The model-theoretic formalism makes the distinctions between the various theories of spacetime particularly easy to define and comprehend. It therefore makes a very useful tool for philosophers of physics interested in spacetime. The basic idea is to split up the elements of a theory of spacetime into various classes of entity: at the most fundamental level we have the spacetime M represented by a differentiate manifold.137 Then over M there are a number of geometric-object fields, relating the points of M in various ways, and assigning properties to them. The geometric-object fields come in two types: 'background' Bi and 'dynamical' Vj. The former are taken to characterize the 'fixed' structure of spacetime, and the latter are taken to characterize the physical contents of spacetime. These object fields define a set of relations on M. Hence, a complete spacetime theory is characterized by a model of the form: M = (M, Bí, Vj). In order to represent a dynamically possible spacetime of a particular theory T, M must satisfy the equations (laws) of T.138 This latter connection highlights the view that the set of models of a theory represent the possibility space of a theory (the kinematically possible histories), and the laws select a subset of this space comprising the physically possible worlds (or 'T-worlds') of that theory (the dynamically possible histories).139 Hence, a model of a spacetime theory would represent a world with a certain spatiotemporal structure as given by the Bi-fields, with Vj-fields distributed over the domain subject to the theory's laws that serve to relate the fields. The interesting fact about general relativity is that its models contain only Vj-fields, so that the spatiotemporal relations over M are always given dynamically, by solving the equations of motion. This is what is meant by background independence: the equations of motion are of the form F[Vi] = 0.
Earman and Norton couch their argument in terms of models, where the models are intended to represent the physically possible worlds of the theory, in the manner I suggested above. Certainly, the responses see it like this, and most physicists seem to view the models in this fashion. How is this association to be understood? It is simply an interpretation of the model: a specification of the possible worlds of the theory; the players (domain) and their roles (properties and relations), and the lawlike constraints they are subject to (equations of motion). So, for a model of general relativity (M,g, T), the domain M is taken to represent the spacetime points (along with absolute properties and relations determined by the topological and differential structure of the manifold) that exist at a world, and g and T defined over M are taken to represent the (dynamical) properties and relations that the points possess. Earman and Norton see manifold substantivalism as
137 We can of course factor this structure down into finer substructures, such as a topological space and a continuum of points. However, such 'low-level' structures are too elementary to use to distinguish spacetime theories: the manifold is the highest level of structure that all spacetime theories will agree upon. This might alter, and it might become necessary to construct a theory that is independent of differential and topological structure too.
138 These equations will, in general, be of the form FDí, Bj] = 0.
139 Note also the connection to the definition of structure that I gave in §1.2. In this context, the structure is a domain of spacetime points, and the relations are geometrical ones. The notion of symmetry given there functions well in this context: the symmetries of a theory are those transformations that preserve geometrical relations. For such transformations, the dynamically possible histories are left invariant, and the theory is said to be invariant with respect to these transformations.
implying that the metric field (represented by g) and the stress-energy field (represented by T) are fields contained within spacetime (represented by M), and that the points of spacetime have their identities fixed independently of these fields.
As ever, for our purposes, the crucial question concerns the nature of the relation between models and worlds. How do the models represent worlds: one-to-one, many-to-one, selectively, not at all, or in some other way? Do we take the elements in the domain of the model to denote the same thing as the elements in the domain of another model? In other words, do the domains Mi of distinct models contain the same points, or are they themselves distinct? If the former is the case then transworld identity can be represented as one and the same individual appearing in different domains in different models (cf. [Melia, 1999], p. 641). But in virtue of what are the points the same? The manifold substantivalist cannot make use of the g and T fields to individuate them, because the points are taken to have their identities fixed independently of these. Yet the points have no distinguishing characteristics independently of g and T, so it would seem that some non-qualitative property must be required to do the job. This line of reasoning leads to the view that manifold substantivalism is committed to haecceities or primitive thisnesses to individuate the points. Earman and Norton trade on this notion to generate the indeterminism, for if the points have their identities fixed independently of the various dynamical fields, then any redistribution of those fields will result in a distinct possibility, even though the possibilities differ purely haecceitistically.
From what I had to say in §2.4 the flaws in this reasoning should be readily apparent. Even if we assume that the manifold substantivalist's position does require haecceities, this does not imply commitment to haecceitism (and vice versa). One can rule out the former implication by denying transworld identities, so that haecceities serve to individuate points within a world only. The latter implication can be ruled out by using 'this worldly' counterparts to ground possibilities.140 Hence, just as was the case in the Leibniz-shift argument, it is haecceitism that causes the problems not substantivalism: these theses can be safely detached from one another.
Let me express what is being suggested by Earman and Norton using the account of structure and symmetry that I presented in §1.2. A model M corresponds to a structure S; the set of spacetime points M corresponds to the domain D; and the tensor fields g and T on M correspond to the set of relations R defined over D. The spacetime points are the objects of the theory. The elements of the domain are indistinguishable with respect to the relations given by g. Hence, we should brace ourselves for symmetry and indifference: we can permute the points of the domain without affecting the relations. In the context of general relativity this property is called diffeomorphism invariance, where a diffeomorphism $ : M ^ N simply corresponds to a smooth permutation of the points. General covariance is then a formal property of models, such that acting on a model by a diffeomorphism generates
140 Thus, for two (same worldly) points x, x' g M, such that P(x) and P'(x') are complete lists of the properties of the points, we can view x! as a counterpart of x,so that x! represents dere of x that it might have had the properties catalogued in P'.
another model—this is a symmetry of the theory; a purely formal requirement.141 Since diffeomorphisms are gauge-type symmetries, they thus generate an inflated possibility space. General covariance concerns the relative admissibility of the elements of this space: if M is a solution of Einstein's field equations (or represents a physically possible world), and M* = $(M), then M* is a solution of the field equations (or represents a physically possible world) too. The reason is symmetry: M and M* will be indistinguishable with respect to the relations encoded in g (and T), differing merely with respect to which points play which role in the relations of the respective structures. Since the field equations (determining physical possibilities) concern g, solutions that respect this will represent physical possibilities. Since dif-feomorphic models are indistinguishable with respect to g, one will automatically represent a physical possibility if the other does. Notice that neither diffeomor-phism invariance nor general covariance commit one to a particular stance with regard to the representation relation between models and worlds.
However, the received view is that particular conceptions of spacetime do underwrite particular interpretations of the representation relation. Earman and Norton argue that since the manifold substantivalist is committed to the view that spacetime points are real and have their identities fixed independently of g and T (the relations), then any permutation of the points (or, equivalently, any redistribution of the fields over the points) will result in an ontologically distinct situation: the points will possess different properties and enter into different relations in each diffeomorphic case. This is so, even though the relations themselves are unaffected by the permutations.142 I think that this shows that Earman and Norton are claiming that for the manifold substantivalist, the identity of the points goes beyond the roles they play in the relational structure of the world. We will see in the responses that this can be denied in a number of different ways. I will deny it too, but not to put a different notion of spacetime point in its place, nor to deny that spacetime points exist; rather, I wish to show that the theory itself does not have the conceptual resources to support any metaphysical conclusion about the ontological status of spacetime one way or the other. Focusing on the relations, and eschewing talk of the natures of objects gives us a firm response to the hole argument. Moreover, it is a response that fits the practice of physicists too (see Chapter 6). However, such a response must be supplemented with an account of symmetries and observables, for the symmetries are still present in the formalism. A firmed up conception of observables can explain how this can be so without leading to problems of the kind outlined in the hole argument.
Clearly there is something to the hole argument; but what it is no different from the similar arguments we have seen in the previous two chapters. It is a general problem concerning gauge-type symmetries. The hole argument doesn't tell us anything about the ontological nature of spacetime, but it does tell us about the conceptual limits of general relativity: the symmetries mean that metaphysical talk
141 There are two ways to act on a model with a diffeomorphism: (1) by applying $ to the points of the manifold; and (2) by dragging along g by 4>*. The two actions are equivalent.
142 A relationalist interpretation is seen as underwriting the view that permutations of points do not result in distinct possibilities, precisely because the relational structures of such diffeomorphs will be isomorphic. I argued in §2.4 this alignment is not a necessary part of the relationalist's position. The same point holds in the present context.
about the objects (configurations, states, possibilities, worlds, or whatever) related by symmetries will be underdetermined by the physics. I will, however, agree that the argument works to this extent: if the substantivalist is committed to the representation relation between models and worlds being one-to-one and is committed to there being the same objects in the different models' domains, and is committed to the independent existence of these objects, then, without another conception of determinism on the table, she is in trouble: many ontologically distinct futures will be compatible with an initial state and the laws. The problem is, as I mentioned, the substantivalist is committed to none of these conjuncts; so regardless of what conception of determinism we have to work with, the substantivalist is safe. Even if all conjuncts are accepted, there are options for taming the indeterminism, as outlined in §3.3.1. I'm getting ahead of myself here, let us now present the argument itself.
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