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

(Not for nothing does Barbour call his theory ‘intrinsic particle dynamics.’) I think this, and not appeals to problems with unobservable entities, is the best motivation for relationalism. It is a motivation that I myself feel, even though I am no relationalist. But—what is important for my purposes—this line of thought motivates relationalism by appealing to (P2). So relationalists should accept (P2).

There is, of course, another well-known motivation for relationalism: the Leibniz shift argument. But I do not think that this motivation is very good. (I discuss this argument, and it successor the hole argument, in more detail in the next chapter.) I have argued that everyone should accept (P1) and relationalists should accept (P2). So relationalists should accept (C): they should accept Purity.

Let me be clear about what this means. If numbers do exist, then material objects surely do bear mixed relations like the distance from x to y is r to numbers.

I only claim that it cannot be relations like these that give the world its spatial structure. When it comes to the fundamental facts about spatial structure, numbers are strictly irrelevant.

**7 Account 2: Pure Distance Relations**

Relationalists cannot characterize the spatial structure of the world by saying that the three-place spatial relation the distance from x to y is r is fundamental. Here is an alternative theory: say that there are inﬁnitely many two-place fundamental spatial relations. These relations have names like ‘the distance from x to y is one,’ ‘the distance from x to y is two,’ and so on. But we should not be misled by the names: although one relation’s name contains the letters ‘o-n-e,’ we are not to think of these as a separate syntactic unit that serves to name a number. We are not to think of these pure relations as derived from the three-place relation the distance from x to y is r by ﬁlling in its third argument place with a number.25 If these relations are fundamental then the way the world’s particles are spatially arranged is intrinsic. So this theory is compatible with Purity and respects the motivation for relationalism. And so long as the inter-particle distances do not violate Euclidean laws, the world’s time-slices will be uniquely embeddable into Euclidean space.

The problem with this theory is that it just cannot be true. For according to this theory there are necessary truths that we must think are brute and inexplicable, but which we ought to be able to explain. For example, it is necessary that if two things instantiate the distance from x to y is one then they do not also instantiate the distance from x to y is two. And not only is this necessary; each distance relation excludes the other distance relations in this way. Or again, it is necessary that if a and b instantiate the distance from x to y is two and b and c also instantiate the distance from x to y is two then a and c do not instantiate the distance from x to y This seems to be the view Melia prefers at the end of his (Melia 1998). It also ﬁts best with Kripke’s discussion of the standard meter (1980). Kripke suggests that we refer to the standard meter stick only to ﬁx the reference of ‘one meter apart.’ Two things can be one meter apart without bearing any relation to the endpoints of the standard meter; they can be one meter apart even in possible worlds in which the standard meter stick does not exist. Account 2 provides the right sorts of distance relations to be the semantic values of expressions like ‘one meter apart,’ as Kripke understands them. So does a version of account 1: if the fundamental spatial relation is the three-place relation the distance from x to y is r then ‘one meter apart’ expresses the derived two-place relation the distance from x to y is n for some number n. But as I mentioned above, while on this second interpretation we can (if asked) tell which things bear the distance from x to y is r to the same number that the endpoints of the standard meter actually do, if asked which things bear this relation to the number one, we have no idea how to ﬁnd out the answer.

is ﬁve. And not only is this necessary; other instances of the triangle inequality are true as well.

The problem is not that these necessary truths are not logical necessities. I do not claim that all necessities involving fundamental relations are logical (as some versions of the combinatorial theory of possibility do). The problem is that these necessities exhibit a striking pattern, and we ought to be able to explain why they exhibit this pattern. But we cannot.26 This argument generalizes to other attempts to turn theories of mixed fundamental spatial relations into theories of pure fundamental spatial relations by substituting inﬁnite families of pure n − 1 place relations for a single n place mixed relation. For example, a theory that says that relations like x and y are two times as far apart as z and w are fundamental fails for the same reasons.

**8 Purity and Substantivalism**

So far I have been using Purity to beat up on relationalists. But (as I remarked in footnote 19), substantivalists should also accept Purity. It may fairly be demanded whether and how substantivalists can produce a theory of the fundamental spatial relations that is compatible with Purity.

Substantivalists have no problem doing this. There is a synthetic axiomatization of Euclidean geometry using just two primitive predicates of points of space, ‘x, y are congruent to z, w’ and ‘x is between y and z.’ (Actually there is one such axiomatization for two-dimensional Euclidean geometry, and one for threedimensional Euclidean geometry, and so on; let’s stick to the three-dimensional Field argues that we would not want to accept a physical theory that used twoplace predicates like ‘the distance from x to y is one’ as semantic primitives, because it would be unlearnable (since it contains inﬁnitely many primitive predicates) and unusable (since it contains inﬁnitely many axioms that cannot be captured in a ﬁnite number of axiom schemas). But this is not enough to show that we cannot accept that two-place relations like the distance from x to y is one are fundamental: for the proponent of this metaphysical view may deny that the physical theory scientists learn and calculate with needs to have semantic primitives that express fundamental relations.

case.) These two predicates clearly express pure spatial relations.27 Field (1980) has produced a similar synthetic axiomatization of the geometry of neo-Newtonian spacetime. A substantivalist in search of a Purity-friendly account of which relations are fundamental will be attracted to synthetic axiomatizations of geometry like this. He need only accept that these synthetic axioms are true of spacetime and make the further claim that the predicates which appear as semantic primitives in the theory express fundamental relations.28 But this account of the fundamental spatiotemporal relations is not available to relationalists. The ﬁrst problem is that it is not obvious what laws the relationalist will propose for betweenness and congruence. He cannot use the same laws that substantivalists use. Those laws entail, among other things, that there are inﬁnitely many points of space; so if a relationalist accepted them and rewrote them to mention particles rather than points of space, he would have to accept that if the world has a Euclidean spatial structure then there are inﬁnitely many particles. And no relationalist wants to accept that.

Relationalists could weaken the laws, only requiring that no set of particles instantiates (at a time) betweenness and congruence in a pattern that is contrary to the substantivalist laws. This law amounts to a guarantee that there is at least one embedding of any time-slice of a relationalist world into Euclidean space. (Whether this law can be stated directly, without mentioning embeddings, is a good question, but I set it aside.) The problem now is that there are (in general) too many embeddings of the particles into abstract Euclidean space. Here is an example.

Suppose that relationalism is true and that there are only three particles A, B There is a theorem that establishes that every model of these synthetic axioms is isomorphic to R3 with a Euclidean geometry. This theorem is only true for the second-order axiomatization; Tarski sketches a proof of a similar result for the ﬁrstorder theory in (Tarski 1959).

And this is an additional claim. If he does not make it, then it is consistent with all that has been said that the unmixed relation expressed by ‘x, y are congruent with z, w’ is analyzed in terms of mixed fundamental relations. (Perhaps it is analyzed as there is an r such that the distance from x to y is r and the distance from z to w is r.) But then Purity is false.

and C, and the facts about the instantiation of betweenness and congruence at a time (other than the trivial ones29 ) are as follows: betweenness is nowhere instantiated;

congruence is nowhere instantiated. Any embedding of this relationalist world into R3 maps these particles to the vertices of a triangle that is not an isosceles or an equilateral triangle; and for any such triangle there is an embedding that maps the particles to the vertices of a triangle like that one. So, for example, there is an embedding of the particles that maps them to the vertices of triangle displayed in ﬁgure 1.1 on the left; and also one that maps them to the vertices of the triangle displayed below on the right. Clearly, no isometry of Euclidean space maps one of

** Figure 1.1:**

these triangles onto the other. So on this version of relationalism, the embedding of this world into Euclidean space is not unique up to isometry. And this is not a problem with this particular choice of pure fundamental spatial relations; the same problem will arise if relationalists use some other choice of primitives for synthetic geometry as a guide to which spatial relations are fundamental.30

**9 Modal Relationalism**

The betweenness and congruence theory has a lot going for it. It escapes the problems that face both of the previous accounts I discussed. Why not use modality to The trivial facts are those like A is between A and A and A, B are congruent to A, B, the universal generalizations of which are theorems.

Other choices of primitives are: congruence alone; the three-place predicate ‘x, y, and z form a right triangle at y’; and the three-place predicate ‘the distance from x to y is less than or equal to the distance from y to z.’ For each choice there are distinct spatial arrangements of three particles that are isomorphic with respect to the primitive predicates. (Royden 1959) surveys choices of primitives for synthetic geometry.

patch up the relationalist version of the betweenness and congruence theory and secure the unique embeddability of time-slices into Euclidean space? After all, some relationalists characterize the view in modal terms to begin with: Leibniz, the archrelationalist, wrote: ‘space denotes, in terms of possibility, an order of things that exist at the same time’ (Leibniz and Clarke 2000, page 14).

The idea is to deﬁne distance relations in terms of betweenness, congruence, and a modal operator, and to ensure that these distance relations obey Euclidean laws. Then the world’s time-slices will be uniquely embeddable into Euclidean space.

One way to do this is to look at how a substantivalist who accepts the betweenness and congruence theory can deﬁne four-place pure distance-ratio relations like x and y are twice as far apart as z and w, and try to convert those into modal deﬁnitions. Substantivalists can deﬁne this relation in terms of betweenness and

**congruence as:**

There is an analogous deﬁnition (R3 ) for x and y are three times as far apart as z and w, and so on.

There are two related problems with this approach.

First, there are problems with the meaning of the modal operator. It cannot express logical or metaphysical or physical possibility. It is logically and metaphysically and physically possible for the distance ratio between two pairs of particles to be anything you please. If the modal operator expresses logical or metaphysical or physical possibility, then, two given pairs of particles will satisfy both (R2 ) and (R3 ). But there must be a unique distance ratio between the pairs of particles.

It cannot express metaphysical possibility restricted to worlds in which the non-modal spatial relations among the four points are as they actually are. Fixing the betweenness and congruence relations ﬁxes all the non-modal spatial relations.

Suppose there are only four particles and that no particle is between any of the others and no two are as far apart as any other two. It is (metaphysically) consistent with this description that the ﬁrst two particles are twice as far apart as the second two. It is also (metaphysically) consistent with this description that the ﬁrst two particles are three times as far apart as the second two. So the four particles again instantiate both (R2 ) and (R3 ), which they should not.31 Relationalists can say that the modal operator expresses ‘geometrical possibility.’32 But what is geometrical possibility? It must be some restriction of metaphysical possibility. I’ve looked at three restrictions; none of them work. I can think of no other candidate descriptions of this restriction. We may wonder whether we understand what this operator means at all.

There are also problems with necessary connections. I rejected account 2 because it asked us to accept necessary truths involving distance relations as brute and inexplicable. This account faces the same problem. Why is it that four particles can only instantiate one of (R2 ), (R3 ), (R4 ), and so on? We cannot derive these necessary truths from the way the modal operator used is analyzed, since it has no analysis.

**The following is the best way for relationalists to respond to these problems:**

I admit that I cannot ‘deﬁne’ the geometrical possibility operator by ﬁnding some sentence and then saying that the geometrical possibility operator acts like a quantiﬁer over just those metaphysically possible worlds in which that sentence is true. But I can tell you something about which worlds (in addition to the actual world itself) are geometrically possible relative to the actual world.