Note: A few minor changes will appear in final journal copy. JEP:HPP

Felice L. Bedford, University of Arizona. 2004

A number of psychological theories assume that one object cannot be in more than one place at one time. For instance, in attention theories, "object files" (Kahneman & Treisman, 1984) are created on the basis of location. Similarly, "indexes" (Pylyshyn, 1989, 1994) are assigned and accessed by location. In both theories, entities in two locations at the same time get two distinct object files or indexes, one for each location. The assumption is even more prominent when the location-based theories are extended to infants' perception of objects: "Because objects are indexed by location, seeing two objects in different locations at the same time forces the assignment of two indexes: therefore two individual objects are inferred." (Leslie, Xu, Tremoulet & Scholl, 1998, pp. 15). Other researchers have been explicit that adult (e.g. Bedford, 1993a,b 1995, 1999) and infant (Narter, 1998; Rosser, Narter & Paullette, 1995; Wilcox, Nadel, & Rosser, 1996; Xu & Carey, 1997) observers possess a constraint that one object cannot be in two places at the same time: "Spatiotemporal criteria are constraints on how physical objects can and cannot exist and move in time and space. For example, one object cannot be in two places at the same time, two objects cannot be in the same place at the same time..." (Narter, 1998, pp. 70). In developmental work generally, two entities that appear in two places simultaneously allow infants to conclude that there are two objects, not one. A one object-one place-one time rule may be a subset of the law that objects must travel on continuous paths in space and time (cf. Spelke, 1991) Such spatiotemporal laws are viewed as the most fundamental of truths and the primary means that even infants have at their disposal for individuating a complex scene (Spelke, 1991; Spelke & Hermer, 1996).

However, is the constraint the universal truth it is assumed to be? First, the proposition as stated is, strictly speaking, false. Not only can one physical object be in more than one place at one time, it must be. The tip of your pen and the base of that same pen are in two places at the same time. Likewise for any two parts of any one object, no matter the nature, size, or nearness of the parts. The only entity that does not have this property is an abstract point, which as Euclid defined, is that which has no part. Physical objects extend in space. Each one must occupy multiple spatial positions simultaneously or it is not an object. This is true even if "part" is problematic, a term that comes with preexisting biases; matter, multiple points, or extent can be used instead. Thus, the constraint as stated is casual shorthand for, at minimum: One point on an object cannot occupy more than one place at one time, as in Marr (1982). I will return to this seemingly trivial change at the close.

Moreover, even with the correction, it may not be a general regularity for sorting the sensory array. It is biased towards vision, as theories often are. Michael Kubovy (1981, see also Kubovy, 1988, Kubovy & Van Valkenburg, 2001) shows how different modalities have different attributes which are indispensable for determining numerosity. In his Theory of Indispensable Attributes, space is used for enumeration in the visual modality. Two patches separated by a spatial gap but otherwise identical indicate two "objects". In Alice and Wonderland, Tweedledee and Tweedledum are counted as two people, not one, even though they seem identical in all regards save for spatial separation (see also Lucas, 1973). However, critically, space is dispensable for audition. Kubovy continues with two sounds coming from two spatially separated speakers. If the speakers play the identical sounds with the same pitch (frequency), observers hear one stimulus, not two, despite the spatial separation. It is only when the two sounds have different pitches that observers individuate two "objects", say both a violin and a flute. Even if the two pitches are in the same location, two distinct objects are discerned. It is pitch which is indispensable for numerosity in audition; space is irrelevant. If space is irrelevant for determining numerosity in audition, then a single "auditory object" (Kubovy & Van Valkenburg, 2001) can be in one, two, or a dozen locations simultaneously and still be one single object, as long as it has one pitch. Kubovy's Theory of Indispensable Attributes suggests then that one object can be in more than one place at one time in audition.

Does this in turn suggest that if our constraint is restricted further by modality, it is now accurate? For vision, (one point on) an object cannot be in more than one place at the same time. Unfortunately, the seemingly intuitive claim is problematic for vision as well. While Tweedledee and Tweedledum may be different people (Kubovy, 1981; Lucas, 1973), Groucho wasn't so sure Chico was separate from himself in the movie Duck Soup. Chico mimicked Groucho's every move. When one image went left, so did the other; when one shook his leg, so did the other; when one picked up a cane, so did the other. There was a spatial separation, but identity otherwise. Was Chico a different person with a different count, or was he merely Groucho seen through a mirror? One (point on an) object reflected by a mirror can appear to be in two places at the same time. I have two identical collie figurines spatially separated from one another and also near a mirror. When my eyes sweep by the display quickly, I cannot tell with certainty how many collies there are - sometimes it looks like two, but other times only one, and other times four distinct individuals. In addition to mirrors, observers with "double-vision" (diplopia) will experience a single (point on an) object as occupying multiple positions because the eye muscles are sufficiently weak as to prevent fusion of the two retinal images into a single image. Another example is a nearsighted eye that leads to a blur circle; that is, one point in the world produces several points on the retina because the lens-eyeball mismatch focuses the image to a single point in front of the retina rather than on it. Consequently, two spatially separated (points on an) object(s) in vision, otherwise identical, does not guarantee their origin is two distinct objects in the physical world.

However, perhaps mirrors, diplopia, and nearsightedness are precisely circumstances where observers infer two objects from two visual samples too often, even if it produces erroneous perception. Someone with double vision may report seeing four fingers, and not the two that were really put before her eyes. Someone who hasn't noticed a mirror may try to walk through to the "objects" on the other side. A mirror causes the appearance of one point in two places simultaneously, but will the two points really be interpreted as only one object?

It is clear that distinctions need to be drawn as to what precisely a constraint about one time, one place, one location is supposed to refer to. At least three meanings can be distinguished. Any theory that makes use of the constraint must, at minimum, be explicit about which meaning is intended. Is the constraint a statement about reality, referring to what is true of the physical world apart from our interactions with it? Is it a statement about appearance or the mapping from the physical world to the sensory arrays: how do the objects we believe to exist in the physical world appear at the sensory surfaces of different modalities? Or does the constraint refer to psychological interpretation: what inferences does the mind draw from the sensory information about the state of the world and how does it does it do so? The constraint as used in the literature can touch on all three meanings. It reflects a belief that the constraint describes reality; that is, we claim objects behave in certain ways independent of our experience of them. In addition, it appears to be a claim that we posses an equivalent constraint psychologically. The psychological constraint may have resulted from "internalization" (Shepard, 1984) of the physical constraint through natural selection, if it produced a survival advantage and reflected a truth that was stable throughout our evolutionary history (Shepard, 1984, 2001). This duplicate constraint, presumably along with assumptions about the rules of appearance, is used to interpret the sensory information.

Yet the different categories can be logically distinguished. For Appearance, we have seen the constraint is false, such as when the visual image of an object has a mirror twin. For Reality, is the constraint that one (point on an) object cannot be in two places at the same time a correct description of the behavior of objects in the world? Putting aside that we have not defined "object", if what we mean by "place" is a holder for a point in the spatial medium, then the "constraint" may say that one point cannot be two points, which may be more of a tautology than a constraint. Does this actually describe the behavior of a point, or what it means to be a point? (See also Bloom, 2000). But this is more of an abstract analysis than a physical one. For the physical world, physicists do tend to believe that bits of matter occupy distinct and definite physical locations. However, in quantum theory, a single component at a very small scale can in fact exist in different locations simultaneously. As a description of the physical world then, the constraint is either obviously true, or it is false.

If the constraint need not be true of appearance, and may or may not be true of the physical world, is it at least true of interpretation? Psychological interpretation of the sensory array is arguably the category of greatest consequence in perceptual and cognitive theories. To return to the issue raised by walking through mirrors, must two simultaneous spatially separated samples involving vision be interpreted as two objects?

No. A counterexample is provided by prism adaptation, a phenomenon that would not occur without an interpretation of one object given to two spatially separated simultaneous samples (Bedford, 1995, 1999, 2001a; Welch, 1972). When observers look through a prism, the visual image of the world appears laterally displaced from where it really is. A (point on an) object that is grasped will appear to be in one position through vision, but in a different position through touch. The spatial discrepancy leads to adaptation (e.g. Wallach, 1968; Welch, 1978), where one modality will change to realign with the other. However, it has been demonstrated both empirically (Welch, 1972) and logically (Bedford, 1993a, 1999, 2001a) that adaptation will not occur unless the differing location signals are believed by the observer (not necessarily consciously) to originate from one object. The mere localization of two places at one time, even one that happens to be through vision while the other through proporioception, is not enough to produce change because two points on two objects are often in two places. It is only if those two places are interpreted to refer to (one point on) one object that there is a genuine discrepancy from expectation, which can lead to the detection of an internal error and its subsequent correction through the process we know as prism adaptation. Bedford argues (1993a, b, 1995, 1997, 1999) that no functionally adaptive change in perception can occur until the sensory information indicates an error in the sensory systems themselves. Prism adaptation may be essential to keep perception accurate in the face of mismatched growth in childhood and random drift in adulthood (Held, 1955; 1965; Howard, 1982). However, adaptation could not occur were it not that the appearance of two different positions simultaneously is interpreted as (one point on) one object, not two objects. The existence of prism adaptation shows that two entities in two places not only can be caused by one object objectively, but they can also be interpreted as one object by the observer.

The counter-example to a one object- one place-one time constraint provided by prism adaptation raises two questions. First, didn't I just sneak back the very constraint as an argument against the constraint? Prism adaptation is error correction, which will only occur if one point on one object appears to be in two places simultaneously, which indicates an error with one of the modalities, because we all know that one object can't be in more than one place at one time! The difference in usage of a constraint is one of processing level. In addition to distinguishing between constraint categories of Reality, Appearance, and Interpretation, there are further distinctions that need to be drawn among levels of processing at which the constraint could operate within the category of Interpretation. An important level concerns the use of spatially separated samples to draw conclusions about numerosity. As the example from prism adaptation shows, if there are two spatially discontinuous samples at the sensory array at the same time, they will not necessarily be processed as two numerically distinct objects. This claim appears to contrast with developmental theories of the object concept in infants and object-based attention theories in adults. It is consistent with the Theory of Indispensable Attributes in perception, if spatial separation does not ever force an interpretation of multiple objects. In general, the presence of two simultaneous spatially separate stimuli is not a sufficient criterion to individuate objects, or automatically open separate object files. However, the counterexample of prism adaptation also suggests that should it be decided by some other criteria that two discontinuous samples at sensory surfaces do refer to one object, then this one-object conclusion will violate expectation at a later processing level.

The second question raised is whether the constraint is really contradicted in circumstances involving only vision? Prism adaptation is based on information obtained from more than one modality. As seen earlier, space is not an indispensable attribute in audition as it is for vision. For proprioception (touch), it is not clear what the indispensable attributes are. If space is dispensable for proprioception too, then getting the one-object interpretation from multiple spatial locations may require that either this modality or audition be involved. What about cases where the only modality involved is vision?

In addition to mirrors, where double images are often correctly interpreted as referring to single objects, consider two simultaneous spatially separated points. They can be processed as one object, and often are. This is precisely what the classic Gestalt grouping principles (Wertheimer, 1923) are about - discerning how many objects are responsible for the sensory array of discontinuous points. "I stand at the window and see a house, trees sky. Theoretically, I might say there were 327 brightnesses and nuances of colour. Do I have "327"? No. I have sky, house, and trees". Two discontinuous points can refer to the same object via the classic principle of "proximity", when they are closer to each other than to any other points in the display. "Common fate" would also link the two points, even if they were far apart. The principles work because we can work backwards from the sensory array to reconstruct properties of objects and how they map onto us; points on an object tend to be near one another; points on an object tend to move together at the same time and rate. The two points do not necessarily refer to one point on one object, but refer to one object nonetheless, the level of analysis that theories on object identity are typically concerned with.

Analogously, one-object interpretations apply not only for individual points but also for spatially extended samples such as two-dimensional closed figures or three-dimensional bounded solids. Consider viewing two fingers simultaneously with a gap between them, but without seeing what they are connected to. If they move in synchrony, an observer will be more likely to conclude that they belong to one hand, not two, than if they did not have common fate. But one may argue that there are still two fingers - that the spatial gap automatically implicates two objects of some sort, even if the objects are on a smaller scale. Now consider two samples that consist of what appears to be the left edge of a finger, and the right edge of a finger, but with nothing between them except empty space. Can they refer to one finger despite the gap? Yes. Perhaps the finger is painted with a chemical that causes invisibility in those regions, or perhaps a visual deficit prevents some retinal signals from further processing. Philosophers are fond of talking about bipartite objects. (For recent philosophical discussion of parts and wholes and their relation to space, see e.g., Casati & Varzi, 1999 especially chapter 6). But what about the interpretation of such a display? We know from the phenomenon of apparent motion that a missing signal on the retina is not an insurmountable obstacle to inferring one object; two stimuli alternating in two different positions are interpreted as one object moving back and forth, even with the missing retinal information in between (Rock, 1983; Shepard, 1984). There is no reason two simultaneous stimuli in two different positions with a gap between them cannot be interpreted as one finger.

But aren't there still two somethings, even if the conclusion implicates one finger? Yes and no. I refer above to viewing two samples. What governs that they should be referred to as two samples, and not one? The answer is that discontinuities can determine samples: whenever anything is different, there are two samples. This suggests that there is an additional level of interpretation at the input for which a "constraint" about one object, one place, one time could be referring to in psychological theories. Two discontinuous points in space (i.e. two points with a spatial gap) indicates of necessity two samples, but not of necessity two objects. The question of theoretical interest is usually whether or not those two distinct samplesomethings refer to distinct objectsomethings.

So, is there or isn't there a constraint for vision that (one point on an) object cannot be in more than one place at one time? To summarize, no for Appearance, unclear for Reality, and for issues of Interpretation, the question needs to be asked separately for at least three processing levels. At the input for Interpretation, spatial discontinuity determines how many samples there are. But there is nothing special about spatial discontinuities that would warrant a constraint. A discontinuity in time also determines different samples. Two spatially separated points/entities that are not simultaneous also produce two samples, as do two samples that are in the same place and not simultaneous. A constraint at this level would not be meaningful. At the next level of determining whether two samples defined by discontinuities do or do not refer to the same object - the issue known as numerical or object identity (e.g. Xu and Carey, 1997; Leslie et al, 1998; Meltzoff & Moore, 1998, Bedford, 2001) - the constraint is false. Two samples do not automatically imply two objects - that is what must be decided. Analysis of the constraint suggests that whenever there are two samples, regardless of whether they are simultaneous or from different times, regardless of whether they are from vision, from audition, or a combination of modalities, and regardless of the size of those samples, the observer must ask whether they refer to one object or two objects. Two simultaneous separated samples in any modality can be interpreted to refer to the same object. A whole set of criteria is needed to resolve identity. But once an object identity decision is reached, then at a third level we do have an expectation: if it is one physical object, each point on that object will occupy a definite and unique position. If information on sensory surfaces indicates otherwise, then there is a violation of expectation. Expectation or surprise leads to learning. At that third level then, we do have the constraint.

Before concluding, note it is clear that constraints need not be propositions or rules that are "applied" but rather built in the architecture (e.g. Manukata, McClelland, Johnson, and Siegler, 1999; O'Reilly and Johnson, 1994 - see Spelke and Newport, 1998 for discussion; Marr, 1982). Yet Paul Rozin and Jonathan Schull (1988) remind us that we would not say that birds are constrained to fly! What we mean by a constraint on perception and cognition, even if it is built into the architecture, and whether it affects phenomenological perception or cognitive beliefs or both requires further explication.

Note also that distinctions drawn here may help solve a puzzle concerning constraints and evolution. Roger Shepard has argued that facts about the world that are general, have been stable throughout our evolutionary history, and convey a survival advantage over having to learn them through "trial and possible fatal error" (1984, 2001) will become internalized as constraints that affect perception and cognition. Yet why does something that is general and true need an a priori constraint? Those should be the very things that are easiest to learn about. I suggest that it is when the appearance of something varies and can conflict with reality, that a constraint is useful. If one point on one object always appears in one place, there may be no need to have a constraint that represents this at any level; we would "discover" it soon enough. But suppose physical growth affects different modalities at different rates (see Held, 1965; Banks, 1988; Bedford, 1999), for example; then, one point on one object will appear in two places simultaneously when surveyed by the two modalities. Were it not for a constraint that this should not occur despite appearances, we would draw a very different, likely inaccurate, conclusion about the physical properties of entities in the world.

Finally, I return to the first point. Is the correction that it is (one point on) the object that cannot be in two places at the same time simply an obvious and picky technicality? First, it should raise the issue of whether objects are critical to the constraint - one point (on an object) cannot be in more than one place at the same time. More importantly, it draws attention to imprecise usage of the idea "the location of objects". Objects do not have locations, points do. One popular misuse may be the distinction between "What is the object?" (dorsal stream) and "Where is the object?" (ventral stream) (Mishkin, Ungerleider & Macko, 1983), a division that has been well known and accepted until recently. "Where is the object?" is shorthand for idealizing the object as a point source, and asking, "Where is the point?" Consequently, a candidate for a major division in perception may be more one of processing a single point vs. an extended contour, a distinction drawn 30 years ago by Richard Held (1970), and not "where" vs. "what" or even "what" vs. "how" (Goodale and Milner, 1992). Once there are spatial extents, as in visual objects, there may not be any hard division between location information and other geometric properties when processing those objects (Bedford, 2001).

I have claimed that an alleged one object-one place-one time "constraint" may not be true of the physical world, is not true of how objects must project to sensory surfaces, and perhaps most importantly, does not prevent observers from perceptual interpretations that violate the "constraint". This is most apparent with Kubovy's examples from audition, where spatial separation is never used to enumerate objects, but is also true of cross-modal perception, and of vision. For cross-modal perception, prism adaptation shows that visual and proprioceptive samples in different locations at the same time are nonetheless interpreted as referring to a single object, and in vision, part-whole Gestalt relations and mirror perception provide counterexamples to the constraint. Additional counterexamples will likely be found empirically now that it is apparent there exist violations of interpreting sensory information in accord with a one object-one place-one time rule.

An important implication of the analysis is that any theoretical framework is wrong if it assumes that simultaneous spatially separated patches/entities/stimuli must be interpreted by observers as indicating two distinct objects. This appears to include the increasing body of adult and infant work that rests on automatically assigning distinct indexes, object files, and objects on the basis of spatially separated samples. It is a question for theoretical and empirical research of how, whether, and under what circumstances two samples, whether they are within or between modalities, simultaneous or successive, will be determined by an observer to refer to one or two objects ("object identity" as broadly defined, see e.g., Bedford, 2001, Xu and Carey, 1996); it is not a matter of assumption. Also open for exploration is how following such an object identity determination, discrepancies between the number of inferred objects and the number of counted positions can be used to drive learning. It is at this level that a constraint that (one point on) one object cannot be in more than one place at one time has the greatest promise - perhaps as a constraint on learning rather than perception . A tangential implication is based on clarification that while localization of an individual point is sensible, whenever there is spatial extent, location is fundamentally intertwined with other geometric properties. Any singling out of "where is an object?" from the more general "what is an object?" is likely to be erroneous. In any event, all perceptual, cognitive, and developmental theories that make use of a one object-one place-one time constraint (either implicitly or explicitly) should re-examine and be explicit about what categories and levels the constraint is referring to and re-evaluate conclusions based on its use.

Thanks to Merrill Garrett, Thomas Bever, William Ittelson, and Michael Kubovy for illuminating discussions, and editor David Rosenbaum for very helpful comments on an earlier version of the manuscript. This work supported by a grant from the Social and Behavioral Research Sciences Institute (SBSRI) at the University of Arizona. Address correspondence to bedford@u.arizona.edu

Bedford (1999) describes three possible reactions an observer can have to a "discrepancy" between localizing a hand in one place and seeing it in another, and adds a fourth in a later work (2001). The differing values from the two modalities could indicate that 1) one hand can be in two places at the same time, which would be a fact about the world ("oh wow") 2) one of her sense modalities is providing erroneous location information because she knows what she is detecting is not possible ("uh oh") 3) there are just two different hands, one in each position ("ho hum") and 4) there is one big hand that extends through both positions, and she is detecting a different part through each modality. The second conclusion is required to get an internal perceptual change that involves a shift in seen or felt locations (adaptation).