Doorway to the dharma of duality

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Doorway to the dharma of duality Monica Leea, John Levi Martinb, a b

⁎

Facebook, Inc, United States University of Chicago, United States

AR TI CLE I NF O

AB S T R A CT

Keywords: Duality Culture Meaning Critical theory

One of the most exciting formal approaches in the sociology of culture involves the exploration of duality–that the meaning of any of one set of elements (for example, choices of cultural consumption) is the subset of persons to which it is linked (for example, the occupations of the persons who make this choice). We here generalize this notion of duality, which we call “Breiger classic,” to the case in which similarities between persons are constructed not on the basis of which cultural elements they choose or deploy, but how these cultural elements are themselves connected, which we call “Mega-Breiger.” We then consider how to look at the duality between cultural and social units within cultural formations, treating each cultural formation as a virtual social structure among cultural elements, which we call “Mondo Breiger.” We finally discuss the “Full Breiger” duality as the complete use of a person-to-person and person-to-cultural element data set to arrange relations among one set of cultural elements. We illustrate with an analysis of all the texts of the members of the first generation of the Frankfurt School of critical theory.

1. Duality 1.1. Spinoza and duality “Then, the Licchavi Vimilakīrti asked those bodhisattvas, ‘Good sires, please explain how the bodhisattvas enter the Dharma-door of duality!”1 Vimilakīrti-Sutra, “The Dharma-Door of Non-Duality” What does it mean to explain a phenomenon? In conventional sociology, this is sometimes taken to mean to exhaust all its explicable variance by linking this to other variances; in other cases, if is taken to mean to determine a cause that is necessary, or sufficient (rarely both), of the phenomenon. It is, however, well understood that such explanations may be formally adequate while failing on what Weber would call a “human” level, in that they do not give us a sense of the meaning of the phenomenon, at least, not the sort that only a human, and not a computer, could grasp. But there is a problem with the attempt to give such meaning: it tends to lead to infinite regress. What is the meaning of some A? According to Peirce, the things that bear meanings are signs, and “the meaning of a sign is the sign it has to be translated into” (1965; CP 4.132). But what, then, is the meaning of this second sign? Clearly, it must be a third, and so on and so forth. The quest for meaning is a pointless, never-ending exercise in futility. But there is another approach to meaning that allows a formal and finite analysis of culture. This is one that is based on the ⁎

Corresponding author at: Department of Sociology, University of Chicago, 1126 E 59th Street; Chicago IL 60637, United States. E-mail address: [email protected] (J.L. Martin). 1 Given that the sutra demonstrates that to say “this is dualistic and that is nondualistic” is itself dualistic, we feel justified in reversing his meaning entirely. Our source is the translation by Thurman (1976: 73–77). https://doi.org/10.1016/j.poetic.2018.01.001 Received 25 June 2017; Received in revised form 16 January 2018; Accepted 18 January 2018 0304-422X/ © 2018 Published by Elsevier B.V.

Please cite this article as: Lee, M., Poetics (2018), https://doi.org/10.1016/j.poetic.2018.01.001

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conception of Spinoza (1998 [1663]; 1930 [1677]). The great debate of the seventeenth century had to do with the relation between body and soul. There was a widespread acceptance that mechanical materialism was on the rise, and that it would soon claim that all was materially determined, including human action. But if that were so, how could one maintain that humans had free will—that is, that the soul affected the body—and that the world of mind has primacy over the world of sensation? Spinoza strode into this debate and in a stroke dissolved the antinomy; he argued that these supposed two worlds were simply two sides of the same coin. The same thing that might comport itself to mind as thought comports itself to sense as experience. Thus while Spinoza is known as a “monist” in contrast to Cartesian “dualism,” what he puts forward is quite different from the classic monisms of Idealism and Materialism: it is, as Breiger (2011) has emphasized, a vision of duality, not dualism. Dualism is when we bifurcate: for example, we decide to double our bait supply by decapitating our worms: heads in this pile, tails in that. Duality is when we realize that every coin has both a head and a tail and that they cannot be separated; tails is what is on the other side of heads, and vice versa. Even more important, if you have a coin in your hand and you look down and see heads, you can be sure you also have tails. To get to tails from heads, you simply look on the other side. So, too, in duality-based approaches to meaning, we do something similar: rather than wander off in a web of mental associations, we look at what is on the other side. On the other side of meaning, of culture, of heaven, we find earth, the social, we find people. It is this core vision of duality that constitutes the basis of our approach.

1.2. Meaningful relations and relations on meanings When we say “our” approach, however, we are a bit misleading; this approach is one that has been initiated and developed by Ronald Breiger, beginning with his seminal (1974) paper on the duality of persons and groups. This notion was emphasized as crucial for sociological theory by Mohr (2000) and Mohr and Friedland (2008) used to good effect by Mohr and Duquenne (1997), Mische and Pattison (2000), Yeung (2005), and others. Our interpretation is greatly influenced not only by Breiger, but by Mohr, Mische, and Pattison. It has also been shown to be theoretically and methodologically central to the approach taken by Pierre Bourdieu by Breiger (2000). Because of the familiarity of Bourdieu’s work, we begin with this for illustrative purposes. Bourdieu (e.g., 1984 [1979]) often collected data in which individuals are asked what activities they enjoy. The resulting data are then subject to an analysis that attempts to put activities “close together” in an analytic space if they tend to be done by the same people or, perhaps, the same kinds of people (whatever this means). From looking at these charts, we get a new sense of what these activities (say) mean. For what tastes “mean” is precisely what social relationships they reference. And we may use these social relationships to interpret the content of these references. Thus mountain climbing “means” teachers as against CEOs (on the one hand) and shopkeepers (on the other) not because teachers may experience a fellow feeling when encountering each other on a mountain (they may hate running into others), but more because the mountain expresses the social distance between the teachers and these other groups. Why do we consider this “meaning”? The answer is simply that we believe that, as humans, there is a basal form of understanding that we reach when abstractions are returned to the world of humans (also Marx and Engels, 1976 [1845-6]). Humans are different from everything else for if they do have “meanings”—if they are signs—then what they signify, as Peirce argued, is nothing other than their own souls (1984 [1866]: 504). In the infinite directed graph of signification, they are termini, or sinks. Mische and Pattison (2000) use this same logic when it comes to the beliefs and programs held by political actors: given that the core to political action is the formation of alliances, to “hold” a belief that another group rejects is inasmuch as to break an alliance with them. Thus what the beliefs “mean” can be said to be the sets of alliances that they facilitate or imply. In the terms of analytic philosophy (see Martin and Lee, this issue), meaning in this sense is equivalent to extension. Specifically, we will call this sort of meaning a sociocultural one: the meaning of cultural elements is resolved into social relations. In contrast, we can speak of the culturosocial organization as the dual (also see Basov and Brennecke (2017) for the same idea), namely, the implied relations between persons established by their pattern of holding or not holding certain cultural elements. We wish to focus on one aspect of this vision of duality, one which has, we believe, rich theoretical implications. This aspect is that of reconceptualization of matrices as storing information about paths that our minds can take. Considering Andre Weil’s attempt to mathematize the rules of descent studied by Lévi-Strauss ([1949] 1969), White (1963) realized that one could treat deterministic rules of descent as permutation matrices, and complex relations could be expressed via matrix multiplication. Just like the famous exchange of women in a system of preferential marriage with father’s sister’s daughter is “there and back”—a man takes a wife from the lineage which received a wife from his lineage in the previous generation—so, too, Breiger duality is a “there and back.” We go from persons to cultural elements, and then back from cultural elements to persons, for example, to arrange persons in terms of cultural similarity. Kovács (2010) and Lizardo (this issue) show that we can improve our estimates of similarity and centrality (respectively) by an iterated process of going there-and-back-and-there-and-back-and-… Here we focus on expanding the walks we can take, and developing parsimonious notation that can allow us to specify three theoretically reasonable classes of questions to ask about the meaning in a dataset. We begin by introducing notation, and then working through examples of increasing depth. We note that we here do not consider issues of inference and refer the reader to the discussion in Martin and Lee (this issue) for our reasoning. Although our core ideas are simple, for purposes of housekeeping, we find it more parsimonious to use a consistent, if awkward, terminology, and provide a glossary as an Appendix A.

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1.3. Basic notation We first must review the basic notation of duality, using the approach which we here call “Breiger classic.” This takes a two-mode matrix X of M cultural elements being held (or not held) by N individuals. Because we assume that all such elements can be either held or not held, X is an N × M Boolean matrix, with rows corresponding to persons, and columns corresponding to cultural items (e.g., beliefs, symbols, terms). Every ‘1′ in this matrix therefore refers to an individual holding a cultural element, and we shall call such observations ‘holdings’ (as opposed to the 0′s in the matrix). The “transpose” of such a matrix X is a matrix Xt such that the (i,j)th element of Xt (denoted xtij) is the same as the (j,i)th element of X (xji). In other words, we flip the matrix on its diagonal. Following Simmel, Breiger (1974) suggested that we use simple matrix multiplication to derive the pattern of shared cultural elements (or group memberships) between persons, and the shared co-holding of cultural elements. Regarding the first, the (i,j)th element of XXt contains the number of cultural elements that person i and person j both hold; regarding the second, the (i,j)th element of XtX contains the number of persons who hold both element i and element j. If we define our operations as Boolean (1 + 1 = 1), then these matrices contain not the number of co-holdings, but rather the logical answer to the question of whether there are any coholdings. Also note that this constructed matrix is a close analogue to a correlation matrix: the (i,j)th element of XtX is equal to the dot product ∑k xikxkj, in turn equal to the correlation of vectors if these vectors are standardized to have unit magnitude. Thus Breiger classic may be seen as a qualitative or discrete analogue to more conventional studies of, say, the constraint of beliefs (in the sense of Converse, 1964). In contrast to the culturosocial relations that are established via XtX, we may also possess direct information on a social structure, such as a set of social ties, which we assume comes in the form of an N × N matrix Y where yij = 1 iff (if and only if) persons i and j have a tie. The comparison of Y to XXt can then be used to examine theories of the constitution of groups or ties on the basis of cultural elements, or the distribution of cultural elements on the basis of social relations, or both. This is a rather common way of thinking in sociology, given that we—quite reasonably—often attempt to explain the distribution of cultural elements X by linking them to patterns of social relationships such as Y (Friedkin, 1998). However, it is also true that if we have accessory information on a cultural structure, which we assume has a form analogous to that of Y, namely an M × M matrix Z where zij = 1 iff cultural elements i and j have a tie (Carley, 1986), the comparison of Z to XtX can be used to examine theories regarding the relation between cultural meaning structures and the social distribution of cultural elements. Such a matrix Z is less familiar than a sociomatrix Y, but Lee and Martin (2015a), for one, construct such a matrix from cultural artifacts using the proximity of two sub-elements in a larger whole. Taking the case in which these cultural elements are words, they create the frequency of the ith word in the kth paragraph of a set of texts (denoted f(i,k)), and then set

z ij =

∑

f (i, k ) f (j, k ). (1)

k

Note that even when we begin to consider more complex datasets, we will assume that links between cultural elements are established when they are used in close succession in a text. Such a Z matrix, containing the degree of co-occurrence of all pairs of elements, is often known as a “concept map,” because it can be portrayed graphically as a network; such networks were first applied to cases of knowledge elements (Novak, 1977). Each Z can be interpreted as a (possibly weighted, possibly binary) graph in which the nodes are the M cultural elements, and the edges are the degree to which the nodes are associated. Were there an authoritative corpus of texts that could be treated as “the” culture from which the M elements were drawn, and which the N individuals were members of, and we constructed a corresponding “grand Z,” we might indeed expect XtX to correlate highly with it, just as if there was an authoritative set of interpersonal actions which could be treated as “the” social structure from which the N individuals were drawn, such that we could create a corresponding “grand Y,” we might indeed expect XXt to correlate highly with it. 1.4. Going mega-Breiger The “Breiger classic” method produces a sociocultural and a culturosocial meaning from an N × M matrix; we have just reviewed well-known ways that we may collect data on N × N social structures or M × M cultural structures. We now go on to generalize the Breiger classic approach to what we shall call the “Mega-Breiger” approach in which we use not the mere holding of cultural elements to arrange persons (or the reverse) using the N × M matrix, but the social distribution of patterns of co-holding beliefs, using the N × M × M array, to construct what we shall call the “mega-culturosocial” meaning of persons, or, dually, the N × N × M array to construct the mega-sociocultural meaning of cultural elements. We illustrate the first here. Consider the case in which we have a set of different Z matrices, each from a different individual (which we here term an “author,” assuming one text per author and one author per text for simplicity’s sake). Following Lee and Martin (2015a), Lee (2017) and Basov and Brennecke (2017), we may construct the agreement of two cultural structures from two authors, i and j, denoted ψij, using the Jaccard similarity, which is the ratio of the number of shared relations, in this case, edges in the graphs, divided by the total number relations in both, or:

Zi•Zj

ψij = jaccard (Zi, Zj ) = ⎛ ⎜∑ z ikl + ⎝ k, l

∑ k, l

⎞ zjkl − Zi•Zj ⎟ ⎠

(2)

where the large dot means summing up over elementwise multiplication. We then may compare the matrix Ψ = {ψij} to Y to 3

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determine whether social relations and the implied agreement of the use of cultural elements coincide.2 Thus two persons are close not because they merely adopt many of the same elements, but rather, they use these elements in structurally similar ways. We therefore make a meaningful network of persons based on the overlap of their network of meanings. For example, taking texts from members of the Frankfurt school, Lee and Martin (2015a) compute Ψ for representative works from the three main generations of the school. While we might imagine that the likeness should be greatest between adjacent generations, we in fact see the greatest overlap between the first and third generation. Interviews with key protagonists confirmed that this had been a deliberate tactic of the third generation, namely, to return to the key ideational structure that had been associated with the founding of the school. 2. Embedded and virtual social structures 2.1. New notation We now wish to generalize, developing an ad-hoc notation that, we think, facilitates what we shall later call “going full Breiger,” namely, considering any dataset a set of interpenetrating sets, the meaning of which only emerges when we can trace from one set to another and from there, to still another. Our approach therefore joins that of Fararo and Doreian (1984) and Kovács (2010) in extending the Breiger formalism. Let A and B be two sets, connected with a binary relation R such that (a,b) ∈ R (also noted aRb) means that element a from set A is connected by R to an element b from set B. Note that (a,b) ∈ R does not necessarily imply that (b,a) ∈ R. R can be expressed as a matrix X in which the rows are the members of A (with cardinality ||A||), the columns the members of B (with cardinality ||B||), and xij = 1 iff the ith element of A has a relation with the jth element of B; alternatively, if R is weighted, X contains those weights. We here, however, suggest the following ad-hoc notation for R, [AB]. The “Breiger classic” duality approach is to examine [AB][AB]t = [AB] [BA], which we propose to denote [AA:B], AA relations mediated by their relations with B, and [AB]t[AB] = [BA][AB], which we similarly denote [BB:A], BB relations mediated by their relations with A.3 This notation has a similarity to that of tensor calculus, and, as we show in Martin and Lee (this issue) can be used for conceptually similar problems involving the mapping of one set to another.4 The virtue of this notation is threefold. First, it means that we do not need to continuously define new matrices for every combination of elements we pursue. Second, we will show how it parsimoniously can be scaled up to deal with trinary and higher-order relations. Third, it highlights that what we are doing is transmitting information about one meaningful way of organizing observations to another. We first begin by using our new notation to expand our approach to duality, and then go on to apply this notion of meaning-transmission. 2.2. Recognition and reference When we think of cultural elements, paradigmatically beliefs, we may tend to think of propositions. Yet this is a rather restrictive and unusual form of element. Other elements—symbols, references, tones, keys, colors, textures, and so on—can also be incorporated in such a study. Consider one particular type of cultural element—the social reference. As Lena (2010) and Lang and Lang (1988) have argued, this is a key way in which position in a cultural sphere may be signaled. This is of particular interest to us, for, if we are attempting to understand the reciprocal meaning of persons and cultural items, we find that, in some cases, our cultural items themselves contain such reciprocal meanings! We may, in other words, observe cultural structures that, considered carefully, turn out to have their own culturosocial and sociocultural meanings. Because we treat each such cultural structure as having its own “world” (its own social world and its own cultural world), we will term the implicit social structure within a cultural structure a “mondo” social structure, and our exploration of the meaning that arises “Mondo-Breiger.” Because we do not see a possibility of an analogous embedding of cultural relations within actual social structures, here our symmetry is abandoned, and we consider the mondo-cultural meaning of persons, but not the mondo-social meaning of cultural items. Let us partition the set of M cultural items, which we will denote C, into two portions, one a set of M* names of persons (P), and the other, a set of M − M* non-person words (W). Once again, we may either treat our matrices as binary (presence/absence) and our operations as Boolean, or as frequency counts and employ conventional matrix operations. Here, we will begin by assuming that we are treating a single text written by a single author, as when we first defined the culture structure Z above. Note that the Z matrix, which is M × M, may be decomposed into three matrices: one, the M* × M* matrix (which, using the above notation, we can denote [PP]) containing relations among the person-elements, another, the (M − M*) × (M − M*) matrix 2 We note that such Jaccard type measures may be difficult to interpret if the two cultural structures draw from different sets of elements (ψ can only be high if the two maps share both nodes and edges). In many cases, we may wish to compute a preliminary and parallel measure, which we will denote ψ'. While ψ looks for the overlap between edges in the culture structure, ψ' is the Jaccard measure of the overlap between the nodes. In many cases, we would want to first examine ψ', and only then look for agreement via ψ conditional on agreement via ψ'. This approach may be necessary where different cases have widely different sets of included elements. While it would be possible to restrict attention to only the shared cultural elements (words), we do not think that this makes sense for most purposes, and that it is entirely reasonable to imagine each text as having a strictly non-zero probability distribution over the set of all elements. Even so, there are issues regarding the comparability of the Z matrices that may require a different weighting, but we defer a discussion of this until we are able to treat it at a more general level below. 3 Note that we do not need to specify the relation R because of our assumption that we are working with a single dataset defining a single such relation; we will relax this assumption below. 4 We are grateful to Jacob Foster for this point.

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Fig. 1. Habermas’s Mondo-cultural Structure [PP|a = Habermas].

[WW] containing relations among the non-person-words, and a third M* × (M − M*) matrix [PW] containing relations between the persons and the non-person-elements. The Z matrix may be understood as composed of these three matrices, plus the transpose of the last. For this reason, we introduce the operation of fusing two matrices through “stacking.” If we have two matrices, C1 and C2, with dimensionalities N1 × N3 and N2 × N3 respectively, the “stacked” matrix C, denoted {C1/C2}, is the ((N1 + N2) × N3) matrix consisting of the rows of C2 appended to the bottom of C1. Given such a C, [CC] = [{C1/C2}{C1/C2}] can be constructed as the set of four sub-matrices [C1C1], [C1C2], [C2C1], and [C2C2], which occupy the upper left, upper right, lower left and lower right quadrants of the [CC] matrix. Mische and Pattison (2000), dealing with a trinary relation, previously proposed to analyze simultaneously [AB], [AC] and [BC], where all three matrices were symmetric, by stacking each of these matrices and thereby creating a matrix we would denote as [{A/B/C}{A/B/C}]. Let us consider [PP], which is, we recall, the ith text’s cultural structure when we restrict our attention to this subclass P. [PP] is thus a sociomatrix indicating those persons who, in the ith text, are tied together by being discussed at similar points. We can understand this as a virtual social network composed of the persons to which a text refers. It is, of course, not necessarily the case that a relation here must be interpreted as a tie of affinity (see Biernacki 2015) but to begin our exploration, we make this simplifying assumption. We assume, in other words, that this mondo-social structure is akin to a virtual social network, something which represents the author’s perception of the intellectual field. Such a network is implied whenever an author positions other persons visà-vis one another while constructing an argument; a different author might construct a different mondo-social structure even though referring to the same set of persons as our first author. An example would be how Habermas portrays Frankfurt School members in his magnum opus Theory of Communicative Action (1984, 1987 [1981]; see Fig. 1). To avoid confusion, when we are writing of a person as an author (and hence a member of the set we will denote A), we give the name in roman type (e.g., Marcuse). When we are writing of a person as a name, a cultural element (and hence a member of the set P), we use italics (e.g., Marcuse).5 Here and elsewhere the size of nodes increases monotonically with the number of times a name is mentioned, the thickness of lines increases monotonically with the number of co-occurrences of two names, and the nodes are positioned according to a spring embedding algorithm. This graph is interesting not only because of who we see as most central, but because of the implied bridging relations. Note that Marcuse is only related to Adorno, and not to any of the

5 Although our data fall short of this, it would be remarkable to consider cases in which A = P, such that we can, first, compare Y to the mondo-social structures of each author (which we define below as [PP|A]) and see which authors arrange the other authors in ways consonant with their actual relations, and, second, compare Y to the [AA] relations of commonalities in each authors’ [PP] graphs (which we define below as [AA:PP]) to see whether the agreement in representation of networks by network members may in fact be used to establish the very network in which the actors are embedded.

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Fig. 2. Adorno’s Mondo-cultural Structure [PP|a = Adorno’s Negative Dialectics].

other school members. We might take from this an implicit hypothesis, namely that Herbert Marcuse is intellectually allied with Adorno—but not with the other Frankfurt School members. We might compare this to information from some Y sociomatrix involving their observed relations. In this case, we can be confident that we would find something quite different, for it is widely known that it was Horkheimer who moderated Marcuse’s relation with Adorno (a divide-and-conquer strategy was integral to his maintenance of control over the school [Wiggershaus, 1995]). Of course, Habermas was not attempting to describe the social relations between these authors, but to say something about their ideas; what is important in the mondo-social structure is what it implies for the cultural structure Habermas is developing. For this reason, we are more interested in comparing his mondo-social structures to those of other Frankfurt theorists than we are to the actual social structure. To simplify our exposition, we first generalize our notation. We began with a two-way matrix denoted [AB] expressing a binary relation R. Let C be a third set, with R* now a trinary relation, such that (a, b, c) ∈ R* means that element a from set A, element b from set B, and element c from set C, are connected. Thus it is a single (trinary) relation, and not three binary relations, that connects them. Correspondingly, while R can be treated as a graph (where the edges are all members of R), R* must be treated as a specific form of a uniform hypergraph (one where every edge has exactly three elements, one from each set). We denote this relation R* as [ABC]; it is clear that this can be extended to quaternary relations and higher. Now use [AB|c] to mean the relation R (as previously defined) only holding within such cases which are defined by the presence of specific element c from C. Given [ABC], this is equivalent to selecting the sub-hypergraph consisting of all edges that include c, and treating it as a graph, given that one mode of the graph is constant (C = c). Finally, use [AB|C] to denote all ||C|| graphs of the form [AB|c], one for each member of C. We can further generalize to four-way and higher relations. It is for this reason that we labeled Fig. 1 [PP|a = Habermas]. Our goal is now to compare members of the set [PP|A], where A = {Habermas, Adorno, Marcuse}. Fig. 2 has the social structure of reference of Theodor Adorno’s Negative Dialectics and Fig. 3 that of Herbert Marcuse’s One Dimensional Man. These two texts lack the focus on Horkheimer, Adorno and Marcuse found in Fig. 1, and so we instead point to a different relationship in these two maps. For here we see two very different visions of the relationships among Marx, Hegel, and Kant. Provisionally accepting our simplifying assumption that co-occurrence can be treated as a tie, we would say that Adorno’s discussion implies a strong relationship between Kant and Hegel, a weaker relationship between Hegel and Marx, an even weaker relationship between Kant and Marx. Marcuse, on the other hand, constructs a strong relationship between Hegel and Marx, while Kant and Hegel are only weakly related, and Kant and Marx are not associated at all. For another example, Marcuse connects Plato strongly to Marx, while Adorno does not have this connection, though Plato is connected to both Kant and Hegel. This suggests that Adorno and Marcuse are, in these texts, constructing different mondo-social structures of preceding thinkers. This also fits what most readers of the Frankfurt school would take as their orientation to these three thinkers.

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Fig. 3. Marcuse’s Mondo-cultural Structure [PP|a = Marcuse’s One Dimensional Man].

2.3. Duality and mondo-culturosocial structure This notion, that we may be interested in the similarity of some texts in terms of the implicit referential social structures that they create, is an intriguing one—as it implies that the different texts may create the same virtual social worlds—but it is somewhat far from our understanding of the meaning of the virtual actors in this social world. Two texts may create the same mondo social structure, but for very different reasons. For example, Marx and Plato may be neighbors because of their similar interest in overcoming social divisions or, as Popper would suggest, their underappreciation of the importance of an open society, or because of the radical disjuncture in their views of the nature of life activity (as Arendt would suggest). We thus need to explore not only which persons are linked via joint reference in text, but the implicit meaning that brings them together—to look at the mondo-cultural meanings of persons. We have focused on organizing the members of one set in terms of meaningful similarity on a second set, but we have bracketed this meaning itself. Following Martin and Lee (this issue), we propose that, given some matrix [AB] we can define the “A meaning” of some element of B, namely b, (denoted ΓA(b)) as the “extent” of b—all a such that aRb (and dually for the B meaning of a, the “intent”). For example, the Mondo-social structures we graph in Figs. 1–3 contains three versions of the “Person” meaning of Persons (ΓP(p)), one for each analyzed text. Let us say that we are interested in taking the matrix [PW] of some text and using this to determine the W-meaning of any person referred to (ΓW(p)). For example, taking only the first five relations for purposes of brevity, Lee and Martin (2015b) had found that the meaning of “Freud” for Marcuse was instincts, civilization, principle, reality, and repressive, while that for Adorno was art, language, like, work, and dream. And this suggests a further extension of our exploration of duality—we examine which authors are similar on the basis of their capacity to construct similar mondo-cultural meanings of persons, which we shall call the mondo-culturosocial structure of authors. Let us define an operation on sets of “vectorization.” This is when we take a larger array (for example, the N by M matrix X) and reshape it as a vector (in this case, one with N × M values). (It is immaterial what order the elements are placed in, so long as the arrangement is consistent.) We use () to denote such vectorization. Thus consider the ||C|| matrices of the form [AB|c]; we use [(AB)C] to denote the binary matrix with ||A|| × ||B|| rows and ||C|| columns. Note that while [(AB)C][C(AB)] ≠ [A(BC)][(BC)A] and [A(BC)] ≠ [A(CB)], still, [A(BC)][(BC)A] = [A(CB)][(CB)A]; for purposes of exploring duality, the order of vectorization is arbitrary. What is key is that we can vectorize multiple dimensions to explore various forms of duality. For example, this simplifies our notating our exploration of mega-culturosocial relations, in which we consider authors culturosocially similar not if they use similar cultural elements, but only if they link them similarly. If (as done in Lee & Martin, 2015) we compare the Z matrices of several authors, thereby constructing [CC|A], we can parsimoniously notate this as [AA:CC] = [A(CC)] [(CC)A] where (CC) is equivalent to a vectorization of the Z matrices discussed above. If we are specifically interested in whether authors agree as to their mondo social structures as defined above, we would be examining [AA:PP]. If we are instead interested in whether authors agree as to the mondo-cultural meaning of persons (ΓW(p)), we would be examining [AA:PW] = [A(PW)][(PW)A]. For example, Lee and Martin (2015b) did this for the particular case of the meaning of “Freud.” They showed that some authors (Fromm and Marcuse) agreed greatly on the general meaning of Freud, in terms of the other cultural elements that were within Freud’s domain of relevance, though they famously disagreed as to the implications of this. In contrast, neither Fromm nor Marcuse 7

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Fig. 4. Non-Metric MDS Scaling of Table 1.

agreed with Adorno as to the meaning of Freud. In a sense, no explicit debate (such as Fromm and Marcuse had) was even possible with Adorno. Here, then, Lee and Martin were using only a single row from the [PW] matrix to construct [AA:PW]; with this new notation there is nothing to prevent us from using the entire matrix once vectorized. There is, however, a complication for the case of weighted graphs (matrices in which we have counts, as opposed to mere presence/absence). As noted above (note 2), we may have a difficulty comparing overlaps in Z type matrices where the number of elements is greatly different across the two cases. We may wish to make use of the implicit weights that come from the frequencies, or we may wish to standardize so that these weights do not drive any results. Let [PW] be an M* × (M − M*) matrix defined as above, and construct the row-normalized version, [PW*], in which all rows have mean 0 and whose squares sum to 1. By construction, the corresponding Ψ″ = [AA:PW*] matrix will be a correlation matrix, with all values −1 ≤ ψ″ij ≤ 1. (This is because the dot-product divided by the harmonic mean of the magnitude of two vectors is their correlation.) It is this form that we use for our results below. Note that correlations between sparse vectors can be surprisingly high if they share only a few of their holdings. The table below shows the results when we compute this for six key authors, using the vector of meanings over the 75 references (our P) who were discussed by more than one author. It is not simply that Marcuse and Fromm agreed on what Freud meant; they agreed quite closely on what most authors meant. In contrast, it will be seen that Lowenthal disagrees greatly with most of the other theorists. A simple nonmetric multidimensional scaling plot6 shows this graphically (Fig. 4): Marcuse and Fromm overlap exactly! (They’ve been spread out for purposes of readability.) Admittedly, there is some undetermination here; the stress is basically zero, because with 6 × 5/2 = 15 observations and 2 × 6 = 12 coordinates, we don’t have many degrees of freedom. But in this case, what is key is that there is no reason to think that Fromm and Marcuse have opposing conceptions of what the authors they used “meant.” It is worth emphasizing that here, in a Mondo-Breiger approach, we construct a set of social relations on the basis of social references which carry meanings, even though we do not make any hazard as to what these meanings are. We have no insight from our analyses as to how Fromm and Marcuse understand the various persons they use as references. We only believe that they share an understanding of this horizon of relevances. Of course, we are assuming that authors make up worlds in their writing, or at least, that they can be treated as such. Thus in our text we arrange these authors as arrangers—we put them in a virtual social world by establishing relations (Table 1) based on how we believe that they make social worlds by establishing relations. If our assumption is incorrect, it would seem that we are more than hopelessly lost. However, we believe that our assumption is valid in a number of cases, and, in particular, that of social theorists such as the Frankfurt school. We chose to focus on the PW relation because of our understanding as to how Critical Theorists constructed their texts (using previous writers to introduce and manipulate concepts). Thus our approach, though formal, is guided by some substantive considerations. We go on to show how our approach can be extended in two ways: one even more formal and content-free, and the other, more substantively guided. We begin with the second.

3. A substantive examination 3.1. Habermas’s Representation of Marcuse and Co. Above we saw that in Habermas’s mondo-social structure ([PP|a = Habermas]), Marcuse’s relations with Horkheimer were wholly 6

We convert the correlations of Table 1 to a distance matrix by taking their inverse and subtracting 1, so that the diagonal is zero.

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Table 1 Mondo-Culturosocial Structure Meaning of Persons.

Adorno Horkheimer Marcuse Lowenthal Fromm Neumann

Adorno

Horkheimer

Marcuse

Lowenthal

Fromm

Neumann

1.000 0.434 0.532 0.290 0.596 0.328

0.434 1.000 0.489 0.265 0.530 0.347

0.532 0.489 1.000 0.302 0.691 0.417

0.290 0.265 0.302 1.000 0.344 0.235

0.596 0.530 0.691 0.344 1.000 0.448

0.328 0.347 0.417 0.235 0.448 1.000

mediated by Adorno. We noted that in their actual lives, it was instead that Marcuse’s relations with Adorno were mediated by Horkheimer, but this is not necessarily relevant. The question is whether, when we consider the cultural elements that are relevant for Habermas’s theoretical purposes, we would find this a reasonable conjunction, in the way that we might imagine that Marcuse’s close linkage of Marx and Hegel is not surprising given his focus on negativity and the dialectic (at least in Negative Dialectics). Thus it could well be that in terms of their use of the concepts in question, Marcuse was in fact closer to Adorno than to Horkheimer…at least, insofar as he was treating the issues that were of interest to Habermas. In other words, if we really wanted to engage with the implied relation between Adorno and Marcuse in Fig. 1, we should not necessarily include all of Adorno’s work on music. If these aesthetic issues were not relevant to Habermas’s own project, we would not expect him to separate Adorno from Marcuse on these grounds. We can pursue this question by examining the mondo-cultural meaning of persons in the texts of different authors. To do this, we propose, first, to determine Habermas’s word-meaning for Marcuse. We will let K = ΓW(Marcuse) for Habermas denote this set. However, we are particularly interested in the words that Habermas particularly associates with Marcuse (and not, say, with the other interlocutors). We then can construct the mondo-cultural structure specifically for the mondo-cutural meaning of Marcuse, or [KK]. If we simply gathered the concepts that are simply most often attached to Marcuse, we will almost inevitably find the most frequently discussed concepts in Habermas’s book overall—concepts that are most often associated with all of Habermas’s interlocutors—not especially with Marcuse. For this reason, we weight every edge in [KK] by the inverse of its prevalence in the graph as whole, and denote this weighted version as [KK]*. Thus, we weight the tie between any elements of this structure by how specific it is to the mondo-cultural meaning of Marcuse, as opposed to Habermas’s more general cultural structure. More specifically, for any two elements i and j in K, from [WW|P = Marcuse] we construct the weighted edge zij|M via equation 1; we also make z•M = ∑ij zij|M, the sum of all these weights. Dividing zij|M by z•M gives us a normalized z*ij|M—the relative weight of this connection in the mondocultural structure around Marcuse. We also do the same for the zij|∼M edges in the [KK| ∼ (P = Marcuse)]* graph formed from the paragraphs in which Marcuse is not mentioned, create similar sums and come up with z*ij|∼M. If we examine the ratio of z*ij|M to z*ij|∼M, we have the relative emphasis that Habermas places on this connection when he is discussing Marcuse. If this is greater than 1, it means Habermas is more likely to link these when discussing Marcuse than otherwise; if it is less than 1, it means he is less likely. We treat cases in which the words are less likely to be linked around Marcuse as not present in the [WW|P = Marcuse] graph, and use the natural logarithm of the ratio otherwise, because of the tendency of the ratio to take extreme values. Thus our edgeweights come from the following:

ρij = ln ρij = 0,

(

z *ij M z *ij ∼ M z *ij M z *ij ∼ M

),

z *ij M z *ij ∼ M

>1

≤1

(3)

With this, we can construct the [KK|a = Habermas]* relation of the concepts that Habermas associates with Marcuse. This is displayed in Fig. 5. Inspection of this graph is itself of interest; it shows that Habermas’s use of Marcuse has three main theoretical directions. The first, represented by the cluster on the left, involves the political and the modern capitalist state. The second, represented by the cluster near the bottom, involves critical theory and problems in the philosophy of consciousness. The third, represented by the cluster in the upper right hand corner, involves social processes such as societal integration and processes of socialization. However, to test the hypothesis that Marcuse and Adorno are treating the concepts in K similarly, we will push to the side Habermas’s own treatment of these concepts, and instead construct the set of all [KK|A] matrices of our authors, thus examining the mondo-culturosocial structure that arises in this restricted world of cultural elements that exist in Habermas’s mondo-cultural meaning of Marcuse. We here exclude Habermas himself, and include the traditional most central four—Marcuse, Adorno, Horkheimer, and Löwenthal. For each author, we chose the two of his works that demonstrate the greatest relevance in terms of the overlap between their mondo-cultural structure [CC] with [KK|a ] Habermas]; we then construct [KK|a] for each. For Marcuse, these works are Reason and Revolution (1941) and Soviet Marxism (1958). For Adorno, they are Aesthetic Theory (1970) and Negative Dialectics (1966). For Horkheimer, they are his 1934 Articles and his 1937 Articles. Finally, for Löwenthal, they are his 1957 Articles and Literature, Popular Culture, and Society (1961). We then investigate the mega-culturosocial relations [AA:KK], which is found in Table 2; all values have been multiplied by 102 for ease of presentation. As we can see, there is no reason to think that there was a stronger relationship between Marcuse and Adorno than between Marcuse and other school members when it comes to the issues that Habermas saw as central to Marcuse. With a maximum of 0.216 similarity between Marcuse’s Reason and Revolution and Adorno’s Negative Dialectics, and an average similarity between Marcuse and 9

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Fig. 5. Concepts Specfically Associated with Marcuse.

Table 2 Socioculturosociocultural Overlaps.

Marcuse (1941) Marcuse (1958)

Adorno (1966)

Adorno (1970)

Horkheimer (1934)

Horkheimer (1937)

Löwenthal (1957)

Löwenthal (1961)

0.216 0.124

0.106 0.063

0.377 0.324

0.198 0.199

0.187 0.167

0.176 0.165

Marcuse-Adorno Avg.

0.127

Marcuse-Horkheimer Avg.

0.275

Marcuse-Löwenthal Avg.

0.174

Adorno’s books of 0.127, Marcuse’s relationship with Adorno seems weaker than Marcuse’s relationship with the controlling member of the school, Max Horkheimer. Marcuse shares with Horkheimer a maximum similarity of 0.377 (between Marcuse’s Reason and Revolution and Horkheimer’s 1934 articles) and an average similarity of 0.275. Marcuse’s overlap with Adorno even appears slightly weaker than his semantic relationship with Leo Lowenthal, whose maximum similarity is 0.187 (between Marcuse’s Reason and Revolution and Lowenthal’s 1957 articles). So we might conclude that Habermas’s Theory of Communicative Action does not reflect the intellectual reality of the Frankfurt School so much as it provides a particularly motivated construction of it. He portrays a special intellectual affinity between Marcuse and Adorno that is not apparent from the semantic relationship between their actual texts. Of course, it is also quite possible that we simply have misinterpreted the cultural structure of the elements that are found in Habermas’s mondo-cultural meaning of Marcuse, or that our procedures were insufficiently sensitive. We should be hesitant to attribute inaccuracy to Habermas, a close reader Marcuse and Horkheimer, on the basis of the cursory overview of a set of simple graphs (see Biernacki, 2015). We therefore return to the original texts. It is true that Habermas at one point links Adorno and Marcuse to oppose them ([1981] 1984: 384), and other times he does link Marcuse to both Horkheimer and Adorno ([1981] 1984: 144, 367). But most important for our purposes, at one point, Habermas ([1981] 1987: 380f) explicitly describes his understanding of the reciprocal organization of the major figures of the Frankfurt School! He provides, in other words, a mondo-social structure. On the one side, Marcuse and Adorno are allied with Horkheimer against Fromm in terms of their (more traditional) reception of Freud. “Another front formed around the question of the ideological character of mass culture, with Adorno [‘(along with Löwenthal and Marcuse)’, Habermas adds in the next sentence] on one side and Benjamin on the other.7 Thus it is not surprising that, as an aside,

7 In contrast to the agreement of the core members on certain themes about the nature of the modern society, the views of Neumann, Kirchheimer, Fromm and Benjamin are, in some respects, hard to place, says Habermas.

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Habermas ([1981] 1987: 250) would speak of “critical theorists such as Marcuse and Adorno.” He sees the two as sharing a location on two analytic dimensions, while Marcuse only shares one with Horkheimer. Our simple procedure seems to have hit the mark. Thus by focusing on a specific question, we can explore one aspect of the meaning contained in a dataset, providing both interpretable cartographic representations and interpretation-free chartings of the relative agreement between certain elements in terms of their distribution across other sets. We end by pointing out how this interpretation-free approach may be generalized. 3.2. Going full Breiger This exercise ends in the “Full Breiger,” a general understanding that the meaning of a set of cultural items manifests in its relations to multiple other sets of other items and persons, and that each one of these dimensions can be modeled mathematically as a vector in a larger adjacency matrix. Thus, duality is not merely a statement about the relation between the social and cultural world but a general formalization of the insight that the meaning of a thing lies in its relationships to other things and people. Consider a “dataset” to be a set of D different linked sets which may be compiled as dimensions of an array Q. It is essential that we be able to define, at least in the abstract, a D-dimensional relation connecting all these sets. It is thus that we see Q as a D-uniform hypergraph. Of course, as in a conventional dataset, each set may be envisioned as a column, and each instantiation a row. For example, a row might be a “qualification,” in which in one text (out of many), one noun (out of many) is associated with one adjective (out of many) and one modality (out of the smaller set “is always,” “is sometimes,” “is rarely” and “is never”). Martin and Lee (this issue) work through the relations between different meanings in a dataset, as well as how to move across datasets via translation, but here we are only interested in how a dataset gives us the full Breiger relationships among one set of elements in a dataset. Given some dataset Q, with D dimensions, which we here denote Q1, Q2, …, QD, for the dth dimension, use Qd to denote the vectorization of Q over all dimensions except d. Then for the dth dimension, we may use [QdQd:Qd] = [Qd(Qd)] [(Qd)Qd] to denote the relations among the members of set Qd implied by all other dimensions of the data set. In other words, it is akin to the culturosocial or sociocultural meanings in Breiger classic, but with one dimension a vectorization of all dimensions other than the focal one. It is this that we call going “full Breiger,” for the internal meanings among the elements of the dth dimension is found in the full crossclassification of all other meanings in the data. Thus we are able to determine the implicit relations of similarity in meaning across a complex data set. In our example above, given that our dataset only had three sets (authors A, person references P and other terms W), when we computed the [AA:PW] relations, we were computing the full Breiger duality for that data set—we have arranged our texts in terms of their shared meanings. A value in this resulting matrix means that the two agree as to the meaning of the terms they treat…whatever that meaning might be. 4. Conclusion “It’s the question that drives us—it’s the question that brought you here. You know the question, just as I do.” “What is the Matrix?” Trinity & Neo We believe that there have been a few true breakthroughs in the human sciences. One was Lévi-Strauss’s conception of structures as sets of transformations that, given different starting conditions, could produce quite different sets of synchronic relations. A second was Harrison White’s (see Lorrain & White, 1971) realization that the structuralist opposition of metonymy and metaphor implied the possibility of examining structural equivalence in informal relations.8 A third, or so we claim here, is the notion of duality, independently pursued by Bourdieu (1984 [1979]), Breiger (1974) and Benzécri (1973) (as well as other French researchers at the time—see de la Cruz & Holmes, 2011). This approach is of special interest to students of culture, as it naturally orients itself to issues of meaning and information, as opposed to, say, marginal treatment effects. We here attempt to pursue to its completion one way of organizing the information in a dataset with an eye to duality. This approach makes it easy to answer certain types of focused questions. While these questions may also be answered using other approaches—there is a way in which our formulation approximates a non-weighted set of analyses that could be conducted using Geometric Data Analysis (Le Roux & Rouanet, 2004)9—we believe that the capacity to interpret datasets as hypergraphs that allow for full Breiger explorations of duality, or reduced representations, facilitates the exploration of the particular problems that are of interest to sociologists of culture. Most important, we seek to develop a simplified notation that can allow us to ask questions about meaning that might otherwise be quite difficult to formulate. “When these teachings had been declared, five thousand bodhisattvas entered the door of the Dharma of [non]duality and attained tolerance of the birthlessness of things.” Vimilakīrti-Sutra, “The Dharma-Door of Non-Duality”

8 A reviewer asks for a reference clarifying the way in which the notion of structural equivalence is rooted in these fundamental structuralist ideas; unfortunately, we know of no one who has written on this other than our brief discussion (Lee and Martin, 2015b), but this statement is God’s own truth. 9 We confess that working out the relation is beyond our capacity and look forward to this being resolved by the more skilled.

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Acknowledgements We are extremely grateful to all the participants of the Bern conference on formal modeling of culture for comments, and especially to three reviewers and the editors for comments and criticisms that greatly increased the clarity of this exposition. Appendix A. Glossary Breiger Classic – using a two-mode matrix to create culturosocial and sociocultural meanings. Cultural Structure – relations established between cultural elements. Culturosocial – relations between persons established by similar cultural distributions. Holding – a connection of a person and a cultural element in which the first holds the second. Mega-Culturosocial – relations between persons established by similar cultural structures. Mega-Sociocultural – relations between cultural elements established by patterns of similar social structures. Mondo-Breiger – the exploration of meaning and duality based on the Mondo level distributions. Mondo-Cultural Meaning of Persons – the two-mode relations connecting persons within cultural structures to non-person elements within these structures. Mondo-Social Structure – the set of relations established within a set of persons treated as cultural elements within a cultural structure. Mondo-Culturosocial – relations between persons established by similar cultural distributions within a cultural structure. Mondo-Culturosocial Structure – The social relations between authors established by similar mondo-cultural meanings of persons. Social Structure – relations established among persons. Sociocultural – relations between cultural elements established by patterns of which persons hold them. Vectorization – the transformation of a set of more than one dimension in a dataset into a single dimension by constructing the Cartesian product. World – a subset of cultural elements used by a person. References Basov, N., & Brennecke, J. (2017). Duality beyond dyads: Multiplex patterning of social ties and cultural meanings. Research in the Sociology of Organizations, 53, 87–112. Benzécri, J. P. (1973). L'analyse des Données. T.2: L'analyse des Correspondances. Paris: Dunod. Biernacki, R. (2015). How to do things with historical texts. American Journal of Cultural Sociology, 3, 311–352. Bourdieu, Pierre, [1979] (1984). Distinction: A social critique of the judgment of taste, translated by Richard Nice. Cambridge, MA: Harvard University Press. Breiger, R. L. (1974). The duality of persons and groups. Social Forces, 53, 181–190. Breiger, R. L. (2000). A tool kit for practice theory. Poetics, 27, 91–115. Breiger, R. L. (2011). Baruch spinoza: Monism and complementarity. In C. Edling, & J. Rydgren (Eds.). Sociological insights of great thinkers: Sociology through literature, philosophy, and science (pp. 255–262). Praeger. Carley, K. (1986). Knowledge acquisition as a social phenomenon. Instructional Science, 14, 381–438. Converse, P. E. (1964). The nature of belief systems in mass publics. In D. E. Apter (Ed.). Ideology and discontent (pp. 206–261). New York: Free Press. de Spinoza, Benedict, (1930) [1677]. Ethic translated by W. Hale White and Amelia Hutchinson, London: Oxford University Press. de Spinoza, Baruch, (1998) [1663]. The principles of cartesian philosophy and metaphysical thoughts, translated by Samuel Shirley, Indianapolis, Ind.: Hackett Publishing. de la Cruz, O., & Holmes, S. (2011). The duality diagram in data analysis: Examples of modern applications. Annals of Applied Statistics, 5, 2266–2277. Fararo, T. J., & Doreian, P. (1984). Tripartite structural analysis: Generalizing the Breiger-Wilson formalism. Social Networks, 6, 141–175. Friedkin, N. E. (1998). A structural theory of social influence. New York: Cambridge University Press. Habermas, Jürgen, [1981] (1987). The theory of communicative action. Volume 2: Lifeworld and system: A critique of functionalist reason, translated by Thomas McCarthy. Boston: Beacon Press. Kovács, B. (2010). A generalized model of relational similarity. Social Networks, 32, 197–211. Lang, G. E., & Lang, K. (1988). Recognition and renown: The survival of artistic reputation. American Journal of Sociology, 94, 79–109. Le Roux, B., & Rouanet, H. (2004). Geometric data analysis. Dordrecht: Kluwer. Lee, M., & Martin, J. L. (2015a). Coding, counting, and cultural cartography. American Journal of Cultural Sociology, 3, 1–33. Lee, M., & Martin, J. L. (2015b). Response to Biernacki, Reed, and Spillman. American Journal of Cultural Sociology, 3, 380–415. Lee, M. (2017). The duality of philosophers’ social relations and ideas. Research in the Sociology of Organizations, 53, 177–209. Lena, J. (2010). Recognition and renown in rap music: Tracing reputation through aesthetic conventions. Paper presented at the annual meetings of the American Sociological Association (ASA). Lévi-Strauss, Claude, [1949] (1969). The elementary structures of kinship, translated by James Harle Bell, John Richard von Sturmer, and Rodney Needham. Beacon Press : Boston. Lizardo, O. (2017). The mutual specification of genres and audiences: Reflective two-mode centralities in person-to-culture data. Poetics [this issue]. Lorrain, F., & White, H. C. (1971). Structural equivalence of individuals in social networks. Journal of Mathematical Sociology, 1, 49–80. Marx, Karl, Engels, Frederick, (1976) [1845-6]. The German ideology, in: Collected works, Vol. 5, New York : International Publishers. Mische, A., & Pattison, P. (2000). Composing a civic arena: Publics, projects, and social settings. Poetics, 27, 163–194. Mohr, J., & Duquenne, V. (1997). The duality of culture and practice: Poverty relief in new York city, 1888–1917. Theory and Society, 26, 305–356. Mohr, J. W., & Friedland, R. (2008). Theorizing the institution: Foundations, duality and data. Theory and Society, 37, 421–426. Mohr, J. W. (2000). Introduction: Structures institutions, and cultural analysis. Poetics, 27, 57–68. Novak, J. D. (1977). A theory of education. Ithaca, New York: Cornell University Press. Peirce, Charles S., [1866] (1984). The logic of science; Or, induction and hypothesis, Lowell lectures of 1866, in writings of Charles S. Peirce, Vol. 1, edited by Nathan Houser et al., Bloomington, Indiana: Indiana University Press, 357–504. Thurman, Robert A.F., translator, (1976). The holy teaching of Vimilakīrti. University Park: The Pennsylvania State University Press. White, H. C. (1963). An anatomy of kinship. Englewood Cliffs, NJ: Prentice Hall. Wiggershaus, R. (1995). The Frankfurt school: Its history, theories, and political significance. Cambridge, Mass: The MIT Press translated by Michael Robertson. Yeung, K.-T. (2005). What does love mean? Exploring network culture in two network settings. 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Monica Lee is a computational sociologist on the Facebook Core Data Science team. A recent Ph.D. from the University of Chicago, her work at Facebook investigates political and civic engagement on and off social media, employing diverse methods from Machine Learning to In-depth Interviews. Her published work on formal and quantitative methods for studying cultural phenomena can be found in PLOS one, Sociological Theory, Poetics, Big Data & Society, and The American Journal of Cultural Sociology, among other journals. John Levi Martin is the Florence Borchert Bartling Professor of Sociology at the University of Chicago. He is the author of Social Structures, The Explanation of Social Action, Thinking Through Theory, and Thinking Through Methods, as well as articles on methodology, cognition, social networks, and theory. He is currently working on the history of the theory of social action.

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