Commentary on Roger Shepard
Abstract: 60 words
Main Text: 1013 words
References: 154 words
Total Text: 1296 words
Department of Psychology and Program in Cognitive Science
University of Arizona
Tucson, AZ
85721
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 it applies to glimpses in time (apparent motion), modality (ventriloquism), and space (Gestalt grouping), has a mathematically elegant solution of nested geometries (Euclidean, Similarity, Affine, Projective, Topology), and is evolutionarily sound despite our Euclidean world.
Not since Hemholtz has the vision for vision been so grand. First, note briefly that Shepard’s approach points the way to a long-overdue task: to compile what can be considered evolutionary constraints on perception. A few of those constraints are: sunlight comes from above (Ramachandran, 1988), one object cannot be in two locations at the same time (Bedford, 1999), babies know they can fall down and not up (visual cliff; Gibson and Walk, 1960), black dots can only match black dots in stereopsis (Marr, 1982), and Gogel’s specific distance tendency (1972) that in the absence of definitive depth cues, an object will appear to be two meters away. They vary from specific to general. Compiling, sorting, weeding, reducing will lead to a set of core constraints on how we perceive and reason about the world.
But the focus here is on Shepard’s goal of integrating generality, mathematical elegance, and evolution, by providing an example inspired by Shepard’s long-standing broad goals (Bedford, in press). Numerical or object identity refers to perceiving and knowing when an object seen at different times refers to different glimpses of the same object. If a rock is thrown behind a dense bush, we usually believe that it is the same rock which emerges out the other side, rather than two rocks, one of which remains behind the bush.
Generality. We suggest the object identity problem is broader. In addition to glimpses separated in time, the decision is required when the glimpses are separated by modality. Suppose you are looking at your pen while writing a note. How do you know that the pen you are seeing and the pen you are feeling refer to one and the same pen? You see the lips of a ventriloquist’s dummy moving and hear the voice of the ventriloquist. You perceive both the sound and the sight, in this case erroneously, as coming from the same object, the dummy. Suppose the samples occur at the same time and the same modality, but differ only in spatial location, e.g. two tennis balls a few inches apart. While usually regarded as definitive for two objects, the object identity decision is required here as well. The two samples could be produced by a single tennis ball viewed with diploma ("double vision") or through a mirror, or could even be a dumbbell properly thought of as a single object. The Gestalt grouping principles are descriptive rules for determining when spatially separated samples belong to the same object, i.e. object identity. Object identity applies to different times, different, modalities, different spatial locations, and even different eyes ("correspondence problem"). It forms the basis of such diverse phenomena as apparent motion, ventriloquism, prism adaptation, Gestalt grouping, priming, and stereopsis, which in turn reflect everyday accomplishments.
Mathematical elegance. Abstracting away from content, the question in its most general form is how are two samples determined to refer to the same or different objects? Following Shepard and Dennett (1996), problems with the same formal structure suggest a common solution. In nature, samples have extended contours, i.e. forms; geometry, the study of form, is a natural candidate. The solution involves a whole set of nested geometries that fit inside one another like Russian nesting dolls; the familiar Euclidean geometry is only the beginning, the smallest "doll" within the set. Felix Klein (1893) showed how different transformations produce different geometries of increasingly larger size within which more and more forms are equivalent. In Euclidean geometry, a square and a displaced square (isometric transformation) are identical, but if the next most radical change to a square is permitted such that it can be stretched uniformly (similarity transformation), this gives rise to a slightly bigger geometry, Similarity geometry, within which a square, a displaced square, and squares of different sizes are all considered the identical form. Next in the hierarchy is Affine geometry, which adds rectangles and sheared squares to the equivalence class, followed by Projective geometry, which broadens to include trapezoids, and finally Topology which is produced by such radical transformations that squares and circles are also equivalent.
For object identity, the more radical the transformation between the samples, the less likely they will be judged as originating from the same object. When there are multiple samples, a mate will be chosen for a sample that comes from the lowest level of the hierarchy. For instance, in apparent motion, if there is a choice between seeing a square move to another square or to a rectangle, object identity will favor the square, but if the choice is between a circle and the rectangle, now object identity will favor the rectangle. Thus, the hierarchy has the desired property that the identical two stimuli will sometimes be judged to refer to the same object, but sometimes not. Transformations from isometric to topological span a range wide enough to apply to nearly all transformations encountered from rocks thrown behind bushes (isometric) to crumpled clothing (topological), as well as image transformations that result from our own moving, tilting and twisting. It is a mathematical solution that has breadth, generality, and elegance.
Evolution. However, doesn’t this solution violate Shepard’s entire thesis that only conditions that prevail in the world will be internalized? As he notes, the space in which we evolved is three-dimensional and Euclidean, yet the above theory uses many geometries that are not Euclidean. Interestingly, there is no violation. All the rules of Euclidean geometry can be derived from Euclid’s original axioms and postulates. Within this axiom approach, removal of axioms produces the more general geometries. For instance, removal of the postulate on angle enables Affine geometry. As Cheng and Gallistel (1984) argue, natural selection would not favor getting an assumption wrong, but could fail to capture all the available principles.
An ingenious evolutionary solution may allow observers to jump between the geometries by alternately giving up and gaining assumptions as the situation warrants. As Shepard argues, in physics, problems have been formulated and reformulated before obtaining generality. We believe removing the restriction of Euclidean geometry is the right reformulation. We are also convinced Shepard would agree.
Bedford, F. L. (1999). Keeping perception accurate. Trends in Cognitive Sciences,3, 4-11.
Bedford, F. L. (in press). Towards a general law of numerical/object identity. Cahiers de Psychologie Cognitve/Current Psychology of Cognition.
Dennett, D.C. (1996). Darwin's Dangerous Idea : Evolution and the Meanings of Life. Touchstone Books.
Cheng, K. & Gallistel, C.R. (1984). Testing the geometric power of an animal's spatial representation. In H. Roitblat, T.G. Bever, & H. Terrace (Eds.), Animal Cognition (pp. 409-423). Hillsdale, NJ: Lawrence Erlbaum.
Gibson, E. J. & Walk, R. D. (1960) The "visual cliff". Scientific American, 202, 67-71
Gogel, W. C. (1972). Scalar perceptions with binocular cues of distance. American Journal of Psychology, 85, 477-497.
Klein, F. (1957). Vorlesungen uber hohere geometrie (Lectures on higher geometry) (3rd ed.) New York: Chelsea. (Original work published 1893)
Marr, D. (1982). Vision. New York, NY: W.H. Freeman and Co.
Ramachandran, V. (1988). Perception of shape from shading. Nature, 331, 163-166.
Thanks to William H. Ittelson, Jason Barker, and Christine Mahoney for helpful discussion. This work supported by a grant award from the Office of the Vice President for Research at the University of Arizona, funded by the University of Arizona foundation.