What Might Cognition Be, If Not Computation?

structure of the target phenomenon, for various reasons researchers in this approach standardly provide a concrete model: an actual computer programmed so that (hopefully) it realizes the same (or a relevantly similar) abstract computational system as is realized by the cognitive systems under study. If the concrete model appears able to perform actual cognitive tasks in much the way people do, then the hypothesis that people are such systems is supported. One reason to provide a concrete model is that the abstract systems themselves are too complex to be studied by purely analytical means. In order to determine whether the model has the right properties, the theorist lets a concrete version run from a variety of starting points (initial conditions), and observes its behavior. Another reason for providing a concrete model is that, given the complexity of the abstract systems, it is very difficult actually to discover that structure except through an iterative procedure of constructing a concrete model, testing it, making improvements, and so on. 2. The dynamical hypothesis. Recall that one suggestion coming out of the discussion of the centrifugal governor was that an interesting alternative to the computational conception is that cognitive systems may be dynamical systems. In order to characterize this position as an alternative within the current framework, we need a definition of dynamical systems as a subcategory of state-dependent systems, a definition which is as useful as possible in clarifying differences among various approaches to the study of cognition. The centrifugal governor is a paradigm example of a dynamical system. Perhaps the most pertinent contrast between it and the computational governor is that the states through which it evolves are not configurations of symbols but rather numerically measurable Is See Marco Giunti, Computers, Dynamical Systems, Phenomena nd the Mind, Ph.D. Dissertation (Indiana University, 1991). 368 THEJOURNAL OF PHILOSOPHY arm angles and rates of change of arm angle. Generalizing this feature, and, of course, looking over the shoulder at other textbook examples of dynamical systems and the kind of systems that are employed by dynamicists in cognitive science, we can-define dynamical systems as state-dependent systems whose states are numerical (in the abstract case, these will be numbers, vectors, etc.; in the concrete case, numerically measurable quantities) and whose rule of evolution specifies sequences of such numerical states. The rule of evolution in the case of the centrifugal governor was a differential equation. In general, a differential equation is any equation involving a function and one or more of its derivatives; informally, for current purposes, it can be thought of as an equation that tells you the instantaneous rate of change of some aspect of the system as a function of the current state of other aspects of the system. Since our interest is in cognition as processes that occur in time, we assume that the function is one of time (for example, @(t)) and that any derivative involved is with respect to time (for example, d9/dt). Because differential equations involve derivatives, they presuppose continuity; hence the "time" set in an abstract dynamical system is standardly R the real numbers. Dynamical systems governed by differential equations are a particularly interesting and important subcategory, not least because of their central role in the history of science.16 But dynamical systems in the general sense just defined might also be governed by difference equations, which specify the state of the system at time t + 1 in terms of its state at time t: