Sophisticated Substantivalism And Unsophisticated Relationalism

The two claims I wish to question are: (1) that the substantivalist is committed to taking I as representing possibility space; and (2) that the relationalist is committed to taking D as representing possibility space. This pair of claims are stated quite forcibly by Belot. Thus, he writes (modified to suit the example of the previous section):

I require substantivalists to maintain that there are a large number of such embeddings [of point particles in Euclidean space, with their relative distances fixed]: place the [three] points as you like; you generate a distinct possible embedding by acting upon this first one by any Euclidean [symmetry]. On the other hand, I require relationalists to maintain that there is a single such embedding. ([Belot, 2000], pp. 276-7)

Thus, Belot is quite explicit that it is possibility counting that distinguishes sub-stantivalists and relationalists; he sees the difference at this level as concomitant with the differences concerning the material dependency thesis that I mentioned earlier. He explicitly ties this to the geometric spaces I presented in the previous section:

Whenever we compare two spaces of possible worlds, one the quotient of the other, we are contrasting two ways of counting possibilities [namely] a relatively haecceitistic means of counting possibilities and a relatively anti-haecceitistic means of counting possibilities. [.] [T]he existence of spacetime

70 The property variable F ranges over all qualitative monadic and relational properties, and the arguments a and b over a domain of objects.

71 At least it does in Leibniz's hands. But see Saunders [2003b; 2003a] for a non-reductive (that is, non-eliminativist) version of the principle. Indeed, Saunders' main thesis in these two works is that relationalism does not imply reduction. However, I think that the position that Saunders ends up with is more like substantivalism than relationalism, for the claim is that relational properties can serve to individuate (absolutely indiscernible) objects; this includes the relations spacetime points bear to one another, hence empty spacetimes are possibilities on this account (see §5.3.3). Also, as I mentioned in the introduction, reduction (elimination) is still implemented at the level of possible worlds, for the simple reason that there are no physical relations between such objects that could serve to individuate them (see §8.3).

points is closely tied up with questions of counting of possibilities—so they are vulnerable to elimination in the transition from a haecceitistic means of counting to an anti-haecceitistic one. ([Belot, 2003a], p. 410)

Thus, Belot essentially argues that one's choice of geometric space is instrumental in one's spacetime ontology. Let me begin by outlining an argument that has been adduced to disprove (1).72

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