In psychology, as in physics, principles approach universality only if formulated at a sufficiently abstract level. Among the most fundamental principles are those of generalization and inductive inference and those of perceptual and mental transformation. With respect to their appropriate abstract representational space, the former principles are well formulated as (Bayesian) integration over suitable (e.g., connected) subsets of points in the space, and the latter as geodesic (hence, least-time) paths between points in the space. Critics sometimes insufficiently appreciate the following: (a) Generality requires such abstraction. (b) Perceptual principles are not themselves given in sensory input. (c) Principles of learning are not themselves learned. (d) Though all such principles are somehow instantiated in the brain, their ultimate, nonarbitrary source must be sought in the regularities of the world – including those reflecting abstract mathematical principles (e.g., of group theory and symmetry). New light may thus be shed on the cognitive grounds of science and ethics.

The universality, invariance, and elegance of principles governing the universe may be reflected in principles of the minds that have evolved in that universe – provided that the mental principles are formulated with respect to the abstract spaces appropriate for the representation of biologically significant objects and their properties. (1) Positions and motions of objects conserve their shapes in the geometrically fullest and simplest way when represented as points and connecting geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of three-dimensional space and the symmetry group of each object. (2) Colors of objects attain constancy when represented as points in a three-dimensional vector space in which each variation in natural illumination is canceled by application of its inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source. (3) Kinds of objects support optimal generalization and categorization when represented, in an evolutionarily-shaped space of possible objects, as connected regions with associated weights determined by Bayesian revision of maximum-entropy priors.

Statistical regularities of the environment are important for learning, memory, intelligence, inductive inference, and in fact, for any area of cognitive science where an information-processing brain promotes survival by exploiting them. This has been recognised by many of those interested in cognitive function, starting with Helmholtz, Mach, and Pearson, and continuing through Craik, Tolman, Attneave, and Brunswik. In the current era, many of us have begun to show how neural mechanisms exploit the regular statistical properties of natural images. Shepard proposed that the apparent trajectory of an object when seen successively at two positions results from internalising the rules of kinematic geometry, and although kinematic geometry is not statistical in nature, this is clearly a related idea. Here it is argued that Shepard's term, “internalisation,” is insufficient because it is also necessary to derive an advantage from the process. Having mechanisms selectively sensitive to the spatio-temporal patterns of excitation commonly experienced when viewing moving objects would facilitate the detection, interpolation, and extrapolation of such motions, and might explain the twisting motions that are experienced. Although Shepard's explanation in terms of Chasles' rule seems doubtful, his theory and experiments illustrate that local twisting motions are needed for the analysis of moving objects and provoke thoughts about how they might be detected.

The notion of internalization put forth by Roger Shepard continues to be appealing and challenging. He suggests that we have internalized, during our evolutionary development, environmental regularities, or constraints. Internalization solves one of the hardest problems of perceptual psychology: the underspecification problem. That is the problem of how well-defined perceptual experience is generated from the often ambiguous and incomplete sensory stimulation. Yet, the notion of internalization creates new problems that may outweigh the solution of the underspecification problem. To support this claim, I first examine the concept of internalization, breaking it down into several distinct interpretations. These range from well-resolved dynamic regularities to ill-resolved statistical regularities. As a function of the interpretation the researcher selects, an empirical test of the internalization hypothesis may be straightforward or it may become virtually impossible. I then attempt to cover the range of interpretations by drawing on examples from different domains of visual event perception. Unfortunately, the experimental tests regarding most candidate regularities, such as gravitational acceleration, fail to support the concept of internalization. This suggests that narrow interpretations of the concept should be given up in favor of more abstract interpretations. However, the latter are not easily amenable to empirical testing. There is nonetheless a way to test these abstract interpretations by contrasting internalization with the opposite concept: externalization of body dynamics. I summarize evidence for such a projection of body constraints onto external objects. Based on the combined evidence of well-resolved and ill- resolved regularities, the value of the notion of internalization has to be reassessed.

Shepard has supposed that the mind is stocked with innate knowledge of the world and that this knowledge figures prominently in the way we see the world. According to him, this internal knowledge is the legacy of a process of internalization; a process of natural selection over the evolutionary history of the species. Shepard has developed his proposal most fully in his analysis of the relation between kinematic geometry and the shape of the motion path in apparent motion displays. We argue that Shepard has made a case for applying the principles of kinematic geometry to the perception of motion, but that he has not made the case for injecting these principles into the mind of the percipient. We offer a more modest interpretation of his important findings: that kinematic geometry may be a model of apparent motion. Inasmuch as our recommended interpretation does not lodge geometry in the mind of the percipient, the motivation of positing internalization, a process that moves kinematic geometry into the mind, is obviated. In our conclusion, we suggest that cognitive psychologists, in their embrace of internal mental universals and internalization may have been seduced by the siren call of metaphor.

Roger Shepard's proposals and supporting experiments concerning evolutionary internalized regularities have been very influential in the study of vision and in other areas of psychology and cognitive science. This paper examines issues concerning the need, nature, explanatory role, and justification for postulating such internalized constraints. In particular, I seek further clarification from Shepard on how best to understand his claim that principles of kinematic geometry underlie phenomena of motion perception.

My primary focus is on the ecological validity of Shepard's kinematic constraint in the context of ordinary motion perception. First, I explore the analogy Shepard draws between internalized circadian rhythms and the supposed internalization of kinematic geometry. Next, questions are raised about how to interpret and justify applying results from his own and others' experimental studies of apparent motion to more everyday cases of motion perception in richer environments. Finally, some difficulties with Shepard's account of the evolutionary development of his kinematic constraint are considered.

Shepard has argued that a universal law should govern generalization across different domains of perception and cognition, as well as across organisms from different species or even different planets. Starting with some basic assumptions about natural kinds, he derived an exponential decay function as the form of the universal generalization gradient, which accords strikingly well with a wide range of empirical data. However, his original formulation applied only to the ideal case of generalization from a single encountered stimulus to a single novel stimulus, and for stimuli that can be represented as points in a continuous metric psychological space. Here we recast Shepard's theory in a more general Bayesian framework and show how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic model of similarity, which is conventionally thought of as the primary alternative to Shepard's continuous metric space model of similarity and generalization. This unification allows us not only to draw deep parallels between the set-theoretic and spatial approaches, but also to significantly advance the explanatory power of set-theoretic models.

A general framework for the explanation of perceptual phenomena as internalizations of external regularities was developed by R. N. Shepard. A particular example of this framework is his account of perceived curvilinear apparent motions. This paper contains a brief summary of the relevant psychophysical data, some basic kinematical considerations and examples, and several criticisms of Shepard's account. The criticisms concern the feasibility of internalization of critical motion types, the roles of simplicity and uniqueness, the contrast between classical physics and kinematic geometry, the import of perceived path curvilinearity, and the relation of perceptual and scientific knowledge.

We present data on colour generalisation by chicks relevant to Tenenbaum and Griffiths' (T&G) Bayesian framework. Chicks were trained with either one or two colours, and tested for interpolation and extrapolation. T&G's framework predicts an observed lack of extrapolation on the red to yellow line in colour space. A modification incorporating stimulus uncertainty deals with a prototype effect, where an intermediate is preferred to exemplars. After training to complementary colours, chicks do not generalise across an intermediate grey as T&G predict. [Tenenbaum & Griffiths]

Object identity, the apprehension that two glimpses refer to the same object, is offered as an example of combining generality, mathematics, and evolution. We argue that it applies to glimpses in time (apparent motion), modality (ventriloquism), and space (Gestalt grouping); that it has a mathematically elegant solution of nested geometries (Euclidean, Similarity, Affine, Projective, Topology); and that it is evolutionarily sound despite our Euclidean world. [Shepard]

The limitations of the concept of internalised kinematic geometry have been recognised by Barlow, Hecht, Kubovy & Epstein, and Todorovič. I am in agreement but I still find the perception of curvature in two frames of apparent motion fascinating and I suggest some new directions. [Barlow; Hecht; Kubovy & Epstein; Shepard; Todorovič]

There is an old joke about a theoretical physicist who was charged with figuring out how to increase the milk production of cows. Although many farmers, biologists, and psychologists had tried and failed to solve the problem before him, the physicist had no trouble coming up with a solution on the spot. “First,” he began, “we assume a spherical cow ... ” [Tenenbaum & Griffiths]

Shepard claims that color constancy needs linear basis-function spectra, and infers the illuminant before removing its dependency. However, of the models of color constancy that have exact (and reasonable) spectral regimes, some do not need linear basis-function expansions of reflectance and illuminant spectra, some do not solve for the illuminant, and some estimate only partial object-reflectance information for single or multiple objects. [Shepard]

Trichromacy may result from an adaptation to the regularities in terrestrial illumination. However, we suggest that a complete characterization of the challenges faced by colour perception must include changes in surface surround and illuminant changes due to inter-reflections between surfaces in cluttered scenes. Furthermore, our trichromatic system may have evolved to allow the detection of brownish-reddish edibles against greenish backgrounds. [Shepard]

We argue that confusability between items should be distinguished from generalization between items. Shepard's data concern confusability, but the theories proposed by Shepard and by Tenenbaum & Griffiths concern generalization, indicating a gap between theory and data. We consider the empirical and theoretical work involved in bridging this gap. [Shepard; Tenenbaum & Griffiths]

Shepard's exponential law provides a functional explanation of generalization. The account complements the more common mechanistic models. The elegant and powerful analyses answer one of Tinbergen's (1963) four whys of behavior: a benefit conferred on the animal by generalizing in this way. A complete account might address evolutionary and developmental questions in addition to mechanistic and functional ones. [Shepard]

Contra Shepard we argue, first, that his presentation of a three-dimensional representational (psychological or phenomenal) colour space is at odds with many results in colour science, and, second, that there is insufficient evidence for Shepard's stronger claim that the three-dimensionality of colour perception has resulted from natural selection, moulded by the particulars of the solar spectrum and its variations. [Shepard]

We criticise Shepard's notions of “invariance” and “universality,” and the incorporation of Shepard's work on inference into the general framework of his paper. We then criticise Tenenbaum and Griffiths' account of Shepard (1987b), including the attributed likelihood function, and the assumption of “weak sampling.” Finally, we endorse Barlow's suggestion that minimum message length (MML) theory has useful things to say about the Bayesian inference problems discussed by Shepard and Tenenbaum and Griffiths. [Barlow; Shepard; Tenenbaum & Griffiths]

Shepard's concept of internalization does not suggest mechanisms which help to understand how the brain adapts to changes, how representations of a steadily changing environment are updated or, in short, how brain learning continues throughout life. Neural mechanisms, as suggested by Barlow, may prove a more powerful alternative. Brain theories such as Adaptive Resonance Theory (ART) propose mechanisms to explain how representational activities may be linked in space and time. Some predictions of ART are confirmed by psychophysical and neurophysiological data. [Barlow; Shepard]

A view is put forward, according to which various aspects of the structure of the world as internalized by the brain take the form of “neural spaces,” a concrete counterpart for Shepard's “abstract” ones. Neural spaces may help us understand better both the representational substrate of cognition and the processes that operate on it. [Shepard]