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Chaos and Computability in Philosophy of Mind Gregory R. Mulhauser University of Edinburgh scarab@ed.ac.uk © 1993 Gregory R. Mulhauser (This page intentionally left blank.) Table of Contents Foreword 5 1. Representing the Problem 7 1.1 Dynamical Systems 7 1.2 Chaos, Graining, and Prediction 10 1.3 Three Representational Spaces 13 1.3.1 l Space 13 1.3.2 w Space 13 1.3.3 y Space 14 1.4 Relating the Spaces 16 1.5 Applying the Schema 18 1.5.1 Existing Theory 18 1.5.2 New Directions 18 2. Psychological Indeterminism 21 2.1 The Argument 21 2.2 Mapping Psychological State Space 23 2.3 Maps and Indeterminism 25 2.4 Psychological Indeterminism: An Overview 28 3. Computability in Chaotic Analogue Systems 29 3.1 Functions and Systems 29 3.2 Recursion Theory 30 3.3 Analogue Chaos 32 3.4 Computability and Behaviour: So What? 34 4. Chaos and Infinite Intricacy 37 4.1 Intricacy in Models and the World 37 4.1.1 Intricacy at the Quantum Level 39 4.1.2 Intricacy at the Classical Level 40 4.2 Applying Intricate Models 41 4.3 Models and Ontological Significance 42 4.3.1 Limited Precision Models and Realism 43 4.3.2 Limited Precision Models and Levels of Description 45 4.3.4 Practice and Theory: Ontological Significance 47 4.4 Intricacy: An Overview 47 5. Prediction 49 5.1 Pinning Down the Problem 49 5.2 Qualitative Unpredictability with Epistemic Determinism 53 5.3 Riddled Attractor and Computability Revisited 56 5.4 Prediction: An Overview 57 6. Complexity in Chaotic and Noisy Systems 59 6.1 Defining Complexity 59 6.1.1 KCS and Its Implications 59 6.1.2 Problems With the Implications 62 6.1.3 Problems With KCS Itself 64 6.2 Other Measures 66 6.3 New Implications 69 6.4 Complexity: An Overview 72 7. Complexity and Representation 73 7.1 Pinning Down the Problem 73 7.2 Complexity in Representations 74 7.3 Complexity Without Representation 76 7.4 Complexity in Naturally Chaotic Systems 78 7.5 Chaos and Complexity in Neural Systems 79 7.6 Complexity and Representation: An Overview 81 8. References 82 The following is a very rough working draft compilation of most of my work to date on chaotic dynamics as featured in neural networks. I outline a framework for understanding the problems and then discuss a number of points we might make about chaos in intelligent systems instantiated by neural networks. Much of this material is inspired either by Peter SmithÕs (University of Sheffield) own work in chaos or by his comments on my work. I am grateful to him for keeping up our several discussions on chaos theory and for providing me with copies of various papers on the topic, as well as to Adam Morton (University of Bristol) and Terry Horgan (Memphis State University) for other papers and discussions. Pat Churchland (University of California at San Diego) and Christof Koch (California Institute of Technology) provided some welcome encouragement on hideous early drafts of the recursion theory work. I am also grateful to conference and seminar audiences and referees in Durham, Sheffield, St. Petersburg, Warsaw, and Edinburgh. Thanks also to my supervisor for the first two years of my Ph.D. research, Stig Rasmussen; to my present supervisor, Alexander Bird, for encouraging me to get more down on paper; and to the Marshall Aid Commemoration Commission, who pay my bills. Although many people have contributed to my thought on these topics, this is primarily a new manuscript in which even existing work has been largely written, and in this form the ideas have yet to be reviewed (even by me!) for completeness or even basic sense. I take full responsibility for omissions, misrepresentations, awkward prose, and other errors. Gregory Mulhauser Edinburgh November 1993 (This page intentionally left blank.) In most of what follows, we shall be concerned with the importance of chaotic dynamics for intelligent systems implemented by neural networks. While its specific r™le is controversial, the capacity of some biological and artificial neural networks to exhibit chaotic behaviour is well established.^{1} We begin by outlining a representational system for understanding chaos in neural networks and an overview of its possible significance. The sensitive dependence on initial conditions of chaotic systems suggests that minute perturbations in the low level dynamics of neural networks could possibly evolve over time into influences on those systems at grosser levels of description. In particular, it might be that microfeatures of low level processes which are not available to coarse grained introspective awareness could yet evolve into significant influences on higher level features of brain dynamics which may be available to introspection. I suggest a new representational schema which provides an economical means of formulating the interactions between dynamics at low, intermediate, and high levels of description. After a very brief introduction to the terminology of dynamical systems and a short but somewhat technical look at chaos theory, I outline the representational framework and consider some of the insights we might gain from it.^{2} The phrase dynamical system has earned a place for itself in the literature of cognitive science and the philosophy of mind. Any classical physical system can be represented in the state space framework of dynamical theory. The state of the system is represented as an ordered ntuplet, with one value for each of the systemÕs n degrees of freedom. The evolution of a system can be represented graphically as a phase trajectory through ndimensional phase space, a curve showing how each of the n variables changes with time. Aside from being a convenient way of representing classical physical systems, phase trajectories allow us to make geometrical observations which might be missed were the systemÕs evolution represented simply as, say, columns of numbers. For instance, it is much more difficult to get an intuitive feel about the dynamics of the pendulum represented in Figure 1 from the table of numbers than from the phase trajectory depicted in Figure 2. In the second figure, we can see that the pendulum is fairly uniform on each swing except that it is gradually winding down under the influence of some damping force. Fig. 1 The phase space framework neednÕt be restricted to representations of real physical systems. We might also represent the evolution of something like public confidence in a government relative to a national inflation rate. Here we might find either a continuous dynamic evolutionÑwhen inflation goes down, confidence goes up by some related amount and vice versaÑor we might find discontinuous dynamics such as a sudden jump in public confidence as soon as the inflation rate reaches a particularly low threshold value. In some cases there is a clear sense in which the dynamics of a system represented at one level are indeterministic despite the fact that those dynamics are based upon deterministic dynamics at a lower level. For instance, consider a three dimensional phase trajectory indicating the relationship between the number of marine research vessels a country has at sea, the number of total marine scientists at sea, and the total funds being devoted to the scientistsatsea programme. We could notice that in general, when the funding goes down, so does the number of vessels and so does the number of scientists, and vice versa for an increase in funding. But the number of marine research vessels is a coarsegrained look at a countryÕs marine research program; a country might have five vessels, each of which carries 100 scientist, or it might have fifty vessels, each of which carries only three scientists. These vessels might also demand different degrees of funding to stay operational. Thus, a country might have a mix of vessels such that sometimes the phase trajectory shows a transition from 94 vessels and 451 scientists to 89 vessels and 430 scientists (with a particular decrease in funding); but on another occasion there might be a transition from 94 vessels and 451 scientists to 89 vessels and 475 scientists (with an identical decrease in funding). If we had lower level information about the vessels themselves and the number of scientist they carried and the cost of keeping each at sea for particular lengths of time, then under the reasonable assumption that there were some mathematical relationship between funding level and the number of scientists a country wanted to have at sea (and assuming there were no other confounding variables), we could predict the impact of any particular change in funding level on which boats would be kept at sea. Fig. 2 But because we lack this lower level information, the dynamics at the higher level are not predictable, even for given changes in funding. There is nothing mysterious about this conclusion: the dynamics at the level of boat numbers and scientist numbers and funding might not be deterministic on the basis of information available at that level, yet the dynamics at the level of specific boats with specific costs and specific scientistcarrying capacities could be entirely deterministic and predictable. Ignoring for the moment the significance of the l labels, we can see in Figure 3 that even if the two trajectories depicted are entirely deterministic, their evolution is nondeterministic at the level of description of grid squares: the two trajectories begin in the same grid square but evolve (deterministically, at an appropriate level) into two different squares. Fig. 3 Shortly we shall explore the application of the dynamical systems framework to states of mind for intelligent systems, but first we take a brief sojourn into chaotic dynamics, a particular kind of behaviour which some dynamical systems display that can make prediction at all but the very lowest levels highly problematic.^{3} The three defining properties of chaotic systems include a dense covering of phase space with periodic points of all possible periods, sensitive dependence on initial conditions, and topological transitivity. (I adopt the terms of Devaney 1988; Barnsley 1988 and BergŽ, et al 1984 are similar.) The first property is the most straightforward: the phase space of a chaotic system at any given time slice (or alternatively, in any PoincarŽ section) is covered with infinitely many periodic points, and between any two of these we can always find another. Periodic points are just points which lie on a closed trajectory; that is, if we ignore the time dimension, the system will keep visiting the same physical locations in phase space. Some systems include, at least in some neighbourhoods, a dense covering of repellent fixed points of all possible periods. Later we shall be concerned particularly with that subset of chaotic systems which includeÑat least in some neighbourhoodsÑa dense covering of repellent fixed points of all possible periods.^{4} A repellent point is simply one from which all points in a local neighbourhood diverge over time. Alternatively, if the system is invertible, trajectories in a local neighbourhood approach a repellent point asymptotically under reverse time evolution. Many strange attractors are densely covered with such repellent periodic points, and in some systems, such as the hyperbolic toral automorphisms, the entire space is so covered. The second property of chaotic systems, sensitive dependence on initial conditions, or SIC, appears when for any state in the phase space of a chaotic system, there is another state within an arbitrarily small neighbourhood which lies on a phase trajectory diverging from that of the first. I give here the definition for the discrete case. A function Ä: J ® J, where J is a metric space, is SIC when there exists a D > 0 such that "x Î J and for any closed neighbourhood N of x, there exists a y Î N and an integer n ³ 0 such that , where n represents the number of times the function is iterated. (The case for continuous systems is directly analogous, with a term similar to n representing the time parameter.) Note that this property occurs for every x Î J. Moreover, every neighbourhood N of x includes at least countably infinitely many diverging y. For one constructive proof, consider any such neighbourhood for an arbitrary x. The definition guarantees us a y Î N whose phase trajectory diverges from that of x, so note this y and consider a new N' which is the previous N minus the y (or, alternatively, a new N' which is the previous N minus a neighbourhood around y which is the limit as the radius of that neighbourhood r ® 0). We are now guaranteed a diverging y' in this new N', which in turn generates a new N'' and a new y'', ad infinitum. The final defining characteristic of chaotic systemsÑtopological transitivityÑindicates that any particular neighbourhood in phase space will eventually be visited by the phase trajectory of some point lying within any other arbitrarily small neighbourhood. Topological transitivity is the property that for any Ä: J ® J as defined above and any two open bounded sets U, V Î J, there exists an integer n ³ 0 such that In other words, the phase trajectory of at least one point in U will intersect V in a finite amount of time, regardless of how small or how far apart U and V are to start with. To put it still another way, given any two open bounded sets in the space of a chaotic system, there will always exist a phase trajectory connecting some point in one to some point in the other. It is because of SIC and topological transitivity that very small errors in approximating the initial state of a chaotic system are magnified rapidly into gross errors about the systemÕs evolution. It is easy to see that if we had only coarsegrained information about a chaotic systemÕs state, and if we were interested in making predictions about the detail of the systemÕs future behaviour, we could not make those predictions over anything but the shortest time scales because of the (normally exponential) expansion of error caused by SIC. This remains true even if we have detailed knowledge of the equations describing the behaviour of the system at a very low level. But there is another kind of prediction which is unproblematic for these kinds of systems, a type which we should keep in mind as we move on to exploring the framework in which we can represent intelligent systems dynamically. The kind of prediction which is relatively unproblematic for chaotic systems is possible when we ask a different kind of question about a systemÕs evolution than what the specific detailed phase space location might be. Instead of asking for details of phase space location, we might be interested only in the attractor towards which a system is tending. Like all dissipative systemsÑi.e., systems in which any given volume of phase space shrinks through time^{5}Ñthe kinds of chaotic systems we are concerned with have attractors, structures in phase space towards which phase trajectories within a particular neighbourhood of the attractor (called the attractorÕs basin) tend asymptotically. (That is, trajectories which begin off the attractor never actually evolve onto it, but they may get arbitrarily near it.) Attractors are also invariant under the operation(s) defining the dynamics of the system. They are the large structure analogue of fixed points: a subspace rather than a point which is invariant. The interesting thing about attractors in the phase space of chaotic systems is that often they are strange; for our purposes, we can simplify the idea of strange attractors by appealing to the property of being infinitely detailed in a nontrivial way, or fractal.^{6} Typically, the dynamics of a system on a strange attractor are chaotic, so we still have, for instance, sensitive dependence on initial conditions for trajectories on the attractor itself. This also holds true in the basins of attraction of these strange attractors. Thus, it is altogether possible for a system to be very difficult to predict in the sense we described above, yet for its activity still to be constrained within a particular basin of attraction. Thus, if we are interested only in knowing which attractor a system is near, it can be a very simple matter to know its future state from a measurement of its present state. Even though we may not be able to comment as to the precise phase space location of a phase trajectoryÕs evolution, we can still be sure that the trajectory has not left the basin of attraction in which it started and that in fact it will always get closer and closer to the attractor itself. Under this kind of complex coarsegraining of phase space, as distinct from the simple coarsegraining we discussed above as in Figure 3, prediction of the time evolution of a chaotic dynamical system is unproblematic. With an understanding of dynamical systems and the basics of chaos theory in hand, we now move to an exploration of a dynamical systems framework in which we may outline the relationships between high, low, and intermediate level descriptions of intelligent systems. I suggest three different metric spaces representing the level of intentional states of mind, the level of computational relevance, and the level of actual physical state. The first space, which we shall call l, is the most straightforward: it is simply the ordinary physical state space representation for every particle which is functionally relevant to the system we are wanting to represent. The metric for this space is the ordinary euclidean distance measureÑthe square root of the sum of the squares of the differences between two points along each dimension. In practice, we would never represent an intelligent system such as a brain in l space because of the extraordinary dimensionality involved. We will, however, use it as the basic physical foundation upon which the other two more complex spaces rest. In the second space, which we shall call w space, we are concerned with Ôcomputational relevanceÕ: instead of representing individual particles, we represent functionally relevant components of the intelligent system. Thus, to represent a neural network, we might include dimensions for each neuronÕs output frequency, its level of fatigue and the efficacy of its synaptic connections, plus dimensions describing neuromodulator distributions and other extraneural factors. The metric for this space can also be something like an ordinary euclidean distance, although we must be aware that the mapping from real state space to w space will not always be straightforward and that nonuniformities in the mapping may mean that transitions between two w points may vary because of differences in the distances between the various sets of distinct l points which may be mapped to the same two w points. Indeed, the topological transforms mapping manifolds in l space to those in w space may be fuzzy, on account of the vague character of components like neurons. (I.e., the precise boundaries indicating which particles should be included in a particular neuron may not be well defined.) We will have more to say about this kind of problem as it relates to the next representational space. Note that something like w space, unlike l space, may serve a practically useful rather than just a theoretical purpose. While it would certainly be difficult to represent the entire human brain in such a space, there is no problem with representing smaller subnetworks directly. For more complex networks, ordinary w space is still useful as the basis for a coarsergrained representation with dimensions for representing the overall activity of populations of neurons (such as columnar arrays in the neocortex) rather than of individual neurons. The final space accommodates a very high level description of what we might call y states, or something like Ôsubjective states of mindÕ, or Ôintentional statesÕ. The parameters defining this space would be largely independent of those describing neural phenomena in w space. They would include every psychological parameter available to the introspection of a subject, such as descriptions of anxiety, happiness, desire, arousal, fatigue, anger, and so forth. It is at this level where we might describe Horgan and TiensonÕs (1992) Ôcognitive transitionsÕ, or changes from one psychological state to another. The metric for such a space might again be based on the ordinary euclidean distance measureÑbut we must keep in mind that the parameters defining the dimensions of the space typically will themselves be vague terms and that it may not make sense to attribute a well defined distance to two distinct points. (It is very difficult to attribute a precise real number description to a measure of something like happiness, for instance.) And as in the case of transforms mapping l space to w space, the topological transforms relating manifolds in l space to those in y space are liable to themselves be fuzzy. Moreover, because subjective states of mind are multiply realisable in neural terms (that is, two distinct w states may correspond to the very same psychological state), transitions between two neighbourhoods in y space may not always take the same amount of time. Whatever the mapping may be from l state or w state to psychological state, there will always be the possibility of this difference in transition times. For instance, given any two distinct physical states which map to the same psychological state, one of them may be closer in l space than the other to another set of physical states which map to a different psychological state. Thus the distance required to traverse the space may also be shorter for the one which is nearer. Multiple realisability, then, suggests that while we might apply a standard euclidean distance to y space, we must keep in mind that our answers must be applied to a fuzzy and temporally variable reality. More significantly, multiple realisability also suggests that y space may be treated as warped: it is a general Riemannian space in which elliptic, hyperbolic, or neutral geometries might apply according to which neighbourhood we are considering. We can understand this just by considering what kinds of topological transforms might be applied to translate a manifold in ordinary physical l space into a manifold in y space. (The explanation could just as easily be phrased in terms of transforms from w space to y space.) A manifold or a set of disconnected volumes in ordinary physical space might map, for instance, to one single point or perhaps a line in y space. This is just what multiple realisability means. Moreover, the topological transform, or the mapping from l space to y space, is liable to vary in detail according to the neighbourhood in question. That is, in some areas of real state space, large volumes might be mapped to single points, whereas in others small volumes might map to large manifolds. The result of such a mapping, apart from the loss of information which goes with any coarsegraining, no matter how complex, is that there is no guarantee of a consistent geometry across y space. We might expect the geometry of y space to look something like that of four dimensional spacetime with massive bodies scattered about. But because the warpage of the space is not caused by anything akin to simple masses, which induce a mathematically uniform deformation of spacetime, the geometry of y space is liable to be far more complex. There are a number of observations we can make about the relationship between these three different metric spaces. In the following I will outline a few of these before moving on to a discussion of one particular argument which has emerged from something like this way of viewing different levels of description of the same intelligent system. First, it seems clear that at any given time slice, the ÔnumberÕ of distinct y states will be less than the number of both w states and l states. Also, while the ÔnumberÕ of w states will generally be less than the number of l states, in particular neighbourhoods of the phase space of chaotic neural networks, the number of w states may be greater than the number of l states. The first observation comes from the multiple realisability of intentional states and of computationally relevant components such as neurons, while the second is an immediate consequence of SIC. This is a refinement on PylyshynÕs (1984) conjecture that the number of computationally relevant brain states is always less than the number of physically discernible states. There is also an important observation to be made here about what we mean by ÔnumberÕ of states in a particular space. As Deutsch (1985b) points out, referring to theoretical work by Bekenstein (1981) on the thermodynamics of black holes, any physical system enclosed by a surface with an appropriately defined area A can have at most an extremely large but finite number N(A) of distinguishable access states: where c is the speed of light and the denominator is four times the product of PlanckÕs reduced constant and the gravitational constant. This reveals something important both about the way we must use the dynamic spaces discussed here and about the way almost all mathematical models should be applied to reality. In particular, if the number of access states of a finite physical system is limited by the equation above to some very large but finite number, then strictly speaking there is only a finite class of points in the state space of a finite physical system which can have measurable significance to us. But a finite set of points marked off next to each other have zero length. That is, only continuous segments made of an infinite number of points have length. Thus modelling any real physical system must require either a discontinuous space made of discrete points and Ôempty spaceÕ between them (as well as discontinuous dynamics in the space) or, alternatively, a continuous space with fuzzy points constituting a continuum. In the latter case, the continuous fuzzy regions surrounding each accessible state provide the infinite class of points to allow for overall continuity. This is a general point which is also significant for the discussion later about modelling Nature with an infinitely detailed numbered system. Fortunately in this particular case we already have a handy way of circumventing the problem; namely, we have already noted that the mappings from l space to the other two spaces are themselves fuzzy.^{7} Thus we can expand y space from a space with essentially no volume to a continuous space with positive volumes; the same applies to w space. We must simply keep in mind when using the dynamical spaces that two points very near each other may simply be indistinguishable: we are operating with models applied to vaguely mapped spaces, and our ÔanswersÕ must be fuzzified appropriately. This is exactly analogous to our later conclusions about using real number models which include an infinite amount of detail. Two other observations about this dynamical systems framework follow on from these kinds of points. The first is that whenever we are modelling a system at a high enough level of description that there is some loss of detail from a lower level where dynamics are deterministic, there is at least the possibility that dynamics at the higher level will be nondeterministic. If there are ÔfewerÕ states at a given level (say, the y level) than at a lower deterministic level (say, the l level), then it follows from what mathematicians call the Ôpigeon hole principleÕ that there will be at least one y description into which more than one l description must go. (Note that this is essentially a restatement of the main point of multiple realisability.) Thus, recalling Figure 3, there may be points which are distinct at the lower level and which have unique time evolutions at that level but which are the same point at the higher level and still have ultimately distinguishable time evolutions at that higher level. The second related point is that because of this loss of information as we move to higher levels, the topological transforms relating manifolds at lower levels to higher levels (which, as noted above, might not themselves even be continuous) may not be symmetric. That is, once we have transformed a manifold in, say, l space, to one in y space, it may not be possible to get the original surface back by inverting the transform. The easy counterexample to symmetry occurs around a singularity where a volume of points in l space maps to a single point in y space. Finally, we move on to some brief comments about applying this dynamical framework to existing theories and explore how new theories might be formulated within the framework. Within this framework, we may formulate succinctly theories about phenomena in intelligent systems which rely on the relationships between different levels of description. For example, one main point of Horgan and TiensonÕs recent work into the computability and computational tractability of cognitive state transitions can be put very economically in the terms of the new schema. Specifically, they suggest that transitions between y states are not always computable solely on the basis of y level information, because the same y state might supervene on two or more distinct w states which could ultimately evolve along trajectories distinguishable not only in w space but also in y space. particularly in chaotic systems, this means the same y state might branch into two different y states on the basis of lower level sensitive dependence on initial conditions. At the least, y state transitions may prove to be computationally intractable even in the case of tractable w computability. In effect, theirs is a comparison of y state overdeterminism with respect to the w and l levels and underdeterminism with respect to the y level itself. We might extend their ideas by asking about the characteristics of w level indeterminism on the basis of underlying l dynamics and chaos in real space. My own thoughts on applying recursion theory to chaotic analogue systems, which appears in a following section, while confined primarily to noncomputability at the l level, might when paired with this representational schema create a foundation for exploring problems of computability at higher levels of description. The schema might also be a useful point of departure for examining further ramifications of our answer to PylyshynÕs conjecture suggested above. As for new ways of using this schema for exploring interactions between lowlevel chaotic dynamics and characteristics of behaviour at the introspectively accessible level, we might gain new insights into psychological questions about creativity and problem solving, philosophical questions about free will and the relationship between reasons and causes, and computational questions about implementing artificial intelligences. For the psychologist, it could be useful to try to describe the relationship between creativity and SIC at the w or l levels. Because of topological transitivity, the trajectory of a system behaving chaotically can eventually visit every neighbourhood in its phase space. This property of chaotic systems might provide some clues about sudden flashes of inspiration or Ôlateral thinkingÕ which, while entirely deterministic at lower levels, may appear at the introspective level as discontinuities in our train of thought. The speed with which a chaotic system may visit different neighbourhoods of its phase space, coupled with the distributed representational abilities of neural networks, might also offer a partial account of what appears to be content addressable memory. That is, a network visiting large areas of its phase space ÔlookingÕ for a pattern to match might be one mechanism subserving content addressable memory.^{8} Understanding the r™le of content addressable memory may well be critical for making sense of creativity, problem solving, and the socalled frame problem. For the philosopher, explorations of interactions between levels might also provide insights into the appearance of free will. Because so much w and l level dynamics are not available to introspective awareness at the y level, it is clear that apparent contracausal behaviour at the y level might be possible on the basis of entirely deterministic l level causation. Yet this apparent contracausal behaviour neednÕt be without precursor reasons at the y level. Very speculatively, we might say that cognitive transitions at the y level represent the reasons for an agentÕs actions, while more complex dynamics at lower levels represent the causes of those same actions. Any sequence of actions would have traceable reasons in y space but causes only in w or l space.^{9} A full treatment of questions about reducing reason to causation must wait for another occasion. But there appears to be considerable philosophical mileage in the idea that as far as we can be introspectively aware, our behaviour is governed by generalisations linking types of y states, generalisations for which exceptions often occur, but that our behaviour is still entirely determined at lower levels in a sort of ÔinvisibleÕ fashion. As far as the y level is concerned, this account is not too different from the way it looks when we introspectively consider our own behaviour. That is, it seems to us (or to me, anyway!) that I do have particular general patterns of behaviour but that I can always violate those patterns when it suits me (i.e., not when IÕm struck by a sudden fit of indeterministic irrationality). Deterministic but chaotic lowlevel dynamics might provide both a possible account of the elusive Ôwhen it suits meÕ as well as a way of reconciling causal determinism with the appearance of reasoned (and not contracausal) free will. I believe this rough sketch of an approach to the problem of free will complements nicely DennettÕs (1984) compatibilist position. Finally, the quest for artificially intelligent systems could be aided by an exploration of the question of what cognitive state transitions at the y level are served by sensitively dependent processes at lower levels, or at least an exploration of what y level transitions are nondeterministic at that level but are deterministic at a lower level. Gaining partial answers to this question would tell us what processes can be modelled at higher levels with lower degrees of detail and which must be implemented at lower levels with more careful attention to detail. For instance, few would dispute that an artificially intelligent system neednÕt model every single behavioural property of every single neuron in a given humanÕs brain in order to display some of the cognitive faculties of the human. Yet it seems nearly as obvious that for many cognitive faculties we might wish to model, attempting to mimic only the highest level behaviour of the human would be unworkable (either nondeterministic or computationally intractable) because of underdeterminism at that level. On the face of it, there should be some boundary area between the two extremes which would allow an artificial system to mimic an acceptable degree of the humanÕs subtlety without unacceptably burdening computational resources by modelling unnecessary details. Until a better understanding of interlevel dynamical relationships is achieved, any choice for this boundary must be at least partly arbitrary. In short, I believe questions about the relationship between the introspectively available level of description and the finer, externally observable levels of description are important ones. The framework I have suggested is substantially ÔunderdeterminedÕ itself, in that it may not yet be fleshed out with enough detail to allow the formulation of any but the most rudimentary theories or observations. But it is a possible starting point for one approach to understanding the mindbrain as a rich dynamical system. In an appendix to the manuscript of his recent Cambridge lecture series on chaos (Smith 1993) and in a commentary at the July 1993 meeting of the European Society for Philosophy and Psychology, Peter Smith has strongly criticised one of the arguments which has emerged from the preceding way of viewing the relationship between psychological states and the underlying neural activity.^{10} In particular, Smith has criticised the notion that any conclusions concerning socalled anomalous monism (Smith and Jones 1986) can be drawn from the sensitive dependence on initial conditions of chaotic neural subsystems of the brain. The problem is related to the various conclusions which might be drawn depending on how we conceive the mapping of y states to sets of points in l space. We begin with the argument Smith attributes to me and claims to be invalid:^{11} 1. A given initial type of psychological state can be realised in a variety of physical ways. 2. But the timeevolution of the physical states in question is sensitively dependent on initial conditionsÑi.e., we may get markedly different physical upshots arising from very similar initial states. 3. Hence we can get (with significant probability) markedly different psychological upshots arising from the same initial psychological state. The argument, he claims, is invalid because it is exactly analogous to the following, which is obviously invalid because it has true premises and a false conclusion: 1.* A given initial type of thermodynamical state (e.g. a certain temperature) can be realised in a variety of physical states characterised by different position/momenta distributions of the particles in a gas. 2.* But the timeevolution of a state with a given position/momenta distribution is sensitively dependent on initial conditionsÑi.e., we may get markedly different distributional upshots arising from very similar initial states. 3.* Hence we can get (with significant probability) markedly different thermodynamical upshots arising from the same initial thermodynamical state. Smith claims this analogy establishes that this general form of argument is invalid and that nothing about indeterminism at higher levels of description can be inferred from the presence of lowlevel chaos. But compare the following argument, which doesnÕt involve chaos at all: 1.** A given type of poker hand (such as one pair or a full house or a royal flush) can be realised by a variety of physical card distributions. 2.** But the timeevolution of physical card distributions (given an appropriate algorithm for discarding cards) is sensitively dependent on the initial conditions of the cards in the handÑi.e., we may get markedly different physical card distributions arising from very similar initial distributions. 3.** Hence we can get (with significant probability) markedly different final types of poker hands arising from the same initial type of poker hand. What makes the thermodynamical argument invalid, the poker argument valid, and the original psychological state argument open to debate concerns our definitions of temperature, poker hand, and psychological state, respectively. The thermodynamical argument is invalid because temperature just is mean kinetic energy of the particles in the gas. No change in the position or momentum of any particle makes any difference to the temperature as long as the mean is the same. The poker argument is valid because a full house, for instance, just is one pair and three of a kind. Changes in cards do change the hand when they bring about a change in the types or sizes of relevant sets that can be formed from the cards. Thus given two distinct card distributions which both have one pair, for instance, discarding the three other cards in both hands and adding identical cards to the hands can yield completely different hands (such as a full house in one and two pair in another). Both of these arguments are easy to understand because we already know how to construe temperature, and we know what makes a royal flush. The psychological argument is open to debate because there are hidden premises concerned with how we are to understand the mapping from physical states to psychological states. Analysing the other two arguments also relies on an appeal to hidden premises, but in these cases those premisesÑhow to construe temperature and how to construe a poker handÑare uncontroversial. Now we must examine some of the ways we might construe this mapping from l space to y space. With respect to what sort of mapping scheme we should adopt, Smith wonders first if perhaps I have fallen prey to the na”ve assumption that psychological states are simply a uniform coarsegraining of physical states and then goes on to suggest, loosely following Freeman and his colleagues (Freeman 1991, 1989; Yao and Freeman 1990; Skarda and Freeman 1987), that psychological states correspond to large structures in phase space which might be strange attractors. Fortunately I havenÕt fallen prey to the uniform coarsegraining assumption, as should be apparent from my comment in the paper under debate that psychological state space should properly be treated as a general Riemannian space. The reason y space should be treated as a general Riemannian space was simply that there might be complex mappings from l space to y space which varied across the space as a whole. This notion is incompatible with a simple uniform coarse graining, which would yield a uniform geometry across the whole space. But letÕs examine the idea that psychological states correspond to states near strange attractors. Smith provides no indication of how we are to construe the word ÔnearÕ here, but for the present purposes I believe it introduces no confusion simply to replace Ôlying near a strange attractorÕ with Ôlying within the basin of attraction of a strange attractorÕ. Smith notes correctly that if a given psychological state corresponds to lying near (within the basin of attraction of) a strange attractor, then we may observe divergent physical phase trajectories from arbitrarily similar initial conditions while retaining the same psychological state (because those phase trajectories, while divergent, remain near the same attractor). This provides what he calls Ômicrochaos but macropsychological stabilityÕ, and, he continues, Ôthe move from one dynamical state (defined by its attractor) to another as control parameters change can in fact be as deterministic as you likeÕ (Smith 1993, p. 76). Such a picture allows for the wild low level behaviour characteristic of chaotic systems without making the high level psychological behaviour similarly wild. I believe Smith has correctly sussed out a description of something like what must happen in response to changes in parameters such as ion concentrations and neuropeptide distributions. Such physical changes correspond to the control parameters to which Smith refers. Some neuromodulators, for instance, influence the way output frequencies of whole cell families vary in response to afferent signals. Thus they change in a global way the shape of the networkÕs possible trajectories through phase space, and they may well lead to just the kind of change in psychological state which we would expect under SmithÕs picture. But to accept this picture as the whole story is to ignore altogether the r™le of perturbations (corresponding to changes in afferent signals) in shifting the state of a neural network from the basin of one attractor to that of a second attractor, coexistent with the first.^{12} It is an elementary observation about even the simplest artificial neural networks (and thus presumably it applies to the far more complex biological neural nets) that the state spaces^{13} of such networks include multiple attractors. In the lingo of artificial neural nets, this is just another way of saying that the output patterns vary according to changes in the input patterns. To use a hideously simplified discrete example, consider any pattern recognition network which fires a single output unit in response to each of ten possible input patterns (the technical details are irrelevant). Then we can put the point very crudely by saying that all the states of the network corresponding to the presence of an input pattern which looks like a Ô1Õ will be attracted to a manifold in state space where the Ô1Õ output unit fires and the others donÕt, all the states corresponding to the presence of an input pattern which looks like an Ô8Õ will be attracted to a manifold in state space where the Ô8Õ output unit fires the others donÕt, and so on. All ten attractors are coexistent in the state space of the network, and the state of the network evolves in response to changes in the afferent signals, not in response to changes in global control parameters.^{14} SmithÕs picture ignores altogether the response of a neural network to such changes in afferent signals. If we accept the hypothesis that psychological states correspond to basins of attraction, the existence of multiple attractors in the state space of a neural network simply means that cognitive states can change in response to changes in neural input. Indeed, it seems almost obvious that this is at least as large a piece of the puzzle as SmithÕs changes in response to modifications of the control parameters: my psychological state changes when presented with a red apple because of the change in visual input, my psychological state changes when the orchestra begins to play because of the change in auditory input, etc. It is not that my neuropeptides arenÕt playing a r™le, but so are changes in the output frequencies of cells responding to environmental input. It is worth noting that once we accept the presence of multiple attractors in the phase space of a neural system, we neednÕt even appeal to chaos to establish something like psychological state indeterminism.^{15} We can illustrate with a trivially simple example. Consider a twodimensional state space with four attractors such that the basins of attraction are marked off by the four quadrants of the cartesian coordinate system. Now suppose we are given an open circle of radius one, tangent to both the horizontal and vertical axes in quadrant I. This possibility is illustrated in Figure 4. Since all the states within this circle are in the same basin of attraction, they all map to the same psychological state. 