sake. Far from being a sign of their barren simplicity, the very conventionality of cross-classifying diagrams makes them ideal candidates for incremental yet real innovation and indicates that, despite appearances, sociological theory does in fact have its own traditional diagrammatic procedures.

The Inferential Limits of the Cross-classifying Diagram

However useful the theorizing procedures embodied in cross-classifying diagrams may be, they nevertheless run up against limits. Some forms of inference are quite difficult to undertake with cross-classifying diagrams. For example, a cross-classifying diagram does not permit one to easily think through implications about the position and distance of the social actors who identify with a set of categories. This difficulty is built in to the very basis of their conventions: In a traditional cross-classifying diagram, all dichotomies are to be considered equally important at a given level of abstraction; one is not supposed to give any particular dichotomy priority over another. Hence, users are enjoined to consider all possible logical combinations. Indeed, this is one of their primary springs of creativity: One discovers combinations that are logically possible that may rarely occur empirically and then can ask why this is so. Generally, the physical distance between concepts in the diagram has little theoretical meaning, and to the extent that it does, it is in terms of their level of abstraction (e.g., higher and lower in a tree) not their social distance (e.g., antagonism and alignment).

While cross-classification procedures elide inferences about distance, position, and space, other diagrammatic conventions make them central. Bourdieu’s diagrams are a prime case in point. While he was profoundly interested in conceptual dichotomies, Bourdieu approached them from a deeply different set of assumptions. For him, abstract dichotomies are not the essence of social life, rather, concrete social spaces and positions are, and it is from these that dichotomies flow. Based on their relative positions in social space, actors are more or less distant, and accordingly become defined in relationship to one another in antagonistic or allied ways: Some are bourgeois, some are bohemian, some are clean, some are dirty, and so on.

Cross-classifying diagrams are exceedingly cumbersome tools for representing and exploring the implications of various positions in social space. Bourdieu accordingly experimented with several diagrammatic forms, most famously with variations on Cartesian coordinate systems developed, for example, in Distinction. These types of graphics are helpful for envisioning actors in terms of their proximity and position in a social space defined by possession of different forms of capital (typically economic and cultural) and to undertake imaginative experiments in manipulating the diagram to think through different ways in which that space could be constructed: How would our understanding of social space differ, for example, in a configuration where artists and business managers moved closer to one another in economic but not cultural capital?

Many of Bourdieu’s graphics are data representations, but some come closer to genuine theorizing diagrams. While less well-known than the Cartesian diagrams, a figure from Outline of a Theory of Practice offers a telling example for specifying more clearly the crucial differences between a more spatial theoretical methodology and the combinatorial methodology of the cross-classifying diagram (Figure 4). This diagram shows the relationship between doxa and discourse (Bourdieu 1977:168).

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Figure 4.

From Outline of a Theory of Practice (Bourdieu 1977:168), this figure illustrates how Bourdieu’s visuals incorporate a set of procedures for making inferences about social life in terms of space and position.

What stands out in this diagram in contrast to cross-classifying diagrams is how actively it makes use of space. The frame makes the space of the figure not merely the neutral page on which the diagram happens to appear but an active field with graphical–social meaning. It matters how much of it is filled. To adapt one of Peirce’s phrases for describing the blank sheet on which his existential graphs are to be written (“the space of assertion”), it is a space of social assertion. The area (another spatial term) between the inner circle and the frame is doxa, a society’s unquestioned taken-for-granted assumptions; the area within the circle is a space of discourse and argument, defined by a (binary) polarity between orthodoxy and heterodoxy.

The content of these positions, however, is irrelevant to the diagram, and it makes no sense to cross-classify them—it is a purely formal and spatial schema. Instead, one is to envision the inner circle expanding and contracting its (spatial) borders and contemplate how this alters the balance of power between orthodoxy and heterodoxy. As the circle fills in the frame, doxa becomes denaturalized, and the legitimacy of orthodoxy diminishes; as it contracts, orthodoxy strengthens. An empty frame is utter chaos, nothing is given and everything is fluid; a single dimensionless point is absolutism, everything is fixed, and change is unimaginable. ⁵ These operations are not possible with a cross-classifying diagram, but they come easily in this spatialized diagram, which reimagines the division of social categories not in terms of abstractions-to-be-combined but as spatial areas, distances, positions, and boundaries to-be-moved.

Inductive Reasoning With Theorizing Diagrams

While Swedberg (2016) has emphasized the interplay of deduction and abduction, the above example indicates that diagrammatic theorizing can also utilize a form of reasoning that could aptly be described as inductive and suggests we can expand and deepen our conceptualizing of graphical theorizing by considering its operations in more detail. This is in line with Peirce’s own approach to reasoning. Once we have adopted a particular set of diagrammatic procedures, we can get on with the business of using the diagram in the open deductive way that Peirce’s philosophy suggests. Yet Peirce did not believe that deduction was the only form of reasoning; he was committed to the proposition that all forms of reasoning operate in all areas of serious enquiry (Paavola 2005; Stjerfeldt 2007).

It can be difficult, however, to recognize this form of induction at work in theory diagrams. As in the case of methodological procedures, we are more accustomed to the notion that inductive reasoning operates in the evaluation of empirical propositions. We have hypotheses on the one hand and data on the other. Inductive reasoning occurs as we experiment with various procedures to determine how well the latter corresponds to the former.

Less common are efforts to examine how well a diagrammatic form of representation fits or can make apparent a given theoretical concept. An exemplary example of this kind of reasoning is J. H. Levine’s (1972) American Sociological Review article, “The Sphere of Influence” (Figure 5). The paper examines interlocking directorates in banks, but its central point is theoretical, in the Peircian sense, about how the various ways we represent a network allow or impede reflection about “the structure of structure.”

“The representation organizes the data, it enables one to “see” relations, it codes the pattern in a way that can be remembered and thought about…All these properties are beyond the data but below the level of formal hypotheses: The value (or deceptiveness) of a representation lies in what it suggests, in its ability to stimulate thought” (J. H. Levine 1972:14).

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Figure 5.

These figures illustrate the use of inductive reasoning to search for a form of representation that best fits a sociological concept. The left side depicts a standard network representation, which Levine (1972) argues cannot represent the notion that each node in a network possesses a certain sphere of influence and that a system revolves around a hidden but common center.

Levine’s article probes various graphical forms, arguing that standard network representations of connectedness are misleading because they cannot make evident the more fundamental phenomenon that around each bank there lies a sphere of influence, and the entire structure revolves about a common, but hidden, center. This is a theoretical point, similar to Burt’s development of stylized figures to make a theoretical point about structural holes. But what stands out in Levine’s article is his explicit experimentation with graphical forms to evaluate their capacity to stimulate thought about this concept. For example, to make the concept perceptible—the hidden structure of structure—he shifts from webs of points to circles and radii, and to a two dimensional projection of a three dimensional space (not shown here) on the model of cartographic representations of a globe. While the data never change, the result is a “primitive analogue to a Copernican change of reference” (Levine 1972:23). The example again is meant to illustrate a larger point, that inductive evaluation of the fitness of forms of graphical representation for the tasks we give them is a type of creative diagrammatic reasoning that can be pursued in its own terms, and to grasp and improve visual theorizing it needs to be recognized and understood as such.

Slotting

Levine’s article suggests that inductive theoretical reasoning with diagrams may have its own characteristic procedures and forms of creativity. These procedures operate on a different register than their deductive counterparts: the referential orientation of our diagrams, the capacity of given graphical form to represent a given phenomenon or enact a given inferential method. The referential orientation of a diagram becomes evident in what we might call its “slotting” (Figure 6).

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Figure 6.

These figures illustrate “slotting,” where elements of diagrams are assigned referential functions. Slotting makes it possible to diagrammatically experiment with the scope of a theory’s application.

Slotting a diagram assigns to certain regions on it a referential function. For instance, before the empirical phase of path analysis begins, the spaces above the causal arrows have to become slots into which numbers can be inserted. The numbers indicate the path coefficients. But the slots make this empirical use possible. Techniques like varying the arrow width or length create similar possibilities. Another common form of slotting involves assigning spaces within boxes to concepts in a cross-classifying diagram. The deductive aspect of these diagrams, as we have seen, is to afford splitting and cross-classifying concepts. That we can do this, and that the diagram asks us to do this, is possible, however, because the empty spaces inside the boxes have become slotted. They are not simply empty; they are supposed to be filled in somehow, they ask to be filled in somehow. They make the diagram able to reach beyond itself.

Considering and experimenting with various slotting practices is thus itself a theoretical exercise, about procedures for capturing phenomena and making inferences that “come before” the data and shape the form in which they appear. When we reason with a theorizing diagram inductively, we investigate how well it allows us to refer to some concept or phenomenon (as in the case of the Sphere of Influence) or set of inferential procedures (as in the confrontation between Parsons and Bourdieu). We experiment with its slotting. This too involves us in the full range of theoretical creativity: searching for an appropriate set of slotting techniques, manipulating diagrams to discern their fitness for a theoretical task, and adding new elements to expand their scope.

These sorts of experiments often involve adopting a kind of double vision, similar to what the art critic W. J. T. Mitchell sees in “dialectical diagrams,” and implicit in Levine’s reference to “Copernican” changes in reference: We have to think about what we want to think through with the diagram, and at the same time, whether the diagram can help us to do that. In this form of reasoning, the diagram, its visual conventions, and its basic theoretical assumptions themselves thus come to the fore, whereas in deduction they (are meant to) fade into the background. In sum, if deductive theorizing with diagrams aids in discovering our theoretical commitments, inductive theorizing examines the fitness of a diagram for the type of reasoning we wish to perform with it.

Abductive Reasoning With Theorizing Diagrams

On Peirce’s triadic conception of reasoning, no account of theorizing practices could be complete without a consideration of abduction, and Swedberg (2016) has accordingly brought this dimension to the fore. Although the notoriously slippery concept raises enormous complications and puzzles (Nubiola 2005; Psillos 2011), in general terms, abduction constitutes that aspect of reasoning devoted to generating ideas (Timmermans and Tavory 2012). This general idea is enough for present purposes. While abduction occurs across all phases of thought, it is also possible to undertake more explicitly for its own sake. When this occurs, reasoning becomes more artful, a free play of the creative intellect relatively unconstrained by fidelity to current conventions, data, and references (Anderson 1987).

This sort of free play has some similarity to the sort of “visual sketches” that Swedberg (2016) discusses, but they are not the same. In Swedberg’s description, visual sketches are often playful and vague, and they help theorists to experiment with various ways of picturing a theory or constructing a diagram. They exist at the nexus of private and public and are made to be thrown away. A diagram explicitly used for abduction is different. It is not a sketch of some future product (a finished picture or functional diagram). It is a practical medium for exercises in the art of getting new theoretical ideas, carried out for their own sake.

To get a more concrete sense of how to undertake this sort of abductive theoretical practice diagrammatically, we can adapt some insights from Swedberg’s (2014) other (pragmatist-inspired) work on theorizing in The Art of Social Theory. There Swedberg discusses practical exercises for getting new ideas. For example, he suggests starting with a verbal proposition and changing the nouns to verbs and vice versa. Using diagrams abductively is similar. The given form becomes plastic and pregnant, and we (seriously) play with the diagram with a view to how doing so can get us to think about some topic—and our procedures for thinking about topics—in new ways. Such play is not random, however, and can be advanced through specific exercises.

Two Exercises in the Art of Diagrammatic Theorizing

Consider Coleman’s “boat” as an occasion for abductive theoretical play. Just as we may get new ideas by changing nouns and verbs in a sentence, we can swap arrows and nodes in a diagram. This is somewhat difficult to envision with Coleman’s own version of the diagram, since he does not seem to have explicitly labeled all the lines and points in one place, and neither have most of his successors. As a starting point for using the diagram abductively, however, we can join and adapt some of the versions gathered by Stolz (2014), Hedström and Ylikoski (2010), Sato (2010), and Esser (1999) (Figure 7).

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Figure 7.

This graphic takes figures in the mode of Coleman’s “boat” depiction of micro–macro transitions (Coleman 1990; Hedstrom and Ylikoski 2010; Sato 2010; and Esser 1999) and swaps the nodes and lines. In the traditional representation, nodes are social “substances,” such as doctrines, actors, values, or emotions, and lines are mechanisms linking them. In this version, nodes are processes, and lines are temporary crystallizations of processes into more stable configurations. The figure illustrates a type of exercise theorists can perform on existing diagrams to get new ideas, by swapping their (visual) nouns and verbs.

This figure starts from the traditional boat and exchanges the labels on the points and lines. It becomes immediately evident that the traditional version embodies the assumptions of the mechanisms approach associated with Coleman, Hedström, Swedberg, and others (Abbott 2007). The active elements in the nodes in the traditional diagram are entities such as doctrines, actors, situations, habits, and values: already formed, substantial agents or situations. These social atoms initiate some process—the lines in the traditional diagram—which in turn generate behaviors that issue in collective outcomes.

Redrawing the diagram with points and lines swapped makes processes the central players in the social drama envisioned by the diagram. It corresponds much more closely to the sort of relational and processual thinking advocating by Abbott (2001). If we work with the transformed version, we start from a process, not an established agent or situation, and ask how actors and situations come to be formed as such. We treat agents and situations as temporary—though often durable—crystallizations of ongoing processes and inquire about how their formation in a particular configuration channels a process in a specific direction and what other processes flow from their having been channeled in this way. In this version, “by making perpetual change foundational…we make explaining change relatively trivial. Explaining stability becomes the central theoretical challenge” (Abbott 2007:8).

Another way to play with an existing diagram to get new ideas is to change its shapes and thus its visual grammar. For example, we could redraw the Coleman boat with overlapping circles rather than points and arrows (Figure 8), adapting Coleman’s own representation of Weber’s Protestant Ethic argument and Sato’s (2010) more schematic version.

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Figure 8.

This figure illustrates another exercise for transforming existing graphics to get new ideas: changing from one set of spatial forms (dots, lines, and arrows) to another (overlapping circles).

This exercise too is an opportunity to revise how we think about micro–macro relationships and action. The traditional points-lines-and-arrows version suggests a series of independent substances, causally acting on one another. Drawing the boat as overlapping circles invites us to question this conception: What if a macro situation (like Protestant doctrine) does not cause individual values, but instead they mutually constitute one another, to varying degrees? A doctrine can completely define a set of values, without remainder, or it can barely play a role, leaving greater leeway for unregulated action. Likewise, we can experiment with theorizing values and actions not as distinct entities connected by distinct mechanisms but as parts of a single configuration that can be more or less integrated. Drawing the diagram in this way would move away from mechanistic thinking and toward the sort of relational and configurational theorizing Simmel pioneered, elaborated (diagrammatically) in Silver and Lee (2012).

In the present context, the point is not that these are necessarily good ideas or to elaborate the full implications of these ways of playing with Coleman’s diagram. That is a work of deduction and induction. It is simply that this kind of abductive reasoning can be carried out as its own form of disciplined practice, not just as a sketch or guide for a finished diagram or picture, and that we can work to develop and codify procedures for undertaking that practice in a more controlled and self-reflective way.

Metaphors in Social Theory

To do so, we can draw on a rich tradition of work examining the role of metaphors in social theory, even though verbal metaphors have dominated these discussions. Brown’s (1978) A Poetic for Sociology, for example, closely examines several root metaphors in sociological thinking, such as society as organism, mechanism, game, and stage. The introduction of a new linguistic metaphor can sometimes itself be proposed as a theoretical innovation. For instance, drawing on Geertz (1983), Ken (2008) proposes that culinary metaphors (sugar, baking, tasting, and digestion) may enhance theoretical understanding of race, class, and gender. Likewise, some verbal theoretical metaphors may stymie thought, as Turner (2014) suggests Latour’s metaphor of the “modern constitution” did.

While these examples indicate the importance of the subject of metaphor in sociological theory, Lakoff and Johnson’s influential work on metaphor (e.g., Johnson 2008; Lakoff 1990; Lakoff and Johnson 2008) provides perhaps the most generative point of reference for illuminating the role of metaphor in theorizing in general. Lakoff and Johnson sparked a revolution in cognitive linguistics that has spilled into many fields including sociology. They highlight how “conceptual metaphors” are at work at the highest levels of abstraction, in examples such as martial or architectonic metaphors for theoretical argumentation (“building” or “winning” an argument). For them, even these seemingly most disembodied domains draw on “image schemas” rooted in concrete, embodied experience such as space, time, motion, building, counting, and the like. Conceptual metaphors work by “mapping” some aspects of the more concrete onto the more abstract, which in turn structures the cognitive assumptions of the latter. It is one thing to undertake an argument on the presumption that it is a battle-to-be-won, another to undertake it on the model that it is a structure-to-be-built. Arguing for the centrality of “analogies, metaphors, and physically instantiated models” to scientific conceptualization, Lizardo (2014) draws on this research tradition to analyze structuralist theories of society. Across this work, the central aspect of metaphors for theoretical imagination involves revealing (often implicit or nonobvious or novel) qualities of one object through some parallelism with another.

Visual Metaphor

By and large, these research streams tend to highlight linguistic metaphors and often turn to poetry as the artistic model for metaphorical innovation. Some scholars in the Peircian tradition, however, have examined visual metaphor. Arnold (2011) elucidates the general Peircian logic: A community establishes conventional relations among shapes and forms, which become associated with specific meanings. Such conventionalized forms may then be inserted into new or unusual (visual) contexts, which in turn calls for a metaphorical interpretation that maps some features of the original context on to the target one. ⁶ For instance, as Arnold (2011) elaborates, David’s The Death of Marat shows the murdered French revolutionary leader lying dead in his bath. The formal configuration of the painting strongly draws on traditional Christian images of the death of Christ. The viewer is thus encouraged to consider the death of Marat through the metaphor of the death of Christ. Metaphorical thought occurs not only in language but also visually.

Metaphor in Theory Visualization

Theory visuals make use of similar metaphorical associations. Baldamus (1976) pointed in this direction in his discussion of the cross-classifying diagram. Sociological theorists’ use of cross-classifying diagrams, he noted, grew up alongside and resembled the statistician’s cross-tabulating devices. The latter establishes conventions whereby an analyst breaks up variables into various subsets—typically dichotomously—and then observes their intersections. These conventions then color the theorist’s cross-classifying diagram, just as the death of Christ colors the death of Marat. We are encouraged to consider concepts through the metaphor of the cross, as variables to be cross-tabulated. Thought conventions from the latter are imported into the former, and it becomes “natural” to treat social life as equivalent dichotomies to be endlessly combined and recombined.

Baldamus’ insight clearly goes beyond the single case of the cross-classifying diagram. For example, Bourdieu’s diagrams of social space trade on their metaphorical association to the Cartesian coordinate system. They make it “natural” to associate social life with spatial distance and position, in effect deploying the metaphor: “society is a space.”

Theories are paths

Baldamus’ work invites us to consider how profoundly our theorizing practices may be shaped by metaphorical associations to mathematical and statistical visual conventions. Joining these insights with ideas from Lakoff and Johnson (e.g., Lakoff 1993; Lakoff and Johnson 2008) suggests that the visual metaphors embedded in theory visuals may run deeper. It encourages us to contemplate the possibility of visual metaphor chains linking our very notions of what a theory is to basic experiences of moving about in the world.

Consider the metaphor of a path. ⁷ This metaphor has been repeatedly discussed by Lakoff and Johnson from their Metaphors We Live By onward (e.g., Lakoff 1993; Lakoff and Johnson 2008). A path defines a direction, a starting point, and a destination. You can stray from a path or get lost. Paths accordingly define a certain teleological logic of inference: “If you are going from A to C, and you are now at in intermediate point B, then you have been at all points between A and B and not at any points between B and C” (Lakoff 1993:10). Lakoff and Johnson show how the path metaphor helps to define arguments as journeys, in contrast or sometimes conjunction with other metaphors such as wars or containers. Lakoff has also suggested that the concept of a quantitative linear scale involves a metaphor of linear scales as paths.

Although perhaps not as institutionalized as Christographical conventions in the history of painting, path-like structures are among the most common visual forms in sociological theory. These theory visuals draw on metaphorical associations to the more formal path diagrams pioneered in sociology by Blalock and Duncan and carried forward in the DAG tradition. Long and repeated use of path diagrams in sociology establishes a conventional association: If you see boxes conjoined by arrows depicting a flow (usually) from left to right, you are probably looking at a path. But as Lakoff and Johnson’s analysis suggests, the path metaphor goes further. As the figure of Christ asks us to interpret Marat in messianic terms, the figure of a path asks us to interpret our objects of study, and our theories, in purposive and teleological terms: A path is from somewhere and to somewhere.

Cornfield’s (2015) Beyond the Beat offers a recent example of this metaphorical extension of “pathway” thinking into a theory of artist activism. ⁸ Cornfield proposes a theory of why some musicians break from corporate trajectories, instead embark on a more entrepreneurial path, and then, having done so, take up a particular form of activism, ranging from more personal, to intrapersonal, to impersonal in orientation. Conveniently for present purposes, he summarizes the “sociological theory of artist activism” in a clear and simple figure (Figure 9).

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Figure 9.

This figure illustrates a set of metaphorical associations, between an actual path, a statistical path diagram (Webley and Lea 1997), to a visualization of a sociological theory as a path (Cornfield 2015).

Cornfield’s theory is not part of a formal path analysis, but its layout and movement resonates with the typical spatial forms of such an analysis and of the actual movement along a path: We move along a course from departure to sojourn to arrival. It transfers pathway thinking into the domain of artist activism. A strategic orientation (toward success as freedom rather than fame or fortune and a primary audience of peers) leads musicians to assume an artist activist role; a risk orientation orients this path in a particular direction (for instance, those with an interpersonal risk orientation are like to become social entrepreneurs). The pathway from assuming a role to enacting unfolds according to a musician’s career inspiration (family-inspired musicians tend toward supportive roles, movement inspired toward artistic). Life is a path, a career is a path, and musicians are on a path toward activism.

The path structure of the figure makes enactment of a role the destination of a journey, strategic orientation its starting point, and assumption of a role a way station that must come between. Obstacles can block the way, such as various risk factors and family support/opposition. But a successful journey overcomes these and reaches the destination. The path metaphor thus transposes a teleological structure to various activities that we might not have otherwise viewed in such terms. For instance, in Cornfield’s elaboration of another path model of “artist activists” in particular, placing artist activist types on a path from more individualistic to collectivist puts them on an implicit teleological trajectory, where individualistic activists are incomplete or preliminary versions of collectivistic. And if the community does not currently show “an overarching organizational framework,” it is because it has not “yet” been able to cohere (Cornfield 2015:121). On a path, “not now” becomes “not yet.” ⁹ As the formal association of Marat with Christ accentuates qualities in Marat that might not otherwise be evident, displaying the theory of artistic activism as a path transfers qualities of the path—purposiveness and teleology—to the theory.

This is the barest sketch of an analysis of the metaphor. One could carry the analysis much further, applying the full range of analytic procedures developed in cognitive linguistics to theory visuals. For example, we might consider how theory visuals employ the path metaphor partially, and how we may imaginatively play at the boundary of the “used” and “unused” parts of the metaphor (Lakoff and Johnson 2008). For instance, what would it mean to visualize a theory as a bumpy or winding path; as an overgrown, twisting labyrinth; as a superhighway; or built from cobblestone? We could also pursue the metaphorical dimensions of other diagrams. For instance, pyramidal diagrams often operate through architecture metaphors—“a theory is a building” (Lakoff and Johnson 2008).

In the present context, however, the point is much more modest. It is simply to highlight metaphorical association as an additional semiotic resource—beyond picturing and diagramming—that gives theory visuals their power to creatively represent the social world and to place the understanding of this resource on the theoretical agenda.