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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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Among the metaphysically possible worlds are worlds in which Machian spacetime exists, all and only the actual particles exist, and the non-modal spatiotemporal relations among the particles are as they actually are. For reasons given above, these worlds do not all agree on Field (1989) makes these points, and extends the argument to all spatial relations, modal or not.

(Belot 2000).

the distance rations between pairs of particles at each time. But the set of these worlds divides up into equivalence classes, where members of any one equivalence class do agree on the distances between the particles at each time. One of these equivalence classes is special. The members of that class and no other are geometrically possible relative to the actual world. But I can’t tell you which class is special, or why it is special and the others aren’t.

(Which worlds are geometrically possible varies from world to world, and since there are worlds that are non-modal duplicates—have the same history of betweenness and congruence relations—but differ in distance ratios, there are worlds which are non-modal duplicates but differ over which worlds are geometrically possible.) This helps with the problems I mentioned: we know how the geometrical possibility operator works, and the necessary connections between distance ratios, while still brute, are not so mysterious.

Still, we should reject this proposal. For one thing, it asks us to believe in brute modal differences: worlds which are isomorphic with respect to betweenness, congruence, and temporal ordering, and so are the same, as far as their non-modal facts are concerned, but which differ over distance ratios—which are modal facts— between pairs of particles.

But the more important problem with this proposal is that it looks like cheating. Relationalists need to guarantee the unique embeddability of the world’s particles into Machian spacetime; the solution here is to use a modal operator to do it by brute force.

If this is how relationalists about motion hold on to relationalism about ontology, then what is attractive about relationalism about motion in the first place?

There are supposed to be epistemic and (related) metaphysical advantages: interparticle distances and relative velocities are easier to observe, and a theory that relies only on them does not need to postulate spacetime. But if we can use these strange modal operators, then relationalism about motion no longer has these advantages over its denial. As for metaphysical advantages: let the world have a Newtonian spatiotemporal structure, so that talk of absolute states of motion makes sense. We can still be relationalists; there is no need to appeal to spacetime to make sense of absolute motion. Just use a new modal operator to guarantee the unique embedding of the world’s particles into Newtonian spacetime, and use this embedding to define states of absolute acceleration. As for epistemic advantages: if inter-particle distances and states of absolute acceleration are equally funny modal facts, why should one be easier to observe?

Like accounts 1 and 2, modal relationalism is a bad move. Even if relationalism about motion is true, relationalists cannot adequately characterize the spatiotemporal structure of the world. We should be substantivalists.

Chapter 2 Modal Arguments against Substantivalism 1 Introduction Two famous arguments against substantivalism—the Leibniz shift argument and the hole argument—turn on substantivalism’s (alleged) modal commitments. Both contain premises to the effect that substantivalists must believe that it is possible that things be just as they are, qualitatively, while differing in some non-qualitative respect.1 Leibniz argued that these the possibilities conflict with some a priori metaphysical and theological principles. Defenders of the hole argument argue that the possibilities in question lead to a failure of determinism. Responding to the hole argument is harder, so I will focus on it. I will say something about how we should think about Leibniz’s argument in the course of responding to the hole argument,

though. The hole argument looks like this:

Qualitative properties are those that ‘make no reference’ to particular individuals. So the property of being red is qualitative, while the property of being in the same room as Ralph Nader is non-qualitative. If we have a language the predicates of which express qualitative properties, and which contains no proper names, then we cannot distinguish between qualitatively indiscernible worlds using this language: every sentence true in one of the worlds is also true in the other. I say more about what the qualitatively indiscernible worlds at work in the Leibniz shift argument and the hole argument look like below.

(1) If substantivalism is true, then (assuming that general relativity is the true theory of the world2 ) it is physically possible that everything be just as it actually is, except that some spacetime points in the (absolute) future ‘play different roles.’ (2) If there is more than one physically possible future consistent with the way things are now, then determinism is false.

(3) Therefore, if substantivalism is true (and general relativity is the true theory of the world), determinism is false.

I’m going to take it for granted (as most do) that accepting the conclusion—that general relativity is not a deterministic theory—is an embarrassment for substantivalists. So there are two ways for substantivalists to respond to this argument. They can deny the existence of the qualitatively indiscernible possibilities mentioned in the first premise; or they can deny the second premise, and so deny that such possibilitties lead to a failure of determinism.

I advocate the second kind of response. Making this response work involves getting clear on just what determinism is. I defend an analysis of determinism that blocks the hole argument and is independently plausible. Although I defend this analysis of determinism in the context of responding to the hole argument, its value is not limited to the use to which I put it. Understanding determinism is something worth doing for its own sake. And, more importantly, determinism shows up elsewhere in the debate between relationalists and substantivalists. For example, Earman (1989), following Stein (1977), argues relationalism about motion entails relationalism about ontology. In the previous chapter I argued that relationalism about motion requires substantivalism. Unless relationalism is inconsistent, we cannot both be right. Earman’s argument contains a premise about determinism;

it presupposes the same analysis of determinism as the one at work in the hole argument. So like the hole argument, Earman’s argument fails for relying on (what I argue is) an incorrect analysis of determinism. I briefly discuss this argument in There is some debate over whether the hole argument applies to theories other than general relativity. Earman and Norton (1987) argue that it does, while Earman (1989) argues that it does not. This controversy won’t matter for my discussion.

section 4.

Before discussing determinism, though, I’ll explain why I accept the first premise of the argument. It has become more common lately for substantivalists to deny the first premise; I’ll discuss why I think this is unwise, and why I think much discussion of the first premise is misdirected.

2 Defending The First Premise

Let’s begin with the first premise. What reason is there to believe it? To answer this question I will first discuss the first premise of the hole argument’s predecessor, the Leibniz shift argument. It is easier to first distinguish between good defenses and bad defenses of the first premise of this other argument, and then apply the lessons learned to the evaluation of the first premise of the hole argument.

–  –  –

The first premise of the Leibniz shift argument is this:

(L1) If substantivalism is true, then (assuming that some theory, like Newtonian gravitational theory, that lives in neo-Newtonian spacetime is true) it is physically possible that at each time, each material object be one foot to the left of where it actually is at that time.3 How might one argue for (L1)? Here’s one way. On one way of formulating Newtonian mechanics, we write down some equations which pick out a set of models.

These models are n-tuples of mathematical objects. A typical model may include R4 and several curves through R4 (functions from R to R4 ).4 The models represent (or correspond to) physically possible worlds: by looking at the models we may figure out what sorts of arrangements of particles in spacetime are permitted by the theory. But of course by themselves the models tell us nothing about what is physically possible. Only when we also put in place some principles of interpretation, principles which assign representational properties to the models, do they yield inLet’s pretend that ‘to the left’ picks out a determinate direction in space.

These curves must satisfy certain constraints, but they won’t be important.

formation about physical possibility. Consider, for example, the model that consists of R4 and just one curve that assigns to each real number r the point (r, 0, 0, 0) in R4. What would the world be like, if this model correctly represented it? We cannot tell, until I specify some principles of interpretation. But when I tell you that R4 represents spacetime, that the first component of points in R4 represents temporal location and the other three represent spatial location, that some of the geometrical relations between points of R4 represent geometrical relations between points of spacetime, and that curves in R4 represent the careers of point particles in spacetime, then we can see that this model represents a universe in which there is just one point particle moving inertially for all time.

Of course I have not given a complete catalog of the principles of interpretation for these models, and there is room to disagree about just what those principles are. (Relationalists, for example, may deny that R4 represents spacetime, and may instead maintain that it is a fictional device for encoding spatiotemporal relations among point particles.) But there is one set of principles which, together with facts about what models there are, entails (L1). Suppose that in addition to the principles I gave in the last paragraph we also accept that each point of R4 represents some actual point of spacetime and that it represents the same point of spacetime in every model.5 With these interpretive principles in place, consider (again) the model in which a particle sits at (t, 0, 0, 0) for all t ∈ R. It follows from facts about the theory that if there is such a model, then there is also another model in which a particle sits at (t, 1, 0, 0) for all t ∈ R. And since in these two models a particle is located at different points in R4 it follows from what I’ve just said that these models represent a particle as located at different points of spacetime. Since each model represents a lone particle moving inertially for all time, these models correspond to distinct possibilities that differ only with regard to which points of spacetime the particle There is an obvious problem with interpreting the models this way, which I will set aside. On this way, if relationalism is true, (and so if there are no actual points of

spacetime), then the substantivalist interpretation of the theory is necessarily false:

not only is every model a false representation, no model could have been a correct representation. (Even if there had been points of spacetime, they would not be the actual points of spacetime, and no model can be correct unless the actual points exist.) occupies. So these possibilities differ merely non-qualitatively. (If we suppose that the direction from (0, 0, 0, 0) to (0, 1, 0, 0) is the left-ward direction in space, then these models differ only in that according to one of them, the lone particle is shifted one foot to the left of its location according to the other model, at each time.) Similar reasoning applies to more complicated models. Assuming that the theory is true, then, there is a model which represents the world as it is, and another model that represents each material object at each time occupying a region one foot to the left of the region it actually occupies at that time. Add the premise that whatever a model represents is (physically) possibly true, and (L1) follows.

But by itself this is not a good way to argue for (L1). For the mathematical models of our theories don’t come with their representational properties built in.

They get their representational properties from us. And substantivalists are not required to give them the representational properties needed to make this argument work. A substantivalist could, for example, interpret the models so that the models yield only qualitative information about the world. On this second way of interpreting the theory, no claims about de re possibility—no claims about which particular points of space a particular particle might be located at—follow from inspection of the models along with the rules for interpreting them. So, since the first premise of the Leibniz shift argument is a claim about de re possibility, it will not follow from inspection of the models along with the rules of interpreting them.

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