Abstract: A defense of the Dynamical Hypothesis against the charge that one of its major supports, the argument from time, is rotten.
 

Suppose you are a being from a very distant planet-Mars, say, or perhaps a Social Studies of Science program. Suppose you have been sent to Earth on a mission to observe the phenomenon earthlings call "cognitive science." In your report you must detail, among other things, the major varieties of cognitive science. Luckily you haven't ever read the standard literature on the matter, and so you don't have preconceptions. Where do you think would you draw the boundaries?

One thing you would surely notice is that some cognitive scientists actually produce models of the processes they are studying. Within this category, there seem to be at least two groups. In one group, researchers specify the behavior of their models by writing programs, sets of instructions for shuffling symbols. The other group of scientists specify their models by writing equations: mathematical rules describing change over time in a set of quantities. You notice that this distinction seems to be at least roughly correlated with many other interesting differences. For instance, the groups differ in their preferred mathematical frameworks. The former group are partial to computation theory and mainstream computer science, whereas the latter group tend to use the tools of dynamical modeling and dynamical systems theory.

In drawing a line between the programmers and the equation-writers, you would have stumbled close by what I believe is currently the single most important theoretical gulf in cognitive science. This is not the distinction between, for example, linguists and psychologists, who differ not so much in theoretical perspective as in their domain of study. Nor is it the distinction between "classical" computational cognitive scientists and "connectionists," though this gets a little closer to the heart of the matter. It is the distinction between what we might call computationalists, on one hand, and dynamicists, on the other.

Computationalists are those who believe that cognitive processes, whether natural or artificial, are a matter of digital computation. The classic statement of the computational approach is Newell and Simon's (Newell & Simon, 1976) hypothesis that "A physical symbol system has the necessary and sufficient means for general intelligent action." Two decades later, we find this slogan more conveniently formulated as the Computational Hypothesis, the simple idea cognitive agents are digital computers (and are best scientifically understood as such). Dynamicists, by contrast, are those cognitive scientists subscribing to the idea that cognitive is in the first instance a dynamical phenomenon. Their central commitment is encapsulated in the Dynamical Hypothesis (DH): Cognitive agents are dynamical systems (and are best scientifically understood as such).

The CH has been relatively well understood for decades, and extensively elaborated in a number of standard philosophical treatises (e.g., Haugeland, 1985; Pylyshyn, 1984). The DH has not been so lucky. Indeed, up until around the early '90s, there was really no such thing as the DH. There was, to be sure, a great deal of dynamical research scattered around cognitive science. As long as cognitive science had existed researchers had been proposing dynamical models. They could be found in pockets as various as connectionist modeling, psychophysics, developmental psychology, synergetics, and ecological psychology. What was lacking was an understanding that, notwithstanding their obvious differences, these diverse efforts actually had enough in common to be regarded as constituting a single research paradigm standing as a coherent and powerful alternative to the mainstream computational paradigm.

The appearance in 1995 of Mind as Motion: Explorations in the Dynamics of Cognition (Port & van Gelder, 1995) was a pivotal event in the emergence of the DH. That book was a kind of debutante's ball for dynamical cognitive science. For the first time one could find, gathered together under one roof, examples of dynamical research from far flung corners of the field, presented as members of one large happy community. Further, the book attempted to articulate the dynamical vision shared by these various efforts. Unfortunately there is not space here to describe that vision in any detail; the interested reader is referred to the introductory chapter, Its About Time. For the moment, a little explication of what the DH means will have to suffice.

When examined carefully, DH turns out to have two basic components. (1) The Nature Hypothesis. Cognitive agents are dynamical systems, i.e., for every kind of cognitive performance exhibited by a natural cognitive agent, there is some dynamical system instantiated by the agent at the highest relevant level of causal organisation, such that performances of that kind are behaviors of that system. (2) The Knowledge Hypothesis: Cognitive agents are best understood scientifically as dynamical systems, i.e., the causal organisation of cognitive agents can and should be understood by producing dynamical models, using the theoretical resources of dynamics, and adopting a broadly dynamical perspective. Here, a dynamical system is best understood as a quantitative system, i.e., one such that there are distances in state and in time, where those distances are systematically relevant to the behavior of the system. A telltale sign that a system is dynamical in this sense is that the state of the system is specified in terms of numerical variables, and the rule governing its behavior is an equation describing amounts of change. (For elaboration of all these points, see (van Gelder, forthcoming).

The claim is that the DH, in something like this form, is a (candidate) "law of qualitative structure" for cognitive science. The DH expresses the basic insight driving dynamical research in cognitive science. It purports to be an open empirical hypothesis. That is, we do not yet know for sure whether the DH is true or not; the extent to which it is true will only be determined by a great deal of hard empirical investigation.

Nevertheless, at this early stage it is possible to articulate some very general supporting considerations. "Its About Time" listed five such arguments, under the headings Time, Continuity and Discreteness, Multiple Simultaneous Interactions, Multiple Time Scales, and Self-Organisation. Of these, the flagship is unquestionably the Time argument. If you ask dynamicists in cognitive science why they use dynamical systems rather than physical symbol systems as their models, much the most common answer is that dynamical models give them superior ability to describe and explain temporal aspects of cognition. Why is this? The following little syllogism roughly captures the underlying reasoning:

• Natural cognitive processes are, in a deep way, a matter of change in real time. They unfold in continuous time; they have intrinsic timings (durations, oscillations, rates, etc.); they have relative timings (synchronies, phase relations, etc.). These temporal properties are not accidental details but are essential to the nature and success of cognition.
• Dynamical systems are defined in real time.
• Orthodox computational systems are defined in discrete, "ersatz" time.

What is meant here by "ersatz" time? Consider an abstract Turing machine. At t₁ the machine and its tape are in one overall state; at t₂ it is in another overall state. These times are essentially discrete; it is nonsense to ask what state the machine occupies at time t_1.523. Note also that for the purpose of understanding the Turing machine as carrying out effective computation, the amount of time it spends in any given state, and the amount of time it takes to change from one state to another, are utterly irrelevant. Durations, rates, phases, accelerations etc. just never enter the picture. We talk about time steps, but the numbers 1, 2, etc., are being used as nothing more than indexes for imposing orders on the states of the system. More generally, the time set of abstract Turing machines is always a mere order. The integers are the most familiar and convenient ordered set we know of, and so we use the integers as the time set. However, none of the properties of the integers, over and above their constituting an ordered set, have any relevance to the Turing machine.

The real time of natural cognitive processes is the set of instants at which actual events can take place. This set is not a mere order. Why? First, it is continuous. In any vicinity (no matter how small) of any one instant of time there is always another instant of time. Second, real time is quantitative. That is, there is an amount of time-a time period-between any two instants of time. In light of these two properties, real time is best modeled mathematically by the real numbers. Real time, and the integers used to order the states of a Turing machine, are both called "time." Clearly, however, the "time" of the Turing machine bears only a passing resemblance to the genuine article. It is "ersatz," a kind of pseudo- or watered-down time.

Now, suppose you have some complex natural process unfolding in real time, and you want to understand that process by constructing an abstract model. Suppose, for example, that you wanted to understand the behavior of a damped mass-spring. Obviously a model defined in ersatz time would generally only be, at best, a kind of clumsy approximation obtained by throwing away much of what is interesting about the behavior in question. It is hardly a coincidence that models of damped mass-spring systems in physics textbooks are defined in real time, not discrete ersatz time.

There are of course exceptions to this general principle. These occur in those relatively rare situations where the system in question, though in fact unfolding in real time, is specially constrained to behave in such a way that some substantial degree of understanding can be obtained by abstracting away from the full richness of real time. Standard digital computers are a case in point: a great deal of what they do is in fact best understood by pretending that they operate in ersatz time. Orthodox computational cognitive science is making a bet that cognitive agents belong to this category of systems.

According to the "time" argument for the DH, this is a bad bet. Cognition is just another complex natural phenomenon unfolding in real time:
 

Cognitive processes and their context unfold continuously and simultaneously in real time. Computational models specify a discrete sequence of static internal states in arbitrary "step" time (t₁, t₂ etc.). Imposing the latter onto the former is like wearing shoes on your hands. You can do it, but gloves fit a whole lot better. ((van Gelder & Port, 1995), p.2.)

Recently, this argument has been challenged by Rick Grush. In a provocative review of Mind as Motion (Grush, 1997), he alleges that many acclaimed dynamical models, including some of more prominent models in the book itself, are actually ersatz imposters. Mind as Motion is guilty of a kind of "bait and switch" operation. The dynamical approach is claimed to be superior because its models are defined in real time, and yet
 

A large portion of the models of 'higher' cognitive processes articulated in the book have exactly the same processing-step character as the vilified computational alternatives, even though the language, mathematics and illustrations used to present the models obscures this fact.

dP(t+h)/h = -S.P(t) + C.V(t+h)

It should be immediately obvious...that this model has nothing at all to do with real time. Indeed, it operates as a sequence of calculations, each iteration of which can take as long or as short a time as the computer running the simulation requires. Of course, if one illicitly equates processing iterations with time, then the model looks like it is in fact doing something genuinely dynamic...

How should the dynamicist respond? The best response to a dilemma is usually to refute the premises setting up the awkward choices. Grush is obviously correct in pointing out that many classic "dynamical" models are defined in discrete time. Does it follow that they are defined only in ersatz time rather than real time? Grush seems to think so, but we need to tread very carefully here.

As a preliminary point, notice that if Grush were correct, many leading dynamicist researchers would actually be living in a rather sorry state of delusion. They describe their models as dynamical, and they believe that their dynamical models are superior in accounting for temporal aspects of cognition. Yet, if Grush is correct these are rather elementary confusions. It is understandable, perhaps, that the editors of Mind as Motion might make such a mistake. It would be surprising indeed if sophisticated mathematical modelers were to fall into the same trap. This should at least give one pause for thought. It suggests that a model might provide some substantial grip on real-time behavior even when defined discretely.

How could this be? Well, it is a commonplace in mathematical modeling that it is sometimes handy to use discrete rather than continuous models even when one is interested in the full real-time properties of the phenomenon under study. In other words, discrete models are on occasion used as convenient approximations to real-time behavior, without damaging loss of explanatory power. Therefore, if you see a discrete mathematical model, it might be one defined in ersatz time, having abstracted away from the subtleties and complexities of real-world timing. It might, on the other hand, be a model of genuine real-time behavior, though defined discretely for mathematical convenience.

We could describe models of the latter kind as being defined in "quasi-real" time. This raises an obvious question. How can you tell whether time in a model is ersatz or quasi-real? Clearly, you have to look at more than just the time set itself. The integers might be functioning as a mere order, or as indexes for points in the real time line. ("Some integers are more equal than others!" one might say.) It all depends on how the time set is integrated with other aspects of the model. Intuitively, a discrete time set is quasi-real time when the difference between one time and the next is not just a matter of succession but is a genuine period or unit of time. Or, put another way, a discrete time set is quasi-real when it makes sense that you could shorten or lengthen the time step, and this would make a difference to the behavior of the model. In such models there are actually units of time; if you change the time unit, you would have to accomodate that by changing some other aspect of the governing equation.

For example, if the size of the time step (h) in the Decision Field Theory model were to be changed, then other parameters of the model would need to be changed accordingly, on pain of delivering absurd empirical predictions. But notice that the very idea of a unit of time has no place at all in an abstract Turing machine. Interestingly, it also has no place in orthodox computational models of cognition defined in ersatz time. These models specify what states the system will go through, and in what order, but tell us nothing interesting about when the system occupies those states.

This somewhat subtle point deserves elaboration. Imagine someone who is, so far, unconvinced. This skeptic notes, quite rightly, that you can get empirical predictions out of an orthodox computational model such as SOAR (Newell, 1991): assume that each processing step takes n milliseconds, count the number of steps the model takes to reach a given state, multiply the result, and-bingo!-a prediction. Each time step then corresponds to a time "unit" of n milliseconds. If the prediction isn't right, then adjust n to bring the model into line with reality. This seems to suggest that quasi-real dynamical models, with their units of time, have no deep advantage in the temporality domain over orthodox computational models.

So, what more can be said about the difference between a genuine dynamical model defined in quasi-real time and a GOFAI model defined in ersatz time? Well, notice that the state spaces of dynamical models are metric spaces. This means that there are distances between any two states of the system. Now, there is nothing remarkable in this alone. Given the existence of certain trivial metrics, any state space will be a metric space. But noting this fact points to an interesting question: is there any systematic relationship between distances in state as measured by the metric, and the behavior of the system? If there is, you could in principle write down a rule telling you how the system behaves in terms of distances in the state set. The rule could, for example, tell you how the system behaves by telling you how much the system changes in each time step, i.e., by specifying the amount of change from any given state. But wait! There's more! You might also ask whether there is any systematic relationship between distances in state and distances in time (durations). If there is, you could write down a rule relating amounts of change in state to amounts of change in time. This would, of course, be a rule describing the behavior of the system.

Suppose we have defined some abstract system intended as a mathematical model of aspect of the actual world-the motions of the planets, or a person's decision-making behavior. Suppose we set up the modeling relationship by specifying a correspondence between certain variables in the model and certain variables in the world. Obviously, in order to set up this correspondence, there has to be a way of measuring the worldly variables. We need some kind of yardstick for assigning numbers to states of the world. Note that we also need a yardstick for measuring time. We can't go about choosing these yardsticks randomly. In order to capture the order that is out there, and set up the proper correspondences with a mathematical model, we must understand the proportional relations between the yardsticks. This is why we use coherent system of units such as the Systeme Internationale d'Unités. Given a choice of units, the behavior of the model must be calibrated to the behavior of the world by introducing constants into the rule; thus, the constant of gravitational attraction differs, depending upon the system of units you are using. So, there is a delicate balance between the way in which distances in state are measured, the way in which distances in time are measured, and the details of the governing equation.

Now we can really understand the deep difference between genuine dynamical models-whether real or "quasi-real"-and ersatz-time models. In dynamical models, there are distances in state, and distances in time, and both are systematically related to the behavior of the system, and so we can describe the behavior of the system by specifying amounts of change in amounts of time. In relating a dynamical model to reality, we have to specify the units of measurement, and we have to make sure the parameters of the model are set correctly. All of this is true in the case of DFT. The difference equations describe amounts of difference in amounts of time, time is measured in seconds, and the parameters of the model are finely tuned to this choice of unit. But none of this is true in the case of Turing Machines, SOAR, or any other model defined in ersatz time. The so-called "units" of time (n milliseconds) relating an orthodox computational model to reality are really just ersatz units. They can be adjusted at will, in order to derive ersatz predictions, but there are no constraints on the way you do this inherent in the model itself. There is no coherent way to change the governing equation to accomodate your in change time "unit;" indeed, the very idea makes no sense.

All this can be summarized in a number of points. First, "real" time in a model is not necessarily the real numbers. Genuinely dynamical models, with superior explantory grip on temporal aspects of cognition, can be discrete but still quasi-real. Supporters of the dynamical approach to cognition can enlist such models as representatives, perhaps even exemplary instances, of dynamical research in cognitive science. (Note that this leaves open the possibility that some models currently described as dynamical may in fact be ersatz imposters. The issue has to be sorted out on a case by case basis.) Second, the time argument for the DH still stands. Dynamical models, in which the quantitative properties of the time set are systematically related to the behavior of the system, have an intrinsic advantage in accounting for the fine temporal detail of natural cognition. Third, Mind as Motion was guilty of no bait and switch. Or, more precisely, convicting Mind as Motion of baiting and switching will take more argument than Grush has provided.
 

Note: I am grateful to Rick Grush and Jerome Busemeyer for helpful e-correspondence.

References

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Author: Tim van Gelder
Last updated: 15-Jul-02
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