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

Field also tries to motivate something like Purity by noting that a theory incompatible with Purity does not give intrinsic explanations of phenomena (1989, pages 192-193). I take it that an intrinsic explanation is one that appeals only to the intrinsic properties of and relations among the things invoked in the explanation; so my focus on the intrinsicness of the way the world’s particles are arranged is not, at bottom, diﬀerent from Field’s focus on intrinsic explanations. But Field just takes it for granted that (P1) is true. I have found many people willing to deny it; I provide an argument for it.

Substantivalists should believe that properties characterizing the geometrical structure of spacetime (like the property of having a Newtonian geometrical structure) are intrinsic. Arguments similar to the ones I give below show that if Purity is false, then these properties are not intrinsic.

6.1 Clarifying the Premises I use ‘the way the world’s particles are spatially arranged’ as a name of a relation.

I take it that talk of the way some particles are spatially arranged is familiar: one way for three particles to be spatially arranged, for example, is for any two of them to be as far apart as any other two. Then they stand at the vertices of an equilateral triangle. In this case, then, ‘the way the particles are spatially arranged’ names the three-place relation x and y are as far apart as y and z, and x and y are as far apart as x and z. (P2) is the claim that relations like this are intrinsic. But what does it mean to say that a relation is intrinsic?

We are familiar with the distinction between intrinsic and extrinsic properties.

Intuitively, an intrinsic property is a property that characterizes something as it is in itself. What intrinsic properties something has in no way depends on what other things exist (things other than it or its parts) or how it is related to them. With extrinsic properties (properties that are not intrinsic), by contrast, other things can ‘get in on the act’ when it comes to determining whether something instantiates them.

Although it is unfamiliar, the notion of an intrinsic relation is a straightforward generalization of the notion of an intrinsic property. Just as an intrinsic property characterizes something as it is in itself, an intrinsic relation (with more than one argument place) characterizes some things as they are in themselves. For example, x is as massive as y is intrinsic (or, at least, seems intrinsic at ﬁrst): if two things are equally massive, that is a matter of how those two things are in themselves. Whether they instantiate x is as massive as y does not depend on what else there is or what those other things are like.20 If x is as massive as y is intrinsic, then it is plausible that it is also internal.

Internal relations supervene on the intrinsic properties of their relata. More formally, a relation R is internal just in case: if x and y instantiate R and x and y are duplicates of x and y (respectively) then x and y also instantiate R. (x and y need not exist in the same possible world as x and y. The generalization to relations with more than two argument places is straightforward.) All internal relations are intrinsic, but not all intrinsic relations are internal. (Fundamental relations (with more than one argument place) are intrinsic but not internal. Some deﬁne ‘external relation’ as ‘relation that is intrinsic but not internal.’ (Lewis (1983) for example.) Here is another way to think about intrinsic relations. Intrinsic relations usually correspond to intrinsic properties of fusions. Consider the property x has two parts that are equally massive. This property is intrinsic. Any intrinsic duplicate of something with two equally massive parts would itself have two equally massive parts. And something instantiates this property just in case it has two parts that instantiate x and y are equally massive. And the latter (as I said) is an intrinsic relation. In general, if an n-place relation R is intrinsic then the property of having n parts that instantiate R is also intrinsic.21 Using the distinction between fundamental and non-fundamental relations I can give a more precise characterization of ‘intrinsic relation.’ All relations supervene on the fundamental relations. But this is global supervenience: if some things instantiate a relation R, they do so in virtue of the global pattern of instantiation of the fundamental relations. An intrinsic relation, by contrast, supervenes ‘locally’ on the fundamental relations. That is, a relation is intrinsic only if it supervenes on the fundamental properties of, and fundamental relations among, its relata and its relata’s parts.

Local supervenience is necessary for a property to be intrinsic. But if there are any things that exist necessarily it is not suﬃcient. Suppose God exists necessarily;

then the property of coexisting with God supervenes on any set of properties. But it is not intrinsic.

The solution is to demand more: a relation Rx1...xn is intrinsic just in case it can be analyzed in terms of the fundamental relations x1...xn and their parts instantiate. An analysis is a kind of ‘deﬁnition’ of the non-fundamental relation: to give an analysis is to display an open sentence that expresses that relation (and so has as many free variables as that relation has argument places) such that every predicate in that open sentence expresses a fundamental relation. We can tell whether a relation is intrinsic by looking at its analysis: Rx1...xn is intrinsic just in case every The distinction between internal and non-internal relations will not play a role in my arguments.

I believe the converse is true for fundamental relations, but can fail for nonfundamental relations. The (non-fundamental) relation ∃z (x and y compose z) is an example: it is not an intrinsic relation, but the property of having two parts that compose something is an intrinsic property.

quantiﬁer in its analysis is restricted to x1...xn and their parts.22

**6.2 Defending the Premises**

I have now explained what the premises mean, and given a more precise characterization of ‘intrinsic relation.’ I turn now to arguing that relationalists should accept the two premises.

Here is my argument for (P1). Suppose Purity is false. Suppose in addition that there is just one fundamental spatial relation, the distance from x to y is r. Then the way the world’s particles are spatially arranged may be analyzed in terms of this relation.

If this is true, then intuitively, the particles get to be spatially arranged in a certain way in virtue of being related in the right way to numbers. So their being arranged in that way is not a matter of the way they are, in themselves; so the way they are arranged is not intrinsic.

To go through this argument in more detail, ﬁx on a particular way the particles might be arranged, and look at its analysis. Suppose there are only three

**particles and that at a certain time they stand at the vertices of an equilateral triangle. Then the way they are arranged may be analyzed as follows:**

∃r(the distance from x to y is r & the distance from y to z is r & the distance from z to x is r) This analysis contains a quantiﬁer that ranges over something other than x, y, z, and their parts; namely, a quantiﬁer that ranges over numbers. So the relation analyzed is not intrinsic.

Someone might object that numbers (and abstract objects generally) should receive a special dispensation in the deﬁnition of ‘intrinsic.’ That deﬁnition bans The deﬁnition of ‘intrinsic’ I am working with is related to the one given by David Lewis in On the Plurality of Worlds (1986b). It follows from Lewis’s deﬁnition that the fundamental properties and relations are intrinsic. So this deﬁnition is appealing only if we accept this consequence. But I don’t think this consequence should be controversial. It is part of our conception of fundamental properties that they are intrinsic: it is usually said that the fundamental properties make for similarity among their instances, and that they carve nature at its joints (Lewis 1983).

quantiﬁers not restricted to the parts of the things instantiating the relation from analyses of intrinsic relations. But perhaps it should allow quantiﬁers that range over only abstract objects into such analyses; so long as no quantiﬁer ranges over concrete things other than the parts of the things instantiating the relation, the relation is intrinsic. (The above analysis can easily be rewritten so that the ﬁrst quantier is restricted to numbers.) We should reject this revised deﬁnition of ‘intrinsic.’ To illustrate why it is wrong, notice that it leads to incorrect results about the intrinsic properties of numbers themselves. If we suppose that numbers exist, then it is plausible to suppose that x is the successor of y is a fundamental relation that natural numbers instantiate.

(Other relations among the natural numbers—like addition and multiplication—can be deﬁned in terms of successor.) Now the property of being the smallest natural number, like the property of being the shortest person in the room, is not intrinsic.

Whether a number has this property depends on what other numbers there are, and whether it is the successor of any of them. But this property’s analysis—‘¬∃y(y is a number and x is the successor of y)’—contains just one quantiﬁer restricted to numbers, and so the revised deﬁnition of ‘intrinsic’ says that it is intrinsic.

In this argument for (P1) I’ve assumed that if Purity is false then the distance from x to y is r is the fundamental spatial relation. It should be clear from my discussion that nothing turns on using this particular mixed relation. The argument works equally well no matter which mixed relation we choose. This is important because, if one were going to chose a mixed spatial relation to regard as a fundamental relation, one would not be likely to choose the distance from x to y is r. It has seemed to many that this relation isn’t really fundamental, but is analyzed in terms of some other fundamental mixed spatial relations. For example, some have thought that the fundamental spatial relation concerns distance ratios, rather than distances. That is, there is just one ﬁve-place fundamental spatial relation, x and y are r times as far apart as z and w. Distances between points are then analyzed in terms of the ratios of their distances to the distance between two reference objects (say, the ends of the standard meter). This theory is appealing because is does away with seeminglymysterious facts about which things are distance one unit apart. (We are very good at determining when two things are one meter apart, and when they are one foot apart; but these facts look like facts about the ratio between the distance between the two things in question and the distance between the endpoints of the standard meter, or the standard foot. Asked to determine whether two things are one unit apart, in some absolute sense that is independent of any standard of measurement, and we do not know what to do.)23 But for all this view’s virtues, my argument for (P1) works just as well if it (or some other theory of mixed fundamental spatial relations) is true.

So much for (P1). What about (P2)? Not everyone will accept (P2). Some substantivalists, in particular, will reject it. For some substantivalists say that the way the world’s particles are spatially arranged is derived from facts about where in space each particle is located.24 The way the particles are arranged, then, is not just a matter of how those particles are, in and of themselves, but is also a matter of the way they are related to space. So their spatial arrangement is not intrinsic.

I claim that relationalists should accept (P2). For (P2) is, I think, one of the claims that motivates relationalism in the ﬁrst place. Distinguish two motivations for relationalism. (There may be others.) One way to motivate relationalism is to complain that points of space and instants of time are unobservable entities, and assert that for that reason we should not believe in them. But another (and better) way to motivate relationalism is to complain that space and time are irrelevant and unnecessary. As I said in the previous paragraph, one role that space plays for substantivalists is to ‘ground’ spatial relations among material objects. I may be a mile above Los Angeles; according to substantivalists, what it is for me to be a mile above Los Angeles is for me to be located at a certain region of space, and Los Angeles to be located at a certain region of space, and for those regions to be a mile apart. That is, substantivalists say that material objects inherit their spatial relations from the spatial relations among the regions of space they occupy. But this detour through spatial relations among regions of space looks unnecessary. Why can’t the After reading Kripke (1980) you might doubt that ‘x and y are one meter apart’ expresses the relation x and y are as far apart as a and b where a and b are the end points of the standard meter. I think the view that best ﬁts with Kripke’s picture is account 2 (section 7 below; see also footnote 25).

Not all substantivalists say this: supersubstantivalists, for example, do not.

Supersubstantivalism is the subject of chapter 3.

‘one mile apart’ relation hold directly between me and Los Angeles? Why does it need to be derived from spatial relations among regions of space? And if it does not need to be so derived, then space is irrelevant. The way the world’s particles are spatially arranged is not a matter of the way they are embedded in space; instead, it is a matter of how those particles are, in and of themselves. It is intrinsic to the

**particles. Here, for example, is Barbour expressing this motivation:**

What is the reality of the universe? It is that in any instant the objects in it have some relative arrangement. If just three objects exist, they form a triangle. In one instant the universe forms one triangle, in a diﬀerent instant another. What is to be gained by supposing that either triangle is placed in invisible space? (1999, pp.68-69).