Tim van Gelder

Cognitive science has always been dominated by the idea that cognition is computational in a rather strong and clear sense. Within the mainstream approach, cognitive agents are taken to be what are variously known as physical symbol systems, digital computers, syntactic engines, or symbol manipulators. Cognitive operations are taken to consist in the shuffling of symbol tokens according to strict rules (programs). Models of cognition are themselves digital computers, implemented on general purpose electronic machines. The basic mathematical framework for understanding cognition is the theory of discrete computation, and the core theoretical tools for developing and understanding models of cognition are those of computer science.

The mainstream computational approach has been highly productive. Hundreds, perhaps thousands of models have shed light on a great many aspects of cognition. In some theoretical treatments, cognitive science is simply identified with the project of understanding the mind as a kind of digital computer. However, cognitive science has always contained an alternative stream of research, wherein cognition is conceived, in the first instance, as dynamical. Here cognitive agents are taken to be dynamical systems; the models are dynamical systems specified by mathematical equations; and the core theoretical framework for developing and understanding models of cognition is that of dynamics (dynamical modeling, and dynamical systems theory).

The dynamical approach maintained a rather low profile until the mid-1980s, when it suddenly emerged, under the umbrella of connectionism, as a serious competitor to the dominant computational paradigm. In the past decade, dynamical theorizing has gone from strength to strength. Dynamical models can now be found in most areas of cognitive science, and are rapidly increasing in sophistication. It now seems fair to say that the single most important theoretical chasm in cognitive science is not behaviorist versus cognitivist, nor bottom up versus top down, nor even connectionist versus classical-it is dynamicist versus computationalist. The essence of the computational approach is often taken to be encapsulated Newell & Simon's famous dictum, the Physical Symbol System Hypothesis:

The essence of the dynamical approach to cognition can also be encapsulated as a law of qualitative structure, one that is similar in form but fundamentally different in content. the dynamical hypothesis (DH):

When the DH is proposed in this blunt form, there is a common tendency-especially among committed supporters of the CH-to adopt some kind of instantly dismissive attitude. It is claimed that the DH is obviously misguided-and then any of a number of standard objections are trotted out. These objections fall into two main categories. In the first are those purporting to show, on purely conceptual or theoretical grounds, that the DH is not really a genuine alternative to the CH. In the second category are objections which grant that the DH is a genuine alternative, but insist that it is manifestly hopeless; its empirical failings are already determined and plain to see.

The primary aim of this paper is to rebut a representative selection of these objections. There are two major reasons for undertaking this project. The first is to provide a partial defense of the DH as an open empirical alternative to the CH. As J.S. Mill put it, "three-fourths of the arguments for every disputed opinion consist in dispelling the appearances which favour some opinion different from it" (Mill, 1975). The second reason is that the DH, as expressed above, stands in urgent need of elaboration and clarification. It is not yet clear exactly what it means to say that cognitive agents are dynamical systems. The standard objections put pressure on any defender of the DH to say exactly what the idea amounts to.

The standard objections can come across as quite plausible. They generally contain important elements of genuine insight. However, on both sides of the debate there is a great deal of conceptual murkiness-confusion, vagueness, ambiguity, and even error. The objections take advantage of this murkiness to transform their insights into powerful but misdirected attacks. Rebutting them demands that we do some basic philosophical spadework, imposing a little order onto the conceptual terrain. Thus, by the end of the paper, an expanded and sharpened version of the hypothesis will be on the table.

Everything is a dynamical system. Cognitive agents must be dynamical systems at some level. The DH is trivially true, and so cannot amount to an interesting empirical alternative to the CH.

The "trivially true" objection is perhaps the most frequent of all. This is somewhat surprising, since anyone with any experience in the area knows that dynamical research is far from trivial. Something has to be wrong with the objection-and yet there is something to be said for its major premise. To deal with this objection we must directly confront some basic philosophical issues concerning the DH. What are dynamical systems? And what are we saying when, in a scientific tone of voice, we assert that cognitive agents are dynamical systems?

One might naively suppose that answering the first question is easy: just look up the textbook definition. But things are not that simple. It turns out that the meaning of the term "dynamical system" varies very widely, both across disciplines and through time. Table 1 illustrates this variety. The table carries an important lesson for anyone engaging in these philosophical debates. There is no one true account of the nature of dynamical systems; there is nothing that dynamical systems "really are." To put the point dramatically, we as a community don't really know what we're talking about when we refer to dynamical systems. This confusion is one of the major causes of controversy, much of it pointless.

Despite the lack of overt agreement, there is among dynamicists in cognitive science a reasonable level of intuitive consensus over the nature of the systems that they study and which strike them as relevantly different from digital computers. To articulate this consensus, we need to start with the notion of a system. In the current context, a system is a set of variables (things that change over time) which are interdependent-each one affects, and is affected by, others. There is a crucial ontological difference between objects and systems associated with them. Consider, for example, the solar system. On one hand we have the sun and planets-vast conglomerations of gases, rock, etc. On the other hand, we have the solar system of classical mechanics, which is the set of positions and momentums of those bodies. These are obviously not identical. Rather, the positions and momentums are properties or aspects or features of the material objects. In what follows, the term "instantiation" will be used for the relation between some object and any system whose variables are features of that object.

What makes a system dynamical, in the relevant sense? It turns out that the unifying theme in dynamical cognitive science is that dynamical systems are quantitative. Informally, they are systems in which distance matters. There are distances between states of the system, and distances between times, and these distances are relevant to the behavior of the system. We can talk usefully about how far it is from one state to another, and how long elapses between one time and another. This makes it possible to talk of rates of change, and of systems whose behavior is described in terms of rates of change (e.g., by differential equations) and even of systems whose variables include rates of change of other variables (systems governed by higher-order differential equations).

For example, one of the most well-known dynamical models in cognitive science is the system governed by the Haken-Kelso-Bunz equation

We can see immediately that it is simply not true that everything is a dynamical system. Many things-rocks, for example-are not sets of variables (though they may instantiate many different sets of variables). Most sets of variables are not interdependent, and of those that are, only some are quantitative. Therefore, the DH is not at all trivial; it asserts that cognitive agents belong (in some sense) to a special subset of all the things in the world.

More needs to be said on this topic, however. We can begin by resolving one apparent difficulty. If dynamical systems are quantitative systems, and systems are sets of variables, how can a cognitive agent-which is surely not a set-be a dynamical system? The claim seems like an ontological howler. This difficulty can be avoided by unpacking the simple slogan a little more. First, the "are" in the slogan is not that of identity, but rather that of instantiation. Second, the claim is not that each cognitive agent "is" some one dynamical system, but rather that it is many-as many distinct systems as are required to exhibit its various kinds of cognitive performances. Putting these points together, to say that Rover the dog is a dynamical system is to say that he can recognize a whistle because there is a set of variables (aspects of his auditory system) whose interdependent change constitutes whistle-recognizing behavior; and he can bark because there is another set of variables whose interdependent change constitutes barking behavior; and so forth. In short, to say that cognitive agents are dynamical systems is to say that for every kind of cognitive performance they exhibit, they instantiate some dynamical system whose behaviors constitute the cognitive performances of that kind.

Next, there is more to the DH than just a strictly ontological claim about the kinds of things cognitive agents are. The reason the DH is scientifically interesting is that it claims we can be understood as dynamical systems-that, in other words, the best way to understand what we are and how we work is to describe us as dynamical systems, using the theoretical framework of dynamics. The DH in this sense is obviously far from trivial. Developing a good dynamical model is hard work in any branch of science, let alone cognitive science. Anyone who thinks otherwise is invited to try their hand at describing some interesting aspect of cognition-natural language comprehension, for example-in dynamical terms. Some of the greatest achievements of natural science have amounted to describing some natural phenomenon as the behavior of a dynamical system, and this presents no less of a challenge in cognitive science than anywhere else. Indeed, the fact that dynamical cognitive science is a relative late-comer to the scientific scene might even be superficial grounds for supposing that a dynamical science of cognition is more difficult to achieve than dynamical sciences in other domains. Newton cracked the nut of the solar system-why didn't he also crack the nut of human decision making behavior?

In short, the DH is trivially true for two reasons. First, it makes a quite specific claim about the nature of cognitive agents (that their cognitive performances are the behavior of quantitative systems they instantiate). Second, it makes a strong claim about cognitive science (that it should take dynamical form). Both these sides of the hypothesis may well turn out to be false.

Ordinary electronic computers are dynamical systems. In general, digital computers are dynamical systems as well. The DH cannot maintain that cognitive agents are dynamical systems as opposed to digital computers. It is therefore not an interesting alternative to the CH.

"Computers are dynamical systems"-this seductive thought causes considerable philosophical grief. There are really three distinct lines of thought tangled up here. When pulled apart, their shortcomings become apparent.

The first line of thought is that digital computers have states, and they change states over time in accordance with some rule, and therefore they count as dynamical systems, in the mathematical sense. Therefore, the mainstream CH already asserts (at least implicitly) that cognitive agents are dynamical systems, and so the DH as formulated is not an interesting alternative. At best, the DH must assert that cognitive agents are dynamical systems of a rather different kind than digital computers, but then it must tell us what kind that is.

After the discussion in the previous section, it should be easy enough to see the problem with this objection. Certainly, digital computers are dynamical systems according to very broad defintions. However the DH is framed in terms of a much more specific definition. It asserts that cognitive agents are quantitative systems. Digital computers do not automatically count as dynamical systems in this sense, just because they have states and evolve according to some rule.

At this point the baton is passed on to the second line of thought. This claims that digital computers really are dynamical in the relevant sense, i.e., quantitative systems. Why? Because there are certain trivial metrics that apply to any set of values whatsoever (see, e.g., Padulo & Arbib, 1974, pp.91-2); therefore, there will always be a way of measuring distances between states and times in the case of digital computers. Further, there are certain not-quite-so-trivial metrics, such as perhaps the Hamming metric, which apply nicely to states of digital computers specifically, giving an even more robust sense in which distances apply to states of computers.

These claims about metrics and digital computers are true. However, it still has not been shown that digital computers really are dynamical systems. This requires not merely that there are distances, but that these distances are systematically related to the behavior of the system. This is precisely what is not (in general) the case with digital computers. Such machines typically bounce around their state spaces in ways which are entirely erractic from the point of view of any measurable distances. This is why, in practice, computer scientists don't bother talking about the distances between states, or about positions in state spaces, or about rates of change, etc.. The non-dynamical nature of digital computers demands a very different (though also very powerful) set of conceptual tools.

Now, I don't know of any proof that distances between states are never systematically relevant to behavior in digital computers. There may well be contrived or coincidental cases in which it is possible to describe the behavior of a digital computer in terms of how far it changes from any given state. However, this would show, at most, that some systems count as both digital computers and dynamical systems. Such cases would be interesting, but they wouldn't demonstrate that digital computers generally are dynamical systems. The basic point here is that digital computers and dynamical systems are two classes of systems defined by different criteria. These classes may well overlap at some points, but they are nevertheless generally distinct.

The third line of thought takes a rather different tack. It takes off from the idea that every actual digital computer is, in some sense, part of the physical world, and every physical thing is (supposedly) made up of the basic dynamical stuff which constitutes the universe. Thus ordinary digital computers, such as the one I am using to write this paper, are very complex conglomerations of electronic circuitry, and at that level are (in principle at least) completely amenable to dynamical description. Therefore, digital computers "are" dynamical systems already.

We have to tread very carefully here. In particular, we must distinguish three kinds of relations-instantiation, identity and implementation. A chunk of the world, such as a mass-spring, a Macintosh, or a brain, can instantiate many different systems at the same time. In the case of the Macintosh, one of those systems is a low-level electronic system. Another is a high-level digital computer. The variables making up these systems are all features or properties of one and the same Macintosh object, but the systems are not identical with that object. Neither are they identical with each other. Systems are sets, and identity between sets requires having exactly the same members. However, the variables in the electronic system are different than the variables in the digital computer system. Nevertheless, it is only because the electronic system behaves the way it does that there is a high-level digital computer, and so the former can be said to implement the latter.

More generally, while it true that whenever we have some concrete digital computer there will be dynamical systems intimately associated with it, there is still an important theoretical and indeed ontological distinction to be drawn between these systems. High-level systems, resulting from "carving up" the world one way, are not the very same things as low-level systems which result from "carving up" the world according to quite different principles.

In sum, as long as we retain sufficient conceptual precision, it can be seen that the simple idea that digital computers are dynamical systems is either wrong or much too simple. Of course, if we relax our grip on the key conceptual tools, all bets are off. Under those conditions, however, anything is sayable and nothing is contestable. We have moved outside the realm of productive philosophical debate.

Much recent research in computation theory has been exploring the computational power of dynamical systems. There is no inherent conflict between dynamics and computation, and so it is nonsense to think of cognitive agents as dynamical systems as opposed to computers.

The major premise of this argument-that dynamical systems can be understood as computing-is true. It does not follow, however, that the DH is not an interesting alternative to the mainstream CH. This time, in order to sort things out, we need to pay more attention to the notion of computation, and the specific nature of the CH.

What is computation? In the most general sense, it is simply the systematic transforming of things of one category into things of another. Informally, we can think of computers as devices which provide "answers" for our "questions." We set the computer running with an input object or start state of some kind (a question), let it do its thing for a while, and then eventually it delivers an output object or final state of some kind (the answer). The function computed by this device is simply the total set of question/answer pairs.

In this very general sense, just about anything can be interpreted in some more or less trivial way as a computer. The notion of computation only starts to get interesting when we place some constraints on the kinds of processes which deliver answers. Classical computation theory has focused on what are known as effective processes. These are processes which amount to a finite number of distinct operations governed by a finite recipe (an algorithm). Any device which carries out effective computation must be digital (so we can count the operations), and so is called a digital computer.

Now, the mainstream CH is the bold claim that cognitive agents are, not computers of some kind or other, but rather digital computers specifically. The exciting idea is that cognition is effective computation.

Classical computation theory, by now the better part of a century old, is well-worked territory. In recent years, mathematicians and computer scientists have increasingly been turning their attention to non-classical computation-i.e., processes which systematically transform questions into answers, but do not satisfy the criteria for effective computation. An obvious place to look is at the computational capacities of one class or another of dynamical systems. Thus we find that in various places and in various ways people are investigating dynamical systems as computers. Some fascinating theoretical results have been achieved in this area. In particular, it has been demonstrated that dynamical systems can have "super-Turing" capacities, i.e., can compute more functions than any digital computer. Such results have naturally attracted speculative interest from cognitive scientists, baffled by the problem of how the brain manages to do such extraordinary things.

Thus, it is certainly true that dynamical systems can be seen as computers, and there is indeed no inherent conflict between dynamics and computation. It does not follow that dynamical systems are computers in the sense that matters for the CH. In fact, the reason research into the computational powers of dynamical systems is interesting is that these systems are not, in general, digital computers. The DH maintains that cognitive agents are, fundamentally, dynamical systems. It does not maintain that they are not computational. It can allow that they can be computers. It happens, however, that in general dynamical (quantitative) systems are not digital computers and vice versa. Therefore, the DH stands as a genuine alternative to the traditional CH-the one that has been the foundation of mainstream cognitive science for around four decades.

The most famous and influential of all critiques of the mainstream computational approach to cognition is surely What Computers Still Can't Do (Dreyfus, 1992). In that book, Dreyfus noted that brains might well be turn out to be "analogue" rather than digital computers. Similarly, as Churchland and Sejnowski have argued at length, biological neural networks can be understood as computing in ways that differ fundamentally from ordinary digital computation (Churchland & Sejnowski, 1992). Like these perspectives, the DH can embrace the idea that cognitive processes are computational, while preserving a contrast with the CH. This does not diminish but rather fortifies the DH, by allowing it incorporate computational ideas without inheriting orthodoxy's excess baggage. However, it must also be stressed that as far as the DH is concerned, the notion of computation is an optional extra. Dynamical systems can be interpreted as computers, but this may not be helpful when understanding how they are responsible for cognitive functions.

There is no good reason to think that any cognitive process is non-computable. Even if cognitive agents are dynamical systems, they will still be computable. Dynamical systems offer nothing outside the scope of digital computation. Therefore, it is misguided to present the DH as an alternative to the CH.

The DH maintains that cognitive agents are dynamical systems as opposed to digital computers. Often, people interpret this as the claim that cognitive processes are somehow entirely beyond the reach of digital computation-i.e., that they are noncomputable, in the classical sense. This then provokes two strong reactions. One is to insist that there is no compelling evidence that cognitive processes are in fact noncomputable, and therefore requiring something other than a digital computer. The second reaction is to insist that dynamical systems are in general computable, and so even if some cognitive processes were in fact noncomputable, it would be a mistake to think that dynamical systems would be the cure. In short, both cognitive processes and dynamical systems are generally computable, and therefore the DH offers no alternative to the CH.

This objection is founded on subtle but critical confusion: blurring the conceptual distinction between computational and computable. These notions are as different as those of employer and employee. An entity is computational if it is a computer of some kind, i.e., if it carries out computation. An entity is computable if it can be computed, i.e., if some computer can carry it out or describe it. Something can be digitally computable even if it is not a digital computer. The DH asserts that cognitive agents are dynamical systems as opposed to digital computers, but is generally quite happy to allow that they are digitally computable.

This deserves a little unpacking. As mentioned above, classical computation theory is concerned with what can be achieved by effective processes, i.e., in a finite number of discrete operations specified by a finite rule. Any system which carries out effective computation is a digital computer. What gets computed , is, in the first instance, a function-a mapping from questions to answers. Early on, one of the central results of the field was established-that the class of effectively computable functions is equivalent to the recursive functions over the integers. This is a very large class of functions, but it also leaves out a great deal-for example, functions defined over real numbers. It didn't take long before computation theorists (including Turing himself) were asking what else might be effectively computable, in some suitably extended or generalized sense. The key notion here is that of arbitrarily good approximation. An entity counts as effectively computable if we can get as close to it as we like given enough effective computation. In this manner the purview of effective computation was gradually extended to embrace real numbers, functions over real numbers, differential equations, and so on (Earman, 1986; Grzegorczyk, 1957; Turing, 1936). Issues of effective computability can be raised for all the standard mathematical constructs of analysis and physics; what is and is not effectively computable rapidly becomes a rather complicated business (see, for example, (Pour-El & Richards, 1989)).

Now, we can regard a system as computable just in case its behavior is governed by some computable function. Currently, as far as we can now see, most if not all dynamical systems of practical relevance to cognitive science are effectively computable (in a theoretical sense). This fact in itself in no way detracts from their character as dynamical. Consider the solar system of classical mechanics-a paradigm example of a dynamical system. This system is computable, and might even be regarded as computational, i.e., a kind of analogue computer. However, it is not itself a digital computer. Similarly, the dynamical systems of interest in cognitive science can be both computable and computational, and the DH still remain a genuine alternative to the CH.

This concludes the discussion of objections of the "not a genuine alternative"genre. We turn now to those which maintain that the DH is obviously hopeless.

Dynamical models are at best descriptions of the data, and do not explain why the data takes the form it does. For explanation, we need computational models describing the underlying causal mechanisms.

Cognitive scientists of a computational bent often regard dynamical models of cognition with suspicion. They concede that a dynamical model really is different from a traditional computational model, but they doubt whether dynamical accounts actually explain anything. They regard dynamical models as really just dressed-up curve-fitting. Such models might succinctly describe certains kinds of change in cognitive processes, but they don't tell us anything about why those patterns of change take place as they do. For real explanation, we need to know about the mechanisms which drive the change-i.e., the computational structure.

There is a certain historical irony here. The Newtonian account of celestial motion was criticised for invoking gravitational "action at a distance,"a force whose operation could not be explained in terms of more basic or familiar mechanisms. Newton himself famously claimed to "frame no hypotheses" with regard to how bodies attract. Yet in time, classical mechanics came to be accepted as providing a genuinely causal explanation of celestial motion, rather just a description of it. The irony is that dynamical models in cognitive science are now being criticised on much the same grounds. The point that seems to have been lost is that good, high-level dynamical description is causal explanation-indeed, some of the most definitive causal explanation in the natural sciences takes this form. (If quantum mechanics provides the "rock-bottom" account of how the world works, then causal explanation is dynamical description even at the most fundamental level.) One cannot fairly criticise cognitive scientists for applying forms of explanation that are perfectly respectable elsewhere in science.

It may well be, of course, that any given dynamical model is in fact little more than a summary of data, and provides no genuine explanation of why things happen as they do. But the problem with such models is not that they are dynamical; it is that just that they fail certain criteria of adequacy which apply to all scientific explanations.

There is an important role for dynamical descriptions in any complete account of the nature of a cognitive agent, but they are invariably pitched at a level much too low to provide adequate accounts of cognition.

A common misconception about the dynamical approach is that it operates solely or primarily at "lower" or "micro" levels of description. In fact, dynamics is not intrinsically limited to any level or domain. In the natural sciences, dynamics finds application at all levels from quantum mechanics to cosmology. It gets its grip wherever sets of interdependently changing quantities are found. Similarly in cognitive science: dynamicists develop their explanations at the level of theoretical interest, whatever that might be.

One significant difference between the dynamical approach and PDP-style connectionism turns on this point. They agree that cognitive performances are behaviors of dynamical systems. The PDP approach, however, takes those systems to be high-dimensional neural networks operating at a level below that of orthodox descriptions (Smolensky, 1988); as expressed in the titles of the famous volumes, they constitute the microstructure of cognition. The dynamical approach is more catholic; it embraces dynamical models of all kinds and at all levels.

Dynamics is a general purpose framework which, when it applies, explains the behavior of systems regardless of whether they are cognitive or not. Dynamics does not focus on the specifically cognitive aspects of systems; it does not explain cognitive performances "as cognitive." One might be able to develop dynamical descriptions of cognitive agents, but these are not the sort of explanations required for cognitive science.

This objection concedes that dynamical explanations are non-trivial empirical explanations, and that they are genuinely different from computational explanations. It targets the nature of the explanation, suggesting that because the same kind of explanation might work just as well for the performances of non-cognitive systems, they cannot be satisfying explanations of cognition "as such." Genuine explanations of cognition would be framed in terms of distinctive features of cognitive agents of a non-dynamic kind.

At the heart of this objection is a deep misunderstanding about the DH. It takes that hypothesis to be asserting that cognition can be explained by using nothing but dynamical tools of the very same kind as are used in explaining other aspects of the world. In fact, the DH asserts that generic dynamics is a central part of the story about cognition, but by no means the full story. There is a direct parallel with the computational approach in this regard. The CH asserts that cognition is digital computation of a particular kind. Analogously, the DH asserts that cognition is dynamics of a particular kind. Computer science is not by itself a theory of cognition; it must be adapted, fine-tuned, supplemented with special resources, etc.. Analogously, dynamics is not by itself a theory of cognition; it must be adapted, fine-tuned, supplemented with special resources, etc.. In short, dynamics plays the same role in the dynamical approach as the theory of computation plays in the computational approach. The proposal is not that cognitive agents are simply dynamical systems of some traditional kind. Rather, it is that cognitive agents are dynamical systems of a quite special kind-and that to understand such systems, we need the resources of dynamics, combined with other theoretical resources as necessary.

Sophisticated cognitive performances require complex internal structures. The dynamical approach is taking a huge step backwards in trying to replace the symbolic representations of the computational approach with simple quantities. To explain high level cognition, dynamical systems have no choice but to implement standard computational architectures.

Almost everyone now agrees that most kinds of cognitive performance can only be explained by reference to complex structures internal to the system responsible for those performances. Still, it remains an open question what form those structures take. Hobbesian cognitive scientists are banking on the idea that they are the kind of structures found in digital computers, i.e., symbol structures (Newell & Simon, 1976) or "classical" combinatorial representations (Fodor & Pylyshyn, 1988). Lying behind this idea is an assumption that the kinds of complex structures required cannot exist in any system except by instantiating digital symbol structures.

However, as dynamical cognitive science has matured, it has become apparent that dynamical systems can incorporate complex structures in various ways without merely implementing their digital cousins (van Gelder, 1990). For example, arbitrarily many structures can be mapped onto states of a dynamical system, such that these states can then be used as the basis of systematic processing (e.g., (Chrisman, 1991; Pollack, 1990)). Other work has found combinatorial structure in the attractor basins of appropriate dynamical systems (Noelle & Cottrell, 1996), or in the trajectories induced by sequences of bifurcations ("attractor chaining", (van Gelder & Port, 1994)). The possibilities have only just begun to be explored. The dynamical approach is not trying to explain cognition without complex internal structures; rather, it is dramatically reconceiving how complex internal representations might be instantiated.

Natural languages are only effectively described by some form of context-sensitive grammar. In the standard Chomskyan hierarchy, languages of this complexity can only be handled by computers at least as powerful as linear-bounded automata (LBAs). Therefore, natural language speakers must be computers at least as powerful as LBAs.

The conclusion of this argument is ambiguous, between computers in general and digital computers. On the former interpretation, the argument is sound, but fails to conflict with the DH. It was pointed out above that dynamical systems can compute, i.e., be computers. The complexity of natural language constrains speakers' computational power, but not the kind of computer they instantiate. It remains an open empirical question whether the computers in question are best thought of as digital or dynamical (Elman, 1995).

On the latter interpretation, the argument simply equivocates. The premises establish that speakers must be computers in some sense; the conclusion claims they must be digital computers. The dominance of digital computers in the theory of computation, cognitive science, and computer technology, has created an unfortunate tendency to confuse computers in general with digital computers. This is what drives the objection.

This short paper has not attempted to cover every general objection to the dynamical hypothesis. Creative opponents will no doubt always be able to devise plenty more. It does purport to have refuted some of the most common and plausible such objections. One general lesson that emerges from the discussion is that resolving these kinds of philosophical issues requires a great deal of care in handling the key concepts such as dynamical and computation. Another is the importance of terminological agreement-those who talk different languages talk past each other.

The paper has not attempted to demonstrate that the DH is true. To do that would require much more than the kind of purely defensive maneovres undertaken here. That kind of positive work is being undertaken by increasing numbers of dynamically-oriented cognitive scientists. The major conclusion of this paper is something that all practising dynamicists take to be the case: the DH is a bold scientific conjecture, one whose elaboration and evaluation is worthy of research careers.

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