Most or all of us have heard or read of the question of how many angels can dance on the head of a pin. This question was introduced to satirize and to dismiss a problem that challenged scholastic philosophers, namely Given that there are things that do not have body but do have location, can two or more of these things occupy the same location at the same time? Now, we might labor the idea of body; but suffice it to say that it were presumed that things with body could not simultaneously occupy the exact same location, but that there were things of another class that could occupy the exact same location as could a thing with body. The question then was whether they could occupy the same location as other things of their own class.
One of the ways in which this puzzle had bearing was on attempts to understand the nature of devils (fallen angels) and how they might interact with ordinary people. The question thus actually had bearing on the witchcraft mania.
But another way in which this question had bearing was in consideration of how we distinguish one thing from another, perceptually and conceptually. In theory, two things might be identical except for location, but the distinct locations perhaps permit us to discern that there are two things, rather than one. And, if the location of an object at any one time is unique to that object, one can combine a description that might fit many other things with that location to identify a singular object, and thus to have an intrinsically singular corresponding concept, as opposed to a concept that might fit more than one thing.
That perhaps seems perfectly sound, but I don't understand space as other than a structure of relationships. For example, when a physicist asserts that objects of mass warp space increasingly with that mass, I take this to be no more or less than a claim that objects of larger or smaller mass have different spatial relationships with things, and cause other things to have different spatial relationships each with others. (The latter implies that a spatial relationship involving the object of mass underlies the spatial relationships amongst the things other than that object.) When we attempt to distinguish objects based upon location, which is a matter of relationships amongst objects, remarkable considerations arise.
First imagine a very small universe, having just one body in it. A universe is not small by virtue of one hitting a wall after travelling some distance; it is small by virtue of having a non-Euclidean geometry, such that after a relatively short amount of travel one finds oneself back where one started. As light travelled from the object, it would eventually find its way back to the object. If one could somehow see, as if within this universe, and looked in various directions, with sufficiently strong vision, one might see the object, seemingly off in the distance, even if the view began as if one were standing right at or on the object. Seemingly beyond the object, one might see the object yet again, and so forth. That's the experience in one sort of universe. Now imagine an infinitely large universe, as if built by tiling duplicate sectors, in which there were infinitely many objects, positioned to given the same experience as in the first universe.
I declared that a universe had just one object; I declared that a universe had infinitely many objects. I don't actually believe that the apparently second universe is intrinsically distinct from the first. I think that we may conceptualize the first universe as the second, and vice versa, and that the count of objects is an artefact of our conceptualization. Of course, if there were no more than indiscernible differences amongst what seemed to be infinitely many objects, then I might claim that there were no practical differences from a universe of one object; but I here make the stronger claim that, if there are no differences beyond perhaps whatever is captured by the two given descriptions of location, then these descriptions are each of the same universe. It would simply beg the question to insist that one universe is different from the other in that one were finite but configured so that it seemed infinitely repeating, while the other truly were infinite and repeating. Granted that viewing as if within the universe seems to locate a means of viewing close to one object. (One might even imagine oneself invisibly located as an additional object in the universe.) But how is the location of that means near one object distinct from its location near all of them? (How is oneself's being located in the universe near one object distinct from the location of a perfect duplicate of oneself being near each of them?)
In a universe more like our own, if we had what seemed to be two otherwise indistinguishable objects at different locations, there would be other discernible objects that seemed to support a distinction. What we might regard as one object would be near to various other objects, and far from still others. Likewise for what we might regard as a different object, with a distinct set of things.
Let's mentally step away from that scenario for a bit, and return to the scholastic problem of things that do not have body but do have location. If two of these things are otherwise indistinguishable, and if they can occupy the same location at the same time, then ex hypothesi there is no way to distinguish one from another when they do occupy the same location. (The space occupied might be different when both moved into it — for example, it might become less translucent — but that doesn't mean that we can distinguish one of two things from another. And if the properties of these things are not in some sense additive or subtractive, but combine according to inclusive disjunction — that is to say that the attribute is either there or not, but has no further possible ordering to it — then we cannot tell how just many of these things are there by discernment of these properties.)
But, when they occupy the same location, I ask whether there are in fact two things. What would be the difference of two such things coming to occupy one location from two otherwise identical things coming to be one thing at one location? Perhaps what seemed one thing might again become two, but that wouldn't prove that they had remained distinct at the one location. Perhaps one thing might have become three, each just like the two from which the one had been formed. My experience (and, as I believe, yours) is that this has never happened. But I know of no logic that prevents it from happening; it merely violates my present best guess of the physical laws (which entail principles of conservation). If what seemd to be two things came together and seemed to be one, and then that one thing seemed to become three, some person might guess that what had earlier seemed to be two things were atually three things, two already in combination. But if all the attributes combined in conformance with inclusive disjunction, then in what sense would that be different from just what had seemed to happen?
If we accept that two otherwise indistinguishable things become one thing when they occupy one location, does that thing continue to exist should it become two things? Is it one of the two things? both of the two things? each of the two things? Is the proper answer different when the two things are indistinguishable except for location from what was one thing?
(If we are transporting some very great criminal by paddy wagon and, upon arrival, find three persons, each indistiguishable from the person whom we tossed into the wagon, and each insisting that he is just that person, do we treat each of them as that person, or charge each with no more than abetting an escape, on the theory that it is most likely of any given one of them that he is not that person? This problem might be primarily epistemological — so that one of the three suspects is our original perpetrator even if we shall never know whom — but that's bad enough; and we can make it still more fundamental if we allow for teleportation and for matter duplication.)
Let's mentally step back to the scenario of two otherwise indistinguishable objects at different locations in a universe rather like our own. Are these actually two objects, or one object that is bi-located, or one object in one location that appears as two locations because of a strangeness of space? Do these three descriptions actually distinguish different realities? If we mark one of these apparently two objects and see an identical mark appear on the other, do we regard it as the same object, or as two objects such that one is some sort of sympathy with the other? If we mark what seems one object and a mark does not appear on the other, do we regard this as proof that there were never one bilocated object nor a weirdness of space, or do we interpret this case as of one object becoming two distinct objects (with an end to the bilocation or with an adjustment of space), perhaps exactly as a a result of our action? (The classic formulation of Ockham's Razor is
entia non sunt multiplicanda præter necessitatem. If we posit that there were always two objects, are we conforming to that prescription?)
Because space need not be as Euclid had insightfully assumed and as Platon and Kant had thoughtlessly presumed, we can interpret any case where we have what otherwise must be a single thing simulatenously occupying multiple locations as in fact that single thing in one location. The practical cost, however, is that we are compelled to identify locations by the things occupying them (and not merely by the things about them); but we had set-out to identify things by the locations that they occupied!
Let's say that we have an object in a strange space, so that it is effectively bilocated, and we point to it and say
this. Assuming that we don't also say
not that, pointing to the other apparent location, is there any problem? It is one thing to incorporate mistaking of one thing for two into an assertion; another simply not to recognize some of the characteristics of that one thing. (There is a problem if Selina Kyle cries
I am in love with Bruce Wayne, not with the Batman! but there would have been no such problem had she simply declared
I am in love with Bruce Wayne!)
But if the case of two otherwise identical but differently located objects (perhaps each in perfect sympathy with the other) and the case of one apparently bilocated object are really just different descriptions of the same situation, then the applicability of
not that — and number more generally — seems in some cases to be an artefact of the descriptive framework. Especially in the context of such implications, some people will insist that one of these descriptions must surely be mistaken, even if as a practical matter we cannot tell which. (Some people will further insist that the description that conforms to simpler spatial relations (that of two objects in perfect sympathy) is the one that is more likely correct; other people will insist that the description that requires fewer objects (that of a bilocated object) is more likely correct.) However, the apparent contradiction isn't internal to either description, and each description may be translated into the other. That one of them is right doesn't make the other wrong.
If I cannot point to something and say
this and thereby distinguish it not merely from everything not there but from everything not it, then how can I have an intrinsically singular concept? To baldly incorporate singularity into a concept is just question-begging. (One ought not to say that two spheres are exactly alike except just in-so-far as one is unique, or is uniquely unique.)
Where, then, is singularity to be found? I think that it is to be found in experience, literally. The raw stuff of experience is sensation and sense-perception, not conception. (We may have concepts of sensations, but sensations are not themselves concepts; we may have concepts of sense-perceptions, but sense-perceptions are not themselves concepts.) Percepts and concepts are constructed to explain sensation and sense-perception. Those percepts and concepts may be perfectly accurate, but they are not intrinsically singular except to the extent that we associate them with sensation or with sense-perception. That is to say, for example, that we have a cluster of sensation or of sense-perception, and we have or build a concept of something by which to explain it, which concept is not singular except in-so-far as we implicitly or explicitly add to it the attribute of causing that particular cluster. And, if we do that, then we must in such case commit to a concept that does not allow co-location of otherwise identical things. That is not to say that we forbid co-location in general; but that singular concepts cannot be fitted to such co-located things. (It is probably a very bad idea to construct an explanatory model that employs co-location of otherwise identical things all of whose attributes combine in accordance with inclusive disjunction.)
In any case, the alternatives to exploring such considerations are dogmatism and nihilism. There is nothing intrinsically practical about dogmatism nor about nihilism, which stand in the way of our understanding the universe as deeply as we might and of our helping those who lose (or never find) their ways in their own attempts to understand the world. The scholastics who worried about the relationship of location to identity during what have come to be dismissed as
the Dark Ages were concerned with foundational questions of what we ought to practice. It is fine to jest about their efforts only if the joke does not hide the truth.