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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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I have said that substantivalists should accept the first premise of the hole argument, and so accept that (if general relativity is true) there is more than one physically possible future consistent with the way things are now: the actual future, and a future that is qualitatively indiscernible in which some future spacetime points have

switched roles. Now, the second premise of the hole argument is:

(2) If there is more than one physically possible future consistent with the way things are now, then determinism is false.

To uphold the claim that general relativity is a deterministic theory, then, substantivalists should deny this second premise. I know this premise looks analytic; but this is merely an appearance. There are reasons to reject it which are independent of its role in the hole argument.

3.1 Determinism: Examples and Counterexamples

It is easy to convince yourself that not just any alternative physically possible future counts against determinism, as we ordinarily think of it. If one accepts that a given world has futures that differ merely non-qualitatively, then it should be intuitive that those futures do not count against determinism.21 But, in the first place, it does not follow immediately: we need an extra assumption—namely that there are no qualitatively indiscernible worlds—which Lewis does not make. And, in the second place, Lewis himself came to think this axiom too restrictive and abandoned it in On The Plurality of Worlds (Lewis 1986b).

David Lewis’s analysis of determinism in (Lewis 1983) entails that alternative futures that differ merely non-qualitatively do not count against determinism. He offers no explanation of why this should be, though. Butterfield’s analysis in (Butterfield 1989) and Brighouse’s in (Brighouse 1997) are equivalent to Lewis’s. I look at Lewis’s analysis in more detail below.

Joseph Melia (Melia 1999) gives an example of a world with possible futures that differ merely non-qualitatively, but which seems, intuitively, deterministic. Consider these simple laws: nothing ever moves; there are two kinds of particles, P+ and P− particles; each P+ particle decays into a P− particle five minutes after coming into existence; the P− particles occupy the same places as the P+ particles that give birth to them. P− particles never decay. (We are also to assume that worlds governed by these laws are relationalist: there is no space or time, only spatiotemporal relations between particles).22 Suppose that the history of the world up to a certain time looks like this: in the beginning there was nothing, and then three minutes ago two P+ particles were created three meters apart. It appears that the future is determined: in two minutes both P+ particles decay into P− particles.

There is no other possibility. But if we are working with the conception of determinism at work in the second premise of the hole argument, then this world comes out indeterministic. Call one of the P+ particles α+, the other β+. Then α+ actually decays into α−, β+ into β−. But α− and β− could have switched roles. So there are two possible futures (differing merely non-qualitatively) for the world; they differ with regard to which P− particles α+ and β+ decay into. But it is absurd to allow these differences count, when asking whether this world is deterministic.

If this line of thought is right, then we have reason to deny the second premise.

Unfortunately there is an example which seems to show that this line of thought is not right, that we do allow the existence of physically possible futures which differ merely non-qualitatively to count against determinism.23 Exert enough downward force on a cylindrical column and it will usually collapse into an elbow shape. We are to imagine a non-actual set of laws of nature according to which cylindrical columns, when they collapse, always collapse into an elbow shape, even when the situation is perfectly symmetric around the axis of the cylinder—the force is applied straight down on the center of the top of the cylinder, the cylinder itself is not But wait: then what do we mean when we say ‘the P− particles occupy the same places as the P+ particles that give birth to them’? Perhaps Melia means to allow for cross-time distance comparisons, so we can say that some P− particle is zero meters from the P+ particle from which it decayed.

The following example first appeared in (Wilson 1993).

weaker on one side, and so on. (We are also to imagine that worlds governed by these laws contain Absolute (Newtonian) space and time.) Now imagine a world containing such a cylinder in which a sufficient force is applied straight down on the center of the top of the cylinder, causing it to collapse, but which is perfectly symmetric before the cylinder collapses. (The world contains only the cylinder sitting on a perfectly spherical planet and a sphere falling directly on top of the cylinder.) Intuitively, this world is not deterministic: the laws do not dictate in which direction the cylinder will collapse. But the different possible futures—the cylinder collapses this way, or it collapses that way—are qualitatively indiscernible.

What does this example show? Even if we agree that the column world is indeterministic, it does not show that the second premise is true. For I, at least, still have the intuitions that in some cases qualitatively indiscernible futures do not count against determinism: unless more is said, the collapsing column example does not suggest that these intuitions were misleading.





So we are left with the task of finding an analysis of ‘determinism’ which sometimes allows merely non-qualitative differences to count against determinism— as in the column world—and sometimes does not—as in the worlds in Melia’s example. And we want an analysis that is appealing for independent reasons, not just because it allows us to defend substantivalism from the threat of the hole argument.

3.2 Analyzing Determinism

There is such an analysis available. Consider the Laplacian picture of determinism:

in a deterministic world, an extraordinarily intelligent demon who knew all the information about the present as well as the laws of nature would be able to deduce all the information about other times.24 But what do we mean, all the information?

We could mean, all the qualitative information; we could mean all the qualitative and non-qualitative information; or we could mean something in between. So this This picture speaks of agreement at a time forcing agreement at all times. There are other varieties of determinism, differing with regard to which regions of spacetime worlds must agree on to force agreement through spacetime (Earman (1986) contains a survey). But this is the one in play in discussions of the hole argument, so it is this one I shall focus on.

gives us several ways to make the Laplacian picture more precise: we can be more precise about what information we give the demon to start with, and more precise about what information we demand that he produce. Here’s one way to make these more precise: we could give the demon all the qualitative information about the present, as well as the laws of nature, and demand that he produce all the qualitative information about other times. Here’s another: we could give the demon all the qualitative and non-qualitative information about the present, as well as the laws of nature, and demand that he produce all the qualitative and non-qualitative information about other times. (This is the conception of determinism relationalists rely on in the hole argument.) The analysis I endorse captures the only natural middle position. We give the demon all the qualitative and non-qualitative information about the present, as well as the laws of nature. (If we were to write down the information he is given in a language the predicates of which express only qualitative properties, some of the sentences we would write down would contain proper names of things that exist in the present.) And we demand that he produce all the qualitative information about other times, and also all the non-qualitative information about other times which (if we were to write it down) can be expressed using only the proper names we have already given him. (So he need not produce any non-qualitative information that can only be expressed using proper names of things that exist only in the future.) If he cannot do this, then the world is not deterministic.

This picture of determinism meshes with our intuitions about determinism.

Consider Melia’s particle decay world. No Laplacian demon is needed to write down the relevant information; we can do it ourselves: given that three minutes ago two P+ particles, α+ and β+, were created three meters apart, we know that in two minutes there will be two P− particles, one having decayed from α+, the other from β+, and they will be three meters apart, for the rest of time. This world is deterministic, even if we cannot deduce which P− particle decays from which P+ particle; even if an extraordinarily intelligent demon cannot, either. For in order to express the required proposition we must use names for the P− particles; and we are not required to deduce information that can only be expressed using such names.25 By ‘names’ I mean names that do not have their reference fixed using descriptions. We can, of course, deduce that α+ decays into α−, because I introduced ‘α−’ The column world is indeterministic, though, because (since the points of space endure through time and exist at the initial time) we must be able to deduce all the non-qualitative information expressible using the names of the points of space. So for any point p we must be able to deduce whether the column will collapse toward point p or not; and this we cannot do.26 This picture of determinism also seems intuitively correct, when considered on its own. I have said that we should demand that the demon produce only that non-qualitative information about other times which can be expressed using only the proper names we have already given him. And if we are going to demand that the demon produce some non-qualitative information, how could we reasonably demand more information than this? Think of proofs from some given set of premises in some formulation of the first-order predicate calculus. It is impossible to derive any sentence containing names which do not occur in the premises you are given.27 How can we ask of the demon something that, as a matter of logic, cannot be done?28 by saying it names the particle into which α+ decays.

The use of Absolute space is not necessary; even if the history of the world unfolds in neo-Newtonian spacetime, we have names for currently existing points of spacetime, so we must be able to deduce whether the column will collapse toward the unique spacetime point that exists at the time of collapse and is co-located with p relative to such-and-such frame of reference. This, too, we cannot do.

Well, not exactly. But it is true that for any derivable sentence ϕ(a) containing a constant a not occurring in the premises, you can also derive ∀x ϕ(x) ; so the non-qualitative information expressible using new names that you can derive is not substantive.

In this argument, I assume that the laws of nature are purely qualitative; they do not, as it were, mention any particular individuals by name. There are theories of laws of nature which may allow non-qualitative laws. The best system theory of laws (Lewis 1983), in particular, may do this: in simple enough or strange enough worlds, the strongest and simplest system may be one containing theorems which mention some particular individual by name. I think my analysis could be amended to accommodate such laws.

3.3 A Formal Definition

I began with the Laplacian picture of determinism and distinguished three ways of making it more precise. The three ways agree that a deterministic world is one in which a powerful demon, when given complete qualitative information about one time and the laws of nature, can produce complete qualitative information about all other times. They disagreed over how much non-qualitative information we give the demon about the initial time, and how much non-qualitative information we demand he produce about other times. I favored the precisification according to which we give the demon all non-qualitative information about things that exist at the initial time, and demand that he produce all non-qualitative information about just those things at all other times.

But this talk of non-qualitative information and of what the demon can produce is still vague. I will now precisify it further. To see how, first consider how to precisify the version of Laplacian determinism that deals only with qualitative information. To give the demon all the qualitative information about a time is to tell him how many things exist at that time, which qualitative properties each of them instantiate, and for each n (where n might be infinite), which qualitative n-place relations any n-tuples of them instantiate. A world that is deterministic in this (purely qualitative) sense, then, is one in which the demon, given all qualitative information about a time and knowing the laws of nature, can tell us how many things there are in total, what qualitative properties each of them instantiate, and so on. And there will be some time slice of the world he describes that will match the initial time we describe to him, where ‘match’ here means ‘be a qualitative duplicate of.’ Although the demon can tell us how many things exist in this world, though, he cannot tell us which things exist, and cannot distinguish between worlds in which two things have switched roles. Dispensing with the demon, this means that a world w is deterministic (in this purely qualitative sense) just in case any other world that is physically possible relative to w and is a qualitative duplicate of w at a time is a qualitative duplicate of w full-stop.



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