The Dynamical Hypothesis in Cognitive Scien ce

Keywords

cognition, systems, dynamical systems, computers, computational systems,omputability, modeling, time.

Abstract

Recent years have seen increasing use of dynamics in cognitive science. If theeart of the dominant computational approach is the hypothesis that cognitivegents are digital computers, the heart of the alternative dynamical approachs the hypothesis that cognitive agents are dynamical systems. This targetrticle attempts to articulate the dynamical hypothesis and to defend it as anmpirical alternative to the computational hypothesis. Digital computers andynamical systems are characterized as specific kinds of systems. The dynamic alypothesis has two major components: the nature hypothesis (cognitive agents are dynamical systems) and the knowledge hypothesis (cognitive agentsan be understood dynamically). A wide range of objections to the generalypothesis are then rebu tted. The conclusion is that cognitive systems may welle dynamical systems, and only sustained empirical research in cognitivecience will determine the extent to which that is true.

1. Introduction

The Humean dream was not the first vision of mind inspired by the emergence of modern science. The new physics had uncovered mathematical laws of great simplicity and elegance, but laborious calculation was required to derive the messy details of actual behaviors. Thomas Hobbes took this calculating acti vity itself as his model of the mechanisms of mental operation. Perhaps thought is symbolic computation, the rule-governed manipulation of symbols inside the head (Hobbes, 1651/1962).

Seventeenth-century speculation became twentieth-century science. Hobbes's idea evolved into the computational hypothesis (CH), that cognitive agents are basically digital computers. Perhaps the most famous rendition is Newell and Simon's (1976) doctrine that "A physical symbol system has the necessary and suffic ient means for general intelligent action." They proposed this hypothesis as a "law of qualitative structure," comparable to the cell doctrine in biology or plate tectonics in geology. It expresses the central insight of the research paradigm which has do minated cognitive science for some forty years.

In recent years, however, the Humean alternative has been gaining momentum. One of the most notable developments has been the rise of connectionism, which models cognition as the behavior of dynamical s ystems (Smolensky, 1988), and often understands those models from a dynamical perspective. Equally significant is the emergence of cognitive neuroscience, and within it, the increasing prevalence of dynamical theorising. Dynamics forms the general framewo rk for growing amounts of work in psychophysics, perception, motor control, developmental psychology, cognitive psychology, situated robotics and autonomous agents research, artificial intelligence, and social psychology. It is central to a number of gene ral approaches, such as ecological psychology, synergetics, and morphodynamics. [Note 1]

The dynamical hypothesis (DH) is the unifying essence of dynamical approaches to cognition. It is encapsulated in the simpl e slogan, cognitive agents are dynamical systems. The aims of this target article are (1) to articulate the hypothesis - i.e., to explain what the slogan means - and (2) to defend it as an open empirical hypothesis standing as a subst antive alternative to the CH. The DH contends for the status of the "law of qualitative structure" concerning the nature of cognition.

One goal in undertaking this philosophical work is to clarify the conceptual terrain. Another is to help clear rheto rical space for dynamicists in cognitive science to get on with the hard work of developing detailed accounts of specific cognitive phenomena. The most important goal, however, is to gain insight into the nature of people - for people are, among other thi ngs, cognitive agents.his paper ploughs an interdisciplinary field. Boulders of ambiguity, vagueness and confusion must be cleared away. Much effort is devoted simply to establishing a single coherent and reasonably precise framework for discussion. Thi s framework involves commitments at terminological, conceptual and even metaphysical levels. Its development requires many choices and stipulations, often somewhat arbitrary in nature. Occasional conflicts with existing intuitions are unavoidable. Still, some such regimentation is essential, for otherwise debating the DH is just a futile exercise in miscommunication. Table 2 in the appendix summarizes the framework by listing key terms and their meanings as deployed here.

2. Some Examples

first task is to sketch some representative examples of dynamicalognitive science, to serve as a backdrop for the following discussion. Spaceimits dictate brevity; readers are encouraged to visit the original sourcesor proper treatment.

Consi der how we come to make choices between actions with various possible outcomes. If we were digital computers, we would symbolically represent to ourselves the various options and their outcomes, together with our estimates of the likelihood of those outco mes and their value to us. Reaching a decision would then be a matter of calculating the most promising option. An alternative Humean account has been proposed by psychologists Jerome Busemeyer and Jim Townsend (Busemeyer & Townsend, 1993). In their "D ecision Field Theory" (DFT) model, relevant aspects of the decision situation are represented not by symbols but by means of continuous quantities. Decision-making is the interdependent evolution of these quantities over time as governed by mathematical e quations (as opposed to algorithms). Decisions are made when certain thresholds are passed. The scientific question then is: Which kind of model best accounts for the actual psychological data on human decision-making? Busemeyer & Townsend claim that t heir model predicts actual decisions better than any "static-deterministic" model, as well as describing temporal properties of decision processes beyond the scope of traditional models (see Figure 1).

For an example of a very different kind, consider how we manage to move our limbs. A Hobbesian would maintain that we calculate how and when to contract muscles, much as a digital computer lands a 747 by calculating engine thrust, flap angle, etc.. A dynamica l alternative has been under development by Scott Kelso and coworkers. His classic example is coordinating the wagging of your index fingers. Performance on this task has some remarkable properties. At low wagging speeds there are two comfortable coordina tion patterns, inphase and antiphase (bistability). As speed is gradually increased, anti-phase patterns start to lose their stability; eventually a point comes where only inphase patterns are stable (bifurcation). As speed decreases, antiphase patterns b ecome possible again, but not until somewhat below the original collapse point (hysteresis). Kelso found that these and other properties can be described and predicted in detail by assuming that a single, continuous, high level, "collective" variable - re lative phase - evolves in a way governed by a suitable form of a simple differential equation (Kelso, 1995, p. 55). [Note 2] Variants of this "HKB" model have been applied in diverse cognitive domains. [Note 3] The basic insight is that coordination is best thought of as explained not as masterminded by a digital computer sending symbolic instructions at just the right time, but as an emergent property of a nonlinear dynamical system self-org anizing around instabilities.

These models purport to provide the best available empirical accounts of phenomena in their domains. Whether they succeed is an interesting question for specialists to address. What matters here is that they nicely illus trate the dynamical approach to cognition.

3. Systems, Digital Computers, and Dynamical Systems

3.1 Systems

For example, the solar system of classical mechanics is the set of positions and momentums of the sun and planets; these are the q uantities whose behaviors are described by Newton's laws. Note that the variables of the solar system in this sense are properties of the sun and planets. We must therefore distinguish objects (parts of the world such as the sun and planets, Macintoshes, and cognitive agents) from the systems they instantiate. Any given object will usually instantiate a great many systems of different kinds.

Concrete systems are those, like the solar system, whose variables are actual features of the rea l world changing in real time in accordance with natural laws. Abstract systems are just sets of abstract variables governed by mathematical rules. Concrete systems can realize abstract systems. For example, two HP61 calculators realize exac tly the same abstract computational system.

Concrete systems are slices of the causal organization of nature. Causal organization comes in many kinds and at many levels. Distinct systems can be intimately related. Compare the classical solar system w ith the system made up of all the positions and momentums of all their constituent subatomic particles. The (macro)variables of the former are built up out of the (micro)variables of the latter. The relationship between these systems is neither identity [Note 5] nor instantiation. In what follows, a lower-level system will be said to implement a higher-level system when the variables of the latter are somehow constructed out of variables of the former. Note that imple mentation licenses us to identify the behavior of the one system with the behavior of the other, despite failure of strict identity between the systems themselves.

Often, change in a system depends on factors outside the system itself (e.g., the force of gravity), referred to here as parameters. Sometimes, changes in a parameter depend in turn on the system itself. For example, the position of the moon both depends upon, and affects, the position of the planets. This kind of reciprocal, direct dependence is known as coupling. System variables and coupled parameters can be regarded as forming a larger system. This illustrates the semi-arbitrariness of systems. It is always up to us to nominate a set of concrete variables as the system we will study. Reality determines whether that set is in fact a system, and how it behaves.

All systems in the current sense change in time. In general, time is just some intrinsically ordered set, or order[Note 6 ] , serving to provide orderings over other things. The real time of concrete systems is the set of instants at which things can actually happen, ordered by temporal priority (before/after). Concrete events are paired with instants or periods of time, and hence stand in temporal relations with each other. Abstract systems are not situated in real time at all, and so must take some other set as their time set; usually, it is the positive integers or the real numbers. The mathematical rule imposes order ings over states of the system by pairing them with members of this set.

3.2 Digital Computers

A computer is simply anything that computes in some way or other. Computing is an informal notion; the basic idea is that of a process systematically transforming "questions" into "answers" - inputs into outputs, start states into final states, etc.. The function compu ted by that process is the set of question/answer pairs themselves, or the set of pairs of entities they represent. In this general sense pretty much anything can be construed as a computer. Computation only gets interesting when significant constraints a re placed on the kinds of processes involved. In classical computation theory, the standard approach has been to require that processes be effective, i.e., produce their results by means of a finite number of basic operations specified by an algori thm (a finite recipe, or set of instructions specifying basic operations).

Digital computers, in the sense that matters for cognitive science, are systems which carry out effective computation over representations. That is, they are systems whose beh aviors are algorithmically specified finite sequences of basic operations constituting manipulations of representations. This characterization can be broken down into four fundamental requirements on a system to count as a digital computer:

(1) Di gital variables and states. First, for each variable there must be some set of discrete values which the variable instantiates digitally for the purposes of system behavior. In the concrete case, this means that the variable must instantiate th ose variables positively and reliably [Note 8]. When all variables in a system are digital, the system's states are also digital. The basic operations required by effective computation correspond to digital state tr ansitions.

2) Time as discrete order. The time set must be a discrete order whose elements are the times at which the system digitally occupies its states. In abstract systems, this is usually the positive integers. In concrete systems, it is t he set of periods of real time at which the machine digitally instantiates its states, as rendered discrete by the flux of transition between states. These are indexed by the positive integers (t₁, t₂, etc.).

3) Algorithm . Effective computation requires basic operations to be specified by an algorithm, i.e., a finite recipe specifying state transitions solely on the basis of digital properties of states. For example, the infinite range of behaviors of a Turing Machine is governed by its machine table, a finite set of instructions expressed only in terms of the digital values of tape squares, head position, and head state. In concrete systems, this rule must capture one level of causal organization. That is, the transition s described by the rule must happen the way they do because the states bear the digital properties in terms of which the rule is expressed

4) Interpretation. The system's states and behaviors must yield to systematic interpretation. Tha t is, there must be some domain, and correspondences between the system and the domain, such that (a) the correspondences are systematic with respect to those digital aspects of the system in terms of which the rule governs system behavior, and (b) the system's states and behaviors make sense in the light of those correspondences [Note 9]

. he distinction made above (S.3.1) between the solar system of classical mechanics on one hand and the sun and planets on the other is mirrored by a distinction between digital computers and the ordinary notion of computers as what you take out of the box and plug into the wall. The digital computer system is the object of theoretical interest. The hunk of silicon, plastic, glass, metal, etc., instantiates some digital computer (system), and of course many other systems as well.

3.3 Dynamical Systems

By comparison with the CH, the DH h as been starved ofttention.[Note 10] Partly as a result, there is nostablished consensus over what dynamical systems are for the purposes of theypothesis. Unfortunately, there is also a wide range of definitions inat hematics and science more generally (Table 1). These range from older, narrow definitions in terms of particles governed by forces to more recent broaddefinitions which subsume all systems in the current sense. There is no singleofficial definition waitin g to be lifted off the shelf. Nevertheless, cognitivescientists do have a good working grasp of the issue. In the vast majority of cases they agree whether a system counts as dynamical in the sense that matters for them. The challenge here is to articulat e that intuitivenderstanding.

n obvious feature distinguishing dynamical models in cognitivecience from standard computational models is that their variables arenumerical. One reason numbers are so useful in science is that they havequantitative properties. This suggests that dynamical systems inognitive science might be defined as quantitative systems. Roughly, a systems quantitative when there are distances in state or time, such thathese distances matter to behavior. This can be true in progressively deeperays, giving rise to progressively more substantial senses in which a systeman count as dynamical.

(1) Quantitative in state. First, there can be distances between any twoverall states of the system, such that t he behavior of the system depends onhese distances. More precisely, a system is quantitative in state when theres a metric[Note 11] over the state set such thatehavior is systematically related to distances as measur ed by that metric.uch systems will be governed by a rule compactly specifying thisistance-dependent change. For example, the difference equations in the DFTodel describe how the system changes by telling us the distance between thealues of variabl es at time t and their values at timet+h.

Standardly, the relevant quantitive properties of state sets are derived from quantitive properties of the variables. Quantitive variables can be either abstract or concrete. For exampl e, the variable f in the HKB model is an abstract mathematical magnitude whose values are real numbers. This variable corresponds (via measurement; see Krantz, Luce, Suppes, & Tversky, 1971) to a concrete quantity whose values are relative phases of os cillation of index fingers. The model works precisely because the quantitative properties of the concrete variable are reflected in the quantitative properties of the abstract counterpart.

(2) Quantitative state/time interdependence. A system i s quantitative intime when time is a quantity, i.e., there is a metric over the time set,uch that system behavior is systematically related to distances as measured byhat metric. At least in cognitive science practice, systems that areuantit ative in time are also quantitative in space, and these properties arenterdependent. That is, the behavior of the system is such that amountsof change in state are systematically related to amounts of elapsedime. Such systems are govern ed by a rule specifying a quantitativeelationship between change in state, elapsed time, and current state. Inoncrete systems, this rule captures causal organization; that is, the systemhanges as it does because system variables have the quantitativ e properties inerms of which the rule is expressed. When both state and time areuantitative, the system exhibits rates of change. Systems that arenterdependently quantitative in state and time are governed by rulespecifying the rate of chan ge in terms of current state (e.g., first-orderifferential equations).

(3) Rate dependence. Third, some systems are such that their rates ofhange depend on current rates of change. In these systems, variables includeoth basic variables an d the rates of change of those variables. The solarystem is a classic example. Systems whose behavior is governed by rules mostompactly expressed as sets of higher-order differential equations areuantitative in this sense.

n what follows, a sys tem is taken to be dynamical to the extent that it isuantitative in one of the above senses.[Note 12]. At least four considerations support this approach.irst, it reflects the actual practice of cognitive scientists in classifyingystems as dynamical or not, or as more or less dynamical. Second, it sitsomfortably with existing definitions. The levels of quantitative characteroughly correspond to definitions 1-4 of . Third, it is cast in terms of deep,heoreticall y significant properties of systems. For example, a system that isuantitative in state is one whose states form a space, in a more thanerely metaphorical sense; states are positions in that space, andehaviors are paths or trajectories. Thus quantitative systems support aeometric perspective on system behavior, one of the hallmarks of a dynamicalrientation. Other fundamental features of dynamical systems, such as stabilitynd attractors, also depend on distances. Fourth, the defini tion sets up aontrast between dynamical systems and digital computers (see Section 6). Forhese reasons, defining dynamical systems as quantitative systems facilitatesrticulation and defense of the DH.[13]

4. The Dynamical Hypothesis

First, some preliminary points. The proper domain of the DH is naturalcognitive agents - i.e., evolved , biological agents such as people and othernimals. It need take no stand on the possibility of artificial cognition inigital computers. Second, the DH is limited in its explanatory pretensions. Its concerned only with the causal organization of age nts insofar as theyxhibit cognitive performances. Other forms of explanation may also be deeplylluminating. For example, evolutionary explanations might best explain whyan agent has a particular causal organization.

What is it to be cogn itive? In the most traditional sense, cognitiverocesses are those involving knowledge; cognitive science would then behe study of knowledge-based processes. However, as cognitive science hasatured it has diversified. Knowledge is now only one indicator of cognitivetatus; others include intelligence, adaptability, and coordination withespect to remote states of affairs. The concept now resists capture in termsf any concise set of strict conditions. This paper simply takes an intuitiv erasp of the issue for granted. Crudely put, the question here is not whatakes something cognitive, but how cognitive agents work.

4.1 The Nature Hypothesis.

irst, the relationship at the heart of the nature hypothesis is not identityut instantiation. Cognitive agents are not themselves systems (s ets ofariables) but rather objects whose properties, etc., can form systems.ognitive agents instantiate numerous systems at any given time. According tohe nature hypothesis, the systems responsible for cognitive performances areynamical.

Secon d, cognitive agents "are," in this sense, not some particular dynamicalystem, but as many systems as are needed to produce all the different kinds ofognitive performances exhibited by the agent. Consider the DFT and HKB modelsrom Section . These mod els invoke quite different sets of variables. Oneuggests that cognitive agents make decisions by virtue of change in valences,references, etc.; the other, that cognitive agents coordinate finger movementsy virtue of change in relative phase. These m odels are not in competition.oth might be complete accounts of phenomena in their respective domains,mplying that cognitive agents are many dynamical systems at once.

Another noteworthy fact about these models is that the variables they posit aren ot low-level (e.g., neural firing rates), but rather macroscopic quantities atoughly the level of the cognitive performance itself. The lesson here is thathe nature hypothesis is concerned in the first instance not with low-levelystems but with how agents are causally organized at the highest levelelevant to an explanation of cognitive performances, whatever that may be.

Finally, notice that the DFT model includes not only "internal" variables such as preferences and valences, but also the "po sition" of the agent. More generally, the dynamical system responsible for a given kind of cognitive performance might include variables not literally contained within the agent itself, on any ordinary conception of its boundaries. For example, ecological psychologists understand visually guided locomotion as change in a dynamical system which includes aspects of both the organism and the environment (e.g., the optic flow; Warren, 1995).

4.2 The Knowledge Hypothesis.

It is one thing for cognitive agents to be dynamicalystems, but it is quite another for us to understand them as such. Thenowledge hypothesis is the bold claim that cognitive science can andhould take dynamica l form. What does this involve?

4.2.1 Dynamical Models

iven something we wish to understand - an explanatory target - aodel is some other thing, relevantly similar but somehow more amenable o investigation. Understanding of the model transfers to the target across theridge of similarity. Note that often the full complexity and detail of thearget will defy human comprehension. In such cases, a model providescientific insight precisely b ecause it is a simplification.

One of the most common strategies in science is the use of abstract dynamicalystems as models. The dynamical approach to cognition follows in thisradition. The performance of interest is taken to be interdepe ndent change inome concrete dynamical system instantiated by the agent. The scientisturnishes an abstract dynamical system to serve as a model by specifyingbstract variables and governing equations. Simple models can be fullynderstood by means of purely mathematical techniques. More commonly, however,cientists enlist the aid of digital computers to simulate the modeli.e., compute approximate descriptions of its behavior). The simulationesults are compared against experimental data fro m the target. To the extenthat the correspondence is close, the target system is taken to be similar intructure to the abstract dynamical model. Note that the digital computer,ince it is not itself a dynamical system (for explanation of this claim, seeection ), is not similar in the relevant sense to the target system, and so isot a model of it. We do not attempt to understand the target byunderstanding the digital computer; rather, we use the computer as a tooln our attempt to underst and the target by understanding the abstractodel.

The distinctive flavor of Humean dynamical modeling is enhanced byuxtaposition with its Hobbesian counterpart (). In both cases, there is aarget system, an abstract model, and a digital co mputer. In the latter case,owever, the target is assumed to be a digital computer; the abstract model isot a dynamical system but a digital computer; and the concrete digitalomputer does not simulate but rather realizes the abstracty stem. Indeed, the abstract model is often specified by providing theoncrete computer which realizes it. Since they are identical in computationaltructure, both will be relevantly similar to the target if either is;herefore, both abstract and concrete systems count as models(Figure 2).

The basic structure of dynamical modeling is nicely illustrated by theusemeyer & Townsend work. There are many parallels with classicalechanics. Such work c omes perhaps closest to realizing the Humean dream.owever, it would be misleading to suggest that dynamical modeling in cognitivecience is stuck in the mold of classical physics. Obviously, cognitivehenomena differ in important ways from ordinary ph ysical phenomena. Dynamicalognitive science has had to generate its own variations on traditionalractices of dynamical modeling. Dimensions along which such variation is foundnclude: (a) To what do model variables correspond? The quantities invoked inynamical accounts often differ fundamentally from ordinary physicaluantities. "Valence" and "preference," for example, do not appear in textbooksf mechanics. (b) At what level is the correspondence with the target? Inhysical models individual va riables are usually taken to correspond directlyo concrete physical quantities. In dynamical modeling in cognitive science,here might be no concrete quantity corresponding to individualariables. The correspondence between model and reality is at higher levels ofynamical structure. Individual units of a connectionist model, for example,ay be significant only insofar as they support attractors which do correspondo aspects of cognition, such as a recognition state. (c) Is the correspondenc euantitative or qualitative? Physical models are generally expected to matchmpirical data in more or less precise quantitative detail. A model of globalarming, for example, should tell us exactly how much average temperature willise. Such virtue i s less common in dynamical cognitive science: as often asot, models match data qualitatively, at some level of abstraction. (In thisespect dynamical modeling apes computational modeling.)

4.2.2 Dynamical Tools

nderstanding cognitive agents as dynamical systems means morehan just using certain kinds of models. Those models, and so the cognitiveerformances themselves, must be understood dynamically. Roughly, thiseans taking the resources of dynamics - as opposed, for example, toainstream computer science - as the basic descriptive and explanatoryramework. But what are those resources?

ithin dynamics there is a convenient distinction between dynamical modeling,n one hand, and dynamical systems theory (DST) on the other. Dynamicalodeling is a branch of applied mathematics; its concern is to understandatural phenomena by providing abstract dynamical models. The skeletaltructure of such modeling was described in the pr evious section. The theory ofynamical modeling is a powerful repertoire of concepts, proofs, methods, etc.,or use in this activity. DST, on the other hand, is a branch of pureathematics. Its domain extends to any kind of describable change, but it ocuses attention particularly on systems for which there is no known way topecify behaviors as functions of time (e.g., systems whose rule is a set ofonlinear differential equations with no solutions). The fundamental move is toonceptualize systems geometrically, i.e., in terms of positions,istances, regions, and paths in a space of possible states. DST aims tonderstand structural properties of the flow, i.e., the entire range ofossible paths.[Note 14 ]. For introductions toynamical systems theory, see or . are chapter length overviews ofynamics for cognitive scientists.

here is no clear line between these two sides of dynamics, but the contrast isignificant. Hume envisione d psychology as dynamical modeling, but that aloneoes not suffice. The distinctive complexities of cognition yield to scientificnderstanding only when dynamical modeling is enriched by the perspective andesources of DST. Poincaré pioneered DS T late last century, but the bulkf it has only been developed in the last few decades. Contemporary dynamicsould be a whole new subject to Newton or even Maxwell. Hume aspired to be theewton of the mind, but in hindsight Poincaré would have m ade a betterodel.

Dynamics plays much the same role in dynamical cognitive science as computer science (the theory of computational systems, particularly digital computers) plays in traditional cognitive science. Computer science is not itself a th eory of cognitive processes. Rather, it provides a powerful set of tools for use in developing accounts of particular aspects of cognition. Therein lies the hard empirical work of mainstream cognitive science. Likewise, dynamics does not somehow automatic ally constitute an account of cognition. It is a highly general framework which must be adapted, supplemented, fine-tuned, etc., to apply to any particular cognitive phenomenon. This typically involves merging dynamics with other constructs (e.g., the sch ema (Rumelhart, Smolensky, McClelland, & Hinton, 1986b)) or theoretical frameworks (e.g., ecological psychology (Turvey & Carello, 1995)). Some authors have argued for even more dramatic reorientations in our understanding of dynamical systems for t he purposes of understanding biological or cognitive systems. See, for example, the work of Robert Rosen on "anticipatory systems" (Rosen, 1985) and George Kampis on "component systems" (Kampis, 1991).

Contemporary dynamics provides powerful resource s for describing general properties of the behaviour of systems. These resources can be brought to bear even in the absence of an actual equation-governed model. If done rigorously, this can buy a qualitative or preliminary understanding of the phenomenon , which may be the best available and forms a solid foundation for further exploration. [Note 15] This approach is useful in situations where, for whatever reason, providing a model is not currently feasible (e.g., Thelen , 1995).

4.2.3 Dynamical perspective

At the highest level, there are a number of general characteristics of aroadly dynamical perspective on some natural phenomenon. The following standut particularl y strongly when the subject is cognition and the contrast isith a computational approach:

4.2.3.1 Change versus state. Change and state are like two sides of oneoin. Nevertheless, theoretical perspectives can differ in their primarymphas is or focus. Dynamicists are interested, in the first instance, in howhings change; states are the medium of change, and have little intrinsicnterest. Computationalists, by contrast, focus primarily on states; change isust what takes you from one st ate to another.

4.2.3.2 Geometry versus structure. How are states of a systemonceptualised? Computationalists focus on internal structure, and inarticular on internal combinatorial or syntactic structure - how basic piecesre combined to f orm structured wholes. Dynamicists, by contrast, understand atate geometrically, in terms of its position with respect to othertates and features of the system's dynamical landscape such as basins ofttraction. In other words, they focus on wh ere the state is, rather than whatt is made up of.

.2.3.3 Structure in time. Sophisticated cognition demands structuralomplexity in the cognitive system. How is that structure realized?omputationalists tend to think of it as laid out sta tically - as all present atne time - and of cognition as simple transformations of static structures. DSTuggests an alternative. Systems with simple states - perhaps just oneariable - can behave in very complex ways. This enables dynamicists to thin k ofognitive structure as laid out temporally, much like speech as opposed to theritten word. Cognition is then seen as the simultaneous, mutually influencingnfolding of complex temporal structures.

4.2.3.4 Timing versus order. Dynamicis ts tend to be interested in howehaviors happen in time, whereas computationalists are interested in what theehaviour is, regardless of timing details. Put another way, computationalistsocus on which states the system passes through, whereas d ynamicistsocus relatively more on when it passes through them.

4.2.3.5 Parallel versus serial. Dynamicists tend to think of systems asperating in parallel, i.e., all aspects changing interdependently at the sameime. Computationalis ts, by contrast, tend to think of systems as serial: mostariables remain unchanged in any given state transition. For a dynamicist,hange is standardly global; for a computationalist, change is standardlyocal.

4.2.3.6 Ongoing versus Input/Out put. Computationalists standardly thinkf a process as commencing with an input to the system. The task for the systems to produce an appropriate output, and it does so via a sequence of internalperations culminating in the system halting with th at output. Dynamicists, byontrast, think of processes as always ongoing, not starting anywhere and notinishing anywhere. The goal is not to map an input at one time onto an outputt some later time, but to constantly maintain appropriate change.

4 .2.3.7 Interaction: state-setting or coupling? How does a cognitiveystem interact with other things, such as the environment? Computationaliststandardly think of interaction as setting state; the system changes ints own way from that s tate, until new input resets state again. Dynamicistsecognize an alternative: interaction can be a matter of parameters influencinghe shape of change. Input is conceived as an ongoing influence on theirection of change, and output as ongoing influence on something else, just as radio set is continuously modified by an incoming signal and at the same times delivering its sound. Sometimes interaction is a matter of coupling - twoystems simultaneously shaping each other's change.

.2.3 .8 Representations. Standard explanations of how systems come to exhibit sophisticated cognitive performances advert to internal representations. Computationalists take representations to be static configurations of symbol tokens. Dynamicists conce ive representations very differently. They find their representations among the kinds of entities that figure in DST, including parameter settings, system states, attractors, trajectories, or even aspects of bifurcation structures ( e.g., Petitot, 1985a). Currently, most dynamicists make use of only the tip of the theoretical iceberg that is dynamics. As dynamical modeling increases in mathematical sophistication, we can expect representations to take even more exotic forms.

.2.3.9 Anti-represent ationalism.Unlike digital computers, dynamical systems are not inherently representational. A small but influential contingent of dynamicists have found the notion of representation to be dispensable or even a hindrance for their particular purposes. Dynamics forms a powerful framework for developing models of cognition which sidestep representation altogether. The assumption that cognition must involve representations is based in part on inability to imagine how any non-representational system could possibly exhibit cognitive performances. Within the dynamical approach, such systems can be not only imagined, they can be modelled and constructed (see, e.g., Beer, 1995a; Beer, 1995b; Freeman & Skarda, 1990; Harvey, 1992; Husbands, Harvey & Cliff, 1995; Skarda & Freeman, 1987; Wheeler, 1994).

4.3 The Dynamical Hypothesis, Exposed.

Summarizing these points yields the following compact formulationf the DH: For every k ind of cognitive performance exhibited by a naturalognitive agent, there is some quantitative system instantiated by the agent athe highest relevant level of causal organization, such that performances ofhat kind are behaviors of that system; in add ition, causal organization cannd should be understood by producing dynamical models, using the theoreticalesources of dynamics, and adopting a broadly dynamical perspective.

5. Considerations Favoring the Dynamical Hypothesis

What can be said in favor of the DH? Specific aspects of cognitionenerate idiosyncratic cases for dynamical treatment, but our interest here isn general considerations. Space limits preclude complete coverage, but the ollowing arguments are among the most important.[Note 16].

Most obviously, there is a kind of empirical success argument,aralleling Newell and Simon's primary argument for the CH.[Note 17]. It starts from the impressive track record of dynamics itself. Dynamics is arguably the most widely used and powerful explanatory framework in science. An extraordinary range of natural phenomena have turned out to be best des cribed as - i.e., to be - a matter of interdependent coevolution of quantitative variables. It would hardly be surprising if dynamics found application in the study of cognition as well. Michael Turvey for one has long been arguing that the proper road to a deep understanding of natural cognition is to strive patiently to extend and apply the tried and true techniques of natural science to incrementally more complex biological and cognitive phenomena (see, e.g., Swenson & Turvey, 1991; Turvey & Care llo, 1981).

The empirical success argument in the form just presented has littleeight on its own, for cognition differs from other phenomena in importantays. Its force really comes into play when combined with evidence of successn cognitive scie nce itself. There is now a considerable amount of suchvidence, some of which has already been cited. Of course, the claim is nothat there is now sufficient empirical evidence to establish theupremacy of the DH. Indeed, there are numerous aspe cts of cognition for which,onsidered in isolation, the case for dynamical treatment is currently weak atest. The argument is that such successes as do exist, in conjunction with theeneral track record of dynamics, augurs well for the DH. The two lev els of thergument require and reinforce each other.

What explains any success the dynamical approach has exhibited thus far?nd what underpins confidence that there will be more? The foremostonsideration is simply that natural cognition happens in real time.his blunt fact is multi-faceted. Every cognitive process unfolds in continuousime, and the fine temporal detail calls out for scientific accounting.oreover, many cognitive structures are essentially temporal: liket terances, they exist only as change in time. Often, getting the timing rights critical to the success of cognitive performance; this is especially so whenn direct interaction with surrounding events.

Hobbesian computational models have made a bet that cognitive phenomena can beescribed in a way that abstracts away from the full richness of real time,eplacing it with discrete orderings over formal states. From a dynamicalerspective, this looks ill-advised. Dynamics, by contrast, takes the nat ure ofhange in time as its primary focus. It is the preeminent mathematicalramework for description of temporal phenomena. Taking cognitive agents to beynamical systems allows scientific explanation to tap into this power.

A third argument focus es on the embeddedness of cognition. Even the loftiest forms of natural cognition are in fact embedded three times over: in a nervous system, in a body, and in an environment. Any account of cognition must eventually explain how it is that cognitio n relates to that which grounds and surrounds it. Now, suppose the behavior of brain, body and environment all turn out to be best described in dynamical terms. Suppose, in short, that cognition is thoroughly embedded in dynamics. The challenge would then be to explain how cognitive phenomena are constituted of, shaped by and interact with those dynamical phenomena. While explaining embeddedness is never trivial, it stands to reason that there will be greater problems in relating systems of fundamentally different kinds than in relating systems of fundamentally the same kind. Mainstream computational cognitive science has for the most part simply shelved problems of embeddedness, preferring to study cognition independently of its neurobiological realizati on, and treating the body and environment as belonging on the far side of occasional symbolic inputs and outputs. When embeddedness is confronted head-on, dynamical accounts of cognition immediately become attractive. For example, one virtue of the Cather ine Browman and Louis Goldstein dynamical phonology (Browman & Goldstein, 1992) is that it integrates directly with Elliot Saltzman's dynamical model of speech coordination (Saltzman & Munhall, 1989). Dynamical cognition sits comfortably in a dynami cal world.

A fourth argument focuses on the emergence and stability ofognition. Investigation of some complex phenomenon can always take at leastwo directions: what is it like? And, how does it get - and stay - that way? Inhe case of cognitive mechanisms and processes, we can address their nature, orow it is that they arise and are sustained. In the long run, our answers tohese questions must hang together. Natural cognitive agents exhibitxtraordinary levels of structural com plexity, yet there are no architects orngineers responsible for building and maintaining that structure. The genericame for the answer to the problem of the emergence and stability of cognitions self-organization. Self-organization of interes ting kinds of complexrder appears to require systems in which there is simultaneous, mutuallyonstraining interaction between large numbers of components. DST is theominant mathematical framework for describing the behavior of such systems. Inhort, the claim is that we must understand cognitive agents as dynamicalystems, since only in that way will our account of what cognition is beroperly integrated with our account of how the world sustains any of it.

Each of these lines of thought was c ast in the form of an attempt toemonstrate that the DH is basically true. With respect to that goal,hey are obviously not "knock down" arguments. They do, however, indicate thathe hypothesis is worthy of sustained empirical investigation of p recisely theind that has been and is being conducted, and which forms the basis of theormulation of the DH presented here.

6. The General Objections

This section considers a selection of general ob jections to the DH as anpen empirical hypothesis. As John Stuart Mill said, "three-fourths of therguments for every disputed opinion consist in dispelling the appearanceshich favour some opinion different from it."[Note 18]Addressing these objections is also a useful way to elaborate and clarify the hypothesis.

The objections considered fall into two main categories: those purporting to show that the DH is not a genuine alternative to the CH, and those pur porting to show that it is not open, i.e., its empirical inadequacy is somehow already determined. All amount to sweeping attempts to dismiss or downplay the DH in advance of detailed empirical investigation. All mix insight with confusion to produ ce plausible but misguided attacks.

6.1 The "Trivially True" Objection

Everything is a dynamical system. Cognitive agents must be dynamical systems at some level. The DH is trivially true, and m akes no substantial claim about the nature of cognition.

On one hand, according to the nature hypothesis, cognitive agents instantiate quantitative systems at the highest relevant level of causal organization. It may be trivial that every cognitive agent instantiates some dynamical system or other. It is certainly not trivial that every cognitive performance is at the highest level a dynamical phenomenon. This is not true of ordinary digital computers, and according to the orthodox CH, it is not true of people.

On the other hand, according to the knowledge hypothesis, cognition can be understood in dynamical terms. If this were trivially true, cognitive science would have been completed long ago. In practice, it is very challenging to establish that some aspect of cognition can be understood dynamically. Patient steps in this direction are the stuff of which whole careers are made. Some of the greatest achievements in science have amounted to describing some natural phenomenon (e.g., celestial motion) in dynamical terms. This activity is no more trivial in cognitive science than anywhere else.

6.2 The "False Opposition" Objection 1 - Computers are Dynamical Systems

Ordinary electronic computers are dynamical systems. In general, digital computers are dynamical systems as well. The DH is therefore not an interesting alternative to the CH.

This objection gains plausibility by mixing together at least three distinct lines of t hought. Each is based on a different reason for thinking that digital computers are dynamical systems. Each has elements of truth but also problems.

6.2.1 Digital computers are state-determined, rule-governed, etc.

6.2.2 Digital computers are quantitative systems.

Digital computers and dynamical systems are two classes of systems picked out by reference to different properties: roughly, effectiveness and interpretation as opposed to quantitativeness. Generally, systems exhibiting the one property fail to e xhibit the other and vice versa. In a typical Turing Machine, for example, there is no systematic relationship between system behavior and distances between states. A tape square's values are different but not relevantly distant from each ot her. System behavior turns only on which values happen to obtain (i.e., type identity), not on how far those values are from any others. Similarly in the case of time. Turing Machine states are indexed by means of the positive integers. There are distance s between integers, but these distances generally bear no systematic relationship to system behavior. The integers might just as well be replaced by any other sufficiently large merely ordered set, such as names in the New York telephone directory.

S ince there are generally no relevant distances in state or time in digital computers, it makes no sense to describe their behavior in terms of rates of change (not to mention dependence on rates of change). This is why in practice computer scientists don' t bother with distances between states, rates of change, etc.

There is a common temptation to suppose that digital computers count as quantitative systems arising from the correct observation that certain metrics apply to any set of values, regardles s of the nature of those values (e.g., (Padulo & Arbib, 1974), pp.91-2). Thus every variable is a quantity, and so even digital computers have metric spaces as state sets. The crucial point, however, is that the distances measured by these trivi al metrics bear no systematic relationship to system behavior. Turing Machines bounce around their state spaces in ways which will seem utterly erratic until one realizes that their order is based on formal properties, not quantitative properties.ranges come in many kinds. Some are valencia, some are expensive; occasionally, an orange is both. Similarly with digital computers and dynamical systems. In coincidental, contrived, or trivial cases, one and the same set of variables might satisfy the conditio ns for both classes. Nevertheless, digital computers and dynamical systems are classes of systems picked out by reference to fundamentally different properties. In general, systems exhibiting one property fail to exhibit the other.

6.2.3 Digital computers are dynamical systems at the hardware level.

6.3 The "False Opposition" Objection 2 - Dynamical Systems are Computers

Much recent research in computation t heory has been exploringhe computational power of dynamical systems. There is no inherent conflictetween dynamics and computation, and so there is no real opposition betweenhe computational and dynamical hypotheses.

It is true t hat there is no inherent conflict between dynamics andomputation, but the conclusion does not follow. Again, the issues must beeased out more carefully.

Recall from Section 3.2 that effective computation is a specific kind of computation, resultin g from a certain kind of constraint on the processes involved. Other kinds of computation result from adopting different constraints. In particular, we can focus attention on some class of dynamical systems (Blum, Cucker, Shub, & Smale, forthcoming; Bl um, Shub, & Smale, 1989; Moore, 1991; Moore, 1996). As long as there is some way to specify the "questions" and "answers" we can see dynamical processes as computing functions. For example, Hava Siegelmann has extensively studied the computational prop erties of one class of dynamical systems, recurrent neural networks (Siegelmann & Sontag, 1994). Indeed, it can be proved that certain classes of dynamical systems are more powerful - can compute a wider class of functions - than Turing Machines.[Note 19] So, dynamical systems can compute, i.e., be computers, without needing to be digital computers. This is why research into the power of dynamical systems an interesting new branch of computation theory!

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 digit al 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 perspec tives, 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 exces s baggage.

6.4 The "False Opposition" Objection 3 - Dynamical Systems are Computable

There is no good reason to think that any cognitive process isot effectively computable. Even if cognitive ag ents are dynamical systems,hey will still be computable systems. Therefore, it is misguided to presenthe DH as an alternative to the CH.

One particularly troublesome mistake is blurring the distinction betweencomputational and computable. Just as employers and employeestand at opposite ends of an employment contract, so computational andcomputable stand at opposite ends of the relation computes. Theormer applies to whatever does the computing; the latter to whatevergets computed. In classical theory, a digital computer does theomputing, and a function over the integers gets computed. The effectivelyomputable functions over the integers are all and only the partial recursi veunctions.

Computation theorists, including Turing himself, quickly turned to asking what else might be effectively computed. Via arbitrarily good approximation, the purview of effective computation was gradually extended to embrace real nu mbers, functions over real numbers, differential equations, and so on (Earman, 1986; Grzegorczyk, 1957; Turing, 1936). In this way, issues of effective computability can be raised for all the standard mathematical constructs of analysis and physics. Just 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 isoverned by some computable function . The solar system of classical mechanicss effectively computable in this sense. Currently, as far as we can now see,ost if not all dynamical systems of practical relevance to cognitive sciencere effectively computable.[ Note 20] This doesn'take those systems digital computers. Digital computers can compute functionsoverning systems which are not themselves digital computers. Thus, theomputability of dynamical models does not destroy the contrast between theyn amical and computational hypotheses.

6.5 The "Straw Man" Objection

Turing Machines are caricatures of computers. The DH is beingatched against a straw man. It is not a substantial alternative to the CH asroperly understood.

There are two issues here. One is whether the CH, as characterized here,s a straw man. Two considerations suffice to dispel this objection. First, theharacterization offered here is just the standard philosophical account, aseveloped in numerous places.[Note 21].econd, a great many models in cognitive science (e.g., those developed withinhe SOAR (Newell, 1991) framework) do in fact conform to that accoun t.

The other issue is whether the standard account misunderstands the "true" CH, i.e., deeply misconceives computers and computational modeling in cognitive science. This may be; Brian Smith, for one, has begun formulating a critique of received wisdo m in this area (Smith, 1996; Smith, forthcoming). These issues go beyond the scope of the present discussion. If and when some superior understanding of the CH clearly supplants the orthodox account, the relationship between the dynamical and computationa l hypotheses will need to be reconsidered.

6.6 The "Description, Not Explanation" Objection

Dynamical models are at best descriptions of the data, and doot explain why the data take the form th ey do. For genuine explanation, weeed computational models describing the underlying causal mechanisms.

Dynamical theories of cognitive processes are deeply akin to dynamicalccounts of other natural phenomena such as celestial mot ion. Those theoriesonstitute paradigm examples of scientific explanation. Consequently, there iso reason to regard dynamical accounts of cognition as somehow explanatorilyefective.

ynamical explanations typically proceed by providing equations d efining anbstract model. Many factors are relevant to the goodness of a dynamicalxplanation, but it should at least capture succinctly the relations ofependency, and make testable predictions. A poor dynamical account maymount to little mor e than ad hoc "curve fitting", and would indeedount as mere description. It's problem, however, is that it is poor, not thatt is dynamical.

raditional computational cognitive science offers explanations of a quite distinctive kind (Haugelan d, 1978), and many cognitive scientists have become so accustomed to such explanations that anything else seems inadequate. The explanations offered in dynamical cognitive science are indeed quite different (Garson, 1996; van Gelder, 1991), but are not fo r that reason inferior.

6.7 The "Not As Cognitive" Objection

Dynamics is a general purpose framework which applies to anyehavior of an agent, regardless of whether that behavior is cognitive o r not.ynamics does not focus on the specifically cognitivespects of systems; it does not explain cognitive performances "as cognitive."enuine explanation in cognitive science must be framed in terms of aspects ofognitive agents other than their purely dynamical properties.

This objection concedes that dynamical explanations are nontrivialmpirical explanations, and that they really are quite different fromomputational explanations. It challenges the natu re of the explanation beingffered. Dynamics is held to be too general, failing to explain cognition inerms of its distinctive features.

nderlying this objection is an important misconception about the DH. That hypothesis asserts that cognitive age nts are dynamical systems of quite special kinds. Therefore, as emphasized in Section 4.2.2, understanding cognitive agents as dynamical systems is not simply the routine application of generic dynamics to systems that happen to be exhibiting cognitive pe rformances. It requires that the resources of dynamics be developed and supplemented in order to provide explanations of those special kinds of behaviors. Thus, dynamical cognitive science always incorporates considerations distinctive to particular kinds of cognition into dynamical frameworks to produce explanations that are fundamentally dynamical in form, but are nevertheless tailored to explain cognitive performances "as cognitive." To take just one example, Jean Petitot merges Ron Langacker's cogniti ve grammar with RenŽ Thom's morphodynamics to yield a thoroughly dynamical approach to syntax (Petitot, 1995).

6.8 The "Wrong Level" Objection

There is an important role for dynamical descriptio ns in anyomplete account of the nature of a cognitive agent, but they are pitched tooow to explain cognition.[Note 22]

A common misconception about the dynamical approach is that it operatesolely or primarily at "lower" or "micro" levels of description. In fact,ynamics is not intrinsically limited to any level or domain. In theatural sciences, dynamics finds application at all levels from quantumechanics to cosmology. It gets its grip wherever sets of interdependentlyhanging quantities are found. Similarly in cognitive science: dynamicistsevelop their explanations at the level of theoretical interest, whatever thatight be (see Section 4.1).

ne significant dif ference between the dynamical approach and PDP-styleonnectionism turns on this point. They agree that cognitive performances areehaviors of dynamical systems. The PDP approach, however, takes those systemso be high-dimensional neural networks operat ing at a level below that ofrthodox descriptions (Smolensky, 1988); as expressed in the titles of the famous volumes,[Note 23]. they constitute themicrostructure of cognition. The dynamical approach is more catho lic; itmbraces dynamical models of all kinds and at all levels.

6.9 The Structure Objection

Sophisticated cognitive performances require complex internaltructures. The dynamical approach is taking a huge step backwards in trying toeplace symbolic representations with quantities. To explain high levelognition, dynamical systems will have to implement computationalechanisms.

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 might take. Hobbesian cognitive scientists are bank ing 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 k inds of complex structures required cannot exist in any system except by instantiating digital symbol structures.

owever, as dynamical cognitive science has matured, it has become apparent that dynamical systems can incorporate combinatorial structu res 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 processi ng (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 chainin g", (van Gelder & Port, 1994)). The possibilities have really only begun to be explored. The dynamical approach is not vainly attempting to do without complex internal structures. Rather, it is in the process of dramatically reconceiving how they might be instantiated.

6.10 The Complexity Objection

Natural languages are only effectively described by some formf context-sensitive grammar. In the standard Chomskian hierarchy, languages ofhi s complexity can only be handled by computers at least as powerful asinear-bounded automata (LBAs). Therefore, natural language speakers must beomputers at least as powerful as LBAs.

The conclusion of this argument is ambig uous, between computers ineneral and digital computers. On the former interpretation, the argument isound, but fails to conflict with the DH. It was pointed out above thatynamical systems can compute, i.e., be computers. The complexity of naturala nguage constrains speakers' computational power, but not the kindof computer they instantiate. It remains an open empirical question whetherhe computers in question are best thought of as digital or dynamical (Elman, 1995).

n the lat ter interpretation, the argument simply equivocates. The premisesstablish that speakers must be computers in some sense; the conclusionlaims they must be digital computers. The dominance of digital computers inhe theory of computation, cognit ive science, and computer technology, hasreated an unfortunate tendency to confuse computers in general with digitalomputers. This is what drives the objection.

6.11 "The Not Cybernetics Again!" Objection

The dynamical approach is just cybernetics returning from theead.

What was cybernetics? Wiener famously defined it as "the science ofommunication and control in man and machine" but it soon developed into an ven wider enterprise: a kind of general, non-reductionistic study ofsystems, particularly self-sustaining systems in their environmentssee, e.g., (Parsegian, 1973)). Throughout its brief ascendancy, cybernetics enthusiastically embraced anythi ng of conceivable relevance to complex systems, including information theory, communication theory, automata theory, neurophysiology, systems theory, game theory and control theory.

Dynamics was certainly mixed up in all this, and the DH is sometimes tracedack to a leading cyberneticist, H. Ross Ashby. Still, the demise ofybernetics implies little about the contemporary dynamical approach, for theyiffer in important ways. The DH is, by comparison, tightly circumscribed. Its concerned with cog nition specifically, rather than systems generally, and isefined in terms of a core commitment to a single framework. The fate ofybernetics as a whole no more attaches to the dynamical approach than it doeso other disciplines with ancestral links to cybernetics, such as computationaleuroscience and artificial intelligence. Moreover, much more powerful toolsre available today. The bulk of DST has been developed in the period sincecybernetics. Also, dynamicists now have on their desks comp uter simulationools (hardware and software) beyond the dreams of cyberneticists. Whereyberneticists could only speculate, dynamicists can now furnish and understandomplex models.

6.12 The "Humans Compute" Objection

Humans can do arithmetic in their heads. At least someognitive activity is specifically digital computation.herefore, the DH cannot be the whole truth about cognition.

If it is granted that mental arithmetic and like processes are,iterally, digital symbol manipulation inside the head, then the DH shouldndeed graciously concede. The general truth of the DH is compatibleith certain special activities counting as excep tions. However, we should beary of granting, in advance, that mental arithmetic is symbolanipulation. Certainly, it seems like symbol manipulation: numerals,ines, etc. are "seen in the mind's eye". It does not follow that there areym bols in the head, i.e., that the states and processes that subserve suchseeing" actually instantiate symbols and their manipulations. Imagining theiffel Tower does not entail that one has the Eiffel Tower, or even a picturef it, inside one's head (R yle, 1984, Ch.8). We must not confuse the content of experience withhe mechanisms implementing it. As usual, the question turns out to be thempirical one: in the long run, what kind of models provide the best account ofhe mechanisms underlying the r elevant kind of cognitive performance?

7. Conclusion

The contemporary dynamical approach to cognition is part of a much widercientific trend. In recent decades, there have been dramatic developments i nhe mathematics of DST, especially the theory of nonlinear systems, complexity,nd chaos. At the same time, there has been exponential growth in availableomputing power, and the arrival of sophisticated programs for exploringynamical systems. The r esult is that dynamical theorising has come to bepplied to a wide range of natural phenomena that were previously eithergnored entirely, or regarded as beyond the scope of standard forms ofcientific explanation. So with cognition. The Humean dream o f a dynamics ofognition can now be seriously pursued. The explanatory umbrella which coverso much of the rest of the natural world so effectively is gradually beingxtended to cover cognition as well.

he DH encapsulates the core commitment of th e emerging dynamical approach.his target article has attempted to say what it means, and to establish itstatus as an open empirical hypothesis standing as a substantial alternative tohe CH. It has not attempted to demonstrate that cognitive agents are inact dynamical systems. There is mounting evidence that certain aspects ofognition are best thought of dynamically, but many others remain completelynaddressed. Only sustained empirical investigation will determine the extento which the DH - as opposed to the CH, or perhaps some other hypothesisntirely - captures the truth about cognition.

APPENDIX Table 2. Key terms and their meanings in the present discussion. This tableas no pretensions beyond partially summar izing the particular regimentationroposed in this paper for the purpose of clarifying the DH in cognitivecience.

ignificant improvements in this paper resulted from discussion with oreedba ck from many people, but among the most influential were John Haugeland,obert Port, Jim Townsend, Dan Dennett, Herbert Jaeger, Tim Smithers, Robertregson, Clark Glymour, Brian Smith, Jeff Pressing, Marco Giunti, Scott Kelso,nd BBS referees.

NOTES

Guiding Idea	Examples
1. system of bodies whose motions are governed by forces. Such systems formhe domain of dynamics considered as a branch of classical mechanics.	"a collection of a large number of point particles." p.215 Webster's: "dynamics...a branch of mechanics that deals with forces and theirelation primarily to the motion... of bodies of matter."
2. physical system whose state variables include rates of change	"In the original meaning of the term a dynamical system is a mechanical systemith a finite number of degrees of freedom. The state of such a system issually characterized by its position...and the rate of change of thiso sition, while a law of motion describes the rate of change of the state ofhe system." p.328
3. system of first-order differential equations; equivalently, a vector field on manifo ld	aynamical system is "simply a smooth manifold M, together with a vectorield v defined on M." p.109
4.apping on a metri c space	"Adynamical system is a transformation f:Z->Zon a metric space (Z, d)." p.134.
5.tate-determination	"aynamical system...is one whose state at any instant determines the state ahort time into the future without any ambiguity." p.188
6.ny mapping, equation, or rule.	Aynamical system may be defined as a deterministic mathematical prescriptionor evolving the state of a system forward in time." p.6
7.hange in time	"A dynamical system is one which changes in time." p.3 The term dynamic refers to phenomena that produce time-changingatterns...the term is nearly synonymous with time-evolution or pattern ofhange." p.1

Term	Meaningn this paper
Variable	Anythinghat changes over time..
System	Aet of va riables changing interdependently.
Instantiation	Aelation between a concrete system and some object or part of the world. Anbject instantiates a system when all the variables of the system are featuresf the object.
Implementation	Aelation between concrete systems, obtaining when the variab les of one systemre somehow built up out of the variables of the other.
Parameter	Somethingutside (i.e., not a member of) a system, but upon whic h change in the systemepends.
Coupling	Mutualirect dependence. Variables x and y are coupled when the statef x shapes chan ge in y and vice versa.
Concreteystem	Aystem whose variables are all concrete features of the concrete world changingn real time.
Abstractystem	Aystem whose variables are all abstract entities.
Realization	Aelation between a concrete system and an abstract one, obtaining when theormer has the same structure as the latter.
Time	Anyntrinsically ordered set, serving to provide orderings over other things. Realime is the set of instants at which things can happen, ordered by prioritybefore/after).
Computer	Anythinghat computes (carries out computation).
Computation/omputing	Transformingome kind of ques tion (e.g., input object or start state) into some kind ofnswer (e.g., output object or final state).
Computational	Anythinghat computes (carries out computation).
Digitalomputer	Aomputer carrying out effective computation over representations. A digitalomputer must have digital variabl es, discrete time, algorithmically governedehavior, and an interpretation.
Effective	Succeedingn a finite number of basic operations governed by an algorithm.
Computable	Capablef being computed; alternatively, being governed by a computable function.
Quantity	Aariable with a metric over its values.
Dynamicalystem	Auantitative system. A system that is at least quantitative in state; may alsoe interdependently quantitative in state and time, or even rate dependent.
Identity	"beinghe very same thing as". Identity is governed by Leibniz Law: identical thingsave all and only the same properties. Identity for sets-and hence forystems-is having all and only the same variables.
Simulate	Compute function approximately describing some process.
Dynamics	Twolosely related kinds of mathematics, dynamical modeling and DST.
Dynamicalypothesis (DH)	Cognitivegents are dynamical systems. See Section .
Computationalypothesis (CH)	Cognitivegents are digital computers.

References

1989) Encyclopaedia of Mathematics. Dordrech t: Kluwer.

braham, F. D., Abraham, R. H., & Shaw, C. D. (1992) Basic Principles of Dynamical Systems. In R. L. Levine & H. E. Fitzgerald ed., Analysis of Dynamic Psychological Systems, Volume 1: Basic Approaches to General Systems, Dynamic Systems, and Cybernetics. New York: Plenum Press,

braham, R., & Shaw, C. D. (1982) Dynamics-The Geometry of Behavior. Santa Cruz CA: Aerial Press.

mit, D. J. (1989) Modeling Brain Function: The World of Attractor Neural Networks. Cambridge: Cambridge Un iversity Press.

abloyantz, A., & Lourenco, C. (1994) Computation with Chaos: A Paradigm for Cortical Activity. Proceedings of the National Academy of Sciences of the USA, 91, 9027-9031.

aker, G. L., & Gollub, J. P. (1990) Chaotic Dynamics: An In troduction. Cambridge: Cambridge University Press.

arnsley, M. (1988) Fractals Everywhere. San Diego: Academic Press, Inc.

eer, R. D. (1995a) Computational and Dynamical Languages for Autonomous Agents. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press, 121-147.

eer, R. D. (1995b) A dynamical systems perspective on agent-environment interaction. Artificial Intelligence, 72, 173-215.

eltrami, E. (1987) Mathematics for Dyna mical Modeling. Boston: Academic Press Inc.

ingham, G. P., Rosenblum, L. D., & Schmidt, R. C. (in press) Dynamics and the orientation of kinematic forms in visual event recognition. Journal of Experimental Psychology: Human Perception and Performanc e.

lum, L., Cucker, F., Shub, M., & Smale, S. (forthcoming) Complexity and Real Computation: A Manifesto. Berlin: Springer-Verlag.

lum, L., Shub, M., & Smale, S. (1989) On a Theory of Computation and Complexity Over the Real Numbers: NP Complete ness, Recursive Functions and Universal Machines. Bulletin of the American Mathematical Society, 21, 1-49.

rowman, C. P., & Goldstein, L. (1992) Articulatory phonology: an overview. Phonetica, 49, 155-180.

ullock, D., & Grossberg, S. (1988) Neural dynamics of planned arm movements: Emergent invariants and speed-accuracy properties during trajectory formation. Psychological Review, 95, 49-90.

usemeyer, J. R., & Townsend, J. T. (1993) Decision field theory: a dynamic-cognitive approach to decisi on making in an uncertain environment. Psychological Review, 100, 432-459.

usemeyer, J. R., & Townsend, J. T. (1995) Dynamic representation of decision making. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press,

asti, J. L. (1992) Reality Rules: Picturing the World in Mathematics (Vols I, II). New York: J. Wiley.

hrisman, L. (1991) Learning recursive distributed representations for holistic computation. Connection Science, 3, 3 45-366.

hurchland, P. S., & Sejnowski, T. J. (1992) The Computational Brain. Cambridge MA: Bradford/MIT Press.

lark, A. (1989) Microcognition: Philosophy, Cognitive Science, and Parallel Distributed Processing. Cambridge MA: MIT Press.

lif f, D., Harvey, I., & Husbands, P. (1993) Explorations in evolutionary robotics. Adaptive Behavior, 2, 73-110.

ohen, A. (1992) The role of heterarchical control in the evolution of central pattern generators. Brain, Behavior and Evolution, 40, 112-124.

ohen, J., & Stewart, I. (1994) The Collapse of Chaos. New York: Penguin.

opeland, J. (1993) Artificial Intelligence: A Philosophical Introduction. Oxford: Blackwell.

esloge, E. A. (1982) Classical Mechanics. New York: John Wiley & Sons.

reyfus, H. L. (1992) What Computers Still Can't Do: A Critique of Artificial Reason. Cambridge MA: The MIT Press.

arman, J. (1986) A Primer on Determinism. Dordrecht: D. Reidel.

lman, J. (1995) Language as a dynamical system. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press,

odor, J. A. (1975) The Language of Thought. Cambridge MA: Harvard University Press.

odor, J. A., & Pylyshyn, Z. (1988) Connectionism and cog nitive architecture: a critical analysis. Cognition, 28, 3-71.

reeman, W. J., & Skarda, C. A. (1990) Representations: Who needs them? In J. L. McGaugh, J. L. Weinberger, & G. Lynch ed., Brain Organization and Memory: Cells, Systems and Circuits. New Y ork: Guildford Press, 375-380.

arson, J. (1996) Cognition poised at the edge of chaos: A complex alternative to a symbolic mind. Philosophical Psychology, 9, 301-321.

iunti, M. (forthcoming) Computation, Dynamics, and Cognition. New York: Oxford U niversity Press.

regson, R. (1995) Cascades and Fields in Perceptual Psychophysics. Singapore: World Scientific.

regson, R. A. M. (1993) Learning in the context of nonlinear psychophysics: The Gamma Zak Embedding. British Journal of Mathematic al and Statistical Psychology, 46, 31-48.

rossberg, S., & Gutowski, W. E. (1987) Neural dynamics of decision making under risk: affective balance and cognitive-emotional interactions. Psychological Review, 94, 303-318.

rossberg, S., & Rudd, M. E. (1992) Cortical dynamics of visual motion perception: group and element apparent motion. Psychological Review, 99, 78-121.

rossberg, S., & Stone, G. O. (1986) Neural dynamics of word recognition and recall: attentional priming, learning, and resonance . Psychological Review, 93, 46-74.

rzegorczyk, A. (1957) On the definitions of computable real continuous functions. Fundamenta Mathematica, 44, 61-71.

uckenheimer, J., Gueron, S., & Harris-Warrick, R. (1993) The dynamics of a conditionally bursti ng neuron. Philosophical Transactions of the Royal Society of London B, 341.

aken, H., & Stadler, M. (Ed.). (1990). Synergetics of Cognition. Berlin: Springer-Verlag.

arnad, S. (1990) The symbol grounding problem. Physica D, 42, 335-346.

ar vey, I. (1992) Untimed and misrepresented: Connectionism and the computer metaphor. University of Sussex Cognitive Science Research Paper No. 245. In

augeland, J. (1978) The nature and plausibility of cognitivism. Behavioral and Brain Sciences, 1, 21 5-26.

augeland, J. (1985) Artificial Intelligence: The Very Idea. Cambridge MA: MIT Press.

irsch, M. (1984) The dynamical systems approach to differential equations. Bulletin of the American Mathematical Society, 11, 1-64.

obbes, T. (1651/19 62) Leviathan. New York: Collier Books.

ock, H. S., Kelso, J. A. S., & Schšner, G. (1993) Bistability, hysteresis and loss of temporal stability in the perceptual organization of apparent motion. Journal of Experimental Psychology: Human Perception and Performance, 19, 63-80.

organ, T. E., & Tienson, J. (1996) Connectionism and the Philosophy of Psychology. Cambridge MA: MIT Press.

ume, D. (1978) A Treatise of Human Nature (1739-40). Oxford: Clarendon Press.

usbands, P., Harvey, I., & Cliff, D. (1995) Circle in the round: State space attractors for evolved sighted robots. Robotics and Autonomous Systems, 15, 83-106.

aeger, H. (1996) Dynamische Systeme in der Kognitionswissenschaft. Kognitionswissenschaft, 5, 151-174.

ampis, G . (1991) Self-Modifying Systems in Biology and Cognitive Science. Oxford: Pergamon Press.

aplowitz, S. A., & Fink, E. L. (1992) Dynamics of attitude change. In R. L. Levine & H. E. Fitzgerald ed., Analysis of Dynamic Psychological Systems, Volume 2: Methods and Applications. New York: Plenum Press, 341-369.

elso, J. A. S. (1995) Dynamic Patterns: The Self-organization of Brain and Behavior. Cambridge MA: MIT Press.

elso, J. A. S., DelColle, J., & Schšner, G. (1990) Action-perception as a p attern formation process. In M. Jeannerod ed., Attention and Performance XIII. Hillsdale NJ: Lawrence Erlbaum Associates, 139-169.

elso, J. A. S., Ding, M., & Schšner, G. (1992) Dynamic pattern formation: A primer. In J. E. Mittenthal & A. B. Baskin e d., Principles of Organization in Organisms. Reading, MA: Addison-Wesley,

rantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971) Foundations of Measurement. New York: Academic.

ugler, P. N., Kelso, J. A. S., & Turvey, M. T. (1980) On the c oncept of coordinate structures as dissipative structures: I. Theoretical lines of convergence. In G. E. Stelmach & J. Requin ed., Tutorials in Motor Behavior. Amsterdam: North Holland, 3-47.

ugler, P. N., Kelso, J. A. S., & Turvey, M. T. (1982) On th e control and coordination of naturally developing systems. In J. A. S. Kelso & J. E. Clark ed., The development of movement control and coordination. New York: Wiley, 5-78.

even, S. J., & Levine, D. S. (1996) Multiattribute decision making in context : a dynamic neural network methodology. Cognitive Science, 20, 271-99.

uenberger, D. G. (1979) Introduction to Dynamic Systems: Theory, Models, and Applications. New York: John Wiley & Sons.

addy, P. (1990) Realism in Mathematics. Oxford: Claren don Press.

cClelland, J. L., & Rumelhart, D. E. (1981) An interactive-activation model of context effects in letter perception: Part 1, an account of basic findings. Psychological Review, 88, 375-407.

cClelland, J. L., Rumelhart, D. E., & The PD P Research Group (Ed.). (1986). Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Vol. 2, Psychological and Biological Models. Cambridge MA: MIT Press.

ill, J. S. (1975) On Liberty. New York: Norton.

oore, C. ( 1991) Generalized shifts: Unpredictability and undecidability in dynamical systems. Nonlinearity, 4, 199-230.

oore, C. (1996) Dynamical Recognizers: Real-time Language Recognition by Analog Computers. No. 96-05-023, Santa Fe Institute.

pitsos, G. J. (forthcoming) Attractor gradients: architects of developmental organization. In J. L. Leonard ed., Identified Neurons: Twenty Five Years of Progress. Cambridge, MA: MIT Press,

ewell, A. (1980) Physical Symbol Systems. Cognitive Science, 4, 135-183 .

ewell, A. (1991) Unified Theories of Cognition. Cambridge MA: Harvard University Press.

ewell, A., & Simon, H. (1976) Computer science as empirical enquiry: Symbols and search. Communications of the Association for Computing Machinery, 19, 113 -126.

oelle, D. C., & Cottrell, G. W. (1996) In search of articulated attractors. In G. W. Cottrell ed., Proceedings of the 18th Annual Conference of the Cognitive Science Society. Mahwah: Lawrence Erlbaum, 329-334.

orton, A. (1995) Dynamics: An I ntroduction. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press,

tt, E. (1993) Chaos in Dynamical Systems. Cambridge: Cambridge University Press.

adulo, L., & Arbib, M. A. (1974) S ystem Theory: A Unified State-Space Approach to Continuous and Discrete Systems. Philadelphia: W.B. Saunders Co.

arsegian, V. L. (1973) This Cybernetic World of Men, Machines and Earth Systems. Garden City N.Y.: Doubleday & Co. Inc.

etitot, J. (1985a) Les Catastrophes de la Parole. Paris: Maloine.

etitot, J. (1985b) MorphogenŽse du Sens. Paris: Presses Universitaires de France.

etitot, J. (1995) Morphodynamics and Attractor Syntax. In R. Port & T. van Gelder ed., Mind as Motion: Ex plorations in the Dynamics of Cognition. Cambridge MA: MIT Press,

ollack, J. B. (1990) Recursive distributed representations. Artificial Intelligence, 46, 77-105.

ollack, J. B. (1991) The Induction of Dynamical Recognizers. Machine Learning, 7, 2 27-252.

ort, R., Cummins, F., & McAuley, J. D. (1995) Naive time, temporal patterns, and human audition. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: Bradford Books/MIT Press, 339-72.

ort , R., & van Gelder, T. J. (1995) Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press.

our-El, M., & Richards, J. (1989) Computability in Analysis and Physics. New York: Springer-Verlag.

ylyshyn, Z. W. (1984) Comp utation and Cognition: Toward a Foundation for Cognitive Science. Cambridge MA: Bradford/MIT Press.

osen, R. (1985) Anticipatory Systems. New York: Pergamon.

umelhart, D. E., McClelland, J. L., & The PDP Research Group (Ed.). (1986a). Paralle l Distributed Processing: Explorations in the Microstructure of Cognition. Vol 1: Foundations. Cambridge MA: MIT Press.

umelhart, D. E., Smolensky, P., McClelland, J. L., & Hinton, G. E. (1986b) Schemata and sequential thought processes in PDP mo dels. In J. L. McClelland, D. E. Rumelhart, & The PDP Research Group ed., Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Cambridge MA: MIT Press, 7-57.

yle, G. (1984) The Concept of Mind (1949). Chicago: University o f Chicago Press.

altzman, E. (1995) Dynamics and coordinate systems in skilled sensorimotor activity. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press,

altzman, E. L., & Munhall, K. G. (1989) A dynamical approach to gestural patterning in speech production. Ecological Psychology, 1.

chmidt, R. C., & Turvey, M. T. (1994) Phase-entrainment dynamics of visually coupled rhythmic movements. Biological Cybernetics, 70.

chšner, G., Zanone, P. G., & Kelso, J. A. S. (1992) Learning as a change of coordination dynamics: Theory and experiment. Journal of Motor Behavior, 24, 29-48.

earle, J. R. (1980) Minds, brains and programs. Behavioral and Brain Sciences, 3, 417-458.

iege lmann, H. T., & Sontag, E. D. (1994) Analog Computation via Neural Networks. Theoretical Computer Science, 131, 331-360.

karda, C. A., & Freeman, W. J. (1987) How brains make chaos to make sense of the world. Behavioral and Brain Sciences, 10, 161-195 .

mith, B. C. (1996) On the Origin of Objects. Cambridge MA: MIT Press.

mith, B. C. (forthcoming) The Middle Distance: On the Foundations of Computation and Intentionality.

mith, L. B., & Thelen, E. (1993) Dynamic Systems in Development: A pplications. Cambridge MA: MIT Press.

mithers, T. (1994) On Behavior as Dissipative Structures in Agent-Environment System Interaction Spaces. In Proceedings of Prerational Intelligence: Phenomenology of Complexity in Systems of Simple Interacting A gents, Zentrum fŸr InterdisziplinŠre Forschung (ZiF), University of Bielefeld, Germany.

molensky, P. (1988) On the proper treatment of connectionism. Behavioral and Brain Sciences, 11, 1-74.

wenson, R., & Turvey, M. T. (1991) Thermodynamics reaso ns for perception-action cycles. Ecological Psychology, 3, 317-348.

abor, W., Juliano, C., & Tanenhaus, M. (1996) A dynamical system for language processing. In Proceedings of the 18th Annual Meeting of the Cognitive Science Society. Hillsdale, NJ: La wrence Erlbaum,

helen, E. (1995) Time scale dynamics and the development of an embodied cognition. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press,

helen, E., & Smith, L. B. (199 3) A Dynamics Systems Approach to the Development of Cognition and Action. Cambridge MA: MIT Press.

hom, R. (1983) Mathematical Models of Morphogenesis. Chichester: Ellis Horwood.

uller, B., Case, P., Mingzhou, D., & Kelso, S. J. A. (1994) The Nonlinear Dynamics of Speech Categorization. Journal of Experimental Psychology: Human Perception and Performance, 20, 3-16.

uring, A. (1936) On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematic al Society, Series 2, 42, 230-65.

uring, A. (1950) Computing machinery and intelligence. Mind, 59, 433-460.

urvey, M. T. (1990) Coordination. American Psychologist, 45, 938-953.

urvey, M. T., & Carello, C. (1981) Cognition: the view from ecolo gical realism. Cognition, 10, 313-321.

urvey, M. T., & Carello, C. (1995) Some Dynamical Themes in Perception and Action. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognition. Cambridge MA: MIT Press,

allacher , R., & Nowak, A. (Ed.). (1993). Dynamical Systems in Social Psychology. New York: Academic Press.

an Gelder, T. J. (1990) Compositionality: a connectionist variation on a classical theme. Cognitive Science, 14, 355-384.

an Gelder, T. J. (1991) Connectionism and dynamical explanation. In Proceedings of the Thirteenth Annual Conference of the Cognitive Science Society. Hillsdale NJ: Erlbaum, 499-503.

an Gelder, T. J. (1995) What might cognition be, if not computation? Journal of Philosophy, 91, 345-381.

an Gelder, T. J., & Port, R. (1994) Beyond symbolic: Towards a Kama-Sutra of compositionality. In V. Honavar & L. Uhr ed., Symbol Processing and Connectionist Network Models in Artificial Intelligence and Cognitive Modelling: Steps Toward Principled Integration. San Diego: Academic Press, 107-25.

an Gelder, T. J., & Port, R. (1995) It's About Time: An Overview of the Dynamical Approach to Cognition. In R. Port & T. van Gelder ed., Mind as Motion: Explorations in the Dynamics of Cognit ion. Cambridge MA: MIT Press,

on Bertalanffy, L. (1973) General System Theory: Foundations, Development, Applications. Harmandsworth: Penguin.

arren, W. H. (1995) Self-motion: Visual perception and visual control. In W. Epstein & S. Rogers ed., Handbook of Perception and Cognition, v.5: Perception of Space and Motion. New York: Academic Press, 263-325.

heeler, M. (1994) From activation to activity: representation, computation and the dynamics of neural network control systems. Artificial In telligence and Simulation of Behavior Quarterly, 36-42.

ildgen, W. (1982) Catastrophe Theoretic Semantics: An Elaboration and Extension of RenŽ Thom's Theory. Amsterdam: John Benjamins Publishing Company.

ak, M. (1990) Creative dynamics approach to neural intelligence. Biological Cybernetics, 64, 15-23.

[Note1] Examples: cognitive neuroscience: (Amit, 1989; Babloyantz & Lourenco, 1994; Cohen, 1992; Guckenheimer, Gueron, & Harris-Warrick, 1993; Mpit sos, forthcoming; Skarda & Freeman, 1987); psychophysics: (Gregson, 1995); perception: (Bingham, Rosenblum, & Schmidt, in press; Grossberg & Rudd, 1992; McClelland & Rumelhart, 1981; Port, Cummins, & McAuley, 1995); motor control: (Bullock & Grossberg, 1988; Saltzman, 1995; Turvey, 1990); developmental psychology: (Smith & Thelen, 1993; Thelen & Smith, 1993); cognitive psychology: (Busemeyer & Townsend, 1993; Grossberg & Gutowski, 1987; Grossberg & Stone, 1986; Leven & Levine, 1996; Tabor, Juliano, & Tanenhaus, 1996); situated robotics and autonomous agents research: (Beer, 1995b; Cliff, Harvey, & Husbands, 1993; Smithers, 1994); artificial intelligence (Jaeger, 1996; Pollack, 1991); social psychology (Kaplowitz & amp Fink, 1992; Vallacher & Nowak, 1993); ecological psychology (Kugler, Kelso, & Turvey, 1980; Kugler, Kelso, & Turvey, 1982; Turvey & Carello, 1995); synergetics (Haken & Stadler, 1990); morphodynamics: (Petitot, 1985b; Thom, 1983; Wildge n, 1982). (Port & van Gelder, 1995) is a representative sampling of the dynamical approach. Note that works cited here are intended as examples and pointers, rather than any kind of exhaustive or definitive listing.

[Note 2]The basic Haken-Kelso-Bunz equation is

Here phi is the single "collective" state variable of the system; in the finger coordination model, it corresponds to the oscillation phase of oneing er relative to the other. The equation specifies how relative phase changess a function of its current value. a and b are parameters ofhis system; their ratio corresponds to the rate of wagging of the fingers. Thequation is such that gradual changes in a and b can yield justhe kind of qualitative changes in relative phase found in the behavior of realubjects.

This simple "frictionless" equation is altered in various ways to generate models with better fit to exp erimental data. For example, fluctuations and symmetry-breaking considerations are accomodated by adding terms for noise and for differences in frequency between limbs and metronome

[Note 3] These include aspects of mo tor skillearning (Schšner, Zanone, & Kelso, 1992), interpersonal coordination (Schmidt & Turvey, 1994), speech perception (Tuller, Case, Mingzhou, & Kelso, 1994), and visual perception (Hock, Kelso, & Schšner, 1993). See (Kelso, 1995) for an overview.

[Note 4] This definition accords with ordinary usage (e.g., Websters Dictionary: "a regularly interacting or interdependent group of items forming a unified whole") and systems theory (e.g., "a set of elements standing in interrelations," (von Bertalanffy, 1973), p.55). The stance on the metaphysical status of sets adopted here is the "set-theoretic realism" elaborated in (Maddy, 1990). In this account, sets of physical entities are themselves physical entitie s, as much part of the ordinary world as planets, people and PCs.

[Note 5] In set theory, set identity is a matter of having exactly the same members. A set of sets is not identical with the set of the elements of thos e sets. Thus, strictly speaking, a set of pairs of socks is not identical with the set of socks belonging to those pairs. Of course, there is still an obvious and important sense in which these sets are the same. In this paper, this sense is captured by t he notion of implementation

[Note 6]A non-empty set X is an order, or is ordered, if there is a relation > over its elements with the property that for each x,y Î X, either x < y, or y < x,, or x = y

[Note 7] For expressions of this consensus see, for example, (Clark, 1989; Copeland, 1993; Dreyfus, 1992; Fodor, 1975; Fodor & Pylyshyn, 1988; Newell, 198 0; Newell & Simon, 1976; Pylyshyn, 1984). The version of this consensus now most widely accepted as definitive is probably that laid out in (Haugeland, 1985). The account of digital computers here is essentially just Haugeland's definition of computers as interpreted automatic formal systems as massaged into the present framework.

[Note 8] See (Haugeland, 1985), Chapter 2. In abstract systems, discreteness of values suffices for digitality.

[Note 9] What is it to "make sense"? This is a difficult issue; see (Haugeland, 1985), Chapter 3, for discussion. Every digital system can be set up in systematic correspondence with some domain (such as integers and functions over them) but not all such systems have an interpretation in the current sense. The ones that do are those exhibiting a further kind of order that does or could seem patterned or reasonable to us (humans); thus, whether something is a digital computer is human-rela tive.

ote that having an interpretation in the current sense may not be enough to guarantee that the system has "meaning" in some stronger sense, (and hence, perhaps, "mind"). For discussion of these issues, see Harnad (1990) and Searle (1980).

[Note 10] Recently, philosophers have begun to repair this neglect. See, for example, (Giunti, forthcoming; Horgan & Tienson, 1996; van Gelder, 1995; van Gelder & Port, 1995) for discussion more or less closely related t o the current issues.

[Note 11] A metric over a set X is a function thatd:XxX ->R that assigns to every pair of elements and y a number d(x,y)>=0 such thatd (x,y)=0 iff x =y,i>d(x,y) = d(y,x), andd(x,y)<= d(x,z) +d(z,y).

[Note 12] This fo rmulation is designed to accomodate some rather special cases of dynamical systems whose behavior is generally quantitative except at certain isolated points (Gregson, 1993; Zak, 1990).

[13] The concept of dynamica l system changes over time, in cognitive science as elsewhere. Future developments might prompt broadening of the current definition. For example, cognitive scientists may come to use as models systems whose state sets are not metric spaces, but do po ssess some other kind of interesting topological structure relevant to system behavior.

[Note 14] For introductions to dynamical modeling, see (Beltrami, 1987; Luenberger, 1979). For introductions to dynamical system s theory, see (Abraham & Shaw, 1982) or (Baker & Gollub, 1990). (Abraham, Abraham, & Shaw, 1992; Kelso, Ding, & Schšner, 1992; Norton, 1995) are chapter length overviews of dynamics for cognitive scientists.

[ Note 15] If done poorly, on the other hand, it is little more than handwaving with impotent metaphors. The jargon of dynamics does, unfortunately, provide all too many opportunities for pseudo-scientific masquerading.

[Note 16] Discussion of a wider range of considerations is found in (van Gelder & Port, 1995).

[Note 17] In their celebrated paper "Computer Science as Empirical Enquiry," Newell and Simon argue for the compu tational hypothesis primarily on the basis of the success of AI in producing intelligent computers, and the success of computational cognitive science in modeling cognition. The only other argument they mention is "the absence of specific competing hypoth eses." See (Newell & Simon, 1976).

[Note 18] (Mill, 1975) Chapter 2. In "Computing Machinery and Intelligence" (Turing, 1950), Turing rebuts nine objections to his stance on whether computers can think; most are not attributed to anyone in particular. This paper follows these august precedents. Except where noted, the objections are not known to have appeared in print; rather, they are based on the author's experience of reactions to the dynamical hypothesis when ex pounded in public presentations or in related work.

[Note 19] The general result that dynamical systems can have "super-Turing" capacities need not be very surprising. Digital computers are a strictly delimited clas s of systems, and it makes sense that classes defined by alternative sets of constraints would allow more powerful processes.

[Note 20] Note that effectively computable is a theoretical notion; it is not the s ame as computable in practice. As chaos theory reminds us, some systems will always outstrip our finite computing resources.

[Note 21] See note 7.

[Note 22] The "Peripheral " objection is very similar, and is dealt with by a similar response. It maintains that dynamical explanations are concerned with peripheral aspects of cognitive agents rather than cognition itself, which is more "central."

[Note 23] (McClelland, Rumelhart, & The PDP Research Group, 1986; Rumelhart, McClelland, & The PDP Research Group, 1986a).