Roger Shepard

"The system of constraints that governs the projections and transformations of... bodies in space must long ago have become internalized as a powerful, though largely unconscious, part of our perceptual machinery."

"In spite of some unresolved issues, the close match we have found between mental rotation and their counterparts in the physical world leads inevitably to speculations about the functions and origin of human spatial imagination. It may not be premature to propose that spatial imagination has evolved as a reflection of the physics and geometry of the external world. The rules that govern structures and motions in the physical world may, over evolutionary history, have been incorporated into human perceptual machinery, giving rise to demonstrable correspondences between mental imagery and its physical analogues."

"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 cancelled 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"

"Students of the human mind have long noted its ability to mimic internally the positive notions and transformations of objects in the external world. In the middle of the eighteenth century, the British empiricist David Hume wrote that to "join incongruous shapes and appearances costs the imagination no more trouble than to conceive the most natural and familiar objects" and that "this creative power of the mind amounts to no more than the faculty of compounding, transposing, augmenting, or diminishing the materials afforded us by the senses and experience.""

"The subject detects the presence and interrelationships of the basic components of one of the two-dimensional drawings - particularly, the variously oriented straight lines, the several types of vertices by which they are connected and, presumably, something of the structural relationships among these components within the two-dimensional pattern. Then, on the basis of some higher-level processing of these extracted features and their interrelationships, an internal representation, code, or verbal description is generated for each picture separately that captures the intrinsic structure of the three-dimensional object in a form that is independent of the particular orientation in which that object happens to be displayed."

"Nor do such theories provide a ready account for the equivalence of the slopes of the reaction-time functions for the picture-plane and depth pairs. For, in order to explain the dependence of reaction time on angular difference, we must suppose that the features that are being compared are the features of the two-dimensional drawings, which differ more and more with angular departure, and not the features of the three-dimensional objects, which are the same regardless of orientation."

"Most studies employing three-dimensional objects as stimuli have used simultaneous presentation whereas most studies employing two-dimensional objects have used comparison of a single visual stimulus with a memory presentation. We suspect that it is this procedural difference rather than the difference in dimensionality that is the principal determiner of rate of mental rotation."

"A psychological space is established for any set of stimuli by determining metric distances between the stimuli such that the probability that a response learned to any stimulus will generalize to any other is an invariant monotonic function of the distance between them. To a good approximation, this probability of generalization (i) decays exponentially with this distance, and (ii) does so in accordance with one of two metrics, depending on the relation between the dimensions along which the stimuli vary. These empirical regularities are mathematically derivable from universal principles of natural kinds and probabilistic geometry that may, through evolutionary internalization, tend to govern the behaviors of all sentient organisms."

"I suggest that the psychophysical function that maps physical parameter space into a species' psychological space has been shaped over evolutionary history so that consequential regions for that species, although variously shaped, are not consistently elongated or flattened in particular directions."

"We generalize from one situation to another not because we cannot tell the difference between the two situations but because we judge that they are likely to belong to a set of situations having the same consequence."

"R. N. Shepard [1978] has argued that a number of highly original and significant creations of the human mind have been produced by a mode of thinking which was essentially nonverbal, involving internal representations which could best be described as images of a largely spatial, and often visual character. Shepard provides an impressive list of creative scholars whose most outstanding achievements have been the result of highly visual thinking: Einstein, James Clerk Maxwell, Michael Faraday, Hermann von Helmholtz, Francis Galton, and Friedrich A. Kekule are on the list as is the mathematician ."