Cahiers de Psychologie Cognitive/
Current Psychology of Cognition
2001, 20, (3-4), 277-293
"A law of numerical/object identity"
Object Identity Theory and the
nature of general laws
Felice L. Bedford
University of Arizona, Tucson, USA
General laws link areas previously regarded as unrelated. Moreover, the greater the number of areas, and the more disparate they are, the greater the contribution of the law. Signal Detection Theory (STD), for instance, applies whenever there are thresholds and even encompasses thresholds in biological systems such as vision and hearing (Green & Swets, 1966) as well as in electronics. Bayes Theorem applies whenever multiple sources of information need to be combined for a single output. It has been used as a model of how humans integrate vision and hearing to perceive speech sounds (Fuzzy Logic Model, Massaro, 1998) how humans integrate depth cues to achieve a single depth judgement, and how different atmospheric conditions predict the weather for tomorrow. Bedford (2001) has suggested a candidate for another such law - let's call it Object Identity Theory for now - open here for scrutiny.
Note first that in any general law, the problem and the solution are often presented as a package, but they are discernable quantities. Thus in SDT, one could characterize the problem as involving the question: Whenever there are thresholds, how can true sensitivity be distinguished from criteria? This is quite apart from the elegant mathematical solution. For Bayes theorem: When there are multiple sources of information for a single parameter, how can they be combined to yield one value? A particular equation applied to each possible output provides a solution. Another example is Pavlovian conditioning in which the problem can be thought of generally as two events that need to be associated (Rescorla, 1972) and the solution has largely been provided by the field of Pavlovian conditioning and the Rescorla/Wagner model. For object identity, I have identified a general problem: Whenever there are two samples, how is it determined whether or not they originate from the same object? I have argued that the problem occurs in multiple, previously unrelated phenomena in cognition and perception such as apparent motion, prism adaptation, Gestalt grouping, priming, stereopsis, and possibly Pavlovian conditioning, McGurk Effect, and contingent aftereffects. It applies to samples in time, space, modality, eyes, and all combinations thereof. I have also offered a general solution that can be applied widely, based on a hierarchy of geometries à la Felix Klein. Both problems and solutions that characterize general laws reflect separate contributions.
As generality both of the problem and the solution is central to Object Identity Theory, I begin there.
Gauker argues that "her problem cannot be described literally as the problem of object identity, and her theory is not a theory of object identity at all". It turns out, however, that all he means is that the theory is a theory of the psychology of object identity (my term) - that it is really concerned with some "mental act" (his term). True, psychologists are egocentric; a theory of x is simply assumed to mean a theory of the psychology of x.
Of greater significance is his claim that there is no single mental act that applies to all the phenomena I describe, that some are judgments, some "seeing as", some "hearing as", and some something else entirely, and therefore there is no common problem like Object Identity Theory claims. I like the claim because it goes to the heart of the theory, and I also like that the claim is false. But it does open a discussion of what it means to have a "general law". His error lies in assuming that there must be a single mental act. Consider a different general law: conservation of energy, in which energy is neither created nor destroyed. It applies to such diverse areas as magnetism, thermodynamics, mechanical energy of a moving object, and even human metabolism. There is no single type of energy just as there is no single mental act. The way in which it applies to each area is vastly different, yet it is still a general law. One expects the same to hold for laws in Psychology.
Gauker's terminological disputes appear to stem from this (erroneous) attempt to find the single mental act that was implied throughout the article. He takes issue with "judgement" and the object identity "decision", for instance, because propositional thinking (conscious or unconscious) is not present for each domain. The theory did not imply that it was, and the term "judgment" was not intended to refer to propositional thinking any more than it was intended to imply to conscious deliberating. But I raise it here to suggest that the search for general laws unavoidably leads to terminological disputes. Separate domains develop separate vocabularies. Neutral unbiased terms of the right generality are hard to come by. The right vocabulary just doesn't exist yet. Perhaps a related example comes from evolutionary biology. "Dinosaurs" and "birds" are different terms that refer to different entities - until now. Their clustering on the evolutionary tree demands a new terminology; "bird" may be dropped and "dinosaur" extended to apply to both. A new vocabulary must be invented for object identity.
Another property of a general law is that it is rarely sufficient to explain each area that it is applied to. Conservation of energy, for instance, is not sufficient to understand thermodynamics. Redding points out the issue of insufficiency for Object Identity Theory. While not a detriment to a general law, he is correct that application to each area requires care, as each area might interact with the identity requirement in unique ways.
Both commentaries by Cheng and by Bloom compare the work to Roger Shepard in its attempt to provide a general law in perception and cognition. Cheng expands that the present work is concerned with functional analyses, like Shepard's theories. He is correct that issues of function are infused throughout the article and in fact, my entry into object identity grew out of an initial question I asked of my empirical research program: but why should a discrepancy between vision and proprioception produce prism adaptation? (See Bedford, 1999, 1993a from target article references). Perhaps this is analogous to one of Shepard's questions: why are there three degrees of freedom in color vision? I hope so. A different point that should not go unnoticed concerns a general point about the mathematics of the solution. Cheng says: "Geometry plays a starring role in the story, but it is not pure geometry, but geometry with particular interpretations applied to concrete and sensory objects, that supplies the constraints". It is too often forgotten that applications of mathematics require there be psychological interpretations to the mathematical concepts; equations by themselves do not suffice.
I am delighted that Reiner & Willingham took on the difficult issue raised by my potentially inflammatory statement that object identity might be the gateway through which consciousness affects perception. They conclude that its role in explaining the influence of beliefs on perception, while compelling, may just be one of many factors that do so. I would like to suggest that the example they give is also special.
They cite a study by Yang, Dixon, and Proffitt which shows that the exact same visual display produces two different estimations of size depending on whether the subjects are led to believe they are viewing a real three-dimensional scene or a two-dimensional picture. These are exactly the results predicted by the recent theory of Ittelson (1996), which argues that there are two fundamentally different perceptual capabilities, regular perception in which 2-D retinal images are interpreted as the 3-D objects in the world that produce them and markings, which are also two-dimensional, but convey nothing about the surface on which they appear (e.g. graphs, pictures, words etc.). "If ordinary perception is about the existence of the real world, markings are about the affectances, the thoughts, feelings, and images of the observer" (Ittelson, 1996, pp. 186). Markings are uniquely human and a human perceiver has the capacity to interpret a 2-D display in either fashion. This theory is also one of the few of its kind in generality and scope.
Could consciousness have a special relation to the major organizing threads that run through perception? I do not know how many such principles there are in perception besides object identity and markings, but I suggest that investigation of how conscious beliefs affect perception may progress in new ways by centering on such principles.
Thus, while I am in general agreement that there may be other (but not arbitrary) influences of conscious propositional thought on perception, at the risk of losing sight of the big picture, there is one issue which should be clarified to prevent any confusion. Reiner & Willingham present a different example which suggests that a case of visuomotor adaptation was not influenced by beliefs about object identity, unlike, they suggest, the Welch finding I had described where "the knowledge that the hand being viewed was not their own erased the participants adaptation in the Welch experiment"(their quote). In brief, they describe that subjects on a virtual treadmill looking at a virtually moving world show adaptation, but their full awareness of the unreality of all components is not influential. However, I suggest that the subject was still able to trace the visual input and the motor input to the same source - even if that source was believed to be virtual; in the Welch experiment, what destroyed the identity conclusion is that subjects believed that the motor input came from one hand, but the visual input came from a different hand. I agree, though, that virtual worlds may be a good way to disentangle some of the elusive issues that conscious beliefs and perception present.
Bertelson, Vroomen, Aschersleben, & Gelder ask at what level of processing the geometry-based identity decisions I theorize are occurring. Specifically, are they occurring within automatic perceptual processing proper or post-perceptual processing including strategic components? They provide evidence from their research that ventriloquism can occur completely automatically at a pre-attentive stage and they further question the accuracy of other claims of top-down influences in ventriloquism. For Object Identity Theory, evidence that ventriloquism (or any phenomenon that I have argued requires identity) can occur automatically is welcome. This would imply that the identity decision itself can proceed automatically; if geometry is at the "core" of identity processing, that it be done automatically is consistent. Core processing should be capable of being automatically elicited and conducted. As I discuss in the target article, core processing should be expected to be fast, less prone to individual differences, and less likely to vary from one context to the next. These are all factors that go along with automatic pre-attentive non-strategic perceptual processing.
As Bertelson et al. and I have both noted, whether post-perceptual factors are nonetheless influential (as opposed to necessary) is a separate and open question. No aspect of Object Identity Theory hinges on whether conscious penetrability of identity or geometry is possible. Note that one could argue that a hard stance against top-down properties, as Bertelson et al. seem to take much of the time, would indeed be good for geometry (though perhaps harder to make the case that identity is required for ventriloquism in the first place, which is largely how I used the literature). If knowledge of teakettles and attention and instructions are irrelevant for identity, then geometry is more likely to be the sole factor for identity. But I have never argued that geometry is the only factor relevant for the identity decision and do not take this stance. Bertelson et al. also ask whether the geometry would be different if post-perceptual conscious deliberation were possible for some phenomena (e.g. prism adaptation), but not others (e.g. they suggest apparent motion). The answer is that, in general, were one to find a difference in conscious access, no, the geometry itself would proceed unaltered in both cases. For these specific examples incidentally, I do not agree that they divide in this way. Apparent motion, for example, is well known to be influenced by what people are led to believe they should see.
Finally, they suggest that "any conspicuous object, cantaloupe or not, held at the right distance and agitated with the right timing, would probably capture the ventriloquist's speech". What the geometric aspect of Object Identity Theory does is explain precisely what "agitated with the right timing means", and moreover links it to other phenomena that at first glance would never be seen as related to synchrony.
Suggested changes to the theory
Two commentators prefer "event" to "object". Cunningham suggests one can "salvage" my object identity theory by changing it to event identity through the integration of time into the theory. While I appreciate Cunningham's enthusiasm for the issue of time, as I agree not enough research has been conducted, let me remind the commentator and the reader that time was a major part of Object Identity Theory, both in characterizing the general problem and in the geometric solution. In the target article, samples can be separated in time as well as in space, and a critical aspect of the solution, which allowed such generality, was to show that geometry can be applied to time. For instance, it is discussed how isometric transformations in time correspond to shifting everything by the same amount of time, as in moving all one's appointments later by half an hour, how similarity transformations in time correspond to uniformly stretching time, as in doubling the time spent at each appointment, and so on through the hierarchy (see pps.148-149). Thus, it is puzzling when Cunningham states things like "...an event identity theory opens some interesting new questions. Rather than introducing a discrepancy between the seen and felt location of an object (a spatial transformation)...one could instead introduce a discrepancy between the seen and felt time of occurrence...(a temporal transformation)". While the manipulation is excellent, these new questions are precisely the sorts that follow from the original theory.
It is possible the commentator is assuming that "object" must exclude time and that whenever temporal factors are involved, one is automatically dealing with events, not objects. However, many views of "objects" have transformations over time as a clear component (most notably Leyton, 1992), plus the original article makes it clear that phenomena the commentator would call "events" are in the scope of the theory. The difference may then be just a semantic one - there is no new theory here. I prefer the term "object identity" to "event identity" in part because of its historical significance.
Similarly, Veres addressing specifically priming, notes that he and Forster encountered the possibility that the prime and the target were treated as the same object, but then came to the opposite conclusion. They concluded: "The reason why masked priming works at all, according to this idea, is that experimenters create an artificial situation in which there is only one perceived visual event that is used to index the information made available by two distinctly presented stimuli: the prime and the target. Because all the information is treated as belonging to the same perceptual event, the target is able to make use of the information provided by the prime." And "The question is, do the two chunks of information refer to the same perceptual event?". This is not an idea different from my idea conceptually - he has substituted the word "event" for the word "object"
As noted above, events defined by temporal extent were included in the theory, though still called object identity rather than event identity. Perhaps both commentators would be content with the more neutral term "source". Perhaps also what they had in mind is the issue raised by the next commentator.
Much deeper is Bloom's question of what is meant by an object in object identity. He points out that "object" has been used to refer to everything that exists to logicians, excludes two-dimensional visual patterns for some psychologists but not others, and can exclude people in layman use. As a working definition "everything that exists" for Object Identity Theory can be used, but only so as to not to risk prematurely omitting any related phenomena with the same formal structure. Better to cast the net broadly. Hopefully, the theory itself will eventually guide how perception psychologists ought to view "object" rather than the other way around.
While I should leave it at that, the bounded three-dimensional entities Bloom discusses - the cups, the pens etc. - i.e., Spelke-objects (the phrase Bloom coined is too good to not worm its way into common usage!) - deserve special attention. He correctly notes that in the article at one point I suggest that rules about object identity may have evolved at the level of Spelke-objects, but then also talk about items that do not fall into this category, such as words priming other words. The idea here was to contrast how things initially evolved with their possible derivative uses afterwards. As I discuss in the article, there is priming involving objects, Gestalt set effects, for example, in which it is adaptive to know whether the ambiguous rat/man is really dinner (the rat) or a companion (the man). Later-evolving language modules may have capitalized on structures already in place for objects. It is hard to shake the idea that Spelke-objects are special. We do grab stuff and pull (à la Pinker). Geometry too has a (2-D) analog to Spelke-objects; the forms often used, circles, parallelograms and so on are extended, continuous, and bounded, and of roughly the same scale as Spelke-objects (smallish with respect to our bodies-is Mt. Rushmore a Spelke-object?). One final note concerns objects vs. parts of objects. Following Marr's analysis, Bloom suggests hands are only parts of objects while detached hands may be (whole) objects. If we consider the prism adaptation case, a hand feels like it is in one place but looks like it is in another. If the observer can see only the hand and no other part of her body (as often occurs in experiments) might the hand in this context be properly thought of as a Spelke-object? It is not physically detached, but may be psychologically detached. Complex issues such as these contribute to the difficulty in defining the object in object identity.
Lachter suggests a different change in which the geometric solution I offer be viewed as "...a theory of shape similarity..." rather "...than as a theory of object identity...", and that the geometries become "...a metric of shape similarity". Note that I do say in the target article that "at issue is a metric of what it means to be 'similar'" and "I suggest here a hierarchy that can serve as a metric of similarity specifically for the object identity problem" (pp. 135) because geometry is a natural way to compare forms. But once that is said, I believe thereafter dropping the vague and over-extended term "similarity" is the more useful path to pursue. The needless difficulties one encounters by not doing so are in fact illustrated by Lachter himself as he explores his suggested change. Of my view of perceived object identity, he states: "The theory is simply that two percepts are more likely to be judged as coming from the same object if they are more similar. Few would disagree" But what it means to be similar for the particular problem at hand is where the weight lies.
Alternative theories - many factors determine identity
Several commentators (Bloom, Huber & Aust, Lachter, Wilcox) drew attention to other factors that are present in object identity decisions, including color, texture, brightness, substance, and accumulated knowledge.
Bloom suggests an alternative theory to the geometry solution in which: "the general point is that judgements of object identity in the real world can draw upon a large body of information, information such as writing on receipts and number of faculty who get the same office furniture". Thus, he suggests, if someone shows you a receipt for a coffee cup he may convince you that, in fact, it is not your cup that disappeared after all, even though it is an unusual cup which otherwise you would have mistaken for your own. However, the specificity of knowledge is problematic - receipts are obviously limited in scope and have no application to varied manifestations of identity, as are specific rules that furniture from office to office tends to be the same, perhaps even qualified by the differences between state universities and rich private ones. As I have said, this is not so for geometry. Bloom may argue that this is the whole point, as he says that we "rely on all sorts of cues". I have no doubt that people do this. They use everything and anything no matter how subtle to determine if that is their cup. People are smart. But they are too smart. This suggests a proliferation of individual independent post hoc overspecialized rules that are ever changing to encompass new information - not the stuff of which general theories are made, short of reinventing logic.
But if people do use general reasoning and statistical information in identity situations, how does one reconcile that with a claim that geometry is special? I believe a clue is provided by analyzing further a nice thought experiment that Bloom offers. He imagines giving a child a piece of paper and returning in 5 minutes to find 1) a paper with a hole in it, otherwise identical and 2) a piece of paper slightly larger, otherwise identical. He suggests that most would conclude that item 1 is the same as the original paper, because of knowledge that paper does not usually expand but can easily be torn. He suggests that this might be problematic for my geometric solution since it is item 2 that is the simpler geometric transformation and should, in Object Identity Theory, be chosen.
Note first that if the result did obtain (if, e.g., the slightly larger paper wasn't interpreted as being the same size paper, but closer), I suggest that if it were a child who returns, and perhaps an adult, she would laugh. It's funny, perhaps precisely because the slightly larger paper should be the right one geometrically speaking, not the broken one, but she knows it isn't. Geometry is overridden, but geometry was there to override. And here too, the rules about paper are reasonably specific; some things can be slightly larger. Fabric expands, such as a stretched out sweater, as does wood, such as when a door gets stuck in the frame with changing climate. To reconcile the two types of information, general geometry on the one hand and context-specific knowledge on the other, I suggest the following manipulation.
Speed it up. That is, don't return after 5 minutes, but return after 1 second. Consider the Bloom example except in a shorter time-frame. Will the original paper prime item 1 or item 2 in a priming paradigm? Or how about an apparent motion display? To which item will the original stimulus be seen to transform to? I hypothesize that when it is faster, geometry will prevail. (Note that non-topological transformations, such as holes, may have special status as reflecting broken objects, but generally the point holds). To disentangle the general from the exceptions, one can generally try the following:
If geometry is still generally overridden by something else under these conditions, then I would begin to question geometry as the core ingredient. Otherwise, as I readily acknowledged in the target article, there are other factors that play a role in object identity. I do not think it is a problem that allowing more time for a decision may allow other local information to override. If the system is stressed, one will see the general geometry at its core. Note also that presence of other factors relevant to identity does not change geometry's suggested privileged status as common to all problems, even to those beyond the scope of what is typically considered "shape".
I can only briefly consider the other example and issue critical to Bloom' theory, that of "substance" because of space and time criteria of a different sort - this journal. If Bloom is correct that substance does "trump" geometry (e.g. chicken cutlet vs. piece of paper), I would point to the arguments above and ask if it does so under conditions where the system is pushed to the limit. But also importantly, how does one determine something's substance? To say that different substances determine different objects sounds good, but substance is something that must be determined in its own right.
Similar discussions are applicable to Wilcox's view that physical reasoning and properties other than geometry are all used. One additional property she raises concerns developmental studies: "They suggest that spatiotemporal information is as important, if not more important, as information about object form". Note first that this misses some of the power of the geometric solution - the geometry is not just about form in the typical sense of the word. It applies to time as well as large spaces, and as shown in Appendix 2, the geometry of time and space can be considered together to reinvent what have been known as spatial-temporal laws; the appendix illustrates for the specific case of Korte's third law. The cornerstone finding for infants and identity that Wilcox discusses is Spelke and colleagues finding in which there are two opaque screens with a gap between them. An object goes behind the first one, and an object emerges out the second one, but there is never an object in the gap between the two screens. Infants appear to conclude there are two objects, not one. Wilcox suggests: "The discontinuity in path clearly signals the presence of two objects; no further analysis is required". In Object Identity Theory, this is false; further analysis is simply always required whenever there are two samples. In the two-screen situation, there was a conclusion that there are two objects, but the same discontinuity can also lead to a conclusion that there is one object. Apparent motion is just such a case; there too, there is a "spatiotemporal" discontinuity, yet the two samples are interpreted as one object.
One manipulation I would suggest which follows from the theory is to have two distinct objects emerge from behind the second screen. In Wilcox's view, I believe the infant would just conclude there were three objects. In my view, both objects will be compared to the original and the closer one geometrically will still be seen as being connected to the original. Also, what would happen if there was a comparison between displays in which one object emerges and in which two objects emerge? Note that another commentator, Narter, also reviews this same cornerstone finding in development and ultimately concludes it does not conflict with Object Identity Theory. I believe Narter is correct that there is no contradiction.
Huber & Aust cover a number of areas discussed above, and also suggest that other factors may be more important than geometry specifically for natural viewing of animate objects. They include a striking picture of silhouettes of a leopard in different poses, and point out that the transformations are often non-topological and hence geometry would be of limited use. I agree with Huber & Aust that natural animal poses will more often than not cluster around the higher levels of the hierarchy, i.e., topological contortions and even non-topological seeming breaks and fusions. But here too, the geometries may be useful for understanding the information we need for processing object identity . Glance at Huber and Aust's silhouettes and it seems reasonable that we would judge them as all coming from one object. But now glance at only two of them, while covering up the remaining cels. I'll stack the deck in my favor and say look at the first row, pictures 2 and 3. If there were just these two, would it be so easy to determine that they were one object? (and are topological pairs easier than non-topological ones?). I developed the solution with respect to one type of frequently occurring decision: if there is a choice, go with the lower geometry. However, there are may be additional ways the rank ordered levels get mapped onto decisions. Consider: the higher the level of transformation/geometry, the more glimpses that are needed to make a confident decision.
Future directions and new ideas
Cheng offers several. He suggests that the other three Tinbergen questions besides function - mechanism, development, and evolution - be pursued for Object identity Theory. I have no doubt that mechanisms can be found; in an earlier version of the article I offered a simple way that groups of neurons could instantiate the geometries (see also footnote 7), but I would prefer to leave this direction to those who have more expertise. I am eager to see developmental pursuit along the lines suggested in the target article that geometry should emerge early. Particularly exciting are evolutionary considerations.
One could regard the question of function as "evolution-lite" - that is, seeking the goal of a behavior, which is really a question of adaptive value of a behavior, but without tracing the phylogenetic origins back millions of years. But a full-species comparison, also suggested by Huber & Aust, would raise new interesting issues. As both comments discuss, it is not obvious all animals would need object identity, or solve it in the same way if they did. This at first seems at odds with my intuition that a "core" geometric solution is evolutionarily old, and hence likely to be present in other species, but the commentators may well be right.
A different direction to pursue is an excellent suggestion by Lachter to apply the geometries to different levels, à la Marr; higher levels would allow overall shape, lower levels would provide detail. He suggests one can look at markings on surfaces, which are just forms but on a smaller scale, and even color. Thus far, my geometric solution has not addressed the issue of the scale of the samples, and this would be a useful development. Likewise for "color space", though I'm not sure that is what he had in mind.
Reinke has an excellent suggestion to explore the relation between object identity and what I like to call "Gibsonian Perceptual learning". Gibsonian Perceptual Learning is a term I use to describe the well-known effect others know in the literature as perceptual learning, which Eleanor Gibson brought to the forefront. (I prefer to save the term "perceptual learning" to refer to the broad class of effects of experience on perception, of which G.P.L is one type; see Bedford 1993a, 1997 from original reference list). For instance, as Reinke notes, two wines may initially be indistinguishable, but with practice, observers can taste the difference. Reinke's insight is that before learning, the object identity decision is for two samples to be judged as "same object", whereas after, they are judged as "different objects" and "...it would be beneficial to have an explanation of how we can overcome the same object identity and produce discrimination learning." It seems to me this is a genuinely new way to think about both Gibsonian Perceptual Learning and about object identity and I am pleased to think my article may have contributed to her insight.
Also notable is Reinke's suggestion that one try to extend the hierarchy of geometries to semantic priming, perhaps by taking advantage of the hierarchical structure of semantic encoding. The spirit of this attempt seems right - to push geometry to the limit. Narter, a developmental psychologist, points out that a few hours after birth, infants already show size constancy, and therefore it should be available as a foundation for later identity judgements when samples are discontinuous.
Finally, Cohen, a physicist, suggests that "...perhaps determination of identity may not always occur in physical space but in domains that are called 'representations' or 'transforms'" and illustrates with an example from speech perception and the Fourier transform. He also generally shows how the Fourier transform effectively solved the problem two hundred years ago of recognizing that an object is the same even when it is in different positions. Position (separation distance) appears only in the phase, allowing the translation factor to be easily ignored if desired. Similarly, he shows that the size factor is in the phase of the Mellin transform, allowing size insensitivity. Two quick points to note here. First, it is interesting that what we can also re-describe as position constancy and as size constancy are achieved here computationally (Fourier transform and Mellin transform) differently than the classic computational solution in psychology, where head position of the observer and distance from observer to object are "taken in to account" to achieve the two constancies respectively. Second, perhaps these transforms are especially useful for situations in which properties from the lower level of the hierarchy (position, size) predominate; it is not clear yet how they might be extended to properties from the higher levels (angle, parallelism, continuity). But as Cohen says: "Thus the concept of 'identity' as discussed by Prof. Bedford may have a richness beyond identity in physical space". It is possible that to achieve true generality of identity, an interdisciplinary approach is called for.
I have extended a recognized concept of "object identity" to a much broader characterization of the problem - but kept the same term. Not surprisingly, this led to a number of terminological disputes. Whether by the name object identity, event identity, identity, numerical identity or source identity, the problem is the same. Likewise for use of the term "object", which was too broad for comfort for some. As discussed earlier, a research area that proceeds independently of another develops its own terminology. When different areas are then unified by a broader framework, there is no adequate terminology. A vocabulary for "object identity" (as broadly defined) does not exist yet.
Many comments centered on the proposed solution, especially the sufficiency of geometry, and drew attention to other factors that determine object identity such as color, substance and context-specific knowledge. Even stronger, it was suggested that when these other factors are put into direct conflict with geometry, geometry is overridden. I have argued that geometry is there to override, and I have suggested manipulations under which I believe geometry will reveal itself, even if originally hidden.
One goal in Psychology is to discover general laws. I want to reiterate the importance of formulating the most general characterization of a problem that links previously separate areas, as I believe I have done, and of having an abstract solution that can be applied widely, such as a set of nested geometries. This does not negate the importance to Cognitive Psychology of having a complete characterization of adult object identity with all its myriad exceptions and accumulated context-specific knowledge. They are both important.
Green, D. M. & Swets, J. A.(1966). Signal detection theory and psychophysics. New York: Wiley.
Ittelson,W.H. (1996). Visual perception of markings. Psychonomic Bulletin & Review,
Kubovy, M. (1986). The psychology of perspective and Renaissance art. New York: Cambridge Press.
Leyton, M. (1992). Symmetry, causality, mind. Cambridge, Mass: MIT Press.
Massaro, D.W. (1998). Perceiving talking faces: From speech perception to a behavioral principle. Cambridge, Mass: MIT Press.
Rescorla, R. A. (1972). Informational variables in Pavlovian conditioning. In G. Bower (Ed.), The psychology of learning and motivation (Vol. 6, pp.1-46). New York: Academic Press.