From Complexity to Creativity -- Copyright Plenum Press, 1997

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Part I. The Complex Mind/Brain

CHAPTER 4. PERCEPTION AND MINDSPACE CURVATURE


CHAPTER FOUR

PERCEPTION AND MINDSPACE CURVATURE

4.1 INTRODUCTION

    The psynet model is a general model of mental structure and dynamics. It does not pretend, however, to exhaust the structure and dynamics of mind. Unlike, say, theories of fundamental physics, it does not pretend to be a complete theory of the domain with which it deals.

    First of all, particular types of minds will have their own peculiar structures and dynamics -- e.g., the human mind inherits a whole host of peculiarities from its specific array of neurotransmitters, its bilaterally symmetric structure, and so forth. And secondly, there may also be other general laws of mental dynamics which are not captured in the basic ideas of the psynet model. At the end of this chapter I will propose one such general law -- the law of "form-enhancing distance distortion," or mindspace curvature.

    The idea of mindspace curvature is a generalization of the idea of visual space curvature, proposed by A.S. Watson in 1978 as a way of explaining the simple geometric illusions. In a recent paper, Mike Kalish and I have shown how this visual space curvature could arise from underlying self-organizing dynamics in mental process space. The picture that emerges is one of subjective visual space undergoing a dynamic, self-organizing autopoietic process -- one which enhances and creates patterns and forms, in the process creating certain logical inconsistencies, that we call "illusions."

    And this notion of visual space creating itself, which arises in the analysis of illusions, turns out to be essential to the psynet model itself. For, the psynet model claims that the mind is an autopoietic magician system -- that the mind, as as a whole, produces itself. This is easy to see in the higher reaches of mind: beliefs support each other, and lower-level percepts lead to higher-level concepts, etc. But it is more difficult to see in the context of the lowest-level mental processes corresponding to the perceptual world. These lowest-level processes, it might seem, come from outside the mind: they are not produced by the rest of the mental system.

    A careful analysis, however, shows that this is not the case. Even the lowest levels of the mind, dealing with raw perceptual information, can be understood as autopoietic systems, producing themselves with the assistance of the immediately higher levels in the perceptual-motor hierarchy. This autopoiesis, which is the cause of perceptual illusions, would seem to have valuable lessons to teach us about the nature of autopoiesis throughout the mind.

4.2 PRODUCIBILITY AND PERCEPTION

    The psynet model states that mental entities are autopoietic magician systems; it also states that mind itself is an autopoietic magician system. This means that, in some sense, each element of the mind is producible by other elements of the mind. This statement is, in Chaotic Logic, called the producibility hypothesis.

    It is not immediately obvious, however that this "producibility hypothesis" is a tenable one. The level at which it would seem most vulnerable to criticism is the perceptual level. Supposing one takes an hierarchical point of view; then it is hardly surprising that higher-level patterns should be emergent from, i.e. producible out of, lower-level ones. But where are the lowest-level processes to come from? Low-level motor pattern/processes are reasonably seen to be produced by higher-level motor pattern/processes. But what about processes carrying low-level perceptual information? On the face of it, this would seem to violate the producibility hypothesis.

    The violation, however, is only apparent. I will show here that the lowest-level perceptual processes in the dual network can, with the aid of higher-level processes, mutually produce each other. Perceptual illusions, I will argue in the following section, are a consequence of quirks in this dynamic of lower-level self-construction -- quirks which manifest themselves as locally-varying curvature of the visual field.

The Construction of Space

    Just as visual forms emerge out of space, so space emerges out of visual forms. This observation is important both philosophically and practically. It is obvious in the case of three-dimensional space, and less obvious, but no less important, in the case of two dimensions.

    The two-dimensional structure of visual space would appear to follow almost directly from the two-dimensional structure of the retina, and the cortical sheets to which the retina maps. There are, admittedly, many other structures to which the 2-D structure of the retina could be mapped besides 2-D lattices. But at any rate, the 2-D lattice is presented to the brain in an immediate and natural ways.

    The step to 3-D vision, on the other hand, requires a far greater degree of abstraction. Somehow the brain posits an additional dimension of lattice structure which is not prima facie there in its input. This additional dimension is an emergent pattern in its binocular input. In this sense, the statement that mind constructs space is not controversial but, rather, almost obvious.

    To understand the emergence of space from visual forms, let L denote a process which produces an abstract spatial lattice structure. I.e., given a collection I of stimuli, L arranges these inputs in a lattice with some specified dimensions. Let P denote the perceptual processes that act on the lattice L*I, producing an higher-level abstracted version of L*I, which we may call Q*I. The idea is that Q*I has already had some segmentation operations done on it -- it is fit to be presented to some of thelower-level cognitive processes.

    In general, we must allow that L and P interact with each other, i.e., that the lattice arrangement of stimuli may depend on the patterns recognized in the lattice. Thus we find Q*I = Pn * (Ln * I), where the higher-order lattice and perceptual processes are defined in terms of an iterative system Ln+1 = Ln * Pn-1, Pn+1 = Pn * Ln-1, where the action of Ln on Pn-1 is a process of top-down parameter adjustment rather than bottom-up information transmission.

    Now, Q*I is not identical to L*I, nor need it be possible to go backwards from Q*I to L*I. But in most cases Q*I will be at least as valuable to the remainder of the mind as L*I. In fact, it will often be more valuable: we need the general picture more than we need the micro-level details. In addition, Q*I will generally be simpler than L*I (a conceptual sketch of a scene being much more compact than a detailed representation of the scene at the pixel level). Thus, the operation Q will, as a rule, be a pattern in the stimulus collection I.

The Construction of Stimuli

    If stimuli can produce space, can space in turn produce stimuli? Not exactly. What can happen, however, is that space and stimuli can cooperate to produce other stimuli. This is the key to understanding the applicability of the producibility hypothesis to low-level perception.

    To make this idea concrete, suppose that one subtracts a particular part of the input collection I, obtaining I' = I - J. Then it will often be the case that the "missing piece" J provides useful information for interpreting the remainder of the image, I'. In other words, at some stage in the production of Q*I', it becomes useful to postulate something similar to J, call it J'. This may occur in the construction of the lattice L, or at a higher level, in the recognition of complex forms in the lattice image. Thus we have, in general, a situation where each part of I is a pattern in Q*I', while Q itself is a pattern in I or I'. In other words, we have a collection of patterns with more structural integrity and resilience than a living organism: chop off any one part, and the other parts regenerate.

    One example of this kind of filling-in, at a very low level, is the maximum entropy principle, or MAXENT, discussed extensively in The Structure of Intelligence. This principle states that, in a lattice L*I' with a missing sublattice L' corresponding to the missing data J, the missing sublattice L' should be filled in with the data that provides the maximum possible entropy. This is a briefly stated and fast-running algorithm for filling-in missing regions, and in many cases it provides very accurate results. To understand MAXENT more concretely, suppose one has a photograph P, and obtains two discretizations of it, D100 which uses a 100x100 pixel lattice, and D400 which uses a 400x400 pixel lattice. If one uses MAXENT to extrapolate from D100 to the 400x400 level of accuracy, one finds that the ensuing image MAXENT(D100) is visually very similar to D400. Viewed the other way around, what this means is that, if a single pixel of D400 is removed, the other three pixelscorresponding to a single pixel of D100 can be used to approximate the missing pixel with reasonable accuracy. Of course, using more pixels in the surrounding region could produce an even better approximation.

Perceptual Autopoiesis and Structural Conspiracy

    I have argued, in this section, for a kind of autopoiesis in perceptual systems, by means of which lower-level perceptual processes are produced by each other, in combination with higher-level perceptual processes. Before summarizing the argument, however, a few ancillary points must be clarified.

    First, it is certainly not true that all possible input stimuli I will be structured so as to lead to this kind of autopoiesis. However, if a collection of stimuli lacking the properties required for autopoiesis is received, it will quickly be transformed into one which does not suffer this lack. This is one of the many ways in which we construct our own world. We grant the world a coherence over and above whatever coherence is intrinsic to it -- and we do this, not only on the abstract conceptual level, but on the level of the processing of perceptual information.

    Next, it should be noted that this same process might occur on a number of hierarchical levels. The output of Q might be taken as inputs to another process Q of similar form, and thus arranged in a lattice. This would appear to be the case in the visual cortex, in which we have lattices of features feeding into lattices of more complex features, and so forth.

    Finally, it is clear that the reconstruction process by which a region I' is produced by its neighbors is not exact. Thus a collection of perceptual processes, in this set-up, does not reproduce itself exactly. Rather, it reproduces itself approximately. In other words, it lies within a stochastic strange attractor in its state space: it can vary, but it varies according to certain hidden regularities, and it is unlikely to vary a great deal.

    The Appendix to this chapter indicates how one may go about elaborating these arguments in more mathematical detail. The main point, however, has already been made. What these arguments show is that the producibility hypothesis is indeed a plausible one. The emergence of higher-level patterns from lower-level ones is plain; and, according to these arguments, the emergence of lower-level processes from higher-level processes and each other is equally unproblematic. The possibility of self-producing mental systems is confirmed. More formally, we are led towards the following conclusion:

PROPOSITION: There exist hierarchical pattern/process systems, involving lattice structure on the lowest perceptual level, which are structural conspiracies (autopoietic, attractive magician systems).

    Admittedly, such an abstract proposition is of rather limited interest, in and of itself. Merely to show that such systems exist, in a mathematical sense, does not say anything about whether such systems are plausible in a scientific sense,i.e. as models of biological, psychological or computational systems. In previous publications, however, I have expended much effort to show that "self-producible" systems -- autopoietic magician systems -- do indeed make scientific sense. The formal notions presented here are, obviously, to be understood in this context.

4.3 THE GEOMETRY OF VISUAL ILLUSIONS

    According to the psynet model, the perceptual world is not just perceived but produced -- and largely self-produced. But what are the consequences of this self-production? Does it make any difference whether we perceive the world or produce it? One obvious consequence of the production of the perceptual world is the existence of perceptual illusions.

    It is worth pausing for a moment to consider the concept "illusion." An illusion is not just a "false perception" -- for there is no absolute, non-perceptual reality to which one might meaningfully compare one's perceptions. An illusion is, rather, an instance where perceptions obtained in different ways disagree. An illusion is a contradiction in the perceptual world.

    For instance, looking at two lines, one judges that line A is longer than line B; but measuring the two with a ruler, one judges that they are the same. Two methods of perceiving the world disagree. The world, as subjectively perceived, is logically inconsistent. This is the essence of illusion. Perceptual illusions are an instance of mental self-organization taking precedence over rational, deductive logic.

Theories of Illusion

    Early perceptual psychologists understood the philosophical power of illusions. Existing illusions were hotly discussed; and the discovery of new illusions was greeted with fanfare. These simple line figures, it was thought, were "windows on the mind." If one could understand the mechanisms of illusions, one would understand the dynamics of perception, and perhaps of mind in general.

    After 150 years of study, however, this early hope has not paid off -- it lost its steam long ago. Today the study of illusions is an obscure corner of perceptual psychology, considered to have little or no general importance. However, despite dozens, perhaps hundreds of theories, the illusions are still not understood.

    The various theories may be, loosely speaking, divided into three categories: physiological, descriptive, and process-oriented. Each of these categories includes a wide variety of mutually contradictory theories. For instance, the physiological theories have covered everything from optical diffraction and eye movements to lateral inhibition in neural networks (Von Bekesy, 1967; Coren et al, 1988). And the descriptive theories have been even more diverse, though they mostly been restricted to single illusions or small classes ofillusions. For instance, it was observed in the middle of the last century that acute angles tend to be underestimated, while obtuse angles tend to be overestimated. Helson (1964), Anderson (1981, 1990) and others have proposed descriptive models based on contrast effects; while theorists from Muller-Lyer (1889) to Fellows (1968) and Pressey (1971) have used the concept of "assimilation" to explain wide classes of illusions.

    Finally, theories of the processes underlying the geometric illusions vary from the simplistic and qualitative to the sophisticated and quantitative. For instance, Hoffman has used a sophisticated Lie algebra model to derive a theory of illusions, the empirical implications of which are, however, essentially equivalent the much older theory of angle distortion. More intuitively, Eriksson (1970) proposed a field-theoretic model of illusions, based on the concept that the lines in the visual field repel and attract one another by means of abstract force fields.

    Cladavetscher (1992) has proposed a general theory of illusions based on information integration theory, which attributes illusions to a weighted sum of contrast and assimilation processes (which, however, are themselves only vaguely defined).

The Curvature of Visual Space

    Out of the vast range of illusion theories, the most concordant with the psynet model is the one proposed by A.S. Watson in 1978, which states that the geometric illusions result from the curvature of visual space. This theory is primarily descriptive, but it also hints at underlying processes. In the end, of course, a thorough understanding of the illusions will have to span the physiology, description and process levels.     

    The concept of curved visual space is most often encountered in the theory of binocular perception. In these models, however, the spaces involved usually possess constant, negative Gaussian curvature, which hypothesizes a non-Riemannian space of fixed geometric structure). Along similar lines, Drosler (1978) has explained aspects of the psychophysics of visual extent by hypothesizing a constant positive Gaussian curvature for monocular visual space. Watson's theory is different from all these in that the proposed curvature of visual space is not constant: it varies locally based on the distribution of objects in visual space, according to a particular equation.

    In its use of locally varying curvature, Watson's theory is a conceptual relative of Einstein's General Theory of Relativity, in which massive objects are considered to curve the spacetime continuum. However, it may also be reformulated as a force field theory, bringing it closer to Gestaltist notions of perception. In this interpretation, rather than curving visual space, objects are understood to attract other objects toward themselves by a certain complex, distance-dependent function.

    What does the curvature of visual space mean? Think about it this way. Nearly everyone can recognize a triangle. And, as Helmholtz discovered, nearly everyone, if asked to estimate the individual angles that make up a triangle, will respond with numbers that do not sum to 180 degrees. This seems paradoxical;but the paradox is removed if one drops the implicit assumption that visual space is Euclidean. The key insight of Watson's (1978) theory of illusions is that all the simple geometric illusions can be dealt with in a similar manner: by introducing the notion of curved monocular visual space.

    According to Watson, each object present in the visual field curves the region of visual space surrounding it. This curving action is carried out in such a way that pairs of small objects near the curving object are perceived as closer together than they actually are, while pairs of small objects at a moderate distance from the curving object are perceived as further apart than they actually are. Pairs of small objects at a sufficiently great distance from the curving object are unaffected. This particular way of deriving curvature from objects is quite different from anything found in relativity or other branches of physics.

    Watson's theory was inspired partly by Eriksson's (1970) unsuccessful field-theoretic model of illusions; it is not surprising, therefore, that it may be recast as a force field model. In particular, Watson's equation for metric distortion is mathematically equivalent to the hypothesis of an attractive force field emanating from an object, with a maximum amount of attraction at some finite nonzero distance.

Some Specific Examples.

    Using his equation for local visual space distortion, Watson (1978) gives detailed explanations for a variety of geometric illusions, including the Poggendorff, Muller-Lyer, Enclosure, Ponzo, Wundt, Hering, Orbison, Zollner and Delboeuf illusions. Extension to other illusions besides these seems unproblematic. Three examples will serve to illustrate the character of the explanations provided: the Poggendorff, Delbouef and Ebbinghaus illusions (see Figure 2).

    The Poggendorff illusion consists of two parallel lines intersected by a diagonal line. The portion of the diagonal line between the two parallel lines is deleted. The remaining segments of the diagonal line then do not appear to be collinear. Watson derives equations for the curvature of visual space induced by the parallel lines. His psychological hypothesis is then that, when trying to continue one of the segments of the diagonal line, a person follows a geodesic in the curved space rather than a straight line (which would be a geodesic in flat, Euclidean space).

    A geodesic is a path with the property that it is the shortest path between any two points that lie on it. It is also the only kind of curve along which one can do parallel propagation of a unit vector. Examples are lines in Euclidean space, and great circles on spheres. There is considerable evidence that geodesics are psychologically natural; for instance, when intuitively estimating the path an object has taken from one point to another, a person naturally assume that the object has followed a geodesic.

To understand the Poggendorff illusion in terms of curved visual space, suppose a person is trying to continue the left segment of the diagonal line. The geodesic which begins at theintersection of that segment with the left parallel line, will intersect the right parallel line at a point above the intersection of the right segment of the diagonal line with the right parallel line. Thus, according to the theory, the person will perceive the right segment of the diagonal line to begin higher than it really does, which is the effect observed experimentally. The parameters of the curvature equations may be set in such a way as to fit the empirical data regarding the amount of displacement.

    Next, the Delbouef illusion consists of two concentric circles (though there are many variants, e.g. concentric squares). The size of the outer circle is underestimated, while the size of the inner circle is overestimated. Here the explanation is more straightforward. One need only assert that the two circles exert an attractive force on each other. The particular form of Watson's force field predicts that, as the two diameters become more and more disparate, the attraction should increases, until a certain point is reached, after which it should begin to decrease. This phenomenon has been observed empirically; it is called the "distance paradox" (Ikeda and Obonai, 1955). A variant on the Delbouef illusion, not considered by Watson, is the Ebbinghaus illusion (Figure 3). This illusion gives particular insight into the nature of Watson's force field. The figure consists of a circle surrounded by four symmetrically positioned context circles. Large context circles make the center circle look smaller than do small context circles.

This illusion was at one point (Massaro and Anderson, 1971) interpreted as a simple contrast effect; however, subsequent experiments reported by Cladavetscher (1990) showed this explanation to be inadequate. Cladavetscher demonstrated a U- shaped curve relating the estimated size of the center circle to the distance between the center circle and the context circles. In order to explain this, he invokes an "information integration" theory, according to which context effects are dominant for large distances, but assimilation effects mute their effect for small distances. This U-shaped curve is taken as evidence for a two- process theory of illusions; Cladavetscher (1990) writes that "no single process seems to provide a satisfactory account of the data." In fact, however, Watson's theory, published 12 years previously, accounts for the U-shaped curve in a very simple fashion. When the center circle is close to the context circles, the center circle is not yet within the range of maximum attraction of the context circles. When the context circles are moved out a little further, the far side of the center circle is within the range of maximum attraction of each context circle, and so the sides of the context circle are pulled closer together. When the context circles are moved even further out, then, the range of maximum attraction of the context circles does not contain the center circle at all, and the perceived size gradually increases again.

    Watson's theory accounts for a wide variety of illusions in a simple, unified way, using only a small number of numerical parameters. A word should be said, however, about the possible interactions between the process identified by Watson and other aspects of vision processing. There are many other processeswhich alter the structure of visual space, and one may very plausibly suppose that some of these act simultaneously with Watson's curvature process. It is possible that the interactions between the curvature process and other processes might, in some cases, confuse the practical application of the theory. In the Ebbinghaus illusion, for instance, Watson's theory leads to the hypothesis that the center circle should actually be perceived as a sort of "diamandoid" shape, due to the non-isotropic nature of the curvature induced by the four context circles. One possibility is that this is actually the case; another possibility is that a higher-level propensity to perceive shapes as circles intervenes here. Similarly, in the Poggendorff illusion, Watson's theory predicts that the diagonals should not actually be perceived as straight line segments, but should be pulled toward the parallel lines by an amount which varies with distance. Again, it is possible that this is the case, and it is also possible that a higher-level "propensity for straightness" intervenes, thus restoring a semblance of Euclidean structure. These questions are interesting and important ones; however, we will not pursue them further here.

Illusion and Self-Organization

    Watson's theory of illusions is descriptive: it says that visual space is curved, but doesn't say why or how. In accordance with the ideas of the previous section, we would like to say that visual space curvature is a consequence of low-level autopoiesis; that it is a quirk in the way the mental processes responsible for visual space construct themselves. In order to do this, however, it is necessary to connect the abstract notion of visual space curvature with some kind of lower-level self-organizing dynamic.

    This has been done in a recent paper by Mike Kalish and myself (Goertzel and Kalish, 1996). We have demonstrated the emergence of the force field equivalent of Watson's equation for visual space curvature from an underlying process of spreading activation dynamics and nonlinear filtering. The spreading activation process may be understood in the context of a continuum model of visual space; or it may, with equal facility, be interpreted discretely, in terms of neural network or magician-type models.

    One considers a two-dimensional lattice of processing elements, each corresponding a small region of visual space. The activation of each element corresponds to the brightness of the corresponding region of space. The network then undergoes a two-phase dynamic: first a phase of spreading activation, in which bright elements pass their brightness to other elements in their neighborhood, and then a postprocessing phase, in which the extra activation introduced by the spreading is removed. These phases are repeated, in order, until the desired amount of motion has been obtained. The result of this two-phase dynamic is a self-organization of visual space; i.e., the active elements move around in a way which is determined by their distribution.

    The dynamics of the motion of the active elements can be modified by adjusting the spreading activation function. Inparticular, as we show, one can construct a spreading activation function that causes active elements to move as if they were obeying a force field equivalent to Watson's equation for visual space curvature. In this way, the geometric illusions are obtained as a consequence of the self-organization of visual space.

The Utility of Illusions

    Finally, an intriguing open question is the utility of the processes leading to illusions. Are the illusions just the result of a shoddily designed visual system, or are they a difficult-to-avoid consequence of some process that is useful for other reasons?

    The spreading activation model of (Goertzel and Kalish, 1996) does not resolve this long-standing question, but it does lead to an highly suggestive hypothesis. For spreading activation and subsequent nonlinear filtering is a dynamic of great potential use in early vision processing. It filters out missing segments in lines, and in general "fills out" shapes. Thus it seems quite plausible to conjecture that the illusions are caused by a spreading activation process which evolved for another purpose entirely: not for distorting things metrically, but rather for constructing comprehensible shapes.

    The mathematical construction of (Goertzel and Kalish, 1996) can be naturally interpreted as a biological model, following the example of Kohonen (1988), who takes two-dimensional lattices of formal neurons to correspond roughly to portions of cortical sheet. Or it can be understood purely abstractly, in the spirit of the previous section: as a particular dynamic amongst lower-level pattern/process magicians. The two-phase dynamic we have identified is a plausible mechanism for the self-organization of that part of the dual network dealing with visual space. It carries out a consolidation and solidification of forms -- and as a consequence, it distorts forms as well. The illusions arise because, in the self-organization of the dual network, autopoeisis and pattern come first, and deductive, rational order comes after.

    Finally, what about perceptual illusions as windows on the mind? Do the processes identified here as being essential to visual illusions have any more general importance? Or are illusions just another example of autopoiesis and self-organization? This question will be addressed in the following section. As I will argue there, it seems quite plausible that the process of local curvature distortion might have implications beyond the study of visual illusions. It may well be a part of the formation of mental structure in general. If this is true, then the conclusion is that the early visual psychologists were right: illusions really are "windows on the mind." They really do embody deep processes of more general importance.

4.4 MINDSPACE CURVATURE

    I believe that the phenomenon of "inner space curvature" isnot unique to visual perception, or to perception at all, but rather represents a general principle influencing the structure and dynamics of the mind. In particular, the concept of mindspace curvature fits in very nicely with the psynet model. It is not a core concept of the psynet model, but it does have the potential to help explain an important question within the model itself: the emergence of the dual network.

Form-Enhancing Distance Distortion

    A less poetic but more precise name for the kind of "mindspace curvature" I am talking about here is FEDD, or "form-enhancing distance distortion." This is what is seen in visual illusions, and what I am proposing to apply more generally throughout the mind.

    FEDD applies wherever one has a mental process P that is responsible for judging the degrees by which other mental processes differ. This process P may be distributed or localized. P's judgement of the difference between x and y will be denoted dP(x,y). Note that this function dP need not necessarily satisfy all the mathematical definition of a "metric" -- neither symmetry, nor the triangle inequality.

    FEDD states that, over time, process P's judgement of distance will change in a specific way -- in the same way observed in visual illusions, as discussed in the previous section. For each fixed x, the distances dP(x,y) will all be decreased, but the amount of decrease will be greatest at a certain finite distance dP(x,y) = F.

    In other words, according to FEDD, each element x pulls the other elements y closer to it, but the maximum amount of attraction is experienced at distance F. This process of adjustment does not continue forever, but will dwindle after a finite period of time. When new entities are introduced to P, however, the process will be applied to the new entities.

    What effect does the pull of x have on the distances between two entities y and z? It is not hard to see that, if y and z are sufficiently close to x, then their mutual distance is decreased; while, if y and z are at a moderate distance from x, their mutual distance is increased. The fabric of the space judged by P is thus distorted in a somewhat unusual way, similar to the way mass distorts space in General Relativity Theory.

    What is the purpose of mental-space-distorting force, this FEDD? As the name suggests, it gives additional form to the processes judged by P. In particular, it enhances clusters. If a large number of processes are all close to each other, the distance adjustment process will bring them even closer. On the other hand, processes surrounding the cluster will tend to be pushed further away, thus making an "empty space" around the cluster. The net effect is to create much more clearly differentiated clusters than were there before. This may be verified mathematically and/or by computer simulations.

    FEDD will not create significant structure where there was no structure before. However, it will magnify small structures into larger ones, and will "clean up" fuzzy structures. Furthermore, it acts in a form-preserving manner. For instance,if there is, instead of a circular cluster, a cluster in the shape of a line, then FEDD will enhance this cluster while retaining its line shape.

    FEDD causes patterns to emerge from collections of processes. Thus it is both a pattern recognition process and a pattern formation process. To obtain more concrete characterizations of what FEDD does, it is necessary to look at the different areas of psychological function separately.

Applications of Mindspace Curvature

    I have already mentioned the relevance of FEDD to visual perception. A FEDD-like process acting on monocular visual space can explain all the simple geometric illusions (Muller-Lyer, Poggendorff, Enclosure, etc. etc. etc.). My own simulations indicate that FEDD can also do some simple pre-processing operations in vision processing. E.g., it fills in gaps in lines and shapes, and generally makes forms clearer and more distinguishable.

    Another likely application is to social psychology. Lewin's theory of force fields in social psychology is well-known. Dirk Helbing, in his book "Quantitative Sociodynamics" (1995) has given a mathematical formulation of Lewin's theory in terms of diffusion equations. Lewin and Helbing's force fields do not necessarily satisfy the form of FEDD. However, it would be quite possible to seek evidence that social fields do, in reality, satisfy this form.

    For instance, I would hypothesize that individuals tend to:

        1) overestimate the similarity of individuals in

            their own social group to one another

        2) underestimate the similarity between individuals

            in their group and individuals "just outside"

            their group

If this is true, it provides evidence that individuals' measures of interpersonal distance obey the FEDD principle. Intuitively speaking, FEDD would seem to provide a very neat explanation for the formation of social groups, meta-groups, and so on.

    Finally, there would seem to fairly strong evidence for something like FEDD in the areas of concept formation and categorization. A "concept" is just a cluster in mental-form space; a collection of mental forms that are all similar to each other. Things which we have grouped together as a concept, we tend to feel are more similar than they "really" are. Differences between entities in the concept grouping, and entities outside, are overestimated. This is FEDD. As in the case of perception, here FEDD serves to create structure, and also to cause errors. Mark Randell (personal communication) has suggested that cognitive errors created in this way should be called "cognitive illusions," by analogy to perceptual illusions.

    Another possible relationship is with our tendency to make systematic errors in reasoning by induction. Human beings tend to jump to conclusions: as a rule, we are overly confident that trends will continue. This may be seen as a consequence of bunching previously seen situations overly close together, andputting new situations in the bunch.

FEDD and the Emergence of the Dual Network

    Now let us see what FEDD has to tell us about the psynet model. The dual network, I have said, is a network of processes which is simultaneously structured according to associative memory (similar things stored "near" each other) and hierarchical control (elements obtaining information from, and passing instructions to, lower-level elements). The only way to combine hierarchy and associativity is to have a "fractal" or recursively modular structure of clusters within clusters within clusters.... Each cluster, on each level, is a collection of related entities, which is govered implicitly by its own global attractor. It is this global attractor which carries out the hierarchical control of the smaller-scale attractors of the entities making up the cluster.

    FEDD is a method for inducing and enhancing clusters; it explains how the dual network might emerge in the first place. Each pattern/process P in the hypothesized mind magician system induces its own distance measure: namely, dP(x,y) is the amount of structure that P recognizes in x but not y, or in y but not x. Here structure can be defined in a number of ways, e.g. by algorithmic information theory. The emergence of the dual network in the mind magician system is explained by the assumption that these distance measures are updated in the manner prescribed by FEDD.

    This idea has interesting implications for human memory. Not only does FEDD explain the nature of concept formation within memory systems but, via the theory of the dual network, it explains the formation of memory systems themselves. What psychologists call separate memory systems are just different clusters in the overall dual network. The clusters all have smaller clusters within them, which is why, having begun to divide memory into separate systems, memory researchers keep coming up with more and more and more different subsystems, subsubsystems, etc. It may be that some of this clustering is there from birth (e.g. episodic versus semantic memory, perhaps). But FEDD could enhance this innate clustering, and help along the more refined clustering that comes along with experience.

Conclusion

    In sum, I contend that the early psychologists, who believed visual illusions to contain deep secrets of mental process, were not far wrong. The illusions give vivid pictorial examples of how we distort things, how we move things around in our minds to make more attractive forms -- to create patterns, in short. We create patterns because we need patterns; because pattern is the stuff of mind.

    Mindspace curvature, distance distortion, is a powerful way of creating new patterns and solidifying old ones. It is a very simple dynamic, and a very general one: it applies wherever there is judgement of distances, i.e., wherever there is evaluation of similarity, or relative complexity. Like gravitational spacetime curvature in general relativity, it is only one force among many. But it is an important force, deserving of much further study.

APPENDIX: PERCEPTUAL PRODUCIBILITY

    In section 2 we have argued that lower-level perceptual processes produce each other, with the collaboration of higher-level processes. Using the notation of that section, I will now pursue this same line of argument in a more specific context.

    Suppose, for concreteness, that I is drawn from the set Sn of collections of n2 real numbers (representing, say, brightnesses perceived at different points of the retina). Let Mn denote the set of nxn real matrices; this will be, also for concreteness, our set of possible lattices, the range of the function L. The lattice process L will be taken as a fixed element of the class of computable, bijective functions that map Sn onto Mn.

    Let Mn;i,j denote the equivalence class containing all matrices Mn which are equal to each other except possibly in the (i,j) entry; let M' denote the set of all such equivalence classes. Let F denote a "filling-in" algorithm, i.e. a computable function that maps M' into Mn. Where J is an element of I, let a(J) denote the corresponding entry (i,j) of L*I. As with M, let I' = I'(J) = I - J. Finally, let P map Mn into a space F of "higher-order features," the nature of which is not important.

    Now, let G be the subset of Sn with the property that, whenever I is in G,

[1+d(I'',I)] [s(Q*I'') + s(K) + C] < s(Q*I'),

where K is an approximation of J, I'' is the member of Mn obtained by inserting K where J should have been, and C is a constant representing the complexity of this insertion. In other words, this class G of input vectors contains only input vectors whose parts are all integral to the whole, in that it is simpler to assume the part has something close to its correct value, than just to ignore the part altogether.

    Clearly, not all inputs I will fall into category G. The nature of category G depends on the process P, the simplicity measure s, the lattice process L, and the metric d. What is required, however, is merely that G is nonempty. In practice one wishes G to be widely distributed throughout the space Sn. The input I may not initially be a member of G, but by a process of successive iteration, the system will eventually produce an input vector I which does belong to G.