«Once Upon a Spacetime by Bradford Skow A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy ...»
terms of space and time; suppose them to be redeﬁned in the way I mentioned above on page 87, so that velocity ends up being change in position along the horizontal axis with respect to change in position along the vertical axis.12 So in these laws the vertical axis plays the time role in the deﬁnition of ‘velocity,’ and also plays the role that I described above: the laws govern the evolution of the world along the vertical axis. It seems to me that when we add these laws to the description of the world depicted by the diagram, we know enough to know that the vertical axis is the time axis, and we rule out Horizontal as a correct description of that world.
Let me address two worries one might have about this example. First, one might worry that my strategy here—maintaining that knowing only how matter is distributed in spacetime, without knowing what the laws are, leaves us unable to tell which axis is time—is incoherent if a broadly empiricist theory of laws of nature is true. According to such theories, the laws of nature supervene on the occurrent (categorical, non-dispositional) facts. Since the occurrent facts surely include facts about the structure of spacetime and the distribution of matter in it, an empiricist will say that by ﬁxing the facts about matter and spacetime I’ve already ﬁxed what the laws are, and so already ﬁxed which directions are timelike.
This may be so; I won’t take a stand here on whether some empiricist theory of laws is true. But even if an empiricist theory is true, it is coherent to ask you to remain ignorant about the laws even though you know about the distribution of matter in spacetime, for it is not always obvious what the laws are in a world with So where the original laws contained ‘...velocity...’ the new laws will say ‘there are two (distinct) directions, x and y, such that x and y play such-and-such geometrical roles, and....change in position in the x direction with respect to change in position along the y direction...’ Given that the horizontal and vertical axes play symmetric geometrical roles, you might wonder why it is the direction along the horizontal axis, rather than the direction along the vertical axis, that gets to play the time-role in the deﬁnition of velocity. But there is no answer to this question. The way the laws are stated guarantees that (in two-dimensional worlds with Newtonian spacetime) one kind of direction will play the time role in the deﬁnition of velocity and the other will not; but it is undetermined which—there are other worlds with the same spacetime geometry and the same laws in which the other plays that role.
a given set of occurrent facts. And the fact that we cannot tell which axis is time when ignorant of the laws but can when we know the laws is still evidence that it is the role they play in the laws that makes timelike directions timelike.13 Second, one might worry about the descriptions in Vertical and Horizontal.
Those descriptions seem to contain more information about the distribution of matter in spacetime than the spacetime diagram alone does. And the points at which the descriptions go beyond the information contained in the diagram are points at which the descriptions disagree. The spacetime diagram tells us only which spacetime points are occupied by material objects. The descriptions contain further information about how many material objects there are, and which points each object occupies. In Vertical I said that there were two particles, while in Horizontal I said there were inﬁnitely many. Now, I claimed that knowing only how matter was distributed in spacetime isn’t enough to allow us to ﬁgure out which axis is time.
But one might complain that by presenting the spacetime diagram I hadn’t given complete information about how matter is distributed in spacetime. Complete information requires the kind of information contained in Vertical and Horizontal.
If I had said that there are only two particles, and each one occupies the points on just one of the curvy lines, then perhaps it would have seemed obvious that the vertical axis is time.
To allay this worry let me make some further stipulations about the world(s) represented by the diagram. I’ll add information so Vertical and Horizontal won’t contain additional information.
Suppose that in worlds represented by the diagram there are uncountably many material objects, that some of them are mereologically simple (without proper parts), and that each simple object occupies exactly one of the occupied spacetime points. Suppose also that for any collection of the simple objects there is a mereologically complex object that they compose. (In slogans, then, I’m saying that An empiricist theory of laws entails that at most one of Vertical and Horizontal describes a possible world. For the descriptions disagree about which axis is time, and so (on my view) disagree about the laws, but agree on the occurrent facts.
Empiricists should read the possibilities at work in my argument as epistemic, rather than metaphysical, possibilities.
four-dimensionalism and the doctrine of unrestricted mereological composition are true in these worlds.) Vertical and Horizontal seem to disagree about how many things there are and which regions those things occupy, but we can suppose that this is mere appearance. These descriptions are not complete; they don’t tell the stories of the spatiotemporal careers of every material object. Instead they only tell the stories of a few salient ones. When we move from Vertical to Horizontal we switch which axis we regard as the time axis; doing this brings about a shift in which of material objects are salient.
So far I’ve presented my proposal and given it some intuitive support. I will now discuss an objection to it.
7 Symmetric Laws and Indeterminacy The claim that geometry distinguishes timelike from spacelike directions ran into problems with spacetimes in which timelike and spacelike directions play symmetric roles. I said that there is still a diﬀerence between timelike and spacelike directions in some of those worlds, and that it is their diﬀerent roles in the laws which distinguishes them. But what about worlds in which timelike and spacelike directions play symmetric roles in both the geometry and the laws? There isn’t any reason to deny that such laws are possible; there are even examples of such laws.
The wave equation for a wave in one dimension, for example, is
(φ is a function on spacetime; it tells you, intuitively speaking, ‘how high’ the wave is at each point of spacetime.) Now v here is the speed of the wave, and we’re free to choose units in which it’s 1. In that case, the equation becomes
It is clear that in this law time and space play symmetric roles: switching t and x leaves the equation the same. Moreover, the roles time and space play in these laws both ﬁt the description I gave above: these laws govern the evolution of the wave both along the time axis and along the space axis.
Laws entails that all directions in this world are timelike, and that is not right.
There are three ways to respond to this problem. First, we could conclude that we do not yet have a complete account of what makes time diﬀerent from space, and we could search for some other feature of the world that is doing work to distinguish spacelike from timelike directions. Second, we could conclude that it is just a brute fact that the timelike directions in this world are timelike, that there is nothing informative to be said about what makes them timelike. Or third, (paralleling the move from Dimension to Dimension∗ ) we could conclude that if the geometry and laws fail to single out one kind of vector as the timelike vectors, then it is indeterminate which directions are timelike.
We should choose the third alternative. It does not seem that it could be just a brute fact which directions are timelike; and in these highly symmetric worlds it is hard to see what else, other than the geometry and the laws, could distinguish timelike from spacelike directions. Worlds governed by the wave equation look the same no matter which axis we regard as the time axis.
To deal with these symmetric worlds, then, amend Laws as follows:
Laws∗ : If a direction is timelike then the laws govern the evolution of the world along it; all vectors that point in timelike directions are of the same kind;
and to the extent that these conditions fail to determine which vectors are timelike, it is to that extent indeterminate which vectors are timelike.
Laws and Laws∗ agree on all worlds except worlds like the one governed by the onedimensional wave equation. In such worlds Laws∗ entails that it is indeterminate which kind of vector is timelike.
How bad is it to admit that it is sometimes indeterminate which directions are timelike? Certainly it’s perfectly determinate in our world which directions are timelike. Earlier I claimed that there are some two-dimensional worlds—complicated ones in which there is a lot going on—in which it is perfectly determinate which directions are timelike. But I don’t think that this must be perfectly determinate in every two-dimensional world, so I see no problem accepting that it is indeterminate which directions are timelike in worlds in which time and space play symmetric roles in both the laws and the geometry of spacetime.
8 Laws That Govern in a Spacelike Direction I turn now to a second objection. My view entails that it is not possible that there be laws that govern the evolution of the world is a spacelike direction. But (so the objection goes), surely this is possible. Surely we can produce examples of possible worlds with this feature.
I have yet to hear a convincing example. Let me go through some of the examples I have heard.14 Example 1: In this world, it is a law that to the left of every apple is an orange, and to the left of every orange is an apple. (Suppose we’ve ﬁxed, once and for all, which direction in space is to the left.) This law governs in a spacelike direction: if I know that there is an apple here, I get information about what is going on to the left.
Example 2: In this world, there is a special plane dividing space in half; and it is a law that the contents of space on one side of the plane are perfectly mirrored on the other side. This law governs in a spacelike direction: if I know that there is a red sphere in a certain place on one side of the plane, I get information about what is going on at the very same time in the corresponding place on the other side of the plane.
I am enough of a metaphysician to take examples like these somewhat seriously.
I have a three-part response. First, I don’t think the laws in these examples are doing enough to govern the evolution of the world in a spacelike direction. Given information about what is going on at one place for all time, they do not give information about what is going on everywhere else at all times. These laws only give I heard many of these examples at the APA Eastern Division meeting, 2004.
Among those who suggested examples are Eric Lormand, James Van Cleve, Philip Bricker, and Jonathan Schaﬀer. There were others, as well, and I can no longer remember who suggested which examples.
information about what is going on in some other places. (The law in example 1 gives information about places to the right of, and some of the places to the left of, the initial place. The law in example 2 gives information about just one other place for all time: the place that is the mirror-reﬂection of the initial place.) So they’re not doing in a spacelike direction what laws like Newtonian mechanics do in a timelike direction.
Second, even if there is a world in which it is true that to the left of every apple is an orange (and so on), and a world with mirror symmetry, I’m not sure why I should believe that it would be a law that to the left of every apple is an orange, or that the world exhibits mirror symmetry. (Certainly the law about apples couldn’t be a fundamental law, but maybe it could be derivative.) For I ﬁnd it hard to have intuitions about what the laws of some world are, given descriptions of the goings-on in those worlds.
Of course, some philosophers can argue that if there is a world in which to the left of every apple is an orange (and so on), and the world is simple enough in other ways, then it is a law that to the left of every apple is an orange. These are philosophers who accept the best system theory of laws. According to the best system theory of laws, those true statements are laws that are theorems of the deductive system that best balances simplicity and strength.15 If the apple world is simple enough in other ways, then the statement that to the left of every apple is an orange might make it in to the best system, and so might be a law.
Earlier I tried to remain neutral between primitivist theories of laws and empiricist theories of laws (like the best system theory). Now it appears that I must take sides. I reject the argument in the previous paragraph because I reject the best system theory of laws.
This is not just an ad hoc move, though. I do not just reject the theory because if conﬂicts with my theory of the diﬀerence between space and time. My primary reason for rejecting it is that it fails to respect our modal intuitions about lawhood.
Brieﬂy, empiricism about laws (and so the best system theory in particular) entails that there cannot be worlds that diﬀer merely in what laws govern them. But I accept (Lewis 1986b).
the counterexamples opponents of the best systems theory oﬀer to this claim.16 For example, it seems that there is a possible world in which an empty spacetime has a Minkowski geometry, and in which special relativity is true; and another world in which an empty spacetime has a Minkowski geometry, and in which general relativity is true. (In the latter world, had there been any matter, spacetime would have been curved; this is not true in the former world.) Empiricists about laws must say that at least one of these worlds is not possible.
There is a third part of my response to examples like 1 and 2. It is part of the description of the possible worlds in these examples that certain directions are spacelike and certain others, timelike. But doesn’t this beg the question? Why should we accept that the directions called ‘spacelike’ in the description really are spacelike?