Tim van Gelder
7 April 1999
Over the past few decades the increasing use of dynamical models and dynamical systems theory has shed new light on many aspects of human cognition. Such is the scope and power of this research that it has become appropriate to talk of a distinctive new approach to cognitive science, the dynamical approach, standing in opposition to the orthodox symbolic paradigm ("GOFAI") which has traditionally dominated the field. In a recent paper  I cast this opposition as a dispute over the nature of the most fundamental "law of qualitative structure" describing cognition: cognitive agents are physical symbol systems (Newell & Simon), or dynamical systems, a dispute which will ultimately be resolved only through an extended slog in the trenches of detailed empirical research.
As we approach the end of the millenium, and a half-century of cognitive science, it is still far from obvious which (if any) of these approaches will emerge as a clear victor. Both approaches can marshal empirical evidence and philosophical support, and can point to apparent weaknesses in the other side. Teed Rockwell's paper "The Modularity of Dynamic Systems"  can be read as attempt to make new progress in resolving this issue. Taking Fodor's well-known thesis of the modularity of mind as his starting point, he observes
So far so good. Rockwell's bold project would certainly be cause for at least garden variety rapture. However I'm not convinced that it can be carried through. The trouble sets in as soon as we try to get a precise fix on Rockwell's proposal. There appears to be a large equivocation at the very heart of the paper. We can see this by asking a simple question: What exactly is the correspondence proposed in the quasi-reduction of classical modules to DST? It sometimes appears that each classical module is to be replaced by something like a dynamical system. Other times, it appears that the primary correspondence is at a lower level, between something like "LISP primitives" and dynamical entities. Unfortunately neither alternative offers much prospect of rapture.
Consider first the mapping of basic symbolic operations or primitives onto bifurcations within a dynamical system, or in other words, the quasi-reduction of LISP to dynamical systems theory. The general idea that symbol processing might be implemented in dynamical stuff is not new. Indeed, it is conventional wisdom that symbol processing "bottoms out" in the dynamical properties of the computer hardware (although nobody I know can actually describe the details).
A more direct correspondence between a high-level language such as LISP and aspects of dynamical systems would be more interesting, though at the end of the day Rockwell has only illustrated what such a correspondence might look like with an informal example. But supposing this kind of quasi-reduction has been achieved: what does it tell us about how classical modules might be replaced with dynamical alternatives? Cognitive modules, of roughly the kind described by Fodor, and the basic operations of a language like LISP are very different fishes; or rather (to turn a rhetorical flourish into an overstretched metaphor) if "left(x)" is a minnow, a module is a whole school. Reducing the parts of a complex entity is not yet reducing the whole, and it leaves open the question: to what, in the world of dynamics, does a classical module correspond?
That leaves the former approach, wherein a given module is mapped directly onto some kind of dynamical entity. The idea is intriguing enough, but here the precise nature of Rockwell's proposal is very hard to pin down. At some points he seems to be suggesting that classical modules be mapped onto attractors; at others, onto dynamical systems; and at others, state spaces. These are importantly different kinds of things, and for the suggestion to have any plausibility it has to be clear (a) which kind is chosen, and (b) to choose the right one. To my way of thinking the idea that the module for low-level visual perception (if there is one) may be quasi-reduced to some complex dynamical system is an interesting speculation; the idea that it be reduced to a state space is an ontological howler.
These are pessimistic observations, and
I certainly hope that in reality they fail to do justice to Rockwell's
proposal. As a card-carrying dynamicist, I'm committed in advance to the
idea that a semi-eliminative reduction to a DST story will eventually account
for whatever elements of truth the Fodorian modularity story happens to
contain. Rockwell's bold speculations may yet turn out to be an important
part of that account, but to achieve the desired phase transition he'll
need to tune the parameters a little more. 
 van Gelder, T. J. (1998) The dynamical hypothesis in cognitive science (with commentaries). Behavioral and Brain Sciences, 21, 1-51.
 Newell, A., & Simon, H. (1976) Computer science as empirical enquiry: Symbols and search. Communications of the Association for Computing Machinery, 19, 113-126.
 Thanks to Tony Chemero for discussion
which helped shape this commentary.