Towards a General Law of Numerical/Object Identity

A theory of object identity, knowing whether or not two glimpses refer to the exact same object, is presented in two parts. Part I argues that apparent motion, prism adaptation, ventriloquism, priming, stereopsis, and Gestalt grouping all require the identity decision. It is argued that object identity is general and required when samples come from different times, places, modalities, and eyes. Part II argues there is a common solution at a sufficient level of abstraction. Sample 1 and sample 2 are regarded as two forms which differ by a transformation corresponding to one of five geometries, Euclidean, Similarity, Affine, Projective, and Topology (Klein, 1893), that nest within each other like a set of Russian nesting dolls. Identity is resolved from the lowest level of the hierarchy available in the situation, producing a flexible solution whereby the same two samples will sometimes refer to the same object and sometimes not. We believe the world contains objects and events that extend in space and endure in time. A general problem for observers is to extract those meaningful individuals. One specific problem is how we determine that a stimulus we are looking at now reflects the exact same item as one encountered before, an issue known as identity, numerical identity, or object identity (e.g. Hirsch, 1982; Leslie, Xu, Tremoulet, and Scholl, 1998; Meltzoff and Moore, 1998; Spelke, et al., 1995; Strawson, 1959; Xu and Carey, 1996). Several challenges await an observer who must make an object identity decision. First, movement of the eyes, head, and body contribute to an ambiguous retinal image. As we move ourselves around, the image on the retina changes. A person you see is first large, then small, on your retina. How do you know you are looking at the same person? You turn sidewise while looking at the closet door and the image transforms from a rectangle to a trapezoid. How do you know it is the same door? Traditionally, the perceptual constancies have described the accomplishment that the object is perceived as staying constant despite the changing retinal image (Helmholtz, 1860; Hering, 1905; see Woodworth, 1938). Second, the objects of our perceptions themselves can move and change (e.g. Shepard, 1988). Rocks can be thrown, sponges squished, leaves on a branch blow in a breeze, tigers pounce, people can twist and turn, and frogs can turn into princes, at least once upon a time. The image cannot be the same before and after each event, yet the pre and post-transformed stimuli often refer to the same object. When an object changes, we need both to perceive those changes to the object as well as know we are still dealing with the same object. Third, identifying enduring objects is further complicated by a lack of continuity of sensory stimulation (e.g. Michotte, 1963; Spelke and Kestenbaum, 1986; Xu and Carey, 1996). If you see Jack’s beanstalk grow before your very eyes, you accept it is the same object despite the unprecedented growth. But stimuli go away and reappear when we blink our eyes, turn away for an instant, attend for a while to something else, or come back later in the day. We do not always have continuity to tell us that a transformation, however bizarre or ordinary nonetheless reflects the same object. While these three problems for object identity are already general and formidable, progress is sometimes obtained, and solutions simplified, by becoming more general still (see Shepard, 1994, 2001). This article makes two claims: 1) Object identity is universal and abstract. Not only is there a problem across time, as it is usually formulated, but also across modalities, spatial locations, and even the left and right eyes. Asked generally, how does the observer determine when any two nonidentical samples arise from the same object? Because much of interacting with the world involves non-identical samples, I claim the same object identity decision plays a role in nearly all domains. 2) There are a core set of criteria abstract enough to apply to all object identity domains. These apply wherever the samples originate (time, space, F. L. Bedford. A general law of numerical/object identity 3 modality, or eye), whatever the scale (short or long time frame, or distance), the level of the computation (preconstancy/retina and postconstancy/postretina) or the size of the sample (point sources, extended contours). We believe this core can be described by geometry not just the familiar Euclidean geometry, but a whole family of at least 5 different-sized geometries (Klein, 1893/1957) that nest within one another like a set of Russian nesting dolls. The probability that two samples will be judged to refer to one object decreases as the smallest geometry within which the they are equivalent gets larger. Part I defends the first claim. It is shown how a number of classic and popular phenomena all depend critically on the object identity decision. Within many of those domains, a principle equivalent to object identity has been independently rediscovered, but the areas rarely interact with one another and are considered to reflect independent modules of mind Part II defends the second claim. It is shown how the mathematician Klein’s hierarchy of geometries is essential for object identity and how it can be applied to the diverse manifestations of identity claimed in Part I, even when they do not appear to involve shape or geometry. Discussion follows each