Varieties Of Relationalism

The final class of response we consider comprises those that see the lesson of the hole argument as implying an endorsement of LE, and see that as implying rela-tionalism. We have already seen that there are forms of substantivalism that fall

176 Though he uses the terminology "Lockean" and "non-Lockean" to denote the anti-haecceitistic and haecceitistic brands of substantivalism.

under this mantle too. Hence, it is possible for both relationalists and substan-tivalists to endorse LE. I begin with a quick look at Einstein's point-coincidence argument. This resolution involves an understanding of points in terms of intersections of particles: quantities defined at these points are deterministic. Stachel's proposal is superficially similar: his central idea is that points of spacetime are dynamically individuated. This can carried out using material objects and fields as Einstein suggested, or one can use the metric itself. There are similarities to Hoefer's response too, but where Hoefer drew a substantivalist conclusion (by denying primitive identities for points, but allowing the metric to individuate them), Einstein and Stachel draw a relationalist conclusion (claiming that the relations are ontologically prior to, or at least on all fours with points). Saunders' response is relational, but not reductive or eliminative in its treatment of spacetime points: the idea is that points can be distinguished by their relational properties.

5.3.1 Einstein's point-coincidence argument

It is now well known that the hole argument was pivotal in Einstein's eventual completion and understanding of general relativity.177 The argument led him to at first reject generally covariant field equations for gravity (on account of the fact that different fields were compatible with the extension of data into an arbitrary region of spacetime), and then led him to reinstate the principle, by understanding the general covariance in a deeper way. The basic idea that led to his change of mind revolves around an idealised conception of measurement according to which any possible observation reduces to the intersection points of systems (observers, apparatuses, and the objects to be measured—though no absolute distinction exists between these categories). This theory of measurement is expressed in what has been called the 'point-coincidence argument'. Einstein states it as follows:

That the requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion. All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurings are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock dial, and observed point-events happening at the same place at the same time. The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. ([Einstein, 1916], p. 117)

Einstein's suggestion superficially smacks of positivism and relationalism: positivism since evidence is reduced to observations, and relationalism since material points are considered prior to spacetime. There is no mention made of vacuum

177 See Toretti [1987] for a nice, highly readable account of this history.

spacetimes; the implication of Einstein's view is, however, that a vacuum spacetime is meaningless! There is a passage of Einstein's that, on the surface at least, looks like a more explicit espousal of relationalism, and that contains an answer to the vacuum spacetime scenario:

If we imagine the gravitational field, i.e., the functions gik, to be removed, there does not remain a space of the type (1) [Minkowski space-time], but absolutely nothing, and also no "topological space." For the functions gik describe not only the field, but at the same time also the topological and metrical structural properties of the manifold. ..There is no such thing as an empty space, i.e., a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field. [1961:155]

This quotation superficially looks like relationalism again. Indeed, it contains a passage that explicitly endorses the material dependency thesis: "There is no such thing as empty space". However, substantivalists have an option to redefine what they mean by 'space', and 'empty space', and so uphold the material dependency thesis. They might say that it is true that a spacetime cannot exist without a metric field, but that the metric field itself is a necessary part of space; we have been here already with metric-field substantivalism, of course: for Hoefer, the metric field just is spacetime (this is how many general relativists speak too). Recall that in the original debate between Leibniz and Clarke the space came ready equipped with a fixed metric. When we said there could be empty spaces according to sub-stantivalism, we meant that there could be empty spaces that had the form of a metric manifold, not a 'bare' manifold. Indeed, it was the symmetries of this metric that allowed for the possibility of the Leibniz-shift scenarios in the first place. It is true that the metric is no longer a background structure in general relativity, but it might nonetheless be seen as an essential component of spacetime; it determines those crucial features that make spacetime what it is. Clearly, on this account we are no longer taking spacetime simply to be a container for matter to occupy; rather, it is something dynamic that interacts with matter. As we have seen, this might be construed as a plus point for the substantivalist; for if spacetime is capable of dynamical action, then surely it is all the more substantial for it. We might also read off a structuralist position from this passage in an obvious way: spacetime is structurally constituted by the metric field. Hence, like the famous passage of Newton's, concerning the individuality of points as depending upon their pattern in a relational structure, we can read off all the major positions from this passage.

Stachel suggests that these passages cannot be considered as independent of the issues taken up in the hole argument. When this is taken into account, Stachel thinks that the view expressed in the above passages points not to an operational-ism, but to the thesis that only a physical process can individuate the events that make up space-time ...a manifold only becomes a space-time with a gravitational field after the specification of the metric tensor field, and that, prior to such a specification, there is no physical distinction between the elements of the manifold ...his comments were not meant ...to indicate that space and time have no physical reality, but that they no longer have any independent reality, apart from their significance as the spatial and temporal aspects of the metrical field. ([Stachel, 1989], pp. 87-8)

Hence, we see Stachel explicitly connecting Einstein's view to a denial of the independent reality of spacetime: relationalism. I consider Stachel's development of this line of thought through his notion of an "individuating field" in the next subsection.

The earlier point-coincidence argument, I think, sits better, not with opera-tionalism, but with a revamped notion of 'observability' and what kinds of observables will be possible in general relativity: general relativity cannot deal with local field quantities (defined at points of spacetime). In fact, although Stachel argues that the second argument is more sophisticated, I think that it is the first argument that most quantum gravity physicists follow. Compare it with this passage from Rovelli:

in a general relativistic theory ...[o]nly quantities that do not depend on the coordinates may correspond to concretely physically observable quantities. Localization with respect to a background spacetime, or with respect to a fixed external reference system, has no meaning. What has physical meaning is only the relative localization of the dynamical objects of the theory (the gravitational field among them) with respect to one another. The physical picture of the world provided by general relativity is not that of physical objects and fields over a spatiotemporal stage. Rather, it is a more subtle picture of interacting entities (fields and particles) for which spatiotemporal coincidences only, and not spacetime localization, have physical significance. ([Rovelli, 2000], p. 3779)

I consider the relevance of the hole argument to the problem of the observables of general relativity in more detail, including the bearing of this issue on spacetime ontology in the next chapter. For now, it will suffice to know where the view expressed by Einstein and Rovelli springs from. For Einstein it was simply general covariance; but, as Kretschmann demonstrated, this isn't quite sufficient to get the desired conclusion.178 Rovelli suggests that it is the active diffeomorphism invariance underlying the general covariance of general relativity that leads to the physical insignificance of the points of spacetime. In other words, it is background independence that leads to this view. This seems to be the 'majority view' of physicists who consider the hole argument and the meaning of general covariance and diffeomorphism invariance.

Note that what is being suggested here is not that spacetime itself is relational, but only that localization is relational (see §6.2). Hence, we cannot consider quantities at manifold points to be observable, but we can consider quantities at materially (possibly including the metric itself) defined points to be observable precisely because they are diffeomorphism invariant. This is a significant departure from previous theories. For example, the observables of, e.g., QED are explicitly local,

178 Recall that Kretschmann's objection is that any (local) spacetime theory can be given a generally covariant formulation. It can, but active diffeomorphism invariance cannot be implemented except in a highly trivial way (the derivatives for metrics and connections must be made to vanish).

they are dependent on a fixed metric on spacetime, as are the observables of most other classical and quantum field theories. The suggestion here amounts to the claim that the observables of general relativity (and its quantization) will not possess such spacetime dependence. Einstein, Stachel, and Rovelli see this as pointing towards relationalism. I say that it doesn't actually have a bearing on the ontology of spacetime vis-à-vis relationalism and substantivalism: both can be made compatible with it—substantivalists can simply adopt the view that the metric field represents spacetime and so, since localization relative to it is possible, relational localization does not rule out substantivalism.179

Since structuralism is fairly neutral (relationalists and substantivalists will agree about the physical relational structure), I believe that it is the best candidate for the ontology if we are trying to read our ontology off the physics. However, my argument is not bound to this ontology; and this stance has to be seen as not strictly flowing directly from the interpretation of the theory. I consider this further in the next chapter, in the next subsection I examine Stachel's take on the hole argument.

5.3.2 Stachel's relationalism and individuating fields

The problem of the hole argument is a result of the general covariance of the field equations for gravity. This property of the equations allows us to derive both g and 4>*g as solutions. But g(x) = 4>*g(x), and so the manifold substantivalist is supposedly forced to admit that these solutions represent distinct possibilities. However, the curious feature of the problem is that if we consider X = 0(x), then we find that g(x) = 4'*g(x'). We want to say in this case that g and 4>*g represent one and the same gravitational field. It is only when we view the points of the manifold as possessing their identity independently of the solutions that we get the result that they represent distinct possibilities. Stachel proposes that we should instead view the points not as independently individuated, but as individuated once a metric tensor has been specified, once the equations have been solved. Thus, he claims that diffeomorphism invariance has the effect of:

precluding the existence of any pre-assigned (kinematical) spatio-temporal properties of the points of the manifold (even locally) that are independent of the choice of a solution to the field equations (no kinematics before dynamics). The physical points of space-time thus play a secondary, derivative role in the theory, and cannot be used in the formulation of physical questions within the theory (they are part of the answer, not part of the question). ([Stachel, 2003], p. 16)

Hence, we should forget about g(x) = 0*g(x), because this implies that we can continue to refer to the same point of spacetime independently of the metric, and we should deal only with g(x) = g(x'). Stachel suggests that we should look

179 Even in the quantum theory this move will be possible. For example, in loop quantum gravity certain objects known as 's-knots' take the place of the classical metric as the best representation of quantum space—see [Rovelli, 2004] for details. Like the metric-field substantivalist, a determined substantivalist could avail herself of the s-knots using the same kind of reasoning. I describe this move in detail in [Rickles, 2005].

upon this latter relation as showing that the identity of the point x 'tracks' the metric, so that it brings its identity over with it.

This all sounds very similar to Newton's position on the identities of space points. Recall that he claimed that if two points were to swap places they would swap their identities. This claim flowed from the fact that he was using the metric to individuate the points, and of course the metric in Newton's theory was a background structure, it was non-dynamical, so it could perform such a function. Presumably Stachel would agree with this, for all such models will be isometric. The points can be individuated independently of solving the equations of motion for the dynamical objects. But in general relativity the metric is a dynamical object, we need to solve its equations of motion, so we cannot individuate the points independently of a solution if we follow Newton's idea—and, in general, distinct models will not be isometric.

The key difference for Stachel lies in what he calls an "individuating field"— a set of properties that can serve to uniquely distinguish the points of a manifold. In the case of Newton's theory, as in general relativity, the metric allows for an in-dividuation of the points of the manifold. However, in the former case the metric is non-dynamical, so it can individuate independently of solving any equations. In this case the metric is what Stachel calls a "non-dynamical individuating field". Not so in general relativity, for the metric is dynamical. One cannot use the metric to individuate until one has solved for it. In this case we have a "dynamical individuating field". It doesn't have to be the metric that performs the function; a physical matter field might do the job just as well. But if we deal with vacuum general relativity then such an option is clearly unavailable and the metric must functions as the unique individuator of points. This goes back to my point in §1.1 that it is background independence that distinguishes general relativity from pre-GR theories of spacetime.

Stachel's response is directed as much at Einstein's hole argument as it is at Earman and Norton's. According to Stachel, manifold substantivalism is tantamount to the possibility of being able to provide a "kinematical coordinatization" of spacetime. The lesson of the hole argument for Stachel is that such a possibility leads to an ill-posed Cauchy problem for general relativity. Since it was manifold substantivalism that allowed for the coordinatization, it should be rejected for the reasons advanced by Earman and Norton. Stachel puts the point as follows:

If the points of the manifold were physically distinguished kinematically (i.e., independently of the solutions to the field equations), we should have to regard these solutions [hole diffeomorphs] as physically distinct. ([Stachel, 2003], p. 24)

Of course, it is just this excess that leads to the indeterminism. But by using the metric field to individuate the points, the indeterminism is eradicated, for the points cannot be labeled before a solution, they don't 'sit' under the metric waiting for a redistribution, they 'come about' in virtue of a certain distribution, and diffeomorphic distributions carry the points identities over. In particular, it makes no sense to speak of the same point existing in different models and worlds. The Kripkean rigid naming of points cannot operate in this context because the refer ents of the names get their individuality from the structure of the dynamic metric field.

Now, Stachel calls this relationalism, for he sees spacetime points as coming into existence relationally. This is clearly a further step from supposing that they are preexisting, but somehow indeterminate (more or less Hoefer's suggestion for the substantivalist). But it is important to note that they are not brought into existence by relations between material objects. They are brought about by relations between parts of the metric field; as Stachel puts it (echoing Einstein):

until a metric tensor field is specified there is no spacetime ...even the topology of the differentiable manifold associated with a model of the theory cannot be specified a priori, but must be chosen so that it is compatible with the metric of that model. ([Stachel, 1993], p. 143)

Stachel isn't explicit as to exactly how the points are individuated by the metric. He mentions that one might use the values of the four invariant scalar fields built from contractions of the Weyl tensor, but leaves out the exact details.180 We saw earlier that there are substantivalists who would like to help themselves to the metric field as forming part of substantival spacetime and applying a similar method to Stachel for understanding the points of spacetime—Hoefer's proposal can be seen along these lines. Stachel is adamant that such a position is at odds with sub-stantivalism; he admits the metric field provides the chronogeometric structure of spacetime, but he argues that they ignore "the second role of the metric tensor field [that it] generates all the gravitational field structures ...and the gravitational field is just as real a physical field as any other" ([1993], p. 144). He views the approach as relational precisely because the spacetime relations and points are derivative from the "metric-cum-gravitational field". This is verging on word mongering: the same object is being used in more or the less the same way to defend opposite positions simply by viewing at as 'space-like' or 'matter-like'. Both parties agree that the metric field is a real physical field: Stachel views this aspect as debarring it from having the status of substantival spacetime and Hoefer views this aspect as all the more in favour of viewing it as a substantival spacetime! I think that a position that eschews or collapses the opposition is better suited, for there really is no opposition here: both parties agree it is real and (ontologically) free standing! Notice that the question is no longer one concerning the reality of spacetime points, but whether or not the metric is spacetime or whether the metric is material and determines spacetime (relations and points). Hoefer opts for the former, and takes spacetime points to be individuated by the metric; Stachel opts for the latter and takes spacetime points to come into existence with the metric. We can see that there is a difference in that Hoefer views the points as existing independently of the metric (but without primitive identity) whereas Stachel does not.181 But this difference is doing no work, and it is not drawn from what general relativity tells us.

180 Dorato and Pauri [2006], have recently filled in these details. They argue, as I do, though for different reasons, that this way of individuating points suits a structuralist conception of spacetime, rather than relationalism as Stachel claims. I consider their proposal in the final chapter. See also the subsection on Saunders' position below, §5.3.3.

181 In fact, it isn't entirely clear that Hoefer endorses this form of substantivalism; presumably he would see the fact that there is always a metric field present as making this independence claim incoherent. If this is the case then the division between his and Stachel's position is ever more blurred.

general relativity can tell us nothing about the natures of the points: all we have to work with is the metric and the relations and properties it has. The question of which points play which role is inconsequential.

5.3.3 Saunders' non-reductive relationalism

The hole argument can be viewed as showing that the particle coordinates at a given time are underdetermined by the field equations; they are arbitrary functions of time—or, in other words: gauge. Likewise for the values of fields. Shifting from a coordinate dependent approach, we can couch the argument in terms of the points of the manifold so that the values of local fields or particle position are underdetermined. According to the hole argument, general relativity cannot predict such quantities. As Saunders ([2003a], p. 152) rightly points out, the hole argument—like its ancestor the Leibniz-shift argument—targets absolute quantities. The manifold is a background structure in general relativity, and position relative to it is an absolute quantity. The natural solution is to shift focus away from absolute quantities, which are not invariant under the transformations of general relativity, to those that are invariant. These happen to be relational quantities, as they were in the resolution to the Leibniz-shift argument. That is, the invariant quantities of general relativity are those that are not defined relative to the manifold, but with respect to physical fields or objects.

Now, Saunders argues that the sort of relationalism underwritten by such a response to these symmetry arguments "has nothing to do with a reductionist doctrine of space or spacetime" (ibid.), i.e. with what I have been calling rela-tionalism and what Saunders calls "eliminative relationalism". He claims further that the response can be applied to "any exact symmetry in physics". I fully concur with Saunders, and indeed much of what I have been trying to get at in my discussion of symmetries is expressed by Saunders. However, I associate the view with structuralism, whereas Saunders advocates a position that resembles a hybrid between Stachel's relationalism and sophisticated substantivalism. More contentious is Saunders' claim that this position is a "natural expression of Leibniz's . . . principle of identity of indiscernibles". This uses a 'modernized' version of PII informed by modern logic: from this version there follows Saunders' position, that he calls "non-reductive relationalism". This move forces Saunders to take a stance concerning the ontological status of those objects related by exact symmetries. I have argued that since incompatible positions can be held with respect to such objects, we should choose a neutral position. Such a position flows from my idea that physics operates with a version of PSR, such that the natures of the objects are irrelevant to the structures that possess the symmetries.

Saunders' method is quite ingenious. He argues that Leibniz was led to his eliminative relationalism because of the logic of his time, based as it was on the notion that propositions were of subject-predicate form. When relations are considered, the proposition is still taken to be of subject-predicate form, and applies to a single subject. The relations had to be reduced to monadic properties of their relata. This view of relations naturally underwrites what Saunders calls "Leibniz's independence thesis", the claim that a description of a thing should be intrinsic, containing no reference to other things or relations (ibid., p. 13). Now Saunders points out that when we consider Frege's logic there is no such privileging of predicates, or "1-place concepts" in the terminology of Frege's Begriffschrift, with its distinction between 'object' and 'concept'. Relations are free standing and propositions aren't restricted to subject-predicate form. Saunders then examines how this shift in logic affects PII. Firstly, he notes that if one deals solely in 1-place predicates then PII says that objects with exactly the same properties are (numerically) identical. Adding higher-order predicates into one's language weakens the principle since then PII says that objects with exactly the same properties and relations are identical—there is another level of 'similarity' the objects have to satisfy. This gives us the strong and weak forms of PII respectively; clearly, Leibniz's logic forced him to endorse the strong form, and it is this overly stringent form that lends itself to easy counterexamples. Working with relations and adding identity to our language, Saunders presents an axiom schema formalising the indiscerni-bility of identicals as follows (ibid., p. 18):

This schema implies that terms with the same reference can be substituted salva veritae. Now Saunders (ibid., p. 19) proceeds to give a definition of identity182 using only terms 'x' and 'y', and unary predicates A (i.e. properties), binary predicates B (i.e. relations), up to n-ary predicates P (i.e. higher order relations), such that x = y iff:

A(x) ^ A(y) B(x, Mi) ^ B(y, Mi), B(mi, x) ^ B(u1,y)

P(x, u1,..., un-1) ^ P(y, u1,..., un-1), and permutations.

The definition simply says that two things are identical if they match up on properties and relations. The relation conditions are defined so that, whatever relations x stands in (for some free variable: u1 in the binary case u1 to un-1 in the n-ary case) y stands in too. I mentioned that for languages with only 1-place predicates one gets a strong principle of identity, such that: [3F, (Fx A —Fy) ^ (x = y)]. For more general languages, admitting higher order predicates, Saunders distinguishes three ways to get x = y, i.e. non-identity (ibid., pp. 19-20). Firstly, he says that two objects are "absolutely discernible" if there is a formula (e.g. P(z, u1,..., un-1)) with some free variable ui that applies to one, x say, but not the other, y. In which case x = y. Secondly, two objects are "relatively discernible" if there is a formula in two free variables (e.g. P(z, u1,..., u2)) that applies to x and y only in one order.183 Thirdly, two objects are "weakly discernible" if B(x, y) is true, B is a symmetric predicate (i.e. B(x, y) iff B(y, x)), and B is irreflexive, so that B(x, x) is always false—

182 He credits the definition to Hilbert and Bernays, and notes that it has been defended by Quine.

183 An obvious example is the 'taller than' relation: Joseph is taller than Dean, but Dean isn't taller than Joseph, hence, Joseph and Dean bear a different relation to one another. Clearly, asymmetry is at the root of this case of non-identity.

this counts as non-identity according to the definition because it implies that there is a u1 such that B(u1, x) is true and B(u1, y) is false, namely for u1 = y.184

This is Saunders' modernised version of PII: "objects are numerically distinct only if absolutely, relatively, or weakly distinct" (ibid., p. 20). With this definition of identity, and with his ways of getting non-identity, Saunders is able to show that the standard counterexamples to PII are in fact examples of weak discernibles, and so do not violate his PII. Black's two qualitatively identical iron spheres in empty space are weakly discernible according to Saunders' account because there exists an irreflexive (distance) relation. Symmetry and qualitative identity are not sufficient to secure indistinguishability, though they are clearly necessary. He notes that there is a counterexample to it in the form of two or more bosons in exactly the same state; so PII is neither necessary nor contingent, it still faces difficulties in quantum mechanics. Though it can accommodate fermions, for even in the most symmetrical scenario—"where the spatial part has exact spherical symmetry, and the spin state is spherically symmetric too" (ibid.)—the fermions will satisfy the relation of having opposite component of spin to one another but not to themselves. This is clearly irreflexive and so any two fermions will be weakly discernible. Saunders draws metaphysical conclusions from the violation of his PII by bosons. He advocates a non-individualistic view according to which bosons are modes or excitations of a gauge field (with the exception of the Higgs boson). Note that given his PII, it is not possible to advocate the 'state restriction' view whereby bosons are individuals whose wave-functions are subject to symmetrization as an initial condition. Now, I am quite sympathetic to this view for it makes a principled distinction between 'matter' and 'force', a difference that seems to occur in nature, but which is conflated on most other conceptions of quantum particles (cf. Saunders [2003b], pp. 294-5). However, Saunders continues to refer to them as "objects", even though he claims that "one cannot refer to any one of them singly", and suggests that they be called "referentially indeterminate" (ibid.). I think as far as bosons go, he'd do better to drop all talk of objects at a fundamental level entirely, possibly in favour of pure structure. The latter might seem vaguer, but at least one can refer to it singly and determinately! As pointed out in French & Rickles ([2003], p. 228), the non-reductive nature of this form of relationalism sits well with the structuralist notion of individuation of relata by relations, according to which the relata do not have ontological primacy over relations but are understood in terms of "intersections of relations".185

Saunders puts his modernised version of PII to work in the context of spacetime theory. The position that results is, as he notes, similar to Hoefer's position; I think it is similar to Stachel's too. His claim is that although spatial points in a homogeneous space cannot be absolutely discerned by that subset of properties that apply to them (for they will share these properties), they can be discerned by relations to material objects and events (ibid., p. 23). This is where he differs

184 An obvious example here is to choose B as a distance relation between two objects: Steve is 10 meters away from Dean, and Dean is 10 meters away from Steve (so they are not relatively discernible), but Dean is not 10 meters away from himself, and neither is Steve.

185 I take this up again in the final chapter, §9, where I argue for a position that I call 'minimal structuralism', based upon the Eddingtonian idea that neither relations nor relata should be seen as ontologically fundamental.

from Hoefer: Saunders points out that his notion of referential indeterminacy corresponds to Hoefer's idea that primitive identity fails for spatial points. It follows that there is no hole-type indeterminism or shift-type underdetermination regarding which points are occupied by a system of matter for "there is no fact of the matter as to which point is occupied" (ibid., p. 25; see also Saunders [2003b], p. 304). However, Saunders claims that the points referred to by a matter distribution can be uniquely picked out, contra Hoefer. The reason: Hoefer denies PII and embraces Leibniz's independence thesis. Hence, Hoefer calls his position 'substantivalist' whereas Saunders calls his 'relationalist'. Hence, Saunders is able to avoid the hole argument by endorsing LE; he believes in spacetime points but in a way similar to Hoefer (they are not absolutely discernible, and do not possess primitive identity, but they are weakly discernible so that empty spaces are possible); appealing to his PII, however, relations to matter can serve to individuate points uniquely. The hole and shift arguments are avoided, but the symmetries are retained. Hence, the relationalism refers to the relations between spacetime points and matter, not to the reductive doctrine that deals only in relations between material objects. In this way, Saunders can be seen as providing a response to Grunbaum's problem of individuating points in homogeneous empty spaces; he writes that:

If there are no extra-geochronometric physical entities to specify (individuate) the homogeneous elements of space-time ...then whence do these elements of otherwise equivalent punctal constitution derive their individual identities? ...I see no answer to this question as to the principle of individuation here within the framework of the ontology of Leibnizian identity of indiscernibles. Nor do I know of any other ontology which provides an intelligible answer to this particular problem of individuating avowedly homogeneous individuals. ([Grunbaum, 1970], p. 587)

Hoefer, of course, simply denied the PII and denied primitive identity, in effect agreeing with Grunbaum as regards the bare manifold. But Hoefer doesn't see that as representing spacetime at all. Saunders, however, has a clear response even for the case of a bare manifold: the points are weakly discernible and so distinct individuals. Clearly, we can simply disagree that Grunbaum is correct in speaking of the points of a manifold as representing spacetime; we can say that the metric field either is spacetime, is part of spacetime, or can serve to individuate the points of the manifold. We have seen that Hoefer opts for the former, Maudlin for the middle option, and Stachel for the latter option. Saunders' position is more subtle, as we have seen.

Saunders considers the Bergmann-Komar idea of using the four non-vanishing invariants of the Riemann tensor to individuate points, as did Stachel (see Bergmann & Komar [1972]). For non-symmetric cases, this procedure will give four real numbers that (hopefully!) differ at distinct points of the manifold. Macroscopic objects, fields and so on, can then be referred to their values, so that for the four invariants ($1, $2, $3, $4), we can speak of the field p having the value X at the point ($1, $2, $3, $4): P($1, $2, $3, $4) = X. Saunders implicitly implies that this works for the case of general relativity, but he rejects it on the grounds that it fails in the Newtonian theory of gravity, for the invariants will be constant along the integral curves of the Killing vector fields. The values of the invariants will then be the same for all points of an orbit of the symmetry group. One must refer to matter again to individuate the points (cf. Stachel [1993], p. 143). Recall that this is slightly similar to the problem of symmetries of the metric faced by Maudlin.

This brings me to a serious problem with Saunders' approach, and with those approaches that use the Bergmann-Komar procedure in general. Classically, the method works, I admit that: relations either to matter or the invariants of the metric can serve to individuate points. However, when we turn to quantum theories, of matter and gravity, we face a problem: the matter and metric fields will be quantized. If we want to individuate the points by the value of some fields, then these fields had better have well defined values. But, generically, they won't in a quantum theory. We could only get a well defined value for the relevant observable if we measured the field. This would require a position whereby the points of space are individuated by making whatever one wishes to use to individuate the points interact with another thing. This is relationalism one level up from Saunders'. I don't say this isn't workable, but it means that if we take the points to be individuated in general by matter or metric, then in non-interacting situations, the points will still be indeterminate. Saunders will still be able to make use of his PII to ground the identities of the points of space, but this will be dependent upon an interaction. That strikes me as a strange position to hold. Again, I say, better to do away with points as fundamental existents a la Saunders, and if one thinks that they are indispensable, to view them as nodes in a network of relations. There will still be quantumness, be we don't view this as applying to independent points: material and points are part of one and the same network.186 I should point out that Saunders does appear to switch to something like this view ([2003b], p. 305); but insists on clinging to a "thin" notion of object. Neither objects nor relations are given full ontological weight over the other; rather, they are on all fours—hence the sobriquet "non-reductive" to describe this form of relationalism. We need to ask just what work is being done by this thin notion of object; indeed, it is so thin as to be almost worthless. Perhaps Saunders is merely concerned to have a framework that accommodates many types of object, and to give them a satisfactory treatment: from spacetime points and quantum particles to cabbages and kings. Fair enough, but if this work can be done without them—using the relational structure and deriving 'objects' from it—then why claim that the objects are to be included in our ontology as well?

186 I suppose there is the possibility of shifting to an interpretation of quantum theory that views position as determinate, such as the Bohm interpretation. Or shifting to an alternative conception of logic, such as quaset theory (this option was suggested to me by Dalla Chiara, private communication). These will have to be shown to be compatible with quantum general relativity: the Bohmian proposal faces problems here since the metric will be quantized, and a classical metric determines certain crucial features of the formalism of that approach. Knowing that Saunders' preferred interpretation is the relative state theory, I would be interested to find out how this would apply in this context. However, this would take me too far afield.

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