Following Sklar, let us define substantivalism as that view that takes "spacetime to be an entity over and above the material inhabitants of the spacetime ...that could exist even were there no material inhabitants of the spacetime" (, p. 8). Rela-tionalism is the denial of this: what the substantivalist calls 'spacetime' is "nothing but a misleading way of representing the fact that there is ordinary matter and that there are spatiotemporal relations among material happenings" (ibid., p. 10). There are two important things to note about these definitions: (1) there is assumed a straightforward distinction between 'matter' and 'space'; (2) the distinction between the positions is grounded in a basic ontological priority claim involving matter or space.48 Let us try to firm up these definitions, by putting the previous chapter's notions of models and structures into play. I focus upon (2) next and then deal with (1) in the next subsection.
Let us assume that the structure S = (D, R) adequately models spacetime (or, if you like, just space).49 We can then get a reasonably clear and sharp distinction between substantivalism and relationalism by simply taking (D, R) to consist of a primitive set of space(time) points with a set of relations defined over them in the former case and to consist of a primitive set of material bodies with a set of relations defined over them in the latter case. Since both descriptions are assumed to model the same empirical structure, we can assume that the structures are isomorphic (with respect to purely empirical properties and relations); what differs is the domain of objects: spacetime points vs material objects (which we can assume are 'point-like').50 However, this equivalence aside, there is an important modal distinction to be made on the basis of these definitions. In the case of substantival-ism it is clear that the spatiotemporal structure of the world is not supervenient on the relations exemplified by material objects; it can exist independently of material objects thus implying the possibility of empty spaces (without change). By contrast, relationalism requires the existence of material objects in order for the spatiotemporal structure to be instantiated;51 there can be no actual empty spaces according to relationalism, whenever there are objects there is space and whenever
48 Auyang notes that "both substantivalism and relationalism presuppose that ...matter is somehow already differentiated into individually identifiable entities" (, p. 206). However, I don't think this is implied by the standard definitions, nor do I think that it is necessary; one can easily extend the definitions to include fields, for example. A problem does occur when the field is the metric field of general relativity, for then we have to make sense of its dual role in determining both gravitational and chronogeometric structure.
49 By "adequate" in this context I simply mean empirically adequate, in the sense that the phenomenal, objective facts of the spatiotemporal structure of the world are captured.
50 There is, of course, an immediate problem concerning regions of spacetime where there are no objects. Clearly, if there are less material objects than there are spacetime points then the structures cannot be isomorphic for the relation will not be one-to-one. If one invokes possibilia to function as places where an actual object might be, then one can restore the isomorphism, though the distinction between the two positions risks being collapsed. I discuss this matter further in §2.1.2.
51 Where spatiotemporal relations are external in Lewis's sense (cf. [1986a], p. 62); namely a relation that does not supervene on the intrinsic natures of relata.
there is space there are objects—let us call this the "material dependency thesis". This brings Sklar's slightly vague definitions in to better focus. Clearly this simple modal distinction between positions that results from the difference in primitive objects paves the way for other conceptual (modal and non-modal) distinctions. In the following sections we examine an important one that has to do with the way in which symmetries are accommodated in the respective interpretations. Before I get to that, I should first like to take a look at how these definitions fit in with spacetime theories in general and show how they face a number of problems in this context.
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