GENERAL USAGE

Of the two meanings of “recursive,” the more general is perhaps more familiar. This usage is also the newer of the two, having entered political science through the work of sociologist Anthony Giddens. He describes “human social activities” as recursive: “That is to say, they are not brought into being by social actors but continually recreated by them via the very means whereby they express themselves as actors” (Reference Giddens1984, 2). This statement is Giddens' answer to the agent-structure problem in sociology and the basis of his “structuration” theory, which has attracted the attention of political scientists occupied with similar concerns. Notable among these scholars is Alexander Wendt; in his paper, “Agent-Structure Problems in International Relations,” he writes:

The text suggests that Wendt uses “recursive” in the same sense as Giddens, and the notes and citations to the article reinforce this suggestion. More recently, other political scientists have taken up this usage as well. For example, Clark places quotes around the term “recursive relationship” (Reference Clark1998, 246), and Imbroscio writes of “an interactive and recursive fashion” (Reference Imbroscio1999, 46); both are referring to Giddens' ideas specifically. Elsewhere, Emmelman (Reference Emmelman1996) describes “recursive decisionmaking” in legal settings. Hopmann, reviewing Jervis's System Effects, writes that interaction effects are “not merely additive or linear but are contingent and recursive” (Reference Hopmann1998, 314).

In each of the above cases, “recursive” means “mutually influential” or “mutually constitutive.” In this usage, to say that A and B are elements in a recursive pair suggests that A affects B while B affects A. As political scientists untangle the nonlinearity of political and social interactions, “recursive” in this sense is a useful term and concept. Many political phenomena comprise mutually influential elements; in studying such phenomena, it is undoubtedly helpful to have a concise way of describing the relationship between these elements.

TECHNICAL USAGE

The problem with the general usage of this term is that “recursive” is used elsewhere in political science with a different meaning to describe certain types of simultaneous equations familiar to quantitative analysis. In recursive simultaneous equations, the dependent variable of one equation is an independent variable in another equation. That is to say, A affects B, but B does not affect A, as shown in the following equations:

(1)

(2)

The term “nonrecursive” is used to describe simultaneous equations in which A and B are mutually influential, as in the following equations:

(3)

(4)

This term is akin to what is implied in Giddens' usage of “recursive.” Whether this description is actually what Giddens means is discussed in the following section. Note that the nonrecursive/recursive dichotomy is specific to the domain of simultaneous equations.

This meaning of “recursive” (and “nonrecursive”) entered political science via econometrics, originally from mathematics (see Kleene Reference Kleene1935, 223). Its origins in mathematics are not relevant, except as they are antecedent to later usage by midcentury econometricians, who first used “recursive” to describe simultaneous equations. Writing in Econometrica, Wold describes models wherein (among other characteristics), “given the development of the exogenous variables and a set of initial values for the endogenous variables, the model allows us to calculate, recursively, the development of the endogenous variables” (Reference Wold1954, 172). Again in Econometrica, Strotz and Wold note “that a recursive, or causal chain, system has the formal property that the coefficient matrix of the non-lagged endogenous variables is triangular (upon suitable ordering of rows and columns) whereas a nonrecursive, or interdependent, system is one for which this is not the case” (Reference Strotz and Wold1960, 417). This distinction will be clear to students of simultaneous equations—but almost no one else. What this means, however, is exactly the distinction illustrated in the previous recursive equations; recursive (in the technical sense) models are not mutually influential, showing instead unidirectional causality. Previously, Bentzel and Hansen (1954–Reference Bentzel and Hansen1955) used the terms “recursive” and “interdependent” to draw the same distinction.

Recursive (and nonrecursive) modeling techniques soon passed to political scientists, especially those scholars concerned with quantitative methodology. Alker describes these techniques in volume two of his Mathematical Applications to Political Science, drawing a distinction between “hierarchical versus circular causation.” He calls models of the former “recursive systems,” but the latter are variously called “reciprocal” or “circular” systems (Reference Alker1966, 15).

Studies applying recursive models began to appear in political science journals soon thereafter; for example, in Reference Nie, Powell and Prewitt1969, Nie, Bingham Powell, and Prewitt employed recursive models in their study of participation. By 1977, recursive models were familiar enough that Sigelman could parody their use, describing this advanced technique for methodological success: “the Recursive Route Restrictor, which simplifies the analysis of complex causal systems by permitting only undirectional arrowdynamic [sic] causation” (Reference Sigelman1977, 304).

However, not all contemporaneous uses of “recursive” demonstrate quantitative sophistication. In their discussion of global diplomacy, Russett and Lamb write:

The parenthetical example illustrates precisely what Russet and Lamb mean by “recursive”—the mutual influence of two elements in a system, very much akin to Giddens' later usage. This is the only appearance of the term in the article, and Russet and Lamb do not use recursive (simultaneous equations) models in their analysis.

As econometricians in the 1950s began to adopt “recursive” to describe simultaneous equations, linguists also began employing the word. In the April–June Reference Bar-Hillel1954 issue of Language, Bar Hillel invoked “special cases of RECURSIVE DEFINITIONS [sic].” From there, the term was adopted by Chomsky, citing Bar-Hillel: “Bar-Hillel suggests that recursive definitions may be useful in linguistic theory; whether this turns out to be the case or not, I agree in this instance with the spirit of his remarks” (Reference Chomsky1955, 45). Chomsky's later work suggests that recursive definitions did prove useful, and a recent paper by Hauser, Chomsky, and Fitch defines recursion as the capacity to take “a finite set of elements and yield … a potentially infinite array of discrete expressions” (Reference Hauser, Chomsky and Fitch2002, 1571). It is possible, but far from certain, that Giddens adopted the term from Chomsky's work.

SIMULTANEITY

As noted, the technical meaning of recursive depends on the simultaneity of the processes described. It is this kernel that differentiates it from the general meaning—but only within the narrow domain of simultaneous equations. That is, if the process in question is not explicitly simultaneous, then the general usage is acceptable and perhaps preferable. The following equations provide a sketch of this distinction. Where the subscript t indicates time, mutual influence between two elements simultaneously can be illustrated as:

(5)

Mutual influence between two elements over time can be expressed as:

(6)

The arrows in these diagrams can indicate influence, constitution, or causality—however one wishes to characterize the interaction of two elements. In equation 5, the two elements are mutually influential. This is recursive in the casual sense, but clearly “nonrecursive” in the technical sense. In equation 6, the same interaction of the elements expands to account for the passage of time; so long as the interaction does not occur simultaneously, equation 6 depicts a recursive relationship in the technical sense.

The trick for political scientists is that the causal chain depicted in equation 6 may unfold at time scales that are difficult to measure by readily available means. In these cases, simultaneous equations provide a shortcut to describing the period-to-period variation in the phenomena of interest. Such phenomena might be strictly recursive while having the appearance of being instantaneously mutually influential—that is, “nonrecursive” in the methodological sense—because they unfold along time scales below the threshold of our ability to measure events. The simultaneity (or not) of the interaction makes all the difference.

This temporal aspect seems to be the source of confusion regarding usage of “recursive.” For example, Russett and Lamb suggest that their trade-consular dynamic is both recursive and a feedback mechanism, which is correct if they mean that the mechanism plays out over time (as in equation 6); however, their description of that dynamic makes it easier for a reader to infer simultaneity (equation 5) as their intended model. Likewise, if Giddens means that actors and activities are “continually recreated” over time—which does seem implicit—then his usage is correct, although perhaps not as specific as it could be.

Even if Giddens' usage is correct, it still unlikely that he has in mind the sorts of models that quantitative methodologists consider recursive. Thus, there might be two correct, slightly different uses for “recursive.” The problem then is not only that the word means something very specific for one subdiscipline and something else for another; if each subdiscipline recognizes the other's usage, they will still be able to communicate effectively. The situation would then be akin to the British and American usage of “billions”—merely an idiosyncrasy of the respective parties and something to watch out for. Instead, the real problem for political science is that neither party understands the nuance of the other's usage. Methodologists tend to assume that any nontechnical use of “recursive” is incorrect, while relatively few nonmethodologists realize that there is a technical use in the first place. This misunderstanding makes it difficult for each party to communicate their ideas across the methodological divide.