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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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We have almost arrived at David Lewis’s analysis of determinism. Lewis’s analysis makes use of an analysis of qualitative duplication; and this analysis itself makes use of the distinction between properties that are perfectly natural and those that are not. The perfectly natural properties (and relations) are the fundamental properties (and relations); wherever any other property or relation is instantiated, it is instantiated in virtue of the global pattern of instantiation of the fundamental properties and relations. The set of perfectly natural properties and relations, then, forms a supervenience base for the set of all properties. (I will also assume that the perfectly natural properties are purely qualitative.) Two things are duplicates, then, iﬀ they share all the same perfectly natural properties, and their parts can be put into correspondence so that corresponding parts share the same perfectly natural properties, and corresponding pairs of parts stand in the same perfectly natural (twoplace) relations (and so on). Lewis’s analysis of determinism, then, is (D0) A possible world w is deterministic iﬀ every world that is a duplicate of w at a time and is physically possible relative to w is also a global duplicate of w.

The column world is a counterexample to Lewis’s analysis. We want a Lewis-style analysis of determinism that corresponds to my modiﬁed version of the Laplacian picture in the way that Lewis’s corresponds to the version that takes only qualitative information into account.

To give the demon all the qualitative information about a time and all the non-qualitative information about things that exist at that time is to tell him not just how many things exist at that time, but also which things exist at that time, and what qualitative properties each instantiates at that time (and so on). So the demon can distinguish between times that are qualitative duplicates in which some things have switched roles.

A deterministic world, then, is one in which the demon, given the relevant information about a time and knowing the laws of nature, can tell us not just how many things there are in total and what qualitative properties each instantiates (and so on), but can also tell us which of them are the things that exist at the initial time. So the demon can tell us what roles the things that exist at the initial time are playing in the global structure of the world. And there will be some time slice of the world he describes that will match the initial time we described to him; and match not just because it is a qualitative duplicate, but also because the same things exist and play just the same roles.

To express this as a Lewis-style deﬁnition I need the concept of a duplication function. Recall that two things are duplicates iﬀ their parts can be put into a correspondence meeting certain conditions. Call such a correspondence a ‘duplication function.’ (Two things can be duplicates according to more than one duplication

function: think of two congruent equilateral triangles.) The analysis is:

(D1) A possible world w is deterministic iﬀ every world that is a duplicate of w at a time and is physically possible relative to w is also a global duplicate of w, under a duplication function that is the identity function on the initial times.29 Worlds at which general relativity is true are deterministic on this analysis, as they were on Lewis’s. (The diﬀeomorphism that generates the hole is a duplication function; since it only changes which spacetime points play which roles in the future, it is the identity on the initial times). But the collapsing column world is indeterministic. Again, let ‘p’ and ‘q’ name points of space that lie in diﬀerent directions from the column. There is a world w in which the column collapses toward p and I’m assuming that we can speak of two things existing in more than one possible world. Counterpart theorists might be nervous about such talk of transworld identity appearing in an analysis of ‘determinism,’ but the analysis can be easily re-written in terms of counterpart relations: say that w is deterministic iﬀ for any world w that is physically possible relative to w and duplication function d, if a time slice of w and a time slice of w are duplicates under d, then w and w are duplicates under some duplication function d∗ that agrees with d on the initial time slices.

Here the function d is a counterpart relation between the two time slices. It provides the standard for identifying things that exist on the time slice in w with things that exist on the time slice in w.

(D1) ﬁrst appeared in (Belot 1995), though he formulates it in counterparttheoretic terms, as does Melia (1999). Belot does not endorse (D1), but Melia does.

Melia does (and Belot does not) attempt to show that there is something natural and intuitive behind this formal analysis, that it is not just an ad hoc device for avoiding the conclusion of the hole argument. But Melia’s intuitive characterization is different from mine. He appeals to branching possible worlds. But at its best this will only allow us to explain senses of determinism in which the past and the laws ﬁx the future. My characterization can be generalized to other senses of determinism.

a world w∗ in which it collapses toward q.30 So a duplication function between w and w∗ must map p to q. But p and q are distinct and exist at t, before the tower collapses. So no duplication function from w to w∗ can act as the identity on t.

4 Earman’s and Stein’s Argument In Chapter 3 of World Enough and Space-Time Earman (following Stein (1977)) argues that relationalism about motion, together with the possibility of determinism, entails relationalism about ontology.

Pause to consider how implausible this is. Suppose relationalism about motion is true, and so that the world contains a Machian spacetime. (The discussion to follow is unchanged if we consider Leibnizian spacetime instead.) Suppose that the one law governing particle behavior is this: inter-particle distances never change.

The law looks deterministic: if I know there are just two particles, and that they are two feet apart right now, I know all there is to know about the future: there will continue to be two particles, two feet apart. Of course I won’t know if they’re in the same place later as they are now, or whether they’re in the same place relative to some frame of reference as they are now. But I couldn’t know these things, because in Machian spacetime there is no sense to be made of ‘same place across time,’ even relative to some frame of reference.

Earman’s argument appeals to (SP2) Any spacetime symmetry of theory T is a dynamical symmetry of T.

Rotations are symmetries of Euclidean space; by performing a (possibly diﬀerent) arbitrary rotation on each instantaneous Euclidean space in Machian spacetime we obtain a symmetry Φ of that spacetime. If we suppose that spacetime is populated There is a debate in modal metaphysics over whether distinct but qualitatively indiscernible possibilities really require distinct possible worlds. (Some defenders of counterpart theory, like David Lewis (1986b), say no.) This makes no diﬀerence to my argument; my argument works even if w and w∗ are the same world, though in some cases I would have to phrase the argument in terms of counterpart theory (see footnote 29 above).

by particles, then this symmetry is a dynamical symmetry of theory T just in case for any model M there is another model MΦ with the same spacetime manifold such that a point x in the spacetime of M is occupied by a particle iﬀ Φ(x) is occupied in the spacetime of MΦ. In his argument Earman presupposes (what I have complained about earlier) that substantivalists must say that distinct models of T correspond to distinct possible worlds.

The argument goes like this. Suppose relationalism about motion and substantivalism are both true. Then spacetime has a Machian structure. Suppose there are some particles obeying some laws. The function Ψ on Machian spacetime which is the identity for all times before and including t but rotates each of the later instantaneous Euclidean spaces through some angle about some axis after t is a symmetry of Machian spacetime. By (SP2), Ψ is also a dynamical symmetry. So the state of the world at t plus the laws fails to ﬁx the state of the world after t: they fail to determine whether the particles are where they actually are after t, or are where they would be if rotated through some angle about some axis. That is, if relationalism about motion and substantivalism are both true, then no theory of particle motion is deterministic. If relationalists about motion want to allow for the possibility of determinism, they must also be relationalists about ontology.

These two futures at work in Earman’s argument diﬀer merely non-qualitatively:

they diﬀer merely over which spacetime points in the future the particles occupy.

Earman’s argument presupposes that the existence of futures that diﬀer in this way is enough to render a theory indeterministic. As I have argued, this is not so. Ψ is a global duplication function that acts as the identity on the original times; so the distinct futures generated by it do not render the theory indeterministic, according to the analysis of determinism I have defended. Relationalism about motion does not require relationalism about ontology; and good thing, too, since I argued the opposite in chapter 1.

5 Conclusion I have argued that the hole argument fails because its second premise is false. That premise may seem true at ﬁrst, but only when read with an incorrect analysis of ‘determinism.’ I have articulated and defended a better analysis of ‘determinism,’ one that is a natural precisiﬁcation of the Laplacian conception of determinism, that has intuitive appeal considered on its own, and that ﬁts our intuitions about several cases. And I have shown how this modiﬁed Laplacian conception can be given formal expression in terms of possible worlds.

Chapter 3 Supersubstantivalism

–  –  –

1 Introduction Substantivalists believe, and relationalists deny, that spacetime exists. Set relationalism aside; substantivalists still have plenty to disagree about.

There is room for them to disagree about the nature of material objects. Some substantivalists—call them ‘dualists’—hold that material objects are distinct from spacetime. Others—‘supersubstantivalists’, as Sklar calls them1 —hold that material objects (if there are any) are identical with regions of spacetime. SupersubIn Space, Time, and Spacetime (Sklar 1974). I believe that Sklar coined this term; it ﬁrst appears on page 214.

stantivalists identify material objects with the regions of spacetime that dualists say those material objects occupy. Dualists, then, like their namesakes in the philosophy of mind, accept additional ‘ontological categories’: just as mind-body dualists hold that minds diﬀer in kind from physical things, so dualists in this debate hold that physical objects diﬀer in kind from points and regions of spacetime. As John Wheeler puts it, dualists regard material objects as ‘strange and nongeometrical objects immersed in spacetime’ (Wheeler and Taylor 1963, page 191).

Supersubstantivalism may initially seem incredible, but that has not stopped some very smart people from believing it. Descartes identiﬁed body with extension, and so looks like a supersubstantivalist. Bennett (1984) attributes supersubstantivalism to Spinoza. More recently, Ted Sider (2001) brieﬂy defends supersubstantivalism in the course of arguing for the doctrine of temporal parts.2 And supersubstantivalism is not just a metaphysician’s fantasy. Respected physicists have held views close to or identical with it: Isaac Newton was a supersubstantivalist, and so was Einstein during one period of his career. And the many physicists who believe that there are only ﬁelds (like the 19th century physicists who identiﬁed charged particles with parts of the electromagnetic ﬁeld3 ) are close to being supersubstantivalists. This is an impressive catalog.

There are several varieties of supersubstantivalism. In the next section I’ll survey these varieties, and then I’ll look at arguments for and against the view.

2 Varieties of Supersubstantivalism

Dualism is straightforward: there is a special fundamental relation, the occupation relation. Every concrete material object bears the occupation relation to some region of spacetime.

Generic supersubstantivalism is also straightforward: there is no such fundamental relation as the occupation relation. Every concrete material object is identical with some region of spacetime. But there are more and less radical varieties of supersubstantivalism.

Other philosophers who ﬂirt with supersubstantivalism include David Lewis (1986b), Hartry Field (1989), and W. V. Quine (1981).

The ‘electromagnetic view of nature’ is discussed in (McCormmach 1970).

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