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The Modularity of Dynamic Systems

by Teed Rockwell

(Published in Colloquia Manilana Vol 6 1998)

A greatly revised version of this paper, entitled Attractor Spaces as Modules: a Semi-Eliminative Reduction of Symbolic AI to Dynamic Systems Theory was published January 2005 in Minds and Machines


What is this Thing Called Modularity?

To some degree, Fodor's claim that Cognitive science divides the mind into modules tells us more about the minds doing the studying than the mind being studied. The knowledge game is played by analyzing an object of study into parts, and then figuring out how those parts are related to each other. This is the method regardless of whether the object being studied is a mind or a solar system. If a module is just another name for a part, then to say that the mind consists of modules is simply to say that it is comprehensible. Fodor comes close to acknowledging this in the following passage.

The condition for successful science (in physics, by the way, as well as psychology) is that nature have joints to carve it at: relatively simple subsystems which can be artificially isolated and which behave, in isolation, in something like the way that they behave in situ. (Fodor 1983 p.128)

If this were really as uncondititionally true as Fodor implies in this sentence, Fodor's Modularity of Mind would have been a collection of tautologies. In fact, Fodor goes to great lengths to show that his central claims are not tautologies, but rather reasonable generalizations from what has been discovered in the laboratory at this point. Fodor gets more specific in passages like this one:

One can conceptualize a module as a special purpose computer with a proprietary data base, under the conditions that a) the operations that it performs have access only to the information in its database (together of course with specifications of currently impinging proximal stimulations) and b) at least some information that is available to at least some other cognitive processes is not available to the module. (Fodor 1985 p. 3)

Fodor sums up these two conditions with the phrase informationally encapsulated, and adds that modules are by defintion domain specific i.e. they deal only with a certain domain of cognitive problems. He also claims that what we call higher cognitive functions cannot be either informationally encapsulated or domain specific. Instead, these higher processes must be "isotropic", (i.e. every fact we know is in principle relevant to their success) and "Quinean" (they must rely on emergent characteristics of our entire system of knowledge.) Because modular processes are by definition neither Quinian nor isotropic, there is "a principled distinction between cognition and perception" (ibid), and given the conceptual tools of cognitive science, it is not possible to have a science of the higher "Quinian-Isotropic" cognitive functions, such as thought or belief. He admits, however, that this limitation may only be true "of the sorts {of computational models} that cognitive sciences are accustomed to employ"(p.128). An examination of the presuppositions of those computational models may reveal this to be a limitation of only one particular kind of cognitive science.

Fodor and the Symbolic Systems Hypothesis

I believe that these limitations do hold for the paradigm that gave birth to cognitive science, which is often called the symbolic systems hypothesis. It's most fundamental claim is that a mind is a computational device that manipulates symbols according to logical and syntactical rules. All computers and computer languages operate by means of symbolic systems, so another way of phrasing this claim is to say that a mind is a kind of computer. The symbolic systems hypothesis is still basically alive and well, but now that it is no longer universally accepted it is often given disparaging nicknames, like Haugeland's "GOFAI" (for good old fashioned artificial intelligence) or Dennett's "High church computationalism". Fodor remains the most articulate preacher of the gospel of high church computationalism, and when his concept of modularity goes beyond the tautologous claim that minds are analyzable, it almost always brings in strong commitments to the claim that minds are really computers. His modules are really what computer programmers call subroutines, which is why he defines modules in the quote above as "special purpose computers with proprietary data bases." GOFAI scientists model cognitive processes by breaking them down into subroutines by means of block diagrams, then breaking those subroutines down into simpler subroutines and so on until the entire cognitive process has been broken down into subroutines that are simple enough to be written in computer code. ( See Minsky 1985 to see this process in action.). Most of what Fodor says about modules follows from the fact that subroutines are domain specific and informationally encapsulated ( i.e. they are each designed for specific tasks, and only communicate with each other through relatively narrow input-output connections), and the fact that Fodor believed (and apparently still believes) that GOFAI is the only game in town for AI. When Fodor says that it is impossible to model Quinean and isotropic properties, what he really means is that it is impossible to model them with the conceptual tools of GOFAI, and in this narrow sense of "impossible" he is probably right.

Dynamic Systems Theory

as an Alternative Paradigm

This paper will deal with whether there are similar constraints on the new sorts of models currently available to cognitive science, which were not available when Modularity of Mind was written. Fodor probably thinks that these constraints are as unassailable as ever, but one must remember that Fodor also believes (at least some of the time) that all of our concepts are innate. If there is any truth to Fodor's semi-serious claim that everyone's philosophy of mind is true for them{1}, it is not surprising that it would be difficult for him to see any "new" concepts as being genuinely new. So perhaps{2} he might be made more sympathetic by the fact that the concepts I have in mind are not strictly new, although their application to cognitive science is a fairly recent development.

Dynamic systems theory (DST) claims that the same basic laws that govern physical systems also govern the laws of cognitive systems, and that therefore cognitive science should use the math of physics (calculus etc.), rather than rules of syntax and computer programming. Described thusly, the claim may seem ridiculous to some. It seems to imply that there is no significant difference between rocks and people, or at best between clocks and people. As Clark puts it " Many scientists and philosophers believe that certain physical systems (such as the brain) depend on special organizational principles and hence require a vocabulary and an explanatory style very different from those used to explain the the coordination of pendula or the dripping of taps" (Clark 1997 p.115). However, recent developments in what is called non-linear dynamics have made it possible to use physics equations to describe systems which have the kind of flexibility that seems to justify calling them cognitive systems{3}.

Because Fodor's modularity theory reveals both the strengths and the weaknesses of the symbolic systems hypothesis, it provides excellent criteria for the evaluation of DST. In order to be on equal footing with GOFAI, DST must enable us to account for the functions and properties that Fodor calls modular. And if it is also able to account for Quinean and isotropic mental process (or show why the distinction between modular and Quinean-isotropic processes is spurious), it would be clearly superior to GOFAI, for whom these processes are, by Fodor's own admission, a complete mystery. Accounting for the properties that Fodor calls modular does not mean accepting everything he says about them. Whenever a new theory replaces an old one, it does so by contradicting some parts of the old theory and accepting others. If it accepts a substantial part of the old theory, the new theory is called a reduction. If it rejects most of the old theory, we say that the new theory eliminates the old theory. The best contemporary theories of reduction claim that there is a continuum connecting these two extremes of elimination and reduction (see Bickle 1998). We decide where on the continuum a particular theory replacement belongs by comparing the old and the new theory, and seeing how much and what sort of isomorphisms exist between the two. I will not try to answer whether the example I am discussing is an elimination or a reduction, because I have argued elsewhere (Rockwell 1995) that the distinction is more misleading than useful. Hopefully, however, my analysis will give some sense of where on the continuum we might place the relationship between DST and the symbolic system hypotheses.

Modules vs. Invariant Sets

I am going to start by describing a concept in DST which I think has many significant isomorphisms with the concept of module: the concept of an attractor which is an invariant set . In Port and Van Gelder 1995, an invariant set is defined as "a subset of the state space that contains the whole orbit of each of its points. Often one restricts attention to a given invariant set, such as an attractor, and considers that to be a dynamical system in its own right." (p.574) The second sentence is encouraging for our project, for if an invariant set can be considered as a dynamic system in it's own right, this seems isomorphic with Fodor's claim that modules are domain specific and informationally encapsulated. But what exactly is this term "attractor", and why would it be helpful to consider an attractor as a separate system?

Port and Van Gelder define "attractor" as " the regions of the state space of a dynamical system toward which trajectories tend as time passes. As long as the parameters are unchanged, if the system passes close enough to the attractor, then it will never leave that region." (p.573). The conditional clause of the second sentence holds the key to the cognitive abilities of dynamic systems. For of course the parameters of every dynamic system do change, and these changes cause the system to tend towards another attractor, and thus initiate a different complex pattern of behavior in response to that change.

The simplest example of an attractor is an attractor point, such as the lowest point in the middle of a pendulum swing. The flow of this simple dynamic system is continually drawn to this central attractor point, and after a time determined by a variety of factors (the force of the push, the length of the string, the friction of the air etc.) eventually settles there. Of course, a system this simple could not possibly be called cognitive in any sense. But consider a dynamic system with the following additional complexities.

1) Instead of making the attractor a point, let us have the system return to a repeating complex motion of some sort. This motion need not be only a motion in physical space. Many of the parameters of the system could be varying in measurable ways, and each of those parameters could be seen as a dimension in what is called state space. State space is not limited to the three dimensions of physical space, for it can have a separate dimension for every changeable parameter in the system.

2) Instead of having the state space contain a pattern that repeats the same way every time, it can mark off multidimensional borders within which we find what are paradoxically described as chaotic patterns. Chaotic patterns vary within certain regions that can be described verbally (eg. when mapped onto Cartesian coordinates, the changes in this system create a pattern of a torus, or a butterfly with three wings etc.) But the exact path through those regions does not repeat,even though it is mathematically predictable. Consequently it cannot be described exactly in geometrical terms, although one can describe the general outlines of the state space that it travels through. Port and Van Gelder describe this situation by saying that even though a system that contains chaotic pattens is unpredictable, it is not unfathomable. (p.576)

3)The system can contain several different attractor state spaces, which are connected to each other by means of what are called bifurcations. This makes it possible for the system to change from one attractor state space to another by varying one parameter or group of parameters.

How Horses Move

One of the most vivid examples of a cognitive dynamic system that uses bifurcations is the ambulatory system of a horse, which divides into four distinct attractor spaces. These are colloquolly referred to as walk, trot, canter and gallop, but each one consists of a set of motions governed by complex input from both the environment and horse's nervous and muscular system. Careful laboratory study has made it possible to map the dynamics of each of these gaits, (see Kelso 1995 p.70) and each map reveals a multidimensional state space that contains a great enough variety of possible states to respond to variations in the terrain, the horse's heart and breathing rate etc., and yet regular enough to be recognizable as only one of these four types of locomotion. There are no hybrid ways for a horse to move that are part trot and part walk; the horse is always definitely doing one or the other. And the primary parameter that determines the horse's utitilization of each of these gaits is how fast the horse is moving. From speed A to B the horse walks, from speed B to C it trots, and so on. There is not an exact speed at which the transition always occurs, for if there were, a horse would wobble erratically between the two gaits whenever it ran anywhere near those speeds. What usually happens, however, is that the horse rarely travels at these borderline speeds (unless it is being used as a laboratory subject). Instead it travels at certain speeds around the middle of each range for each gait, because those are the ones that require the minimum oxygen and/or metabolic energy consumption per unit of distance traveled. This means that a graph correlating the horse's choice of gait with its speed usually consisting of bunches of dots, rather than a straight line, because certain speeds are not efficient with any of the four possible gaits

If we were to make a computer model of the horse's ability to adapt its gait to its speed, one of the best computer languages for the job would be LISP. LISP models cognitive processes by means of commands that tell the program how to behave when it comes to a branch in the flow of information, which seems isomorphic in this respect to a bifurcation in the flow of a dynamic system from state space to state space. According to Haugeland 1985,(pp.154-5) all LISP programs are built out of six primitive functions.

1) left (x)

2) right(x)

3) join(x,y)

4) equal(x,y)

5) atom(x)

6) If (x) then (y) else (z)

If a program contains the command "left(x)" it activates the subroutine on the left side of a branch, if it encounters "right(x)" it activates the subroutine on the the right, and if it encounters "join(x,y)" it activates both subroutines. "Equal(x,y)" returns a special token "true" if if x and y are the same, otherwise it returns a token called "false". "Atom(X)" returns "true" if x is a single atomic token, otherwise it returns "false". Both of these terms are useful when combined with "If (x) then (y) else (z)"{4}, which chooses between y and z depending on whether x is true. (There is also a seventh primitive called "Nil", which Haugeland mentions on the previous page, which is used to plug any empty branches of the program that don't go anywhere.)

We'll start by positing four subroutines we'll call walk, trot, gallop, and canter, as well as a fifth subroutine we'll call "CURRENT SPEED" which measures how fast the horse is moving. Because we are only modeling the decision making process, rather than the entire dynamic system, we will accept those as unexplained primitives. With these five subroutines and two primitives from LISP {"Equal(x,y)" and "If (x), Then (y), else (z)"}, we can create a program that essentially duplicates the decision function of the horse's dynamic ambulatory system. We will posit convenient speeds for each gait of 5,10, 15, and beyond 15, and use the mathematical functions of "greater than" and "less than", because according to Haugeland, all such mathematical functions can be made from LISP primitives. Because LISP must define new words to connect programs together, we will call this program "GO". (see the next footnote for further explanations.)

GO =def


(if [ equal(CURRENT SPEED, O), Then 0, else

(if[ CURRENT SPEED <5, then walk, else

(if [CURRENT SPEED<10, then trot, else

(if[ CURRENT SPEED<15,then canter, else gallop]) ]) ]) ])


To underscore that this entire program is made from "bifurcating" LISP nodes, we can draw it as a tree diagram.

To some degree, this example{5} is an extension of Tim Van Gelder's "computational governor" thought experiment. Van Gelder's thought experiment showed that if a computer were to duplicate the function performed by the device which controls the speed of a Watt Steam engine, it would require fairly sophisticated computations. Van Gelder claimed that this task was clearly cognitive, that the Watt governor performed this task without computations, and that the same kind of physics which underlies DST was the best explananation for the Watt governor's cognitive abilities. This prompted some to say that the Watt Governor really was computational after all (Bechtel 1996) and others to say that the task was too simple to be called cognitive, and therefore the analogy was spurious. (Eliasmith 1997). Others pointed out that the Watt Governor was merely a feedback loop, and therefore DST must be (in Van Gelder's own words summing up this criticism) "Cybernetics returned from the dead." (Van Gelder 1999) My Horse LISP example is meant to be a partial answer to these last two criticisms. The paradigm cognitive ability in computer science is often considered to be decision making i.e. choosing between alternatives. An If-then-else command is certainly more of a decision making device than a feedback loop, and this example shows that the ambulatory system of a horse is a dynamic system that, among other things, performs the function of an If-then-else command.


These functional isomorphisms certainly don't permit us to think we can replace dynamic systems theory with LISP. In a real LISP program, all of the subroutines would be written in LISP as well. There is no reason to believe that the level of real time efficiency and environmental sensitivity possessed by the dynamic systems we called "walk" and "gallop" could ever be duplicated in LISP or any other symbolic system, and many reasons to think that they couldn't. The sword, however, might cut the other way. If the horse's ambulatory system is capable of making the kind of cognitive distinctions that we ordinarily associate with high level computer programs like LISP, and dynamic systems theory can explain how this is done by equations that show bifurcations connecting sets of attractors, then perhaps we have something like a reduction of certain aspects of the LISP computer language to dynamic systems theory. And if it were possible to duplicate the other five branching primitives in LISP with other kinds of bifurcations in dynamic systems, this could mean that certain dynamic systems would in principle be capable of performing all of the functions of the LISP computer language. From this, it would be tempting to conclude that the symbolic system hypothesis had been reduced to being a subset of DST, in much the same way that Newtonian physics was reduced to being a subset of Einsteinian Physics.

This is unquestionably a very big "If", which could not be resolved without a tremendous amount of research. But it does open the possibility of an interesting treasure hunt for those who claim that DST has the ability to be the next paradigm in cognitive science. What I hope to do in this paper is to lay some of the ground rules for such a treasure hunt, and discuss what some of the implications might be if it is successful.

First of all, no matter how succesful this treasure hunt is, it would not put LISP programmers out of work. Symbolic systems would not be invalidated, only the hypothesis that organic minds are fundamentally symbolic systems. There are plenty of excellent engineering and economic reasons for continuing to employ the formal model of human thought which is computer logic, even if it is revealed to be only a mechanical stylization. The fact that we do not think like computers is one of the main reasons that computers are so valuable; they are good at many things at which we are bad, and vice versa. The only real casualty of a DST reduction of symbolic systems like LISP would be the philosophical position that we need computer languages to tell the whole story about why we behave the way we do. If DST can in principle model all of the functions performed by humans with the same level of efficiency (or lack thereof) that humans perform them, then symbolic systems theory would be an important branch of engineering, but only a brittle metaphor for how organic minds actually work. For this reason, we should not say that our goal is to reduce the LISP language to a certain kind of dynamic system. More accurately, what we are trying to do is to reduce the similarities between minds and computer languages to such a dynamic system. The differences will remain, and anyone that relies on computers to do things that people cannot do (or vice versa) will rightly exclaim "Viva la difference!"

In fact, if DST is incapable of emulating symbolic systems in the same ways that human organisms are incapable of emulating them, it would make DST an even more compelling paradigm for organic cognitive science. However, one way to find out the similarities and differences between the two is to search for (and possibly construct{6}) dynamic systems that bifurcate in the same way that LISP trees bifurcate. Given that there are only six LISP primitives, it would be very convenient if we could simply find styles of dynamic bifurcation that corresponded to each of them. Then we would have a perfect reductive identity between LISP and those particular dynamic systems, and our job would be complete. But the chances of things working out exactly that neatly are very slim, for a variety of reasons.

For one thing, it is far more likely that most cognitively effective DST bifurcations will require several lines of code, or even whole programs, to be modeled effectively. Our model of the horse ambulatory system, for example, contains several elements that could not have any parallells in a dynamic system, because they presuppose a computer's need to search and choose before each action. The recursive terms in our horse LISP subroutine made it possible to compare the value of the incoming speed variable to each of the gait subroutines in sequence until the correct one was found. Dynamic systems do not have any need for this kind of comparing function. They shift among different sets of attractors when certain parameters change in value, but in no sense do they "consider" other alternatives before they shift. They do it right the first time. A connectionist net, for example, does need a training period to adjust its weights to perform the proper output. But unlike a computer program, it does not need to reconsider all of the wrong choices after it has been trained.{7} A case could be made that the recursive subroutines are functionally irrelevant when we are considering the abstract similarities between LISP and DST, and that therefore there is a real functional identity between the if-else-then commands and the dynamic bifurcations that control the ambulatory system of the horse. But it will always be a problematic judgement call to decide what is and is not functionally relevant in each case.

An even bigger problem is that there will probably not be a single isolatable element, whether equation, or state space or anything else, that will perform the same function in every dynamic system. For although dynamic systems are modular in the loose sense that they are comprehensible, they don't fit easily into many of the constraints that Fodor superimposes on the concept of module. In a sense, the attractor regions of state space in a dynamic system are both informationally encapsulated and domain specific to some degree. But the "boundaries" that separate attractor sets are not "hard-wired", and consequently there is little hope of separating out the bifurcations as separate nodes that could be plugged in like computer chips to connect attractor sets together. '"Translating" a dynamic system into a computer program would suffer from all of the indeterminacies of translation that Quine discussed in his essay "ontological relativity". There would rarely be a one to one correspondence between parts, even when there was a strong functional similarity between both wholes. In fact, the very question of what sense dynamic systems have parts at all is a very confusing one, and the most plausible answers to that question stretch the concept of module to very near the breaking point.

Can There be Distributed Modules?

Fodor claims most of the time that his modules are not organs with concrete locations in the brain, but rather abstract faculties defined by the functions they perform. A module is thus "individuated by its characteristic operations, it being left open whether there are distinct areas of the brain that are specific to the function that the system carries out" (Fodor 1983 p.13). In the breach, however, Fodor usually speaks as though his modules probably are organs in some sense. This is most noticeable on p.98 of Fodore 1983, which has the heading "Input systems are associated with fixed neural architecture". I can see no difference between an organ and fixed neural architecture. Although Fodor admits that there might not be distinct areas of the brain for each function , he apparently does not take this possibility really seriously. The only real cash value of this assumption for Fodor is to permit him to describe the function abstractly, and ignore as mere "hardware problems" exactly how the function is physically embodied.

This strategy became more obvious a few years later when connectionist AI began acheiving notable successes with what are called distributed systems. Fodor's response to distributed systems was basically to say that they were distributed only physically, that functionally they were still modular. However, he has never really explained how a system could be physically distributed yet functionally modular. I think, however, that if DST does deliver on its promise as a cognitive science paradigm, there is a sense in which distributed systems can be modular, although with several important qualifiers.

Van Gelder 1991 claims, I think correctly, that the essence of distribution is summed up in a concept he calls superposition. For our purposes, I think Van Gelder's concept of superposition is effectively illustrated by the following series of examples. Let us consider a set of 26 cards, each of which has a letter of the alphabet on it. In this case, the representation of the alphabet is completely modular and undistributed. Each card represents exactly one letter of the alphabet, without any reliance on the other cards. Now let us suppose that instead of twenty six cards we have only 10 cards. We lay the cards out on the floor in a set pattern, and paint each card white on one side and black on the other. We then represent "A" by turning a certain combination of black and white surfaces face up (Odd cards black, even cards white, for example), another combination is posited as representing "B", and so on for all twenty six letters. In this case the representation of the alphabet is superposed on all ten cards, because no one card represents any one letter. Despite the fact that we have only ten cards, and all of them are used to represent a single letter, we do not end up with less representational power, but far, far more. There are, in fact, 1,024 (2 to the 10th) possible combinations of black and white, leaving us 998 (1,024-26) possible other combinations to represent whatever else we like. (the Russian and Sanskrit alphabets, perhaps). If we used six sided cubes with different colors on each side, instead of cards with only two sides, we would have 6 to the 10th possible combinations.

In the one-card-per-letter system, each card could be seen as a module both physically and functionally. one physical piece of paper performs exactly one linguisitic function, no more no less. In the superposed system with the black and white cards, however, no one card is a letter. Instead, each card functions like an axis in a Cartesian state space, and each axis has exactly two points on it. (one for the black side of the card and one for the white) When the cards are replaced with six sided cubes, each cube functions as an axis with six points on it. And because the cards are performing this Cartesian function, they make it possible to conceive of each of these letters as a point in the 10 dimensional state space defined by these 10 cards or cubes. Consequently, we can functionally represent all twenty six letters without needing twenty six distinct physical cards.

The Abstract and the Material

Nevertheless, to keep the labels "functional" and "physical" is seriously misleading if we extapolate this analogy over to dynamic systems theory, because DST is a branch of physics. The attractor state spaces and invariant sets of a dynamic system are every bit as physical as gravity and electromagnetism. And yet they are doing the job that we ordinarily ascribe to functional concepts in cognitive science. Consequently, I think we need new words to describe both sides of this distinction.

Even though gravity and magnetism are physical entities, I think all would agree that they are also abstract entities. In the same sense, state space, attractors, and bifurcations are also abstract entities, and so are those properties of symbolic systems which are referred to by the word "functional". ( Churchland 1989 refers to functional properties as "abstract causal roles" on page 23.). But because abstract entities like bifurcations and gravity are also physical in a meaningful sense, we're going to need another word to refer to what are usually called physical properties.

The replacement word I like is "material", because it does not imply the physics laboratory, but rather the kind of vulgar "stuff and particles" ontology championed by Hobbes, Democritus, and the more unvisionary breed of man-on-the-street. Physics has not been limited to an ontology of this sort for some time. In fact, a case could be made that even Newton compromised this ontology when he introduced gravity's ability to effect things at a distance{8}. And it became even more compromised as physics progressed to Maxwell's electromagnetic fields, the ghostly incoherencies of relativity and quantum mechanics, and finally to a general sense that you never know what physicists are going say next. Consequently, most philosophical physicalists admit under duress that by "physical" they mean whatever physics believes the world is made of. In the breach, however, they usually return to the materialism of Hobbes and Democritus, which sees the real as being the stuff that we can see and touch and trip over, rather than the abstractions of theoretical physics. My distinction between the "abstract" and the "material" (which subdivides the physical, among other things, if there are other things) is designed to prevent the conflation of these two very different concepts. One can believe that everything is physical, while acknowledging that the material is a subset of the physical.

With this distinction in place, I would say: a cognitive dynamic system can be materially distributed, and yet abstractly modular. Furthermore, there is no scientific reason to say that the abstract modules created by state spaces are somehow less real than the grey slippery material stuff through which they are distributed. The position that calls itself "scientific realism" has always maintained that the fundamental abstract entities described by theoretical science are realer than manifest entities like tables and chairs. As far as I can see, all of the arguments of the scientific realists are equally effective in defending the claim that so called distributed systems are fundamentally modular ontologically, even if they appear to be distributed to us. One could take a more nominalist position, and say the grey slippery stuff is real, and the state spaces are just ways of talking about it. But one couldn't say that the science itself compells such a position. Van Fraasen's instrumentalism is such a position, and to some degree, so is the radical empiricism of James and Merleau-Ponty. But all of those positions presuppose that we have access to some kind of experience which is more fundamental than scientific knowledge. As far as the science is concerned, the state space modules described by DST are as real as anything can be.

DST and Connectionism

It may be that connectionist AI was guilty of a kind of misplaced concreteness by conceiving of these state spaces as organlike neural structures, rather than dynamic patterns of motion and change. I am tempted to think of the connectionist modules used in contemporary AI as little dynamic systems imprisoned like birds in cages, so that they can communicate with other modules only by means of input-and-output devices. Computationally, a connectionist net is rather like an AI "toy world" version of a dynamic system. The fundamental computational tool in both is state space transformations. But in a connectionist module, the state spaces are relatively easy to isolate. One can simply measure the inputs and weights of each neuron. Of course, real neurons really do have inputs and outputs with reasonably exact voltages and weight summations. And by replicating those in silicon, it becomes possible to create modules that perform state space transformations on specific inputs. But only a subconscious Cartesian Materialism prompts the assumption that the state space transformations in the cranium are telling the entire cognitive story.

When one measures the dynamics of a complete dynamic organism interacting with a world, there is no reason to doubt, as Kelso puts it, "that both overt behavior and brain behavior, properly construed, obey the same principles." (Kelso 1995 p.28. Quoted in Clark 1997 p.116). Of course, there are too many factors at work in an embodied living system to delineate the state space with the same level of precision that can be done when we are measuring or building a single connectionist network. But this is only an epistemological problem. The fact that it is harder to measure the parameters of the state space and their possible variations does not make any difference to the fundamental nature of the cognitive process.

For this reason, I feel that if we accept Clark's claim that "connectionist approaches. . . are nevertheless both computational and (usually) representational" (p.151), we must also accept that dynamic systems can be computational and representational. If we make a distinction between "representational" and "symbolic", using the latter to refer to the subspecies of representations that constitute symbolic systems, I think that even Van Gelder would accept that neural nets are representational in this narrow sense. If so, DST could conceivably reduce the concept of representation in organic systems to certain kinds of attractor sets, but it would not eliminate the concept altogether. If neural nets are computational and representational, then so are dynamic systems.

The Dynamics of State Spaces

Once we recognize that state spaces can be dynamic, they can be free of many of the limits that Fodor proscribed on modules, even though they are capable of performing the same kinds of functions. Strictly speaking, a dynamic state space is much more like an event than an object. It endures longer than most events, but in this respect it resembles events like tornadoes or waterfalls, which seem object-like because the flow of their constituting processes cohere in a stable pattern. Nevertheless, a state space is a far more volataile entity than an organ, and its boundaries and functions are far more flexible.The concept of organ is certainly a useful one, and will probably continue to be so. But it is based on a possibly misleading assumption: that morphology is always an accurate guide to function, because the body, like a Hi-Fi set, is supposedly divided up into distinct modules with materially delineable borders. Those who accept this assumption acknowledge that it may require microscopes, or sophisticated staining techniques, to find those borders. But the assumption remains that once one has marked out those borders, one has carved the brain or body at it's fundamental joints, and that the purpose of neuroscience is then to answer questions like "what is the hippocampus for?".

We must not discount the possibility that morphology is not the essential factor in determining function. The grey matter of the brain could be seen as primarily a medium through which dynamic ripples eddy and coalesce into state spaces. The fact that people can often relearn skills lost after brain damage, even though the "organs" supposedly responsible for those skills have been surgically removed, could offer support for that possibility. So does the fact that brain scans of different people (or even the same person at different times) often reveal brain functions taking place at radically different locations. On the other hand, we also shouldn't assume that we must make an either/or choice between a dynamic brain and a brain assembled from organs. To continue the ripple metaphor (if it is only a metaphor), even a river's dynamic flow is shaped by the contours of the riverbank. In a similar way, the biological structure of the brain would make it more likely that certain dynamic patterns would stabilize into recurring patterns, just as certain kinds of eddys and tidepools are more likely to form if the river banks contain the appropriately shaped inlets.

The connections between state spaces in embodied dynamic systems are not hard wired the way they are in AI connectionist systems. The bifurcations between attractor spaces are not specific connective neurons, they are only abstract measurements of forces. As the parameters that shape these forces shift, so do the cognitive characteristics of the attractor. For this reason, it is highly implausible that attractor sets in dynamic systems are domain specific and informationally encapsulated in the way that Fodor claimed modules must be. Many of Fodor's arguments for informational encapsulation (for example, the fact that modules must have fast response times), require only that the module be informationally encapsulated at the time it is performing its function. Fodor is correct when he says that "in the rush and scramble of panther identification, there are many things I know about panthers whose bearing on the likely pantherhood of the present stimulus I do not wish to have to consider" (Fodor 1983 p. 71 italics in original). After the rush and scramble have subsided, however, there is no reason that the module which enables us to instantaneously identify panthers shouldn't receive all sorts of input from other sources, and reconfigure itself so as to make more effective responses the next time we see a panther. One could learn, for example, to tell the difference on sight between a male panther and a female panther, and thus learn to initiate two different sets of fast reflexive behaviors. (For example, we could develop one flight strategy designed to avoid hunters, and another to avoid a creature protecting her cubs.) Anyone who has ever learned to sight read music knows that sets of quick responses can be constructed that are very effective for one style of music, but do not yield the necessary speed for another style. Even if one can sight read Bach fluently, it is likely that difficulties will arise the first time one tries to sight read Duke Ellington until one has learned several new pieces in his style. But because the sight reading "module" is not informationally encapsulated, it is possible for it to take in new information the more one studies a new style of music, and thus learn the reflex like speed that makes fluent sight reading possible the next time around.

Dynamic systems have the kind of speed that makes these quick responses possible, and for them, informational interpenetration is probably the rule, rather than the exception. We must remember that "invariant set" is a highly conditional term in DST. "Invariant" really only means that there is a pattern that stays stable long enough to be noticed and (partially) measured. Given the number of parameters that must reach some kind of equilibrium for an invariant set to emerge, it is highly unlikely that they will always remain stable enough to produce anything that could be called informational encapsulation. This is exactly the opposite situation that would arise for modules that were constructed out of fixed neural architecture. As Fodor points out "if you facilitate the flow of information from A to B by hardwiring a connection between them, then you provide B with a kind of access to A that it doesn't have to locations C,D,,E, . . ." (Fodor 1983 p.98). But although this is true for hardwiring, it is not true for bifurcations in a dynamic system. The slightest flicker in the parameters that hold an invariant set stable could bring in information from almost anywhere else in the system, which could change the system (hopefully for the better) when it restabilized{9}.

In a dynamic system, domain specificity could also be every bit as flexible. Kelso 1995 points out that almost anyone can pick up a pen with their toes and write their name the very first time they do it. If this ability were stored in a rigidly domain specific module, it would be hard to see how this were possible. The actual neurological signals for the commands that make our arms do this would have very different values and relationships from the same commands when sent to our legs. Ont the other hand, if the module that made this skill possible were an invariant attactor set, it would be much easier to bend the vector transformations in the attractor set just enough to do the different but related task.


All of this is highly speculative, of course, but Fodor's claims about modularity are also highly speculative. (as Fodor himself admits.) It is, however, important to dream of things that might become and ask why not, for all research is shaped by some sort of speculative assumptions. Once we put these assumptions on the table, it is possible to speculate about their fruitfulness. Doubtless there will not be consensus about which paradigm is the best to follow, and I think that Clark is right that this is a good thing. But if further research reveals that attractor sets are as good at emulating modular processes as they appear to be now, without the inflexibility of hard wired modules, then it might be possible to bridge the nasty moat that Fodor posits between modules and Quinean-Isotropic processes. And this might justify a meta-modification of Fodor's closing exclamation of "Modified rapture!" to just plain garden variety rapture. Or at least hope.

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{1} quoted by William Lycan, in conversation. [Back]

{2} or perhaps not [Back]

{3}This improved descriptive ability has ironically sometimes been bought at the expense of some predictive ability. See the upcoming discussion on Chaotic patterns [Back]

{4}this is usually written as "If (X,Y,Z)", with only commas to mark the difference between the subwords "then" and "else". Haugeland labeled all three components of the word when he defined it, which seemed to make it much easier to follow the flow of the progam. There is another computer language called Forth, which always adds the "then" and "else" for this word, but for all I know LISP would be completely derailed if a real computer encountered these additions. [Back]

{5}Strictly speaking, we also need to take each of our posited ambulatory subroutines and combine it with "GO", the name of our LISP routine. This is called a recursive definition, which means that the definition of the routine contains itself. This is a bad thing in philosophy, but a very good thing in LISP because it enables LISP to repeat functions until a job is done. Thus the complete program would have to contain four definitions that looked something like this: Walk =def (make the horse walk) (GO) And so on for the subroutines trot, canter, and gallop. Once each subroutine is defined this way, the program will continue to go back to the beginning after each subroutine, so it can run CURRENT SPEED again. (the program returns a zero when the horse stops.) I did not include the addition of "GO" to each gait subroutine (or the extra branches that would show the subroutines returning back to the beginning of "GO") because they are of no significance for the point I am making, and would be unnecessarily confusing. Thanks to Barry Smiler of Bardon Data Systems for advice on LISP programing, and for pointing out that this program did not need to have ways of dealing with negative numbers because "The horse isn't going to run backwards". [Back]

{6}The fact that the symbolic system hypothesis is a good engineering metaphor for the mind does not mean it is the only good engineering metaphor for the mind. Much robotics these days uses dynamic systems theory, and is highly critical of the symbolic systems paradigm see Brooks 1991, and several other citations and descriptions in Clark 1997. [Back]

{7}This is also a helpful argument against Fodor's claim that a fast system cannot be Quinean and isotropic. Fodor says "if there is a body of information that must be deployed in. . .perceptual identifications, then we would prefer not to recover that information from a large memory, assuming that the speed of access varies inversely with the amount of information that the memory contains (Fodor 1983 p.70) This assumption is unavoidable only if we also assume that every system must follow the GOFAI procedure of considering several possible options before acting. [Back]

{8} This was Descarte's main objection to Newton's physics, for Descartes' own physics posit a universe with no empty space, which made mysterious non-material forces like gravity unnecessary. One could thus say, ironically, that Descarte was one of the last true materialists, at least when he wasn't talking about minds. [Back]

{9}Those who admire John Dewey's prophetic abilities might enjoy this passage from Dewey 1896(!) The 'stimulus. . . is one uninterrupted, continuous redistribution of mass in motion. And there is nothing in the process, from the standpoint of description, which entitles us to call this reflex. It is redistribution pure and simple; as much so as the burning of a log, or the falling of a house or the movement of the wind. In the physical process, as physical, there is nothing which can be set off as stimulus, nothing which reacts, nothing which is response. There is just a change in the system of tensions. (italics mine) To my knowledge, Dewey never used the mathematics of dynamic physics to understand the behavior of living creatures. But it is impressive that over a hundred years ago, he was able to conceive of psychological processes as being best understood as stablized patterns of interlocking dynamic forces. [Back]

{10} [Back]


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