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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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So for example Carl Hoefer writes, ‘given the identification [of spacetime with the manifold; that is, given that the four-dimensional manifolds that appear in the models represent spacetime], and a straightforward interpretation of the mathematical apparatus, the substantivalist is committed to an infinity of qualitatively indistinguishable possible worlds’ (Hoefer 1996, page 7; italics mine). Hoefer supposes that an interpretation of the models according to which they yield nonqualitative information (an interpretation according to which each point of a manifold represents a particular spacetime point, and represent the same spacetime point in each model in which it occurs) is more straightforward than an interpretation according to which models yield only qualitative information. I don’t see how one interpretation is any more straightforward than the other. It may be that one fits better with the antecedently established modal commitments substantivalists make;

but (as I remarked above) this justification is question begging in the current context. Again, arguing for the first premise in this way by looking at the models of the theory does not work.

Part of the trouble is that the first premise is often stated as a claim about the representational properties of the mathematical models. So Carolyn Brighouse writes, ‘The central claim [of the hole argument] is that the substantivalist has to view diffeomorphically related models as representing distinct situations’ (Brighouse 1994). Similarly, the central question that Jeremy Butterfield thinks the hole argument raises is, Do models related by a hole diffeomorphism represent the same possible world (1989, page 12)? Stating the first premise in these terms leads to confusion. Brighouse’s claim is stronger than the first premise of the hole argument as I have written it down; it is the conjunction of my premise and some claim about the representational properties of models. So her claim could be false while my first premise is true. Arguments against the first premise as Brighouse states it, then, are not sufficient to block the hole argument.

Although Earman in his book (1989) also presents the first premise as a claim about models and their representational properties, Earman and Norton (1987) do not seem to make this mistake. They first assert (in the context of the Leibniz shift argument) that substantivalists must accept the possibility of shifted worlds, and then ‘translate’ this claim into a claim about the representational properties of models. Claims about the representational properties of models do not appear as premises in an argument for the first premise.14

2.4 Defending the First Premise

We still have no argument that substantivalists need to accept the first premise.

When I discussed the first premise of the Leibniz shift argument, I gave an argument for it that relied on certain combinatorial intuitions. Can we appeal to those intuitions here?

It’s not clear that we have the combinatorial intuitions needed in this case. It is obvious (to me anyway) that (assuming points of spacetime exist) the spatiotemporal locations of material objects are independent of each other. But the indiscernible possibilities at work in the Leibniz shift argument do not differ with regard to the geometrical role that each spacetime points plays, as do the possibilities at work in the hole argument; and it is not nearly so obvious that a point of spacetime could have played a different geometric role from the one it actually plays.

One can try to make the two cases look similar, so that the same intuitions

that support the Leibniz shift also support the hole construction, as follows:

In the context of the Leibniz shift argument, you accept that material objects could be located in regions other than the regions at which they Though it is true that their paper contains no arguments for the first premise;

they seem to think its denial is inconsistent with substantivalism.

are actually located. But as we are understanding general relativity there are no things other than points of spacetime. (This is not to say that according to general relativity people do not exist, or that according to general relativity there are no tables; they are just not what we thought they were.) So in the context of general relativity the possibility according to which you are located in some region other than the one in which you are actually located is not correctly described, at a fundamental level, as one in which certain particles bear the location relation to certain points of spacetime. Instead, it is correctly described as one in which certain non-geometric properties that certain spacetime points instantiate have been changed around in an appropriate way. So what your combinatorial intuitions are telling you, in this context, is that there is no impossibility in the idea of changing which non-geometric properties certain points of spacetime instantiate. But there is no principled distinction to be made, in the context of this theory, between the geometric and the non-geometric properties. So you ought to admit that it is possible to shuffle the geometric properties as well.

Why is there no principled distinction to be made? There are several reasons that could be offered here. For one thing, just as there are many physically possible ways to distribute non-geometric properties in spacetime, in general relativity there are many physically possible geometries for spacetime. For another, in general relativity different distributions of non-geometric properties in spacetime require as a matter of physical law different distributions of geometric properties.15 This line of reasoning has some plausibility. But it is not nearly as strong as the combinatorial support for the first premise of the Leibniz shift argument.





Here’s one reason. The combinatorial intuition I cited was not an intuition that the spatiotemporal locations of material objects can be changed around, whatever the correct physical and metaphysical theory is. Rather, my intuition was this: if this is the correct way of describing the world—there are material objects, there is Something like this line of thought occurs on page 519 of (Earman and Norton 1987).

spacetime, material objects are located in spacetime—then it seems intuitive that the locations of material objects can be changed around.16 So the support for the first premise of the Leibniz shift argument does not transfer automatically to the first premise of the hole argument.

Still, I accept the first premise. I think it becomes clear that substantivalists should accept it when we see what is involved in denying it. Denying the first premise involves some appeal to essentialism. Since the first premise is about futures in which spacetime points switch roles in the geometry, first-premise deniers often appeal to some form of geometrical essentialism.17 There are various ways to formulate geometrical essentialism: one could claim that (actual and possible) points of spacetime have their qualitative geometric properties (including their curvature properties) essentially; or one could claim that points of spacetime have their qualitative properties and their non-qualitative relational geometric properties (like being ten feet from point p) essentially.

I think it clear that one’s intuitions about possibility can depend on one beliefs about fundamental metaphysical matters. Here’s an example. Forget about time for a minute; and suppose space is Euclidean. And suppose it is Euclidean in virtue of the points of space (along with numbers) instantiating a three-place relation, the distance from x to y is r, in a certain pattern. Then it seems perfectly possible that every pair of points of space be twice as far apart as they actually are. For this simply involves each pair m and n instantiating the distance from x to y is 2r iff they actually instantiate the distance from x to y is r, where r is some real number. But now suppose instead that space is Euclidean in virtue of the points of space instantiating two relations, betweenness and congruence, in a certain pattern.

Then it seems impossible that every pair of points of space be twice as far apart as they actually are—because there are no absolute facts about how far apart two things are, there are only comparative facts about whether two things are the same distance apart as two other things. (One might assert that ‘the congruence relation can hold between points of space in different possible worlds,’ and that this allows one to make sense of the possibility in question; but this assertion only makes sense if one is a modal realist, which I am not. And even modal realists will hesitate to say that the congruence relation can hold between points of space in different possible worlds: Lewis defines ‘possible world’ as ‘maximally spatiotemporally related concrete object.’ It follows from this definition that parts of distinct possible worlds bear no spatial relations (and congruence is a spatial relation) to each other.) (Maudlin 1990).

We should reject these versions of geometrical essentialism. In general relativity, the geometry of spacetime depends on the distribution of mass-energy.18 So, if general relativity is true, then if I had raised my hand a moment ago, the geometry of the region of spacetime around me would have been different. So, if these versions of geometrical essentialism are true, then if I had raised my hand a moment ago, (part of) the region of spacetime I actually occupy would not have existed. But certainly it was up to me whether I raised my hand a moment ago; so if these versions of geometrical essentialism is true, it was up to me whether a certain region of spacetime exists. But that is absurd.19 There is a weaker version of essentialism that does not entail that what points of spacetime exist depends on what I do. It is an instance of the more general doctrine that there are no possible worlds that differ merely non-qualitatively. On this view it is impossible that (a) all the actual points of spacetime exist, (b) the geometry of and distribution of matter in spacetime is just as it actually is, and (c) some points of spacetime play different roles than they actually play. But this view does not entail that a given spacetime point have any particular curvature property essentially, or that it must be any particular distance from some other point. So this view evades the objection above: it is consistent with this view that even if I had raised my hand, all the actual points of spacetime would still have existed.

What motivates those who appeal to geometrical essentialism is the idea that a point of spacetime cannot be separated from its geometrical properties. This idea The argument to follow depends on special features of general relativity; analogues of the hole argument in the context of other physical theories are immune to it.

Maudlin replies to an argument like this one by ‘appealing to counterpart theory’ (1990, page 550). That is, after arguing that ‘spacetime (the actual spacetime) could have had a different geometry’ is false, he proposes a counterpart-theoretic semantics for our modal vocabulary on which ‘spacetime (the actual spacetime) could have had a different geometry’ is true. But for this response to work, he must claim that my objection seems plausible when we understand the modal vocabulary that occurs in it in the new way, but does not seem plausible when we understand it in the old way. But I don’t think this is so; when I wrote the objection down, I meant to be using the modal vocabulary in just the way Maudlin does when he defends essentialism; and the argument seems plausible when read that way.

does not motivate the weaker essentialism I am now discussing. I gave my reasons for rejecting its motivation, the doctrine that there are no merely non-qualitative differences, above on page 32: it leads to implausible essentialist claims. So I look elsewhere for a response to the hole argument.

2.5 Counterpart Theory

Butterfield asserts that one can deny the first premise of the hole argument without being an essentialist by appealing to counterpart theory (1989, page 22). How might this work? Essentialism is a collection of de re modal claims. Counterpart theory is not; it is (part of) an analysis of de re modal claims. (Roughly speaking, de re modal claims like ‘Nader could have won’ are analyzed not as ‘There is a possible world in which Nader wins,’ but as ‘There is a possible world in which a counterpart of Nader (someone sufficiently similar to him) wins.’) And counterpart theory is not incompatible with essentialism. There are counterpart relations that make certain essentialist claims true: there is a counterpart relation, for example, which makes true the de re modal sentence, ‘I could not have been a poached egg.’ Of course there are other counterpart relations that make this and many other essentialist claims false. To use counterpart theory to deny the first premise without embracing essentialism, then, one must find one of these anti-essentialist counterpart relations that makes the first premise false. But this cannot be done. There are no such counterpart relations because the denial of the hole argument’s first premise is equivalent to a version of essentialism. The first premise just is the claim that certain spacetime points can switch their geometrical roles; for this premise to be false is for the points to have (some aspect of) their geometrical roles essentially. This is so whether you analyze de re modal claims in terms of counterparts or not.20 In Lewis’s original version of counterpart theory, as presented in ‘Counterpart Theory and Quantified Modal Logic’ (1968), he made it an axiom that each thing has at its own world only one counterpart: itself. This seems to lead immediately to the result people like Butterfield want: for (reverting for the moment to the Leibniz shift argument) you might think that the truth of ‘Each thing could have been one foot to the left of where it actually is’ requires each point of space to have a counterpart at its own world other than itself. This is still essentialism: but one might think that it is made more palatable by following immediately from Lewis’s theory.

This argument shows more than that appealing to counterpart theory is not a way to avoid essentialism while denying the first premise. It shows that the only way to deny the first premise is to embrace some form of essentialism.

3 The Second Premise



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