1. Introduction.
Work by a host of philosophers over the last three decades has bequeathed us two distinct conceptions of biological function, causal role (CR) functions and proper etiological or evolutionary (ER) functions, each with a menu of bells and whistles. While some have suggested that a unification is possible, none is at present available. Those who use functional notions in practice move back and forth between the conceptions, often inferentially. It is with such inferences that I am nominally concerned. I say nominally because I think attention to them illuminates the importance of modularity, at least for those interested in the evolution of behavior, and this, as much as anything else, is my aim here. In any case, the points I wish to address concern functions of each kind with a particular set of bells and whistles, outlined below. If those bells and whistles exclude functional categories about which you care, or include too many, no matter: my concern is with inferential practice rather than proper definition.
Functional claims are causal claims, of a sort. I know how to think about causal inference only when causal relations are held to obtain between variables; the particular set of bells and whistles deployed below allows translation of talk about the function of objects or properties into talk of the function of variables and their values. I claim no originality for the ideas; the causal background is in Spirtes, Glymour and Scheines (2000), and nearly all of the ideas about function and modularity can be found or are intimated somewhere else, e.g. Faber (1984) and Magwene (2001). The presentation is of course mine, as are one or two extrapolations.
I take a causal system to be a causal structure over variables and a set of mathematically expressible dependencies between variables in that structure. Sets are represented by bolded letters, variables are italicized, and variable values are capital letters in plain font. Variables are taken to be measured on individual units in some population of units; a causal system over variables characterizes one or more units in the population. Causal structures are represented by directed graphs. Directed edges between nodes represent asymmetric relations of direct causation, relative to the set of variables in the graph. If there is a directed path from V to Y (a sequence of edges, all lying head to tail, out of V and into Y), then V is a cause, simpliciter, of Y. It is assumed that the joint probability density over S factors according to the Causal Markov and Causal faithfulness conditions.
2. Causal Roles with Bells.
Many advocates of CR functions have been content that a variable should have a function just in case it has an effect. I am less generous, if only because the inferences with which I am concerned involve CR functions in a specific kind of causal system. Graphically, a variable is functional only if it is on a cycle, i.e. a directed path from the focal variable back to itself (Fig. 1).
[if !vml]
[endif]
If a directed graph is diachronic, variables will be time-stamped. Suppose two time stamped variables Vt and Vt’ have values that represent an identical range of properties of a unit, and are measured at distinct times t and t’. Then we may say that the values of Vt and Vt’ represent the same variable V measured at different times. Say that a t-cycle is a directed path in a diachronic graph between two such time-stamped variables (Fig. 2). T-cycles and cycles share certain features that will be important in what follows, so though they differ in others I will call both cycles.
[if !vml]
[endif]
Systems with cycles form an interesting class because they can generate peculiar phenomena, among them: one or more focal variables can equilibrate, or may exhibit a cyclical pattern of values over time. Either phenomena may be phase-dependent, e.g. the focal variable exhibits different cyclical patterns in different contexts. For a name, call either sort of phenomena patterned behavior. For another, say that a system S of variables is an economy if 1) the associated causal graph is connected, 2) there is at least one cycle in the graph, and 3) there is some variable P exhibiting patterned behavior P in S. An example is given in Fig. 3.
[if !vml]
[endif]
An n-function is specified by the causal role of a variable V on some cycle in an economy E producing patterned behavior P. Exactly what a ‘role’ is depends on context and interest, but may include at least the following: the immediate causes and effects of V, the mathematical form of the dependencies between V and its immediate causes and effects, and the set of cycles in E on which V occurs.
A single variable may be connected to itself by several cycles. If a variable lies on two or more cycles, the cycles may have no variables in common other than the focal variable, or they may share several variables in common. If the cycles are nested, as in Fig. 4, I will take them to be part of a single economy. If cycles are not nested, and each
[if !vml]
[endif]
cycle generates a distinctive patterned behavior, as in Fig. 5, then I take it to be a matter of choice whether we diagnose one economy defined over variables in all cycles, or a distinct economy for each choice of cycle and patterned behavior. But, if the latter choice is made, cycles not included in a given economy will constitute constraints on the functional organization of that economy. E.g., if in Fig. 5 an economy E is defined over S={V, M1, M2, P}, then the economy E’ defined over S’={V, I1, I2 , P’} constrains E.
[if !vml]
[endif]
3. ER Functions with Whistles.
Individual units in a population may differ in the causal system over some set S of variables so that the population includes a mixture of units with different economies over S. The economy over S for a unit may then be regarded as the value of a non-quantitative variable E defined over units in the population, with values E1,…En, each representing a specific causal system over S. If in such a population E is a cause of survival and reproductive success (hereafter, SRS) through P, then an n-function of V with respect to E and P is also an evolutionary function (hereafter an e-function) of V provided the usual further conditions are satisfied. Units characterized by the variables in S must reproduce; the value of E for a unit must be heritable; there must have been variation in E in the lineage to which a unit belongs; this variation must imply variation in fitness among units; and it must be that the fitness of units with E=E was higher than that of units with different values of E.
Given this definition, it will not follow from the fact that V has an e-function in E with respect to P that this e-function is evolved. E can be selected for against alternative economies even if the causal role of V is identical in all instantiated values of E. Put differently, that V has an e-function in E with respect to P does not imply that V’s n-function in E is optimized, or even improved, with respect to the influence of P on SRS. V’s n-function may fail to be optimized for trivial reasons, e.g. the optimal variant never occurred, or for structural reasons. Among the structural reasons is the possibility that V has an effect on SRS that is not mediated by P. A special case occurs when V is in distinct but not alternative economies, both influencing SRS through different patterned behaviors. The secondary n-function of V in such cases represents a constraint on the evolution of E.
In what follows some terminology will be helpful. Consider the graph (Fig. 6) of two economies below. Let us call any variable which is in both economies a shared variable (here, V), any variable that is on a cycle that contains a shared variable a constrained cyclical variable (M1, M2, I1 and I2), any variable that is a cause of a
[if !vml]
[endif]
constrained cyclical variable a constrained non-cyclical variable (C1 and C2), and any other variable an unconstrained variable (C3).
4. Inferences.
Space prevents any sustained discussion of inference from n-function to e-function. But it is important to note that the strongest available inferences, inferences that can be formalized and quantitatively assessed with respect to reliability, are inferences from the fact that some evolutionary force will tend to remove a trait from population to the claim that the trait is e-functional. But such inferences do not permit a characterization of the relevant e-function. Inferences to more fully characterized e-functions typically depend essentially on a characterization of current n-functions, with the strength of the inference and the specificity of the inferred e-function varying directly with the specificity of the known n-function. As a consequence we are commonly in a position to diagnose at most only the e-functionality of traits involved in generating patterned behavior.
My concern here is with inferences from e-functionality to n-functions. It has seemed plausible to some that evolutionary considerations can illuminate the n-function of a trait. The idea seems to be that even if we do not know the exact e-function of a trait, we can sometimes plausibly infer that the trait is e-functional in virtue of its influence on some patterned behavior P=P. Supposing the trait is e-functional, we can then ask what the causal role of that trait would be in an economy that optimized P with respect to SRS. An answer tells us something about the causal structure characterizing the economy generating P, and about the n-function of the trait in that economy. Various extensions are possible, and occur in the literature. Arguments of these kinds nowadays nearly always presuppose modularity, and for good reason. Modularized phenotypes are supposed to be subject to independent evolution under natural selection. It is only given such independence that the assumption of optimization is at all plausible.
Reasons to doubt the cogency of such arguments are legion. I wish to focus on just one: optimization requires the absence of constraint, but standard tests for modularity are not tests for the absence of constraint. Let P be the optimal value of P with respect to SRS; let E be an economy that produces P=P with maximal efficiency; and let V be some variable in E. E will necessarily be selected for over alternatives only if every path from V to SRS goes through P. Any such path that is not mediated by P imposes a constraint on the evolution of E: the optimal value of E is that which maximizes fitness, and this in turn depends on all paths from an economy to SRS. The causal role of V in E will have been selected to optimize P with respect to SRS only if V itself is unconstrained. In particular, if V is a constrained cyclical variable, e.g. is on cycles in distinct economies E and E’, then the causal role of V may be relatively stable under selection on P. Changes in E which involve new causes of V will modify the dependence relations between V and its causes in E’. So if E’ is the optimal value of E’ relative to P’, these changes have a fitness cost not born by values of E that leave the causal role of V intact. Further, since the effects of a variable on a cycle are its own (future) causes, this cost is born by values of E that change the effects of V, when those effects are on the cycle between V and itself.
This is not to say that constrained cyclical variables are immune to evolution (indeed, there is evidence that shared cyclical variables are especially liable to evolved change, c.f. Fraser, 2005). It is to say that such variables do not evolve in ways that optimize their role in any particular economy. And it is to say further that if economies containing such variables do optimize the patterned behavior they produce, much of the evolution by which they came to optimize this behavior will have involved changes in the causal role or value of unconstrained variables. There are then two relevant notions of ‘independent evolvability’. Economies are independently evolvable in a weak sense if natural selection can lead to the evolution of changes in one economy without also leading to changes in others (the ‘quasi-independence’ of Lewontin, 1978). This requires only the presence of at least one unconstrained, unshared variable in the target economy. Economies are independently evolvable in another, strong, sense if they contain no constrained variables. The arguments from e-functionality to n-function glossed above presuppose the strong sense of independence. So the question is whether standard tests for modularity can establish anything like such independence. As it happens, they cannot.
I set aside tests employing genotype-phenotype maps (e.g. Mezey et al. 2000) and functional specialization. The first (and much the best) fails to consider the functional role of phenotypes in systems of phenotypic variables, and the second reduces to the more standard test of dissociation. I take it that two phenotypes are dissociated if there is some context in which they vary independently of one another. Double dissociation occurs if there is some context in which the first phenotype can be modified without a concomitant modification in the second, and some other context in which the second can be modified without concomitant variation in the first. Suppose the phenotypes in question are patterned behaviors P and P’, generated by economies E and E’. P and P’ are dissociable just in case there is some intervention on some variable that modifies P, so P¹P, without also modifying P’, so P’=P’. That is possible so long as E contains any one unshared, unconstrained variable. Double dissociation requires only that E’ also contain an unshared, unconstrained variable. Clearly, dissociation is really bad evidence for the claim that E, and the role of any arbitrary variable V in E, are unconstrained by E’. Hence, dissociation is also really bad evidence for strong independent evolvability.
Tests for modularity, then, do establish independent evolvability in the weak sense. Dissociation establishes the presence of an unconstrained variable; that variable and its causal role are subject to evolution by natural selection, and so dissociated modules can be modified by natural selection without changes in other economies. But standard tests, and dissociation in particular, do not establish independent evolvability in the strong sense, which requires the absence of constraint. So much, perhaps, differs from common knowledge only in the causal details by which constraints are characterized. I rehearse the result because I think the details bear further consideration.
5. Conclusion: Independence, Modularity and Function.
I do not know whether some particular conception of modularity will turn out to adequately serve the interests of theoreticians and experimentalists, or whether, as with functions, no univocal notion will do. I suppose that any adequate notion will require at least the weak sense of evolutionary independence. I do claim, however, that conceptions of modularity that do not also require independence in the strong sense will not underwrite inferences to optimization. I also claim that dissociation is an inadequate test of modularity if modularity is taken to underwrite inferences to optimization. I wish to suggest something further.
Modules can be defined so that they are independent in the strong sense. Suppose we say that variables have n-functions relative to modular economies, and take a set S of variables to be modular relative to a larger set U of variables if S is connected and there is some proper subset N of S such that conditioning on N renders every other member of S probabilistically independent of every member of U not in S. Then there will be no shared cyclical variables in modular economies. However, the economies operative in modules so defined may generate more than one selectively relevant patterned behavior. To see why, consider Fig. 6. There are here two economies, E and E’, but only one modular economy, E+E’. This is because the set S={V, C1, C2, C3, M1, M2, P} over which E is defined is not modular: if V is omitted from N then I2 and V are associated conditional on N; if V is included in N then I2 and M2 are associated conditional on N. To induce the relevant independence relations, we must collapse economies that share cyclical variables. Consequently, V has an n-function in E+E’ relative not only to P and P’ individually, but also to their conjunction, which is itself a patterned behavior. It is this n-function, and no other, that is identical with its e-function: the e-function of V in E+E’ must be defined with respect to the optimal tradeoff between optimal values of P and optimal values of P’.
I think this is revealing. In order to validate the assumption that evolution optimizes a given behavior we must define modules so that they are strongly independent, else the possibility of evolutionary tradeoffs between otherwise optimal behaviors threatens to undermine the assumption that evolution optimizes any one of those behaviors. But modules so defined may potentially generate several evolutionarily relevant behaviors, internalizing, as it were, the tradeoff. And realizing that tradeoff is, in some sense, the function of cyclical variables shared by distinct non-modular economies operating within the larger modular economy.
So my suggestion. Dissociation is essentially a test for presence of unconstrained variables in an economy. Insofar as the notion of independence used in conceptualizing modularity is tested by dissociation, a focus on modularity is a focus on the causal role of such variables. But if one is interested in the evolution of behavior these are perhaps the least interesting of variables. As the above definition of modularity reveals, it is shared variables rather than unconstrained variables which embody tradeoffs between components of fitness. Characterizing those tradeoffs is, of course, one of the really hard problems in the evolution of behavior, e.g. the evolution of life-histories. So for the evolutionary behaviorist, at least, the action would appear to be in shared cyclical variables. If modularity is not useful for illuminating their evolutionary role, it may turn out not to be so terribly interesting after all.
References
Faber, R. (1984): “Feedback Selection and Function: A Reductionistic Account of Goal-Orientation”, in Methodology, Metaphysics and the History of Science, R. Cohen and M Wartofsky (eds.), pp 43-136, D. Reidel Publishing, Dordrecht, Holland.
Fraser, H. (2005): “Modularity and evolutionary constraint on proteins”, Nature Genetics, 37 (4):351-2.
Lewontin, R. (1978), “Adaptation”, Scientific American, 239(3); 156-159.
Magwene, P. (2001): “New Tools for Studying Integration and Modularity”, Evolution, 55(9):1734-1745.
Mezey, J., J. Cheverud and G. Wagner (2000): “Is the Genotype-Phenotype Map Modular?: A Statistical Approach Using Mouse Quantitative Trait Loci Data”, Genetics, 156:305-311.
Spirtes, P. C. Glymour and R. Scheines (2000): Causation Prediction and Search, 2nd edition, MIT Press, Cambridge, Mass.
