# «Once Upon a Spacetime by Bradford Skow A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy ...»

**objects, or only concrete objects—are unmixed, or pure. The constraint is this:**

Purity: Necessarily, the fundamental spatiotemporal relations are pure.8 Here and throughout I limit this claim to qualitative, non-modal relations.

Again, I am working in the context of pre-relativistic physics. But the arguments generalize to relativistic spacetime structures.

If one believes that all possible relations exist necessarily and that necessarily, if a relation is fundamental then it is necessarily fundamental, then the modal operator at the beginning of Purity is redundant. In what follows I will make these assumptions.

Purity presupposes that some fundamental relations are spatiotemporal relations.

Leibniz denied this; resemblance nominalists (who think that the fundamental relations are relations of resemblance) presumably deny this; some advocates of string theory or of some other theory of quantum gravity may deny this. My arguments need not presuppose that such views are false. Even if no fundamental relation is Here is an outline of the rest of this chapter. In section 3 I state a necessary condition on relationalist characterizations of the spatiotemporal structure of the world, and in section 4 I explain in detail what the spatiotemporal structure that corresponds to relationalism about motion looks like and how it diﬀers from Newtonian and neo-Newtonian spacetime. In section 5 I look at the most obvious relationalist characterization of this spatiotemporal structure and reject it because it is inconsistent with Purity. I argue for Purity in section 6, and in the remainder of the chapter I look at relationalist accounts of the fundamental spatiotemporal relations that are compatible with Purity. I argue that they are unacceptable.9

**3 The Embeddability Criterion**

Suppose a relationalist answers (3) by saying that the world has a Newtonian spatiotemporal structure—not a likely answer, but this is just an example—and that he tells us a story about the fundamental facts in virtue of which it has this structure (a story that, as I said, will include a list of the fundamental spatiotemporal relations that material objects instantiate and laws that those relations obey). I claim that this relationalist story is correct only if it guarantees that relationalist worlds in which it is true are uniquely embeddable in Newtonian spacetime, up to the symmetries of that spacetime. (And similarly for other possible spatiotemporal structures the world might have.) More precisely: let R4 with the spatial distance between any two points (x1, x2, x3, x4 ) and (y1, y2, y3, y4 ) given by the standard Euclidean formula d s (x, y) = (x2 − y2 )2 + (x3 − y3 )2 + (x4 − y4 )2 and temporal distance given by dt (x, y) = |x1 − y1 | be our canonical model of Newtonian spacetime geometry.

Then the relationalist laws are correct only if there is a one-to-one function f from spatiotemporal, still (I take it) there are some spatiotemporal relations that are most fundamental: no spatiotemporal relation is more fundamental than they are. Purity may then be cast as a principle about the most fundamental spatiotemporal relations. In the body of this paper I will ignore this complication, since nothing turns on it.

The argument I give here is similar to the one Field gives in ‘Can We Dispense with Spacetime?’ (1989). While I think that much of what I say is in the spirit of the arguments Field gives, there are important diﬀerences between our arguments.

I will mention these diﬀerences in the relevant places.

the pointsized things in the relationalist world into R4 such that

• f preserves spatial and temporal distance: for any pointsized instantaneous10 things x and y in the relationalist world, d s (x, y) = d s ( f (x), f (y)) and dt (x, y) = dt ( f (x), f (y)), and

• every other such function diﬀers from f by a symmetry of Newtonian spacetime.

(I use ‘d s ’ to name both the spatial distance function deﬁned on the material objects in the relationalist world and the spatial distance function deﬁned on points of R4.) The symmetries of Newtonian spacetime are the one-to-one functions from spacetime onto itself that preserve spatial and temporal distance. There are similar deﬁnitions of symmetry and embeddability for other spacetimes. (Not all of these deﬁnitions need be given in terms of a spatial and temporal distance function.) Of course, the relationalist’s story may not explicitly mention a spatial or temporal distance function. But these functions must be deﬁnable from the story he does tell, for certainly there are facts about the spatial and temporal separation of any two pointsized instantaneous objects in a Newtonian relationalist world (given a choice of units of measurement), whether or not those facts are fundamental.11 The embeddability criterion is accepted by many who discuss the debate between relationalists and substantivalists.12 I take it that it needs little defense.

Clearly, if a relationalist says that a world has a Newtonian spatiotemporal structure, but there is no way to embed that world in Newtonian spacetime, then he is Relationalism is easiest to defend if is conjoined with the doctrine that every material object is composed of pointsized instantaneous parts. (So this doctrine entails the doctrine of temporal parts.) If this doctrine is true, then to specify the spatiotemporal relations among the material objects in the world it is enough to specify the spatiotemporal relations among all the pointsized instantaneous material objects.

In fact it is possible to formulate the arguments in this paper in terms of the deﬁnability of a Euclidean spatial distance function in relationalist worlds, instead of in terms of the embeddability of (parts of) relationalist worlds into Euclidean space. I think framing the arguments in terms of embeddability makes the issues clearer.

(Belot 2000), (Earman 1989), and (Friedman 1983) are three examples.

lying. And if the embedding is not unique up to a symmetry of the spacetime, then the relationalist’s story underdetermines the spatial relations among the material objects. (For example, if the relationalist says that the world has a Euclidean spatial structure, but there are three instantaneous pointsized material objects, all simultaneous, and there are two embeddings of them into Euclidean space, one which maps them to the vertices of an equilateral triangle, and one which maps them to the vertices of a triangle that is not equilateral, then the relationalist has not adequately characterized the spatial structure of the world.) Embeddability is a necessary condition that a relationalist characterization of the spatiotemporal structure of the world must meet. But it is not suﬃcient. There are two commonly cited obstacles to its suﬃciency.13 First, any relationalist world that is embeddable in a four-dimensional Newtonian spacetime is also embeddable in a ﬁve-dimensional (or higher) Newtonian spacetime with an extra spatial dimension. The embeddability criterion alone does not ﬁx the dimensionality of space.

Second, any relationalist world with (as a substantivalist would say) large enough empty regions of spacetime that is embeddable in a four-dimensional Newtonian spacetime is also embeddable in a four-dimensional spacetime in which space is Euclidean in all the occupied regions, but is non-Euclidean in some of the empty regions. So the embeddability criterion alone does not always ﬁx the geometry of unoccupied regions of spacetime.

I mention these problems with treating the embeddability criterion as suﬃcient only to set them aside. I will only appeal to its status as a necessary condition.

**4 Machian Spacetime**

To say that all motion is relative motion of bodies is to say something about the spatiotemporal structure of the world. I can now be more speciﬁc about just what this structure looks like. (To do so I speak as a substantivalist.) Earman (1989) lists six possible (non-relativistic) spacetimes and orders them by how much structure they have. All the spacetimes Earman lists consist of a stack of three-dimensional Euclidean instantaneous spaces. The order in which they are stacked gives their (Earman 1989).

ordering in time. Machian spacetime is the weakest of these: all there is to that spacetime is a stack of three-dimensional Euclidean instantaneous spaces. We can say how far apart in space points on the same instantaneous space are, but not how far apart in space or in time points on diﬀerent instantaneous spaces are. So in Machian spacetime the only meaningful questions are questions about how far apart particles are at any instant, and qualitative (not quantitative) questions about their relative motion: whether two particles are getting closer together, for example. Machian spacetime, then, is a spacetime in which all motion is the relative motion of bodies.14 Barbour designed his relationalist replacement for Newtonian mechanics to live in a Machian world. (By contrast, Newtonian spacetime has more structure: it comes with a way to determine how far apart in both space and time points on diﬀerent instantaneous spaces are. In this spacetime, we can tell which trajectories through spacetime are trajectories of particles at absolute rest, and we can ask whether a single particle is in absolute motion, and if so what its numerical speed is.) Can relationalists say what it is for the world to have a Machian spatiotemporal structure?

To guarantee the embeddability of the world’s particles into Machian spacetime relationalists must guarantee that each ‘time-slice’ of the world (each equivalence class of the world’s pointsized instantaneous material objects under the simultaneity relation) is embeddable in three-dimensional Euclidean space. It might seem at ﬁrst like this is easy to do: all a relationalist needs to do specify the spatial distances between the particles in that time-slice, and ensure that those distances obey Euclidean laws. Then the time-slice will be uniquely embeddable into Euclidean space.

But what are the fundamental spatial relations that give the distances between the particle-slices? There are several choices here, and the problems relationalists face do not come out until we are more clear about which choice we have in mind.

It is not the only such spacetime; Leibnizian spacetime, which diﬀers from Machian spacetime only by having a temporal metric, is also a spacetime in which all motion is the relative motion of bodies.

** 5 Account 1: One Mixed Fundamental Distance Relation**

One account of the fundamental spatial relations a relationalist might use to account for the distances between particles is this: say that there is one fundamental spatial relation, the distance from x to y is r. Then state laws for this relation that guarantee the embeddability (up to uniqueness) into Euclidean space of the particle-slices instantiating it. I suspect that when relationalists think that a Machian spatiotemporal structure is perfectly acceptable they do so because they think that this is all that need be done.

**The problem with this account is that it is incompatible with Purity:**

Purity: Necessarily, the fundamental spatiotemporal relations are pure.

It is now time to argue for this principle.

6 Arguing for Purity There is one group of philosophers who will ﬁnd Purity appealing: nominalists, those who deny that there are any abstract objects.15 For if Purity is false, because (say) the distance from x to y is r is the only fundamental spatial relation, and if in addition there are no numbers, then no spatial relations are instantiated, and so the world is not spatial at all. Since nominalists do not think their view entails that spatiality is an illusion, they will embrace Purity.

But I think that even anti-nominalists should accept Purity. I myself ﬁnd the following argument convincing: even granting that there are numbers, the following

**counterfactual is true:**

(4) If there were no numbers, the world would still be spatial, and in fact the spatial structure of the world would be just as it actually is.

But if Purity is false because the distance from x to y is r is the only fundamental spatial relation, then (as before) if there were no numbers no spatial relations would Some nominalists deny that there are any properties or relations. They will deny that Purity is literally true. But nominalists should have some ﬁctionalist reading of sentences containing property talk, and they will accept Purity when read that way.

be instantiated. And any world in which no spatial relations are instantiated is a world that is not spatial at all. And that means that (4) is false.16 Many anti-nominalists will be unconvinced by this argument, though. They will say that numbers exist necessarily, and so that (4) is a counterfactual with a necessarily false antecedent. So either (4) makes no sense at all, or it is vacuously true; in either case, there is no problem for an anti-nominalist who accepts Purity.

**So I will take a diﬀerent approach. My argument for Purity is this:**

(P1) If Purity is false, then the way the world’s particles are spatially arranged at any one time is not intrinsic to the particles.

(P2) The way the world’s particles are spatially arranged at any one time is intrinsic to the particles.17 (C) So Purity is true.18 This argument is addressed to relationalists. I mean to convince relationalists to accept its conclusion. (I think substantivalists should also accept its conclusion, but for diﬀerent reasons.19 ) I do not claim that its premises are true (its second premise in particular), just that relationalists should believe them. Before I explain why, I will clarify what the premises mean.

If Purity is false because some, but not all, fundamental spatiotemporal relations are mixed, then it does not follow that if there were no numbers the world would not be spatial. But in this case (4) is still false: if there were no numbers then the spatial structure of the world would not be just as it actually is.

From now on I omit the reference to time.