## Three Types Of Structuralism

Most flavours of structuralism are united on the point that relations are important; where they differ is on the issue of the extent of their importance. We have seen in our discussion of the hole argument that many responses work precisely by invoking some relations; not all of these were relationalist, strictly speaking. However, much of the division between relationalists and those who call themselves structuralists, and indeed substantivalists, turns on the question of the status of the metric field: is it spacetime or is it matter? Putting this subtle matter aside, I think it is fair to say that the three structuralist positions I consider below see themselves as advocating an alternative position to both relationalism and sub-stantivalism. This they do in quite different ways: I take Stein as advocating an 'even handed' position with respect to matter and space, taking both to be different manifestations of the same totality; French largely follows suit, though he

270 One might suggest that Ockham's Razor—not to multiply types of entity beyond necessity—provides a clear cut basis. However, that is not an internal basis and is itself steeped in metaphysical presuppositions. Regardless of this, it is not clear what types of entity are necessary. Necessary how? Empirically, modally, what?

271 For example, this matches Shapiro's notion of "realism in ontology" ([1993], p. 455). Perhaps some confusion arises because of an attempted assimilation of the spacetime debate to the debate between scientific realists and anti-realists (see Boyd [1984] for a 'four-tier' set of conditions that have to be met in order to count as scientific realism). I have shown why I think the two should be held apart.

sees his position as involving the elimination of objects (traditionally conceived) from his ontology; Dorato et al. adopt a 'hybrid' view combining aspects of Hoe-fer's metric field substantivalism with aspects of Stachel's relationalism and that retains objects. I expand on these summaries in what follows and show where each position fails.

Stein is concerned to detach the ontology of physics from the mathematical representations used to model physical systems. Thus, although the setting up of the spacetime models requires that we distinguish levels of mathematical structure (point sets, topologies, manifolds, metrics, etc.), we should not read this as implying levels of ontological priority, or as implying any ontological division at all. Indeed, the models of general relativity, and spacetime theories, simply represent some spatiotemporal structure; or, rather, a substructure or "aspect of the structure of the world" ([1977], p. 397). That is, the fact that we model spacetime by setting up divisions between manifolds and metrics does not thereby imply that the two types of object represent some distinct physical objects standing in some relation to each other. This simple, and I think correct point underlies much of Stein's structuralist position. However, he is ambiguous on the scope of this structuralist position; for example, in his seminal paper on Newtonian spacetime, where he discusses the rough usage of the terms "ideal" and "real" by Alexander and Toul-min ([Stein, 1967], p. 276), he makes the following remarks (often wheeled as the example par excellence of spacetime structuralism by structural realists):

What exactly do these authors mean by "ideal entities which it is helpful to consider in theory", or by a notion of theory that "has a physical application",—as opposed to entities that "exist in reality," or to "the objective existence of a cosmic substratum"? If the distinction between inertial frames of reference and those which are not inertial is a distinction that has a real application to the world; that is, if the structure [Newtonian spacetime] ...is in some sense exhibited by the world of events; and if this structure can legitimately be regarded as an explication of Newton's "absolute space and time"; then the question whether, in addition to characterizing the world in just the indicated sense, this structure of space-time also "really exists," surely seems to be supererogatory. [.] [T]he notion of structure of space-time cannot, in so far as it is truly applicable to the physical world, be regarded as a mere conceptual tool to be used from time to time as convenience dictates. For there is only one physical world; and if it has the postulated structure, that structure is—by hypothesis—there, once and for all. if it is not there once and for all ...then it is not there at all; although of course it may still be ...that a structure is there that approximates, in some sense, to the postulated one. ([Stein, 1967], p. 277)

Dorato reads Stein as "claiming that the traditional dispute between substan-tivalism and relationalism is completely analogous to that between realism and antirealism ...: neither position is tenable" ([2000], p. 1614). Realism and antire-alism about spacetime are then to be cashed out as a denial that "the world of events "really exhibits" a certain geometrical structure" and the claim that "the spatiotemporal structure "really exist[s]", [in the sense of] independent existence"

antirespectively (ibid.). He then takes Stein as seeing the (substantivalist) realist's claim of 'reality' as "supererogatory" and the (relationalist) denial as simply false. Dorato is unhappy with this reasoning, and well he should be, but I think that it is in fact a misreading of Stein's original paper.272 It isn't that Stein sees the relationalist position as false; rather, he sees that position (and substantivalism) in a different light from most authors (in particular, from Alexander and Toulmin). Thus, he takes both Leibniz qua relationalist and Newton qua substantivalist as equal on the existence on the reality of space and time; or, as he puts it on "the reality of space and time as an objective framework of the phenomenal universe" ([1967], p. 277). Where they differ boils down to the nature of the relations that constitute their structure. For Leibniz the relations are grounded in "what actually is ...[and] anything that could be put it its place" (that is, actual and possible material bodies), whereas for Newton "the relations that constitute space and give its parts their individuality are . . . internal relations"; and unlike Leibniz "Newton ...is content to postulate the entire structure of space, without attempting to derive it from or ground it in the relations of non-spatial entities" (ibid., p. 278). Thus, I have to disagree with Dorato: there is no analogy to be made between the realism vs anti-realism and substantivalism vs relationalism debates (at least, not in this piece of Stein's opus).

French sees elements of Hoefer's metric field substantivalism in Stein's structuralism; and sees the structural characterization of spacetime points concomitant with this view as motivating a general denial of haecceitism, and so as offering a response to the hole argument. French endorses this structuralist line, but he takes such an implication as pointing to the acceptance of Leibniz equivalence ([2001], p. 27)273—of course, this is taken by Earman and Norton as eo ipso implying relationalism. Yet he claims that this does not amount to relationalism, for there is no reduction of the spacetime manifold to relations between material bodies. Rather, French sees the view that results as according "ontological status"274 to the spacetime relations themselves and he concludes that one should be realist about "the relevant structures" (ibid.). It is these structures that "we should be realist about ...and it is to them that we should direct our philosophical attention" (ibid., pp. 27-8). However, French is not forthcoming on the question of what form these structure take on: just what exactly are the "relevant structures"? Presumably they are things composed of the relations he mentions; but neither is he clear about what the "relevant relations" are. This is an important point if we wish to assess the relationship between this view and relationalism and substantivalism, and to properly distinguish it from these latter positions. I disagree too with the claim that it is relations that get assigned the ontological weight in spacetime theories; at least, I cannot see how this could be drawn from the physics in any case. Moreover, it goes clean against the basic structuralist ideal that a primitive notion

272 Note that Dorato bases his reading on a much later paper of Stein's [Stein, 1989]. I don't think the two positions match in the way suggested by Dorato.

273 Note, however, that Cover and O'Leary-Hawthorne [1996] argue, convincingly, that PII (underpinning Leibniz equivalence, of course) does not go hand-in-hand with anti-haecceitism. Indeed, the anti-haecceitist simply cannot make any sense out of the claims of permutations of individuals that PII and Leibniz equivalence require to function at all.

274 Presumably by "ontological status" he means reality; and reality of a non-supervenient and primary sort, for many would agree that spacetime relations are elements of reality in at least some sense.

of individuality is not to be had, for the relations are assumed to be individuals of a sort, they can stand on their own so to speak. I think Eddington had the right idea: one should treat the relations and the relata as mutually dependent ontologically speaking, they come as a package deal—see, e.g. [Eddington, 1923; 1928]; see also [French, 2003] for a discussion of this aspect of Eddington's work (see, especially, p. 235). It was this move that enabled me to sidestep both the hole argument and Unruh and Kuchar's objections to the correlation view. More importantly, for the theme of this section, it sticks to the physics, for the symmetries do not allow us to attribute independent reality to either relations or relata; this is why we couldn't simply consider reduction options as self-evidently the correct move.

Dorato has outlined a position called "structural spacetime realism" that he claims is a "tertium quid ...between classical substantivalism and relationalism ...[that] sides with the latter doctrine in defending the relational nature of spacetime, but argues with the former that spacetime exists, at least in part, independently of particular physical objects and events" ([2000], pp. 1607-8). The relationalist aspect follows from Dorato's endorsement of Stachel's idea that the points of spacetime are individuated by the metric field; in his words: "spacetime points can only be identified by the relational structure provided by the gravitational field" (ibid., p. 1610). The substantivalist component is captured by the thesis that "the geometrical structure used to represent them [space and time] is "really" mind-independently exemplified by the physical world" (ibid., p. 1612). Finally, he claims that the position that results is a "synthesis" of substantivalism and re-lationalism in the sense that the "metric field is both matter and spacetime" (ibid.). Now, from these few passages it is quite clear that Dorato is defending something very different from both French and Stein. The details are spelt out in a more recent paper written with Pauri [Dorato and Pauri, 2006] in which they attempt to provide a more explicit defense of structural spacetime realism by essentially mixing elements of Hoefer's abandonment of primitive identity for spacetime points with Stachel's idea of the dynamical individuation of points—where, for the latter idea, they implement Bergmann and Komar's idea of using the four (scalar) eigenvalues of the Riemann tensor to define an intrinsic coordinate system.275 They label their view "point-structuralism"; however, they note that the resulting position is "entity-realist" as regards "both the metric field and its point-events" (ibid., p. 123). This suggests that their position is in fact simply Hoefer's metric field substanti-valism in disguise. The similarity is indeed close, as the following passage makes clear:

we believe that it is not at all clear whether Leibniz equivalence really grinds corn for the relationalist's mill, since the spacetime substantivalist can always ask: (1) why on earth should we identify physical spacetime with the bare manifold deprived of the metric field? (2) Why should we assume that the points of the mathematical manifold have intrinsic physical identity independently of the metric field? ([Dorato and Pauri, 2006], p. 128)

275 Recall that Saunders too considered the possibility of points admitting determinate reference by being localized in the manner of Bergmann and Komar's suggestion.

The first question is common to Stein's and French's structuralist approaches, and we saw in §4.1 that most substantivalists about the spacetime of general relativity would distance themselves from simple manifold realism; Hoefer too concluded that the metric field was an essential part of spacetime and would have to play a part in any defensible substantivalism. The second question paves the way for sophisticated substantivalist positions, and we have seen variations in Hoefer, Saunders, and Stachel, though each sees the question as pointing in a different di-rection.276 However, Dorato and Pauri are concerned to show how a coherent and robust notion of spacetime point can emerge from the formalism of general relativity. The agreement with Hoefer goes as far as metric field realism and the denial of primitive identity; the divergence of positions springs from the fact that Dorato and Pauri specify how the points get their individuality from the metric field and it here that the position becomes formally analogous to Stachel's. Thus, the idea is to use the metric as an individuating field for the spacetime points. Clearly, however, since the metric is not itself invariant, we need to extract the invariant information from its ten physical components. It turns out, as I have mentioned earlier, that a set of four invariant scalars277 can be constructed and Bergmann and Komar's idea was to use these scalars to produce a kind of invariant chart for a vacuum solution of the field equations. Any quantities localized to the so defined points will be diffeomorphism-invariant, as will the points themselves, and so the hole argument problem is resolved at a stroke.

Pauri & Vallisneri also follow this Stachelian view: "it is impossible to consider the points of the space-time manifold as physically individuated without recourse to dynamical individuating fields" ([2002], p. 1). They argue that the metric field serves the latter role by separating out the "metricalfingerprint of point-events from the gauge variant components" (ibid.). Again, we find Bergmann and Komar's idea involving the intrinsic set of coordinates constructed out of the curvature invariants. However, they ground the necessity of dynamical individuating fields in the idea that "the points of a homogeneous space cannot have any intrinsic individuality" and support this with a quote from Weyl to the effect that such points do not have any distinguishing objective properties with respect to which they could be distinguished (ibid., p. 4). They are quite clearly conflating the inability to distinguish and the lack of individuality. However, the two notions are quite distinct, as Saunders as brought out well in his analysis of identity.278 The conflation is a common one in the literature on the hole argument (and, indeed, elsewhere); however, it need not detain us from the main thrust of their program. The technical details are based on Pauri & Lusanna's extension of the Bergmann-Komar approach to the Hamiltonian formalism (as developed elsewhere in this book)—see [Lusanna and Pauri, 2002]. In moving to this formalism the distinction between gauge vari

276 In brief: Hoefer denies primitive identity for the points and claims that the metric field individuates them; Saunders claims that relations to other points individuates them and relations to matter and the metric field can allow them to be referred to; Stachel, by contrast, sees the points and the metric as entangled so that not only are the points individuated by the metric field but without a metric field there are no points.

277 These invariant scalars are eigenvalues of the Weyl tensor Cabcd, where the Weyl tensor is the trace free part of the

Riemann tensor defined by: Rabcd = Cabcd + n-l (ga[cRd]b - gb[cRd]a) - (n-1)2(n-2) Rgga[cgd]b (where n is the dimension of the manifold—see Wald [1984], p. 40).

278 Substrata theorists or defenders of haecceities will also clearly wish to reject this conflation.

ant and independent variables becomes apparent, and the latter are vital in the construction of the individuating fields. However, the details of their approach fall foul of my arguments from the previous section. In particular, they make essential use of the Shanmugadhasan transformation (Shanmugadhasan [1973]), which induces a gauge fixing such that the Dirac observables are restricted to the reduced space—in fact, they form a Darboux basis for the space (Pauri & Vallisneri [2002], p. 14). In projecting from the constraint surface to the reduced space they are making a decision as to which states and observables are physically real, a decision that is not read off the physics.

The first problem with these 'Italian' structuralist approaches is that, as with any Bergmann-Komar style approach, it is required that the spacetime admits no non-trivial symmetries—in order to work, the spacetime must be inhomogeneous and asymmetric to force the functional independence of the scalars. Only if this condition is satisfied can the four scalars be constructed.279 A further problem is that I mentioned in relation to Saunders' non-reductive relationalism. If we consider the scalars to be 'physical' then they will surely be among the things that are quantized. If they are indeed quantized than it is difficult to make sense of the coordinate system. If they are not quantized, then it is difficult to make sense of it as a physical thing, for it will be dynamically decoupled from any quantum fields. Thus, the transition of this framework to quantum theory will most likely lead to an incoherent conceptual structure (if not in fact technical structure). The problem is that, since the metric field will become a quantum operator in quantum gravity, the scalars will undergo fluctuations (the six gauge variant degrees of freedom will not, of course: they get erased). If these scalars are the metrical fingerprint then quantization will inevitably smudge them, and the notion of a local event (with respect to these scalars) is lost. Thirdly, even if we ignore the complications that quantum theory will bring, we have the result that this approach is a 'reduced space' approach and this means that all of the interpretations will be able to access it.280 Thus, it cannot be seen as offering unique support to structuralism.

In slightly different ways, each of the authors discussed above make the same mistake Belot makes: they assume that there is a connection between possibility counting and spacetime ontology. Since there is some connection between phase spaces and possibility counting, the issue of spacetime ontology is seen to be tied to phase spaces. It is true that if one desires an 'all out' anti-haecceitism, as, for example, French appears to, then the reduced space is the most obvious space to choose: the symmetries have been removed and it was just these that lead to the problems concerning individuals and indistinguishables. But this space is compatible with structuralism, relationalism, and substantivalism alike. Clearly, when mutually incompatible interpretations can occupy the same space, that space is not relevant in the characterization of these interpretations. But the underlying motivation for French is not the elimination of haecceitism per se, he sees this as

279 This might, of course, be met in physically realistic models; but as a constraint on interpretation, it is, I fear, too stringent. To make an interpretation dependent on these features when the theory it is an interpretation of admits symmetries unproblematically is surely missing something about that theory.

280 Note that Lusanna explicitly assumes that the gauge invariant view of observables requires the reduced space: "Leibnitz [sic.] equivalence is nothing else than the selection of the gauge-invariant observables" ([2003], p. 6). I hope I have said enough in the previous section to dispel this common myth.

part and parcel of his position; rather, he draws on the underdetermination of metaphysics by physics that, as we have seen, is rife in considerations of symmetry in physical theories. I go along with French's insistence on the importance of this underdetermination for the question of ontology; however, I think it is deeper than French gives it credit for. Just as both 'standard' sides of debates concerning ontology are underdetermined—i.e. substantivalism/relationalism; particles as individuals vs non-individuals, etc.—so structuralism in French's sense is implicated in this underdetermination: a fourth way is needed.281 One's structuralism should be grounded in the cause of the underdetermination, rather than in the underdeter-mination itself; and that cause is symmetry. I the next section I attempt to pinpoint where the stumbling block is in these approaches before explicating a more honest form of structuralism that respects these points.

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