Four Views On Reduction

Belot believes that the existence of spacetime points is bound up with possibility counting. For example, he claims that

[s]ubstantivalists count each possible embedding of a set of N particles into R3 as (being capable of) representing a distinct possibility—which is just to say that they will work with the standard 6N dimensional phase space when constructing mechanical theories. Relationalists about space will deny that embedding related by rigid motions can represent distinct possibilities; so they will identify points in the standard configuration space so related; thus they will employ that 3N — 6 dimensional configuration

254 The subject of the ontological status of ghostly variables is in need of investigation. Physicists are often ambiguous on the matter of their physical status, alternating between viewing them as a heuristic crutch and having direct physical significance (see, for example, Henneaux & Teitelboim [1992], p. 166 and Ch. 11). Unfortunately, this issue is too complicated to tackle here—see Weingard [1988] for a detailed analysis based on connections between the interpretation of ghost fields and virtual particles.

space (parameterized by the relative distances) and the 6N - 12 dimensional phase space (parameterized by ...relative distances and velocities). ([Belot, 2000], p. 580)

Thus, according to Belot, substantivalism is bound to the extended phase space (high possibility count) and relationalism is bound to the reduced phase space (low possibility count). And his argument is that if one moves to the reduced space—as he believes one generally should—then one is committed to the non-existence of spacetime points. But we have seen that the proposed connection between possibility counting and spacetime ontology is based upon a hidden assumption about modality: the substantivalist-extended space connection requires haecceitism and the relationalism-reduced space connection requires anti-haecceitism. Thus, possibility counting has got nothing to do with spacetime ontology; it is the intrusion of modality that underwrites the supposed connection to possibility counting and particular representation spaces. The argument I presented in Chapter 2 involved showing how both the substantivalist and the relationalist could occupy both deflated and inflated possibility spaces respectively. I return to this argument again in §7.6, where I use it to dampen the expectation that quantum gravity might be decisive in matters of interpretation as regards spacetime ontology.

We need to concede, however, that the distinction between the reduced and extended spaces is connected to differences in possibility counting: the latter contains points that simply are not contained in the former. But the latter can, with suitable contortions, incorporate anti-haecceitistic possibility counting; so too can the reduced space incorporate the haecceitistic possibility counting of the extended space. This is simply a result of the formal and empirical equivalence of the formulations. Given this equivalence, what are the reasons for choosing one over the other? Belot argues for reduction along the following lines:

The trick is to allow the absolutist to specify a large space of possibilities which fall into equivalence classes ...The advocate of PSR can then claim that the true space of possibilities arises by identifying equivalent absolutist possibilities, so that there is exactly one possibility corresponding to each of the absolutist's equivalence classes. ...we can always use this trick to protect PSR against refutation by Clarke's sort of examples, where indifferent possibilities are generated by the application of symmetries. ([Belot, 2001], p. 4)

Thus, as Belot notes, a direct interpretation of a theory with symmetries (of the relevant kind) will risk violating PSR and his answer is to shift to a reductive interpretation that puts orbits in to a direct correspondence with possibilities: "by always choosing interpretations ...which "factor out" symmetries ...we can ensure that our interpretations will always respect PSR" (ibid., p. 7).255 With this

255 Belot claims that "the techniques and results of this literature [on symmetry in geometrical mechanics] promise to offer a unifying perspective on a number of classic problems in philosophy of physics (the relation between the nature of space and the nature of motion in Newtonian physics, identical particles, the nature and significance of gauge freedom and general covariance)". I agree with this statement as it stands, but Belot takes the claim too far and attempts to create alignments between philosophical stances regarding the nature of individuals and the treatment of symmetries in the areas

I don't disagree: we know—given the formal and empirical equivalence of the reduced and extended spaces—that we will always have the option of shifting to the reduced space (at least in principle) and we know that this space will have any points related by symmetries removed; since these points were responsible for the potential violations of PSR, we will indeed have resolved the difficulty. The technical foundations of Belot's proposal are impeccable, as one would expect. However, the question is whether this approach is necessary and, if not as I have been arguing, whether it is worth the various technical pitfalls that such approaches inevitably must face—i.e. the difficulties with construction mentioned above.256 This is not to mention the chunk of possibility space that we will have jettisoned without any good physical reasons! In other words, the decision to reduce in the manner suggested by Belot is a purely metaphysical decision that, quite literally, makes worlds of difference.

There are two further 'technical' problems with Belot's proposal: (1) rarely do we construct 'intrinsic' reduced phase spaces for theories, generally beginning with the extended space with symmetries and then factoring them out; (2) although the extended and reduced spaces are classically equivalent, they in fact lead to distinct quantizations, and so physics might be decisive in choosing one over the other. The first point is simply that if Belot's PSR wielding theorist is to hold his head up high, he should be able to construct the reduced space form of a theory directly; as he points out himself, "the reduced theory knows where it came from" ([2001], p. 14).257 The second point is more complicated and arises out of studies at the intersection of geometric mechanics and quantization (see Gotay [1984] or Plyushchay & Razumov [1995] for details). The upshot, however, is simply that the choice between extended and reduced spaces cannot simply be a matter of policy. As to Belot's underlying desire to show how PSR can always be protected by imposing PII on symmetrical worlds, there is another option that always works too: one simply views the symmetries as expressing an indifference concerning the states and observables of physical systems entering into them (i.e. along the lines of Brighouse's anti-haecceitism concerning physics). Thus, there is always a sufficient reason for the world's being where it is in a universe with a homogenous spacetime: the world is indifferent to where it placed; one position is as good as any other! Given this rather obvious possibility, Belot's account seems to be somewhat unmotivated. Further, his account faces a serious problem when one considers quantization, for certain states factored out via Belot's method might be required to fluctuate in the quantum theory.258 Saunders offers an alternative defense of the PSR based on his idea that the individuals entering into symmetrical of physics he mentions. The equivalences and underdetermination I have shown to hold in such contexts outlaws such alignments.

256 There are also the problems—mentioned by Belot ([2003a], p. 407)—concerning the ad hoc removal of certain points (those representing symmetrical configurations and those representing collision points) from the extended phase space in order that the reduced phase space can be constructed. Technical details involving differences between discrete and continuous symmetries are crucial here—see Belot [2003b] for more details.

257 Compare this with Earman's point that the relationalist should be able to construct his theories in relationally pure vocabulary, rather than hitching a ride on the substantivalists formulations ([1989], p. 135).

258 Belot is clearly well aware of this, of course (cf. [2003b], p. 221); indeed, it informs his and Earman's taxonomy of interpretations of general relativity. (See the end of this section for further details of this problem and §7.6 for a critique of Belot and Earman's taxonomy.) Note that Belot seems to shift to the view that one must await an answer from quantum theory to the question of how best to deal with symmetry [2003b]. However, if quantum statistical mechanics is anything relations of the kind we are interested in will be weakly discernible (and absolutely indiscernible) but referentially indeterminate; the symmetries fail to get their teeth into the PSR. However, I argue that the end result, as regards the question of reduction, is the same as with Belot's proposal: there are no indiscernible possibilities.

We have already seen Saunders' view at work in the context of the hole argument. However, as Saunders points out himself, the view that emerges can be applied to "any exact symmetry in physics" ([2003a], p. 153). Saunders calls his view 'relationalism', but he sharply distinguishes this from what is usually labeled relationalism, and his main argument is that the kind of symmetry arguments I have considered in this book have "nothing to do with a reductionist doctrine of space or spacetime" (ibid.).259 Saunders calls this latter form of relationalism "eliminative relationalism" and his form "non-reductive relationalism" (ibid.). Let us recall Leibniz's original shift argument. This was supposed to cause problems for PSR: space's being homogeneous, there was no reason why a system should be located at one part of space rather than some other. I suggested that Leibniz's use of PII can be thought of as rescuing PSR from the grip of the argument (much as Belot suggests). That does not mean PII is ad hoc, simply that Leibniz thought that commitment to PII was part and parcel of being committed to PSR. However, we cannot forget Leibniz's notion of object as given intrinsically, and its description as giving a 'complete concept'. Saunders calls this aspect of Leibniz's philosophy the "independence thesis": roughly, an object's identity is independent of anything else 'external' to that object. Saunders' claim is that Leibniz understood PII as entangled with the independence thesis: without the independence thesis, PII might allow external reasons to come into play in its protection of PSR and, in particular, in the individuation of the homogeneous parts of space, and thus bring the shift argument to a halt without recourse to reductive (i.e. eliminative) measures. With external reasons not playing a role, and internal reasons absent, the symmetry arguments are in clear violation of PSR. But the 'PII + independence thesis' package can be divided, and in so doing Saunders argues that a version of relationalism follows that is non-reductive precisely because it denies the independence thesis, and thus allows external factors to enter into the definition and individuation of an object. However, it remains committed to PII. I shan't go over the details again (for which, refer to §5.3.3), but simply wish to show how this scheme fits into the question of reduction vs non-reduction. The connection is clear: "relations, for Leibniz, had to be reducible—derivable from the monadic properties of their relata" (ibid. 168); when these monadic properties are equivalent so is the relational structure—the corresponding possibility space is represented by rred.

In brief, we have the following chain of reasoning leading to Saunders' view. Leibniz's relationalism involves three components: PSR, the independence thesis, and PII. The independence thesis filters into PII, and restricts the latter principle to internal factors, so that relations to other things are not to be included in the to go by, even quantum theory cannot determine the correct geometric space of the classical theory: as Huggett [1999] and French & Redhead [1988] have demonstrated, the reduced and extended formalisms are compatible with both classical and quantum theories. (Though, as I mentioned above, a lot can hang on the nature of the symmetries in question—discrete versus continuous—what goes for one type will not necessarily hold for the other.)

259 With this I am in complete agreement as my arguments from previous chapters should make clear. I differ with Saunders, however, in the way I argue for this position, and in what I think the conclusion signifies (see below).

description of an object. PSR faces trouble from the symmetry arguments, since it seems that objects related by certain symmetries count as identical in all internal respects, i.e. in all respects that matter in this case. PII, informed by the independence thesis, enters the analysis and is used to identify any such objects (points, worlds, etc.). This means that only internal (i.e. intrinsic) qualitative differences count towards numerical differences so that the differences generated by symmetries do not imply genuine physical differences. Saunders denies the independence thesis thus allowing any physical relations to individuate and, though absolute quantities—represented by e.g. gauge dependent variables—are eliminated in favour of relations between objects, his analysis still allows for spacetime points (and any weak discernibles) to be distinct and individuated. The upshot of this vis-à-vis the PSR is that the problems posed by the symmetry arguments dissolve; one uses relations to matter and events to specify points of space:

Absolute positions disappear; under the PII points in space, considered independent of their relations with other point and with material particles, all disappear. But points in space considered independent of matter, but in relation to other points in space, are perfectly discernible (albeit only weakly), for they bear non-reflexive metrical relationships with each other. There is no problem for the PSR in consequence; there is no further question as to which spatial point underlies which pattern-position, for they are only weakly discernible. ([Saunders, 2003a], p. 174)

Saunders sees this as motivating an "even-handed approach to matter and space": things from either category can serve to individuate other members from their own and from the other category (ibid., p. 176). Now, my claim was that reductive relationalism follows from the symmetry arguments considered in this book only if it is coupled to PII (construed as an anti-haecceitist principle). Saun-ders, however, claims that PII is not necessarily anti-haecceitist, nor is it necessarily reductive and, therefore, that the symmetry arguments do not imply reductive re-lationalism. His path was to deny the independence thesis and retain PII, whereas I argued both that PII wasn't a necessary part of the relationalist's position and nor was it not a part of the substantivalist's position. This latter point leads naturally into the sophisticated substantivalist positions; and, indeed, Saunders mentions the similarities between his own self-styled relationalist approach and these other self-styled non-relationalist approaches. Both approaches accept PII but the latter see reduction (i.e. identification of equivalent worlds) as concomitant with this, whereas Saunders does not; rather, he claims that spacetime points can have well defined identities in the absence of matter, and can be uniquely referred to in the presence of matter. One might think that this is even more substantivalist than sophisticated substantivalist positions! But Saunders agrees with me that it is on-tological priority that counts when it comes to the definition of these positions, and his approach is neutral on this: each can be used to individuate the other. In this sense I would say that Saunders' position is more naturally understood as a structuralist one; indeed, he ends up in more or less the place I wish to end up, but he gets there by a different route and for different reasons.260

260 I expand on the similarities and differences in the next chapter.

Thus, Saunders protects PSR by implementing but modifying PII. The result looks non-reductive, but on closer inspection the non-reductive aspect concerns objects within worlds and not worlds themselves. Since the geometric spaces correspond most closely to possibility spaces, rather than singular possible worlds and their contents, we have to inquire as to what Saunders' version of PII says about possibility space. The first thing to note is that Saunders restricts the application to worlds that have the same physical laws as our own: for different laws there may be different PIIs ([2003b], p. 297). Then, since there are no physical relations that hold between distinct possible worlds,261 Saunders' PII reduces to Leibniz's PII and we are left with what is essentially a reductive version: "Given that possible worlds bear no physical relations to one another, it follows from the PII that numerically distinct worlds will be absolutely (and in fact strongly) discernible" (ibid., p. 298). Furthermore, his discussion on the relationship between symmetry and observables shows that he in fact endorses a rather extreme reductive view. Although the PII itself does not appear anti-haecceitist or reductive at first sight, countenancing as it does physical relations, it is both of these things when applied in the context of possible worlds. Thus, although Saunders can retain spacetime points (and any weak discernibles) with his PII, he is lead to back the connection between relationalism and possibility counting that I denied in Chapter 2; according to this account, the relevant geometric space for physical theories is rred. Hence, in the final analysis, PSR is preserved a la Leibniz-Belot, simply by implementing reductive PII. However, Saunders has shown us that reduction at the level of worlds does not imply eliminativism of indiscernible entities within worlds.262 As I mentioned earlier, this is simply yet another flavour of sophisticated substan-tivalism, albeit one clothed in relationalist garments.

Rovelli has recently outlined an interpretation based on the full, unconstrained (extended) configuration space (along with its associated extended phase space). I presented some of the details of this program in the previous chapter; let me here simply address how it fits in with the concerns of the present section. Recall that Rovelli's claim was that a number of thorny problems from general relativity and quantum gravity can be cleaned up or resolved by utilizing his distinction between 'partial' and 'complete' observables. Partial observables are taken to coordinatize extended configuration space Q and complete observables coordinatize reduced phase space rred; the "predictive content" of some dynamical theory is then given by the kernel of the map f: Q x rre¿ ^ Rn. The relevant aspect from this program for this section is captured by his claim that "the extended configuration space has a direct physical interpretation, as the space of the partial observables" ([2002], p. 124013-1). This space gives the kinematics of a theory and the dynamics is given by the constraints, $(qa,pa) = 0, on the associated extended phase space T*Q. Both are invested with physicality by Rovelli. Thus, whereas, for example, Stachel argues that the kinematical state space of a background independent theory like general

261 I think Saunders' reasoning is sound on this point. He notes that a world "is a system that is physically closed" ([2003b], p. 297), and that simply means that any physical relations that hold at that world are contained in it too.

262 Belot, on the other hand, sticks to the original PII to protect PSR. He doesn't consider Saunders' version of PII, and goes along with the idea that both absolute quantities and spacetime points (more generally: the things with respect to which absolute quantities are defined) are eliminated.

relativity has no physical meaning prior to a solution (so that only the dynamical state space is invested with the power to represent; kinematics being derivative), Rovelli appears to take both kinematic and dynamical spaces as equally capable.

The view Rovelli defends has some immediate philosophical interest since it is non-reductive and yet Rovelli is a self-proclaimed relationalist. Thus, prima facie, we seem to have an instance of a break between possibility counting/geometric spaces and spacetime ontology. However, it quickly becomes evident that there is a conflict between his relationalism and his choice of representational space. As regards the former, we saw that a rather naive verificationism was responsible for Rovelli's views: only measurable things are real and since spacetime location is not measurable but relations between objects are measurable, space and time are not real but are instead defined by correlations between objects. We can agree with Rovelli that the physically measurable quantities are those that are invariant under the symmetry group of a theory, i.e. the gauge invariant quantities. It is quite another matter to then say that these are the only physically real things, that they exhaust physical reality. Clearly, in Rovelli's view, however, there are plenty of physically real objects; namely, those things entering in to relations that are not themselves measurable. Any relationalism will require a definite set of material objects to generate the required relations (particles, fields, etc.). Rovelli's view is not that there are no objects per se, but that there are no objects corresponding to those that ground absolute (non-measurable, gauge variant) quantities. Thus, he moves from the fact that we never measure position in spacetime to the non-existence of spacetime points. However, his work on partial observables suggests something very different to this rather crude verificationism. Let me develop some more of the details of this latter approach.

Rovelli distinguishes between two extremes of interpretation with respect to the formal variables of a theory for a system with constraints (I have changed the notation to suit my own):

It is sometimes claimed that the theory can only be interpreted if one finds a way to "deparameterize" the theory, namely, to select the independent variable among the variables qa. In the opposite camp, the statement is sometimes made that only variables on the physical phase space rred have a physical interpretation, and no interpretation should be associated with the variables of the extended configuration space r. ([Rovelli, 2002], p. 1240137)

By contrast, Rovelli invests elements of r and Q (including gauge variant quantities) with physical reality; indeed, elements of the latter are taken to be "the quantities with the most direct physical interpretation" (ibid.). Complete observables— i.e. the quantities we actually measure and are able to predict uniquely (i.e. Bergmann/Dirac observables)—are dynamically determined a la Stachel:

Such a quantity can be seen as a function on the space of solutions modulo all gauges. This space is the physical phase space of the theory rred. ...Any complete observable can thus be expressed as a function on rred. ([Rovelli, 2002], p. 124013-3)

Crucially, Rovelli notes that there is an equivalent description of any complete observable "as a function on the extended phase space having vanishing Poisson brackets with all first class constraints" (loc. cit.; my emphasis). Thus, we see again the formal equivalence between reduced and extended spaces even at the level of observables. In this approach, then, Rovelli distinguishes between what is observable and what there is (i.e. ontology), whereas before, in arguing for his relationalism, he assumed a direct connection between the two.

However, I think it is clear that Rovelli does not want to imbue what are physically impossible states with physical reality—that is, r isn't Rovelli's space of choice. That would clearly be crazy. Though he often speaks as if he means to endorse this 'crazy' metaphysic, we can best understand his view, I think, as being based upon the constraint surface C. Thus, he speaks of a mechanical system as being completely determined by the extended, unconstrained phase space plus a set of constraints (if necessary). We should, therefore, view the constraints as physical 'reality conditions' and only those quantities that satisfy them as invested with reality. Nonetheless, Rovelli is still avowedly realist about non-gauge invariant quantities, quantities that do not commute with the constraints (partial observables). However, he avoids any 'hole-type' problems by defining (complete) observables as functions on the reduced space, quantities that are constant along gauge orbits of the extended space. Though nowhere near as crazy as the above metaphysics involving impossible states, the position Rovelli presents is a metaphysics nonetheless. In imbuing the gauge variant quantities with physical reality he is putting in his metaphysics by hand, for it is not being read off the physics. This is quite evident from the fact that a structuralist position can also be adopted, as I demonstrated at the end of the previous chapter. Clearly, too, one could, as Rovelli did previously, adopt the view that only the complete observables are physically real; one doesn't require the reduced space for this sort of position.

Finally, we have the expansionist option. Redhead outlines such a liberal view of symmetries: "forget all about gauge symmetry in the original Yang-Mills sense, and impose BRST symmetry directly as the fundamental symmetry principle" ([2003], p. 137).263 The idea, as Redhead describes it, is to "allow non-gauge invariant quantities to enter the theory via surplus structure ...[a]nd then develop the theory by introducing still more surplus structure, such as ghost fields, anti-fields and so on" (ibid., p. 138). He claims that this is the method that is most in line with the practice of physics. He also notes that, given the mathematical nature of the surplus structure, "this [approach] leaves us with a mysterious, even mystical, Platonist-Pythagorean role for purely mathematical considerations in theoretical physics" (ibid.). However, though it may be of value in the quantum gauge field theories of the electromagnetic, electroweak, and strong forces, I don't see that it is at all applicable in the context of classical and quantum general relativity in which the gauge symmetries are directly connected to the dynamics. Even if it could be shown that the BRST method is applicable, the suggested enlargement

263 He may not actually wish to be associated with this view, it isn't fully clear from the text which of the methods he endorses. However, the fact remains that this is a possible interpretive option to take with regard to gauge symmetries. If he doesn't in fact endorse this view then let us say Redhead refers to some 'other-worldly' philosopher of physics who does endorse it.

of phase space is a purely classical affair: one reduces by the BRST operator once the quantum level has been reached. Thus, the device of BRST appears to be a purely heuristic one, and cannot be seen as underwriting any unique interpretive stance. Indeed, in the final (quantum) analysis, the resulting picture matches, more or less, the Dirac quantization methods in that reduction is carried out at the quantum level. Classically, of course (as Redhead points out), the problem is to make sense of the auxiliary variables that are employed, and this would require considerable work. In particular, I think analysis is needed on the differences between the various senses of 'surplus structure' that come into play here: ghosts, impossible states, and indiscernible states. If it can be shown that ghosts and impossible states are of the same kind then I think this would give us grounds to reject the BRST expansionist approach.

Thus, we have four diverging views on the question of whether to reduce or not: Belot argues that we should, as a matter of general practice, reduce and get rid of the symmetries;264 Saunders says that 'all out' reduction (i.e. elimination) is not necessary to get the kind of deflationary conclusion Belot wants, but nonetheless implements reduction at the level of worlds;265 Rovelli says that we should utilize the constraint surface (or, rather, the extended space plus a set of constraints); and Redhead argues that we should expand rather than reduce or constrain. We saw above and in the previous chapter that Rovelli advocates a view whereby no reduction or gauge-fixing is carried out: the extended space and the set of constraints is sufficient to determine a sensible interpretation. With this I agree, but the interpretation I give differs from both Rovelli's and Saunders' 'non-reductive' methods. I will outline this in the next two sections. Before I get to that, let me quickly sound a warning note for hasty reduction proposals, followed by a brief summary of what I hoped to have shown thus far.

It is clear that any choice of space must come about as a result of experimental confirmation; and this can only come about at the level of quantum theory. Even then, whether or not this will choice will be possible—i.e. whether it could ever be shown that a certain way of counting possibilities is the correct one—is far from obvious. On a purely conceptual level I suggest that the extended space is to be preferred over the reduced space. The extended is ontologically neutral in that it allows for large and small possibility counting in a fairly unproblematic way. It leaves intact (and manifest) properties to do with symmetries, such as covari-ance and locality. It makes no prior assumptions about what degrees of freedom should be quantized and allowed to fluctuate (cf. Plyushchay & Razumov [1995], pp. 248-9). The resolutions I defended regarding the various symmetry problems considered in this book are best expressed in the extended space, where that space is minimally taken to encode a network of relationships. The symmetries mean that only the network, and correlations expressed within the network, are observable. Claims about the nature or existence of the 'nodes' of the network (i.e. the

264 Elena Castellani [2004] too has recently defended a similar view in her analysis of Dirac's theory of gauge systems and constrained Hamiltonian systems.

265 Essentially, Saunders' argument is that the fact that PII amounts to a reductive principle when imposed at the level of the worlds themselves does not, as if often believed (by Belot, for one), imply that PII involves elimination within worlds. Denying the independence thesis allows one to individuate what would have otherwise been indiscernible entities by using the relations they bear.

symmetry operands) are radically underdetermined. The reduced phase space, of course, takes a stance on what is physically relevant, and this choice is carried over to the quantum theory. Thus, there will be elements of the extended space that will not be subject to quantum fluctuations, but will be eliminated instead. Though I think the reduced phase space can be given a well-motivated structuralist defense (encoding, as it does, the supposedly physical (invariant) structure), I think that it should be a part of the honest structuralists manifesto that stances taken regarding to the individual elements entering into gauge-type symmetries should be avoided.266 This, of course, includes Rovelli's partial observables realism.

I hope to have shown (or at least suggested) the following in this section:

• Theories are not bound to either reduced or extended spaces (they admit PII and non-PII-type formulations while still respecting PSR). Reduced and extended formulations are empirically equivalent.

• Reduced and extended spaces (with their associated possibility counting) are not bound to particular spacetime ontologies.

• The reduced and extended spaces are bound up (in a certain sense) with possibility counting: they contain different cardinalities of possibilities. But haecceitism and anti-haecceitism are nonetheless compatible with theories formulated in both spaces.

• If we choose the reduced space (without pressure from quantization) then we are cutting out possibilities in a way not dictated by the physics itself—i.e., the metaphysics of possibility counting that results is not 'read off' the physics in this case.

• Since extended spaces allow 'all the options' (conceptual elbow room, as it were) we would be better off choosing such a space as the neutral base. We should, more properly, view the constraint surface as our base, for in this case the metaphysics of outlawing physically impossible states is easily read off the physics.

I think these points allow us to infer three more conclusions: (1) that PSR (respecting, e.g., gauge invariance) does not imply PII (i.e. PSR does not link up to reduction, quotienting, or eliminativism); (2) that substantivalism and relational-ism are not linked to the denial and endorsement of PII (reduction/elimination) respectively; and (3) that theories with gauge symmetries (or similar) cannot give decisive reasons for interpretive options concerning the nature of the individual elements connected by the symmetries. In particular, symmetry arguments like the Leibniz-shift and hole arguments, and the problem of time, cannot be used

266 Thus, I diverge quite radically from French & Ladyman's 'ontic' version of structural realism (see their [2003]—I discuss this version in §9.2). The reason: they see the underdetermination as applying to the 'individualistic' and 'non-individualist' packages only, and not as involving the eliminativist views; for this reason they drop the former package entirely and opt for an ontology of pure structure (not involving objects). These latter views are most naturally expressed in the reduced space, and it is that space that the structural realist (of the ontic brand) will wish to be aligned with: it encodes all and only the invariant and, they will say, physical structure. Incidentally, I think the fact the reduced space does encode this structure, and can be associated with an elimination of objects—though this is underdetermined, of course; hence my desire to stick to the extended space—, offers a quick and easy answer to the question: 'What is structure?' It is given precisely by the variables that separate the points of this space; I suggested that these should be understood as structural (i.e. non-reducible) correlations. Moreover, the factorization (by the gauge group) that leads to the reduced space also offers a response to Cao's objection concerning the distinction between mathematical and physical structure (on which, see French & Ladyman [2003], pp. 45-6): it is that which is invariant under this group of transformations.

to defend relationalism or substantivalism.267 In the following sections I outline and defend these points more forcefully, and show how they impact on questions of ontology. The conclusion I draw is that if we want to read off ontologies from physical theories with symmetries then the best we can hope for is a structuralism; anything else involves auxiliary metaphysical assumptions. I turn to the connection between the findings of the present section, symmetry and structuralism in the next chapter.

267 Indeed, the symmetries give us an explanation of the interpretive underdetermination that occurs in theories with such symmetries. Since a non-reductive interpretation can occupy all of the conceptual spaces open to a reductive interpretation (but not always vice versa) on account of the nature of the symmetries, any taxonomy that aligns such ontological stances to these moves is sure to fail.

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