There’s a story — probably a myth — about Pythagoras killing one of the members of his math cult because this member discovered irrational numbers (Choike 1980). (He also either despised or revered beans).

The Greeks spent a lot of time arguing about arche, or the primary “stuff.” Empedocles argued that it was the four elements. Anaximenes thought it was just air. Thales thought it was water. Pythagoras and his followers figured it was numbers (Klein 1992, page 64):

They saw the true grounds of the things in this world in their countableness, inasmuch as the condition of being a “world” is primarily determined by the presence of an “ordered arrangement” — [which] rests on the fact that the things ordered are delimited with respect to one another and so become countable.

For the Pythagoreans the clean, crisp integers were sacred because they conveyed a harmony — an orderedness — and there is an undeniable allure to this precision. (Indeed, such an allure that Pythagoras and his followers were driven to do some very strange things.)

Looking at even simple arithmetic, it does seem obvious that classical categories do in fact exists: there is a set of integers, a set of odd numbers, a set of even numbers, and so on. If we continue to follow this line of thought to pure mathematics in general, there is an almost mystical, quality of the “objects” of this discipline.

When thinking about mathematical objects like geometric forms, however, there is a fundamental difference between squares or circles or triangles as understood in our daily life (i.e. as having graded similarities to certain exemplar shapes we likely learned about in grade school) and the kind of perfectly precise shapes in theoretical geometry. That is, as far as we know, a perfect circle does not exist in nature (even though an electron’s spin and neutron stars are pretty damn close), nor has humankind been able to manufacture a perfect shape.

And this is the main point: precision is weird. If “crispness” is really only found in mathematics (and pure mathematics at that), then we should be skeptical of the analytical traditions’ use of discrete units as an analogy for knowledge in general.

But, sometimes, thinking with classical categories is useful.

Property Spaces

While we can be skeptical of the Chomskyan program presuming syntactical units must necessarily be classical categories, this does not mean we can never proceed as if phenomena could be divided into crisp sets.

Theorists commonly make something like “n by n” tables, typologies, or more technically, property spaces — for the classic statement see Lazarsfeld (1937) and Barton (1955), but this is elaborated in (Ragin 2000, page 76-85), Becker (Becker 2008, page 173-215), and most extensively in chapters 4, 5, and 6 of Karlsson and Bergman (2016). In this procedure, the analyst outlines a few dimensions that account for the most variation in their empirical observations. This is essentially “dimension reduction,” as we take the inherent heterogeneity (and particularity) of social experience and simplify this into the patterns that are the most explanatory (if only ideal-typical).

For example, Alejandro Portes and Julia Sensenbrenner (1993) tell us that there are four sources of social capital (each deriving conveniently from the work of Durkheim, Simmel, Weber, and Marx and Engels, respectively). These four sources are then grouped into those that come from “consummatory” (or principled) motivations and those that come from “instrumental” motivations. Thus the “motivation” is the single dimension that divides our Social Capital property space into a Set A and a Set B: either resources are exchanged because of the actor’s own self-interest, or not. More often, however, these basic property spaces based on simple categorical distinctions are the starting point for more complex (or “fitted”) property spaces.

Consider Aliza Luft’s excellent “Toward a Dynamic Theory of Action at the Micro Level of Genocide: Killing, Desistance, and Saving in 1994 Rwanda.” Luft begins with a critique of prior categorical thinking: “Research on genocide tends to pregroup actors—as perpetrators, victims, or bystanders—and to study each as a coherent collectivity (often identified by their ethnic category)” (Luft 2015, page 148). Previously, analysts explained participation in genocide in one of four ways: (1) members of the perpetrating group were obedient to an authority, (2) responding to intergroup antagonism, (3) succumbing to intragroup norms or peer pressure, (4) and finally, ingroup members dehumanize the outgroup. While all are useful theories, she explains, they are complicated by the empirical presence of behavioral variation. That is, not everyone associated with a perpetrating group engages in violence at the same time or consistently throughout a conflict (and may even save members of the victimized group).

 

What she does to meet this challenge is to add dimensions to a binary property space which previously consisted of a group committing murder and a group being murdered. Focusing on the former, she notes that (1) not everyone in that group does actually participate, (2) some of those who did (or did not) participate eventual cease (or begin) participating, (3) some of those who did not participate not only desisted but also actively saved members of the outgroup. Taking this together, we arrive at a property space that can be presented by the spanning tree shown above. Luft then outlines four mechanisms that explain “behavioral boundary crossing.”

In this case, previous expert categories lead to an insufficient explanation for the perpetration of genocide, and elaboration proved necessary. Attempting to create classical categories — with rules for inclusion and exclusion and the presumption of mutual exclusivity in which all members are equally representative — is likely a necessary step in the theorizing process. Much of the work of developing theory, however, is not just showing that these categories are insufficient (because, of course, they are), but rather pointing out where this slippage is leading to problems in our explanations, and how they can be mended, as Luft does. 

The Scholastic Fallacy

Treating data or theory as if they can be cleanly divided into crisp sets is like the saying “all models are wrong, but some models are useful.” But taking for granted these distinctions can also lead analysts to commit the “scholastic fallacy.”

This is when the researcher “project[s] his theoretical thinking into the heads of acting agents…” (Bourdieu 2000, page 51).  This, according to Bourdieu, was a key folly of structuralism: “[Levi-Strauss] built formal systems that, though they account for practices, in no way provide the raison d’etre of practices” (Bourdieu 2000, page 384). This seems especially obvious for categories, as discussed in my previous two posts. It is one thing to say people can be divided into X group and Y group for Z reasons, and it is another to say people do divide other people in X group and Y group for Z reasons (see Martin 2001, or more generally Martin 2011)

Categorizing for the “acting agent” is not a matter of first learning rules and then applying them to demarcate the world into mutually exclusive clusters. It is, for the most part, a matter of simply “knowing it when I see it” —  a skill of identifying and grouping that we have built up through the accrued experience of redundant patterns encountered in mundane practices. Generally, rules, if they are used, are produced in post hoc justifications of our intuitive judgment about group memberships. It is here, however, where expert discourse is likely to play the largest role in lay categorizing: as a means to justify what we already believe to be the case.

This is not to say “non-experts” cannot or do not engage in this kind of theoretical thinking about categories. But, again Bourdieu points out, most people do not have the “leisure (or the desire) to withdraw from [the world]” so as to think about it in this way (Bourdieu 2000, page 51). More importantly, relying on expert categories for most of the tasks in our everyday lives would not be very useful because categorizing is foremost about reducing the cognitive demands of engaging with an always particular and continuously evolving reality.

References

Barton, Allen H. 1955. “The Concept of Property-Space in Social Research.” The Language of Social Research 40–53.

Becker, Howard S. 2008. Tricks of the Trade: How to Think about Your Research While You’re Doing It. University of Chicago Press.

Bourdieu, P. 2000. Pascalian Meditations. Stanford University Press.

Choike, James R. 1980. “The Pentagram and the Discovery of an Irrational Number.” The Two-Year College Mathematics Journal 11(5):312–16.

Karlsson, Jan Ch and Ann Bergman. 2016. Methods for Social Theory: Analytical Tools for Theorizing and Writing. Routledge.

Klein, Jacob. 1992. Greek Mathematical Thought and the Origin of Algebra. Courier Corporation.

Lazarsfeld, Paul F. 1937. “Some Remarks on the Typological Procedures in Social Research.” Zeitschrift Für Sozialforschung 6(1):119–39.

Luft, Aliza. 2015. “Toward a Dynamic Theory of Action at the Micro Level of Genocide: Killing, Desistance, and Saving in 1994 Rwanda.” Sociological Theory 33(2):148–72.

Martin, John Levi. 2001. “On the Limits of Sociological Theory.” Philosophy of the Social Sciences 31(2):187–223.

Martin, John Levi. 2011. The Explanation of Social Action. Oxford University Press, USA.

Portes, A. and J. Sensenbrenner. 1993. “Embeddedness and Immigration: Notes on the Social Determinants of Economic Action.” The American Journal of Sociology.

Ragin, Charles C. 2000. Fuzzy-Set Social Science. University of Chicago Press.