Recall that the manifold substantivalist was supposed to get into bother with inde-terminism because of the alleged commitment to the existence of worlds that are qualitatively identical but differ with respect to how the geometrical properties are spread over the points of space. The general covariance of general relativity implies that the equations of the theory cannot uniquely determine this spreading of the geometrical properties over the points. If the manifold substantivalist is so committed then the indeterminism surely follows. We have just seen two ways in which the substantivalist can escape this commitment, one manifold sub-stantivalist (Butterfield's) and one involving metrical properties (Maudlin's). Both, however, deny LE. In this subsection I outline a proposal that is both substan-tivalist and endorses LE. It does this simply by denying primitive identities for spacetime points.166

To the best of my knowledge, the approach is question began with Maidens [1993], and variants have since been defended by Stachel [1993], Brighouse [1994],

164 Recall that in some cases, due to the Gribov obstruction, the gauge conditions will intersect some orbits more than once. Such a scenario would correspond to Maudlin's response.

165 For Lewis' expression of the problem see ([1975], p. 39).

166 Primitive identities are non-qualitative individuating properties, such as a = a—the basic details were given back in §2.4. An immediate problem with the denial of primitive identities is, then, that it is unclear how one is able to support set theory on which general relativity's models are based (I owe this point to Steven French). There are ways of accommodating the denial of primitive identities through the use of 'quasi-set theory' in which the identity relation is not a well-formed formula for indistinguishable objects (see French & Krause [1999] and Krause [1992]). However, how this is supposed to be implemented in the context of general relativity is beyond me. There are perhaps two ways to accommodate this problem though: (1) claim that one's usage of standard set theory, with identity, is merely heuristic; (2) claim that standard set theory can be applied just fine given that the points have been individuated by, e.g., the metric field or some matter field.

Hoefer [1996], Saunders [2003a], and Pooley [in press]—Earman [1989] mentions it, only to reject it.167 Since the points of space are absolutely indistinguishable it must be a non-qualitative property of this kind that individuates them. According to this approach, it is not manifold substantivalism per se that causes the indeter-minism of the hole argument; the real culprit is an extra unnecessary component regarding the identity conditions of spacetime points. In particular, the hole argument assumes that the points can be identified, distinguished, and labeled from world to world and within a world. That is, it assumes a notion of transworld identity for spacetime points allowing us to speak of this point having certain properties in this world and having different properties in another world, and of this point being different from that point. Therefore, if we jettison this component from manifold substantivalism, the hole argument is avoided and substantivalism is rescued.

Anna Maidens suggests that whether or not a notion of transworld identity holds depends upon physical theory. She compares the role of spacetime points in the hole argument with the case of indistinguishable particles in quantum statistics. In the classical scenario we can speak of reversing the results of outcomes, since the physics is based upon Maxwell-Boltzmann statistics. Hence, it makes sense to speak of the world in which this coin was heads and that coin was tails. This cannot be the case with indistinguishable particles like electrons. For if we say of a pair of electrons that their states could have been swapped, we would have to count this in the statistics, but this would simply yield Maxwell-Boltzmann statistics instead of Fermi-Dirac statistics. Redhead and Teller [1992] argue that the case of quantum statistics highlights a crucial difference between the identity claims that can be made in classical and quantum mechanics. In the former we can assign labels to the particles that pick out the same particle in many possible worlds, they designate their referents rigidly. They call this property "transcendental label identity" (p. 203), this is, roughly, another way of saying 'primitive identity'. This labeling is, they say, not workable in the quantum scenario.168 Maidens suggests that the points of spacetime are analogous to quantum particles in terms of their transworld properties, we can't think of them as having names that function as Kripkean rigid designators. Crucial to this way of thinking is that there is no denial that spacetime points exist. They do exist, just as much as quantum particles exist; only, they don't possess that kind of identity that make transworld identifications (or, indeed, independent intra-world distinctions) possible. In the case of quantum particles, it was the puzzling quantum statistics that pointed to this conclusion, in the case of spacetime points it is the hole argument that points to this conclusion.169

167 Note, however, that Stachel and Saunders associate their view with relationalism rather than substantivalism as the others do.

168 In fact it is, and Redhead and Teller admit that it is possible to think of the particles as individuals in a robust classical sense, provided we impose certain conditions on the form of their wave-functions (as argued in French & Redhead [1988]). (Cf. French and Rickles [2003], for an overview of this and other philosophical aspects of quantum statistics.) Note that Redhead and Teller, though acknowledging the underdetermination, defend the view that particles should be viewed as 'non-individuals' (field quanta) on "methodological grounds" (ibid.). I'm not convinced by their transition from methodology to ontology, but I cannot go in to this here.

169 There are many points of contact between the issues surrounding the hole argument and those of quantum statistics. A natural alignment would seem to be between an individualistic and a substantivalist package on the one hand and a non-

Let us get clearer on what the ascription of primitive identity to objects amounts too. For this, I switch to Hoefer's discussion [1996]. He writes that the commitment to primitive identity is well illustrated by the following example:

Suppose I have two dice and name them A and B by pointing to them. I now ask: Does it make sense to ask whether die A could have been located where die B is with all B's actual properties? ([Hoefer, 1996], p. 14)

The example matches, more or less, the Leibniz-shift scenario. Clearly, if this 'property swap' scenario were a possibility it would be one qualitatively identical to the actual one. Hoefer thinks that the question is senseless, that the "question rests on the presupposition that the names 'A' and 'B' can be used to talk about particulars that have a primitive identity wholly independent of the properties these particulars actually possess" (ibid., pp. 15-6). Consequently, if it can be shown that "primitive identity is a mistake, then names cannot in fact be used in this way" (ibid., p. 15). Hoefer calls upon Lewis definition of haecceitism to further illustrate the notion of primitive identity. Recall that Lewis' definition amounted to the claim that there are cases of haecceitistic differences between worlds, where haecceitistic differences are cashed out in terms of worlds differing in what they represent de re of some individual, but do not differ in any qualitative way—anti-haecceitism is simply the denial of this claim (cf. [ ^ewis, 1986a], p. 221). Hoefer claims that primitive identities are a necessary condition for haecceitism, though the converse is not true—hence, reject primitive identities and you rule out haecceitism. Now, the crucial claim is that the diffeomorphic models of the hole argument can only represent different physically possible worlds if spacetime points exist and have primitive identities. Why? Because the models do not represent any qualitative difference, and the difference amounts to a swapping of geometrical properties between actual spacetime points.170 Hence, Hoefer sees primitive identities as responsible for the haecceitistic differences that make up the multiple futures responsible for the indeterminism. If this layer of metaphysics can be stripped away from sub-stantivalism, then we have a solution; for without primitive identities we have no worlds that differ by haecceity, and without those there is a unique world, as required by determinism. In other worlds, the way is opened up for the substanti-valist to endorse LE.

The argument that Hoefer gives to detach primitive identity from substanti-valism is rather brief: The fact that two objects (of the same type) have contingent properties does not license the conclusion that a property swapped situation is a possible situation distinct from the actual one. If by "possible situation" Hoefer means "possible world" then I think we can agree. Lewis, for example, will want individuals and relationalist package on the other (see, for instance, Stachel [2002]). However, on Maidens' account we see that the substantivalism and non-individualistic packages are aligned. This is an aspect of the transformability of positions by revising conceptions of identity and modality that I mentioned earlier: interpretive positions can help themselves to features generally assumed to be unique to their opposites by tweaking certain parts concerning modality and identity. 170 Recall that Maudlin avoided the problem by arguing that the swapping operation is not metaphysically permissible because the metrical properties of spacetime points are essential ones. The swapping only gets a purchase on contingent properties. Thus, only the model representing the actual world represents a physically possible world. Butterfield avoided the problem by arguing that swapping operation should not be construed as generating a physical possibility for the actual points, because the actual points cannot figure in the domain of any other world. Thus, if we fix a model to represent the actual world, then no other model represents a physically possible world. Both options deny LE like good Earman and Norton defined substantivalists.

to say that the actual world contains the possibility that Hoefer outlines: the actual object B represents de re of the actual object A that it might have had all of Bs properties. If, however, he means "possibility", then I think that he is wrong. We are asked to suppose that the properties of A and B are contingent, and that there exists A and B in the actual world, and we wish to consider them having their properties switched. Surely this is a distinct possibility, a different way that A and B could have been, given contingency? I think we should side with Lewis in saying that it is a possibility, though not a possible world: B is As counterpart and represents of A that it might have had all of the properties B in fact has. There is no proliferation of possible worlds, but there is a proliferation of possibilities, many to each world. Let us assume that Hoefer intends the former reading, then we have severed primitive identity from substances, for A and B can be substances capable of possessing properties and being named, but we aren't forced into the property swapped situations representing distinct possible worlds. This latter feature is taken by Hoefer to go hand in hand with the notion of primitive identity. There is a second aspect to Hoefer's argument, based upon another use of primitive identity, namely to ground the identities of distinct qualitatively identical objects, such as spacetime points. Hoefer rejects the need to use primitive identity in this context because he doesn't believe that one needs a principle of individ-uation to individuate such objects. He cites Black's counterexample to Leibniz's PII involving two qualitatively identical iron spheres to make his case. The reason there are two spheres rather than one is "because, as we stipulated, there are two of them and not one" (ibid., p. 19). Clearly, PII is being rejected here because that requires that distinct objects must differ in some qualitative (intrinsic or extrinsic) respect, yet the spheres, nor spacetime points (in the vacuum case) do not. PII prevents us from speaking about qualitatively identical objects that differ numerically. Hoefer rejects it within worlds, for we can stipulate such worlds. There is, I think, a better way of grounding the distinctness between the spheres, and that is simply that there is an observable relation between them, being two miles apart, for example.171 Saunders [2003b] has recently described a new way to understand (Quine's) PII that does not admit cases like Black's spheres as counterexamples. The idea is that the two spheres are distinguishable because, although the two objects share their monadic properties, and a symmetry holds between them preventing us from using relations they bear to other objects to distinguish them, they do satisfy an irreflexive relation: they are two miles from each other, but not two miles from themselves. Such objects Saunders calls "weakly discernible". But since Hoefer explicitly rejects PII such moves are not necessary.

Hoefer is satisfied that he has stripped primitive identity from spacetime points, and substances in general. But the view that results has been associated with a structural role theory of the identity of spacetime points. Hoefer admits that this theory is problematic for reasons outlined by Earman ([1989], pp. 198-9), but seeks to distance the denial of primitive identity from such a theory. What exactly does this theory say? Hoefer claims it is the view that

171 This relation will not be part of the ontology of a Hacking style redescription in terms of a single object in a highly curved spacetime. See French [1994] for a critique of Hacking redescriptions.

the identities of points are determined by some subset of the properties and relations ascribed to the points by a model of the space-time theory. So, to be point A in a world described by model (M,g, T} is just to have the metrical (or metrical plus material) properties and relations to other points that A has in the model. A's structural role in the model constitutes what it is to be A ..."Identity follows isomorphism," in this case diffeomorphism: two models related by the right kind of isomorphism have the same points, and the isomorphism shows how to identify the points in one model with points in the other. ([Hoefer, 1996], pp. 20-1)

A problem of 'multiple isomorphisms' faces any such account of the points' identities. Earman describes the problem as follows:

If : W ^ W' and ^2 : W ^ W' are relevant isomorphisms, total or partial as the view of identity requires, and if i is an individual of W, it follows that i is identical with ^i(i) and with f2(i). And so by transitivity of identity, f1(i) = f2(i), which gives a contradiction if and f2 are distinct. ([Earman, 1989], p. 199)

Hence, if identity does indeed follow isomorphism, then domain points will be identical with its image points for any isomorphism. If there are multiple isomorphisms, as in general relativity, then it will be identical with multiple image points. But when the range is the same for the multiple isomorphisms then we get the absurd conclusion that distinct image points are identical. To escape this problem Hoefer refers back to Lewis' anti-haecceitism with its qualitative notion of transworld identification: "no two worlds can represent de re different things about the very same objects, unless the worlds differ qualitatively in the properties and relations ascribed to the objects" (ibid., p. 21). This is the same in all essentials to Brighouse's response to a similar problem. Speaking in terms of counterpart theory, she says that the substantivalist will say that the isomorphic models will represent one and the same world because "the counterpart of any point in any of the qualitatively indiscernible worlds will have all the same qualitative properties as that point has" ([1994], p. 122). However, I think Saunders ([2003a], p. 25), defending a similar position (see below) says the right thing here (if we eschew counterpart theory): the isomorphisms if distinct will map the point into nominally distinct worlds that are related by isomorphism, not the same world in which two objects are related by isomorphism.

This leads me to a problem with Hoefer's approach. He wants to help himself to tools from Lewis' workshop, but he nowhere claims that he advocates counterpart theory. He believes that denying primitive identity gives him what he wants: anti-haecceitism. However, if Hoefer is a closet counterpart theorist, then he needs to come out and say so. For denying primitive identities isn't enough to escape the possibility of haecceitism; I showed as much in my discussion of the Leibniz-shift argument. French ([2001], p. 21) also draws attention to this problem, noting that "[a]ccording to Lewis, a belief in haecceities is neither necessary nor sufficient for haecceitism [in Hoefer's sense] ...one might assert haecceitism but deny primitive identity". Presumably French has in mind Lewis' notion of "haecceitism on the cheap", according to which the case where A and B swap their qualitative properties is a possible world, but is the same world as the actual world; i.e., B de re represents of A that it might have had all of Bs properties. French suggests that it is not so much primitive identity that needs to be rejected, but haecceitism simpliciter (ibid., p. 22). But I showed earlier that this can be done just as easily.

A crucial question with this proposal is whether or not this haecceitism (assuming that this is really what Hoefer wants rid of) can be jettisoned from substanti-valism without thereby destroying the latter. Oliver Pooley has recently argued at great length that "haecceitism is not an obvious concomitant of viewing space as a genuine, substantival entity" ([in press], p. 191). Pooley calls the anti-haecceitist substantivalist position that results "sophisticated substantivalism" (rescuing the term from Earman and Norton's derogatory usage and meaning anti-haecceitistic substantivalism). For the most part I agree with Pooley and Hoefer that haec-ceitism is not a necessary part of the substantivalist's position, and that such a modification offers a way out of the hole argument. I argued in my discussion of the Leibniz-shift argument that the substantivalist's position can be disentangled from all manner of metaphysical baggage, and that the relationalist can be saddled with it. However, I believe that there are independent reasons—primarily to do with the nature of observables in general relativity—to reject both substantival-ism and relationalism in the case of general relativity and its quantization. I make a start on these reasons in the next section.

I agree with Hoefer's general conclusion that primitive identities are not a necessary part of the substantivalist's position. I agree also with the claim that what results might nonetheless be construed realistically (i.e. robustly).172 We have, then, a variety of substantivalism that endorses LE. However, as we saw earlier, Hoefer does not believe that the manifold of points, primitive identities or not, possesses sufficient structure to represent spacetime. Instead, he defends the view that "[w]hen it comes to representing spacetime, literally and to the best of our current abilities, the metric field of GTR is the only game in town" ([1998], p. 464). It is only the metric that has all "the usual spatiotemporal features" ([1996], p. 24). I agree for the reasons Hoefer gives, and the reasons I presented earlier. Given this, he characterises substantivalism about general relativity as follows173:

The metric field as presented in GTR literally describes a substantial, quasiabsolute entity that interacts with ordinary matter. It is 'quasi-absolute' because, unlike earlier absolute spaces, its structure is partially determined by the coarse-grained material contents, in the way specified by Einstein's field equations. It merits the term 'absolute', however, because according to GTR it can exist without any material contents, and with a variety of structures; it merits the term 'substantial' for this reason and because our causal-explanatory understanding of (gravitational) mechanics involves spacetime both acting on matter and being acted on in turn by matter. ([Hoefer, 1998], p. 464)

172 I differ from Hoefer in that I think that both substantivalists and relationalists can be said to be realists about spacetime, merely differing in how they think that spacetime exists, qua independent substantial entity or relational entity.

173 In the paper that this passage comes from, Hoefer is at pains to set up a decent formulation of the substantival-ism/relationalism debate in a bid to overcome Rynasiewicz's objection that it has become outmoded.

Relationalism is then characterised by the contrary claim that "[t]he basic constituents of our universe do not include an independent, substantial space or spacetime" (ibid., p. 465). I think that there are some terminological problems here: for "absolute" I would say "dynamical"; absolute implies fixed or model independent. This aside, I think that the way Hoefer sets it up loads the dice heavily in favour of substantivalism, for in identifying the metric field with a substantival spacetime, he is forcing the relationalist out of a position he would naturally like to occupy, viz. treating the metric field along the lines of a matter field. The relationalist must adopt the absurd view that the universe does not contain a 'real' metric field, for he doesn't believe in a substantivalist spacetime, and in this case that just is the metric field. Substantivalism is sure to win! But Stachel advocates a very similar view to Hoefer's metric field substantivalism and calls it relationalist. Hoefer admits too that Teller outlines a 'relationalist' position that "arrives at essentially the same place" ([Hoefer, 1996], p. 25). What gives? We seem to be right back in Rynasiewicz's problem.

Now, I agree that the best chance that substantivalism has in general relativity is by treating the metric field as representing spacetime. Only it has all of the properties that we expect a spacetime to have. Recall that Newton was not a manifold substantivalist; he too was a substantivalist along similar lines: that is, his attributions of independent reality concerned E3 x E not R3 x R. The reasons are clear: it is the metric that provides the structures that we associate with spacetime, viz., past, future, distances between events, and so on. Of course, his spacetime (space and time) was not dynamical; the metric was a background structure; it does not possess energy in any kind of way. Hence, the problems we face in deciding what represents spacetime were not applicable in that context (remember Earman's witty remark about having the luxury of knowing what they were talking about). The relationalist was not in a position to help himself to the metric as an object, because it didn't have any properties associated with material fields. But the spacetime relations encoded in the metric had to be reduced to relations between objects not including the metric. Hoefer is right to think that the metric is where the debate should be focused, but the formulation is unfair: it is lopsided, and the poor relationalist is almost certain to lose. A question is surely being begged when Hoefer claims that the "metric field of GTR ...[does] not seem to be eliminable in favour of some set of primitive relations holding among material things (whether fields or particles)" ([Hoefer, 1998], p. 463). Surely the relationalist's relationalism will involve the metric field as the "material thing" that determines the "set of primitive relations". For example, Stachel writes that

[s]everal philosophers of science have argued that the general theory of relativity actually supports spacetime substantivalism ...since it allows solutions consisting of nothing but a differentiable manifold with a metric tensor field and no other fields present (empty spacetimes). This claim, however, ignores the second role of the metric tensor field; if it is there chrono-geometrically, it inescapably generates all the gravitational field structures. Perhaps the culprit here is the words "empty spacetime". An empty spacetime could also be called a pure gravitational field, and it seems to me that the gravitational field is just as real a physical field as any other. To ignore its reality in the philosophy of spacetime is just as perilous as to ignore it in everyday life. ([Stachel, 1993], p. 144)

No wonder Hoefer suggests that the "debate ...may well be settled"! But Stachel is no better, for he claims the metric field all to his relationalist self by treating it as a physical field like any other: clearly it isn't quite like any other, for we can set any other fields to zero (i.e. T = 0), but not the metric field. A fairer way would be to focus attention on the metric field, as Hoefer suggests, but to allow the relationalist access to it as a physical field. But not in the way that Hoefer implies, for what he seems to be calling for is a reinstating of the doctrines of "material metric field" versus "physical metric field". According to the former the metric of general relativity spacetime is reducible to the behaviour of material objects, such as clocks and rods and so on; according to the latter view the metric field is an irreducible physical field (cf. Weingard [1975], pp. 426-7).174 Hoefer takes the physical metric thesis as corresponding to substantivalism. But no relationalist in their right mind will accept this: the material metric thesis is simply false, because there are solutions even when the stress-energy field is set to zero. I suggest formulating the debate precisely in terms of whether or not the metric field is 'space-like' or 'matter-like', whether the chronogeometrical properties outweigh the gravitational properties. Then many of Hoefer's arguments concerning the difference of the metric to the other fields might prove themselves successful.

Hoefer's claim, then, is that the metric field is the best candidate for a substan-tivalist's spacetime.175 He suggests that such a view can be attributed to Einstein (loc. cit.: 6). This interpretive option does not face the hole argument because it is natural on this account to say that the members of an equivalence class of diffeomorphic models represent one and the same state of affairs: all invariants are preserved and the identities of the points are carried along with the metric transformations—remember there are no primitive identities. The view that it is the metric field that represents spacetime is well motivated too by the fact that there is no such thing as spacetime until a metric has been defined—he draws this point from Einstein—and it is the metric that does all the explanatory work in general relativity. But once again, the relationalist can agree with both of these and still use them to support their own view, as indeed Stachel does, drawing on the same passages from Einstein that Hoefer uses.

As I have intimated, depending on one's views about the ontological status of the metric field, Hoefer's position might come out as a rather uncomfortable looking relationalism, more than a substantivalist position—indeed, Belot ([2000], p. 576, fn. 36) characterizes him as a "crypto-relationalist". Belot and Ear-man [2001] see (LE-endorsing) sophisticated substantivalism of this sort as being aligned with relationalism. This is certainly how Stachel sees his position, and it is virtually identical to Hoefer's aside from terminological niceties. More importantly than this word mongering is what I take to be a crucial aspect that Hoefer does not properly discuss: the transition to quantum theory. Hoefer believes that the time is not yet right; but as I argued in the introduction to this

174 This is largely the debate as conceived of by Grunbaum [1973] who himself advocated the material metric field thesis.

175 Note that Hoefer himself is a Machian relationalist.

book, we can apply our interpretive skills just fine in a number of proposals. Especially important, I think, is the assessment of how well a position does, and what kind of approach results when the interpretation is applied in quantum general relativity—canonical quantum gravity being the best candidate. On the front runner from that approach, the metric is a derived object and it is the connection that forms the basic ontological category. On the canonical approach in which the metric does appear, it is quantized, and this will surely have a bearing on whether or not it is substantival or relational. In fact, I don't think it definitively supports either, but it should still be a consideration for any conception these days. I will have more to say on this in later chapters.

Before I leave this approach, I should point out that there are other LE endorsing substantivalist options related to this proposal. One such is the view that the manifold points represent spacetime points, but which spacetime point a particular manifold point represents is dependent upon how the geometric fields are distributed over the manifold (see Brighouse [1994]). Each diffeomorph will then represent the same possibility because the properties and relations of spacetime points are carried over with the diffeomorphisms. Hence, we are not necessarily dropping primitive identities, but we are dropping haecceitism, for it is qualitative similarity that counts. This is grounded in a counterpart relation. We see that both are also very closely related to Stachel's idea that the points of spacetime are individuated dynamically by solving for the metric field, so that the points of the manifold cannot be considered to be named rigidly a la Kripke. (I turn to Stachel's view in the next subsection.) Belot has argued against this form of substantival-ism176 on the grounds that it is a mere "variation on relationalism" ([1996], p. 184). As he goes on to say:

Lockeanism is not the saviour of substantivalism: rather it is a pallid imitation of relationalism which should be of interest only to those substan-tivalists who are too cowardly to wager that quantum gravity will carry full-blown non-Lockean substantivalism to a decisive victory over relation-alism. ([Belot, 1996], p. 184)

The resolution to the hole argument comes about because the formally distinct spaces that result from diffeomorphisms are seen as numerically identical. Models differing only in terms of which points of space possess which geometric properties are identified. Our substantivalist claims are to be applied to this intrinsic space, rather than the individual diffeomorphs. Hence, the substantivalist can accept Leibniz equivalence, as long as haecceitism is ruled out.

Was this article helpful?

## Post a comment