A Puzzle About Coinciding Modes

Consider once again the example of a particular rubber ball's sphericity. It might be thought to be wholly unproblematical that we can determinately identify something as being this particular sphericity, something which qualifies as a determinately distinct mode of precisely this ball. One question which we could ask here, though, is whether, this being so, numerically distinct sphericities can exist in the same place at the same time. That question could arise for the following sort of reason. The ball is wholly composed by a certain piece of rubber and yet very arguably we cannot identify the ball with that piece of rubber, because they differ in their modal properties and very probably in their histories. For instance, the piece of rubber could be reformed into a ring and yet continue to exist, whereas the ball would cease to exist in these circumstances. Again, the piece of rubber pre-existed the ball, if it was synthesized before being shaped into a sphere. But so long as the piece of rubber composes the ball and the latter is spherical, both of them are spherical—they are shaped in exactly the same way. Does this mean, however, that they possess one and the same sphericity—that they share a mode—or does it mean that they possess numerically distinct but qualitatively indistinguishable and exactly coinciding sphericities? Neither answer seems altogether easy to accept and neither seems defensible in a principled and non-arbitrary way.

If we say that two numerically distinct modes can exactly coincide, what is to prevent us from saying that any number of distinct modes could, in principle, exactly coincide? After all, why should the ball and the piece of rubber be restricted to having just one sphericity each, once we have admitted that numerically distinct sphericities can coincide? (Nor are coinciding modes to be compared with superposed electrons in a helium atom, being countable but not determinately distinct: for nothing akin to the Pauli Exclusion Principle can be invoked to justify such a description in the case of coinciding modes.) Moreover, even if the ball and the piece

of rubber each have only one sphericity, what determines which of these sphericities belongs to the ball and which to the piece of rubber? After all, the sphericities in question are intrinsically indistinguishable.

Suppose, on the other hand, that we say that two numerically distinct modes cannot exactly coincide and consequently that the ball and the piece of rubber share one and the same sphericity (though nothing would seem to support this judgement in the way that the identity-criterion for parts of stuff underwrites the judgement that distinct parts of stuff cannot exactly coincide). Then we should have to say, paradoxically as it seems, that the piece of rubber's sphericity could survive the ceasing-to-be of that piece of rubber: for if an interior part of the rubber composing the ball were to be destroyed, without affecting the spherical shape of the ball, then that original piece of rubber would no longer exist, even though, presumably, the ball would not acquire a new sphericity in the process.

Now it may be that this example is not liked for one reason or another. For instance, it will not be liked by those philosophers who would simply deny that a ball and the piece of rubber composing it are numerically distinct objects existing in the same place at the same time.¹²³I shall discuss that and related issues more fully in Chapter 9, but for present purposes it may be helpful to modify the example somewhat so as to bypass the contested point. At the same time, I shall try to bring into sharper focus the problem which the question about coinciding modes creates.

Let S be a solid sphere of matter, such as the piece of rubber which we have been talking about hitherto. Then, if we acknowledge the existence of modes at all, we must surely acknowledge the existence of the following mode: S's sphericity. And let S* be the spherical parcel of matter constituting S's outer shell to a certain depth d from S's surface (remembering, here, our earlier distinction between ‘pieces’ and ‘parcels' of stuff, this being that while both pieces and parcels are connected, pieces are distinctive in being maximal). Then, likewise, if we acknowledge the existence of modes at all, we must surely also acknowledge the existence of the following mode: S*’s sphericity. Clearly, S and S* are numerically distinct objects (they do not even occupy the same place at the same time, although they do overlap).

Moreover, S can exist without S*, and S* can exist without S. The proof of this last claim is as follows. If S's inner core up to a distance d from its surface is removed or destroyed, then S ceases

123 See, for example, Michael B. Burke, ‘Preserving the Principle of One Object to a Place: A Novel Account of the Relations Among Objects, Sorts, Sortals, and Persistence Conditions’, Philosophy and Phenomenological Research, 54 (1994), 591–624. I discuss Burke's position in my ‘Coinciding Objects: In Defence of the “Standard Account'”’, Analysis, 55 (1995), 171–8.

to exist, but S* continues to exist; on the other hand, if some matter is exchanged between S* and S's inner core, then S* ceases to exist (since a parcel of matter ceases to exist when any of the matter composing it is removed or replaced), but S continues to exist.

Now I ask: is S's sphericity identical with S*'s sphericity? In favour of a positive answer is the fact that these sphericities are empirically indistinguishable, occupying exactly the same region of space at the same time. In favour of a negative answer, however, is the fact that S's sphericity belongs to S while S*'s sphericity belongs to S*. Modes are existentially dependent entities, depending for their existence upon the objects which ‘possess’ them (a notion which I shall explore more fully in Chapter 6). Thus, if S ceases to exist, then S's sphericity should cease to exist too; whereas if S continues to exist and does not change in shape, then S's sphericity should likewise continue to exist. The same applies to S* and its sphericity. However, if we were to say that S's sphericity is identical with S*'s sphericity, then we should have to deny that the foregoing existential dependencies obtain. For these dependencies imply that if, say, S ceases to exist while S* continues to exist and does not change in shape, then S's sphericity ceases to exist while S*'s sphericity continues to exist—and this is plainly incompatible with the thesis that these modes are identical.

So the identity thesis is incompatible with the view that modes are existentially dependent entities in the sense explained. But the latter view is very hard to abandon. If modes do not depend for their existence upon the objects which possess them, why should they not ‘float free’ of such objects altogether? And yet the thought that they might do so is seemingly absurd. Moreover, if modes were to have determinate identities, it seems that these would in fact have to depend upon the identities of the objects possessing them: how else could we identify a particular sphericity other than as the sphericity of such-and-such an identifiable object? Perhaps, then, one should prefer to say that S's sphericity and S*'s sphericity are numerically distinct modes, despite the fact that these modes are not empirically distinguishable. But this position is quite as hard to defend as the view that the modes in question are identical. For once we admit that there are two distinct but empirically indistinguishable sphericity modes occupying exactly the same region of space at the same time, it is clear that we must admit infinitely many such modes. This is because, in addition to S*, there are infinitely many (indeed, continuum many) other spherically shaped shells of matter located within S, each with a different depth d; and each of these will, by parity of reasoning, have to be assigned its own distinct sphericity mode—for all of these shells, like S*, are distinct and existentially independent objects. I take it, however, that such an uncountable infinity of empirically indistinguishable sphericity modes would be

regarded as an embarrassment of riches even by the most extravagant ontologist.

We see, then, that it is difficult to defend either the view that all of these sphericity modes are identical or the view that all of them are numerically distinct. The lesson, I suggest, is that we should not think of entities such as modes as possessing determinate identities at all. They are not themselves ‘objects’, somehow related to the objects which

‘possess’ them. Rather, they have an ‘adjectival’ status: they are, quite simply, particular ways those objects are.

This conclusion can be reinforced by other examples. Here, for instance, is another conundrum that we might pose concerning the sphericity of the rubber ball discussed earlier. What happens if the ball is temporarily squashed out of shape but subsequently returns to being spherical again? Does it acquire a new sphericity, numerically distinct but qualitatively indistinguishable from its previous sphericity, or does its original sphericity come back into existence after a period of non-existence? How many sphericities would the ball possess in the course of a vigorous game of tennis?

What begins to emerge from these and our preceding reflections is a suspicion that it really doesn't matter how we answer these questions about the identities of sphericities and how to count them, because there are no real facts of the matter which determine what the ‘right’ answers to such questions are. We can, I think, include sphericities and other modes in our ontology without needing to address or even to countenance most such questions. Only a prejudice in favour of the Quinean dictum ‘No entity without identity’ could tempt us to suppose otherwise. We can say, for instance, that the ball has a sphericity and that the piece of rubber composing it has a sphericity and simply refuse to admit as legitimate any question as to the identity or diversity of these items—even if we also decide to say that the ball and the piece of rubber composing it are definitely not identical with one another. Sometimes, it seems, the only questions of identity and diversity which we can sensibly address in the case of modes are questions of qualitative, not numerical, identity and diversity. We can always sensibly ask, thus, whether two numerically distinct objects have exactly similar sphericities: but, having answered this question positively, there may simply be no further question of substance to be raised as to whether or not the sphericities of these objects are themselves numerically distinct. The principle of the identity of indiscernibles can indeed seem to fail in such a case for want of significance. (Of course, modes which are not indiscernible—such as a ball's sphericity and that same ball's redness, or the sphericity of one ball and the sphericity of another ball—must for that very reason not be numerically identical; what I am suggesting is that in the case of modes which are

indiscernible—such as a ball's sphericity and the sphericity of the piece of rubber composing the ball—there is no fact of the matter as to their numerical identity or distinctness.)