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«Once Upon a Spacetime by Bradford Skow A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy ...»

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To clarify this second way of interpreting models, return to our two models, one in which a particle sits at (t, 0, 0, 0) for all t ∈ R, the other in which a particle sits at (t, 1, 0, 0) for all t ∈ R. On the second way of interpreting models both of these models say the same thing about the world: they both say that there is just one particle moving inertially for all time. Neither says anything about just which (actual) points of spacetime that particle occupies. (Indeed, we cannot even say whether it is one and the same particle that both models represent.) So the above argument for (L1) has no force without some argument that substantivalists must give the models one set of representational properties rather than another. What might such an argument look like? The most obvious way to argue that substantivalists should assign models the first set of representational properties goes like this: substantivalists believe that it is possible that everything be one foot to the left of where it actually is at each time; so they should interpret the models in such a way that there is a model for each of these possibilities. But such an argument is question-begging in the current context: we are trying to argue for (L1), so we cannot use (L1) as a premise.

How else might one argue for (L1)? The only real ‘argument’ for (L1) is an appeal to one’s modal intuitions. It seems intuitive to many people that some restricted combinatorial principle governs possibility.6 In particular, the following seems intuitive: any way of distributing particles across spacetime is a possible way of distributing particles across spacetime. That is, as far as (metaphysical) possibility goes, the spatiotemporal location of a given particle is independent of the location of any and all other particles. But this principle (together with facts about what the laws are) entails (L1).

(L1) also seems intuitive when we consider what happens when a substantivalist denies it. Denying (L1) is objectionable for the same reason that the more general denial of the existence of possible worlds that differ merely non-qualitatively is objectionable: it entails implausible essentialist claims.7 (L1) concerns a world in which everything is shifted one foot to the left at each time. But presumably anyone who denies (L1) will also deny similar premises asserting that substantivalists must believe in worlds where everything is shifted two feet to the left, or one foot to the right, and so on. For short, let’s say that such a person asserts that there are no shifted worlds. Now, for simplicity, let’s consider the case where there is no time, only (three-dimensional Euclidean) space. Suppose that there are actually only two point-particles, Joe and Moe. Joe and Moe are two feet apart, and they occupy points p and q. Then if there are no shifted worlds it is necessary that if Joe and Moe are two feet apart (and nothing else exists), they occupy p and q. And we can say something stronger. According to (L1) substantivalists must accept possible worlds that are shifted relative to the actual world. But any reason to believe that The unrestricted combinatorial principle is much more controversial. It entails that anything could have had any combination of fundamental properties at all. So (given plausible assumptions about which properties are fundamental) it entails that I could have been a positron, or a point of spacetime, or the successor of four.

(Adams 1979), (O’Leary-Hawthorne and Cover 1996).

would be a reason to believe that substantivalists must accept possible worlds that are shifted relative to any physically possible world. The negation of this claim is this: for any specification of the distances between n point particles there is exactly one possible world in which only those n point particles exist, and they have those inter-particle distances.8 But how could this be? In the case of Joe and Moe, what is so special about p and q that makes them the only possible locations of Joe and Moe, when Joe and Moe are two feet apart? Of course there is nothing special about them; if you think that there are no shifted worlds, you must believe that this is just a brute modal fact.9 That’s all I’m going to say to motivate (L1). What further debate there may be about (L1) can easily be translated into debate about the first premise of the hole argument; since that argument is my focus, I’ll save the further debate for my discussion of it.

Since I accept (L1), I will briefly explain where I think the Leibniz shift argument goes wrong. What is supposed to be wrong with recognizing the possibilities in (L1)? The possibilities at work in (L1) are qualitatively indiscernible; but what I’m continuing to assume that there is no time, only space, and that that space’s existence is physically necessary.

Admittedly, there is something special about p and q: Joe and Moe actually occupy them. I find it hard to believe that this explains why they have the modal property in question. And there are other brute modal facts that someone who denies (L1) must accept that cannot be explained in this way. There could have been three particles, instead of two, each two feet from the others; if (L1) is false then there are three points of space that are the only possible locations of these three particles, when they have those inter-particle distances. What makes them so special? Not, in this case, that they are actually occupied.





One might claim that there are no shifted worlds and try to avoid these consequences by asserting that it is indeterminate where each material object is located.

(Presumably it will still be determinately true that there are points p and q such that necessarily, if only Joe and Moe exist and they are two feet apart, then they occupy p and q. But there are no points p and q such that it is determinately true of them that necessarily, if only Joe and Moe exist and are two feet apart, then they occupy p and q.) But facts about where things are located are supposed to be fundamental facts; and I don’t believe that there could be indeterminacy where such fundamental facts are concerned.

is bad about qualitatively indiscernible possibilities? Relationalists mention several distinct problems. There is an epistemic problem: we can never know which of the indiscernible possibilities is actual. There is a theological problem: God could have no reason to actualize this world rather than a possibility qualitatively indiscernible from it. The theological problem worries no one these days, and the epistemic problem is a pseudo-problem.10 What remains is just the bare claim that (4) There are no qualitatively indiscernible possibilities.

Some find (4) intuitively plausible. I do not, and I also think there are reasons to reject it. Above I argued that substantivalists who accept (4) and deny (L1) must accept implausible essentialist claims. As I mentioned above, I believe more generally that anyone who accepts (4), substantivalist or not, must accept implausible essentialist claims. But I won’t argue for this more general thesis here.

2.2 Back to the Hole Argument

As in the Leibniz shift argument, the first premise of the hole argument asserts that substantivalists must believe in possible worlds qualitatively indiscernible from the actual world. But just what do these alternative possibilities look like? In the context of the Leibniz argument, it was easy to say: those were possibilities which differed with regard to where material objects were located. In the context of the hole argument things are not so easy. Now general relativity is our background physical theory. And general relativity (when given a substantivalist interpretation) differs in ontology from Newtonian mechanics. According to the latter theory (or the version of it that I had in mind) there was spacetime, on the one hand, with its geometrical properties; there were material objects, on the other hand, with their intrinsic properties (properties like mass and charge); and material objects and points of spacetime ‘interacted’ by the former being located at the latter. But general relativity is a field theory. In the mathematical models of the theory, there are vector Earman (1989) finds both problems in Leibniz’s letters to Clarke, though Earman puts the epistemic problem in verificationist terms. Maudlin (1993) dissolves the epistemic problem: it is a priori that each particle is where it actually is at each time, rather than shifted one foot to the left.

and tensor fields (like the metric and the stress-energy tensor) defined on the fourdimensional manifold which represents spacetime. But what is this theory telling us about the world? Of course, if we’re substantivalists, we think it is telling us that spacetime exists. But what else is there? Are there in addition a bunch of mathematical objects related to the points of spacetime? That is a strange view. Or are there in addition a bunch of very large material objects—fields—the properties of which vary from point to point? Or is there instead just spacetime, which in addition to its geometric properties, has non-geometric intrinsic properties? (Or is some fourth interpretation correct?) We can’t settle these questions here. But we need to have some way to characterize the possibilities mentioned in the first premise, if we’re going to look at arguments that substantivalists need to believe in them. We can of course characterize the possibilities in very general terms—we can say, they are possibilities which are qualitatively indiscernible, but which differ non-qualitatively because some spacetime points ‘play different roles.’ But without further information, we don’t really know what this means.

Let’s suppose we give general relativity the last ontology I mentioned—the one according to which there is just spacetime, with geometric and non-geometric properties. (Nothing will turn on this choice.) Then the possibility we are asked to consider is this one: in some future region of spacetime (the ‘hole’), the geometric and non-geometric properties of points of spacetime in that region are ‘pushed around’ smoothly (so that nearby points stay nearby11 ), leaving things qualitatively as they actually are. Of course, putting it this way is slightly misleading, because we are ‘pushing around’ geometrical properties. Since the geometry of spacetime is changing, there is no sense in which the points of spacetime are ‘staying put’ while the properties are ‘moving.’ We could just as truly characterize the possibility this way: in some future region of spacetime, the geometric and non-geometric properties are left where they are, but the points of spacetime are ‘pushed around’ underneath them in a suitably smooth manner. And indeed this characterization is The ensures that the function we’re using to move the points around is continuous, though actually the function must meet stricter requirements: it must be a diffeomorphism.

somewhat more accurate. For consider two points inside the hole, p and q, such that q ‘gets pushed to where p used to be.’ Now take any point r outside of the hole.

r is actually some distance d from p.12 In the possibility we’re contemplating, the distance from r to p is (probably) not d; instead the distance from r to q is d. Similar facts hold for other points. So in this non-actual possibility q is playing the ‘geometric role’ that p actually plays. (Here is a picture of what is happening: Imagine God looking down on the spacetime manifold with the properties distributed across it as one might look down at the island of Manhattan; imagine Him ‘lifting up’ these properties as one might lift up the buildings of Manhattan; imagine Him then focusing on some future region of spacetime underneath the properties, and pushing around the spacetime points in that region as one might then push around the dirt underneath some region of the upper west side (say, the region under Columbia University); imagine him finally putting the properties back down, ‘just the way they were.’ After God’s activity, things are just as they were before, qualitatively speaking.)13

2.3 Confusions About The First Premise

What reason is there to believe that the first premise is true? In section 2.1 above I discussed a bad argument for the first premise of the Leibniz shift argument. An analogous bad argument often appears in discussions of the hole argument. As I’m speaking loosely here. Take r to be a point such that there is a unique (either timelike or spacelike) geodesic connecting r and p, and let d be the ‘length’ of that geodesic.

These possibilities are somewhat complicated, because continuum-many points are being ‘moved around.’ Are there simpler possibilities that work just as well? Why not take just two spacetimes points in the (absolute) future and have them ‘switch roles?’ That is, why not just take two spacetime points in the future and have them exchange all of their geometric and non-geometric properties (including relational properties like the property of being ten meters from point q)?

This possibility is certainly easier to imagine: we simply imagine God focusing his attention on two spacetime points in the future, reaching down and removing them from the spacetime manifold, and then placing each in the hole left by the other.

Melia (Melia 1999, section 2.1) argues that these simpler possibilities do work just as well.

before, this argument appeals to facts about the models (in this case, models of general relativity) together with contentious principles for interpreting them. A model of general relativity is a four-dimensional manifold together with vector and tensor fields defined on it. It is a fact about the theory that if M is a manifold in some model then there is another model that also contains M but in which the vector and tensor fields have been pushed around, so that they make the same pattern on M but the points in M play different roles. (The mathematical device that does the pushing around is a function from the manifold to itself called a ‘diffeomorphism’ and the two models are said to be ‘related by a diffeomorphism.’) We are asked to accept the first premise because the theory contains diffeomorphically related models.



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