## Looking Ahead To The Modern Debate

I have been speaking liberally about symmetries being at the core of Leibniz's complaints. But I haven't yet properly unpacked this idea. This brief section does just that. The question to be asked is: What allows Leibniz's shift argument to get its grip? Belot suggests this:

80 Note also that just such an operator occurs in the context of quantum statistical mechanics, where it can be understood precisely as shuffling particles over states. A similar problem to that raised by the shift argument occurs in this context too. Since no observable can distinguish between permuted states (i.e. they commute with the permutation operators) the question arises as to whether we should allow worlds corresponding to permuted states into our ontology. The debate gets an extra kick, however, since it appears that the statistical behaviour of quantum systems can be understood as slicing these worlds out of possibility space, and thus as underwriting an anti-haecceitism. Naturally, there is more to the story than this; indeed, as is the case with spacetime ontologies, there are empirically equivalent interpretations that can support both haecceitism and anti-haecceitism in both classical and quantum mechanics. Hence, a general picture is beginning to emerge about the relationship between ontology and symmetry.

Clarke begins with a single possible configuration of matter in space, and generates many more by allowing the symmetries of Euclidean space (translations, rotations, and compositions thereof) to create new possible dispositions of this system in absolute space, such that the spatial relations between objects are the same in each of the possible configurations, although the disposition of the objects in absolute space will differ. Leibniz retorts that Clarke's many possibilities are in reality one: the bodies, in space, with such and such relations between them. The trick is to allow the absolutist to specify a large space of possibilities which falls into equivalence classes—where two possibilities are considered to be equivalent if and only if they arise from one another by the application of a symmetry. 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. ([Belot, 2001], p. 4)

What Leibniz has drawn attention to, albeit in different terms than Belot, is the fact that Newton's theory, formulated on absolute space, possesses symmetries, and these symmetries introduce potential redundancy in the theory in the form of distinct yet indistinguishable states. In reifying absolute space, lifting it from formal to physical, Newton supposedly thereby commits himself to the redundant elements, to their physical existence. It is this redundancy that the shift arguments trade on.81 The symmetries that Leibniz utilizes can be viewed as the Galilean symmetries of a Newtonian spacetime. As Belot points out, these are: symmetry under spatial transformations; symmetry under rotations; symmetry under boosts, and symmetry under time translations. Ten symmetries in all, giving the group of symmetries the structure of a 10-parameter Lie group GN (I use the "N" to highlight the fact that the group is associated with Newtonian mechanics). Newton's laws of motion are covariant with respect to these symmetries, and systems are invariant under the symmetries. What this means is that, if we act on a system (e.g. the material contents of a world) with one, or some combination, of these ten symmetries, we get a state that is indistinguishable from the original. There is no way of telling whether we are in a universe that is uniformly rotating and moving with a uniform velocity in some direction or in one that is at rest: the laws of physics are indifferent with respect to these cases.

We can connect this up to the general account of symmetry outlined in the previous chapter as follows. We begin with a space S such that the points x e S are taken to represent physically possible states of affairs according to Newton's theory. A set of curves y e S, as picked out by Newton's equations of motion (as derived from the action), will represent the dynamically possible histories of this theory—in this case there will be one through each point of S. The idea is

81 Clearly, the claim that a certain structure is redundant depends upon whether or not we can formulate an equally successful, empirically adequate theory without that structure. Newton thought that absolute space was indispensable since he believed that absolute accelerations mandated it. The burden of proof clearly rests on the relationalist to prove him wrong. Leibniz made the assumption that the distinct possibilities were redundant simply because they were indistinguishable and, therefore, in violation of PSR and PII, without actually backing it up with a workable relational alternative. This task wasn't achieved until Barbour and Bertotti formulated their theory of 'intrinsic dynamics' [1982]. Pooley [in press] contains the best discussion of these issues that I have seen.

that the points represent the possible instantaneous states of a system and the histories represent possible sequences of these instantaneous states. Of course, this is simply the notion of a geometric (possibility) space and dynamics that I introduced in the previous chapter. Recall also from that chapter that an interpretation of this setup involves the explication of a representation relation between the formalism (qua instantaneous states and dynamical histories) and a set of possible worlds (qua ontology). This brings us back full circle to the nature of the representation relation between formal and physical: do we look for an interpretation of the formalism that lines up the states and dynamical trajectories one-to-one or many-to-one with physically possible states and worlds? Leibniz argues that Newton is committed to the former interpretation and that this leads to inflation via the symmetries of the theory: he is therefore guilty of violating PSR and PII. Leibniz himself appears to be committed to the latter interpretation on account of his commitment to PII which he believes is enforced by potential violations of PSR by indiscernibles.82

How exactly does this work? I have already mentioned that Newton's theory possesses certain symmetries, of which there are ten in all. These symmetries are manifested on S as structure preserving maps from states to states and histories to histories. Consider two states x, y e S, such that y = g(x) and g e GN, so that they are related by a symmetry. Suppose that g corresponds to one of Leibniz's shift operations, the symmetry transformation that shifts the material contents of the world some distance in some direction, 10 meters to the West, say. Then x and y will be isomorphic and so will yx and Yy, the histories containing x and y, since g is structure preserving. If we consider yx and Yy to be parameterized by t e R, we will find that for any choice of t, x(t) and y(t) will be related by g. The worlds corresponding to Yx and Yy will be indistinguishable, differing solely with respect to the location of matter in absolute space. A one-to-one interpretation of the elements of S will result in an infinity of indistinguishable physically possible states and histories, for there are infinitely many ways to choose g: inflation indeed! We have seen, though, that substantivalists are not necessarily committed to such interpretations, nor are relationalists necessarily committed to many-to-one interpretations.

The same argument can be put in more technical terms, using notions that arise in the context of gauge theory. I deal with this subject in detail in the next chapter. For now let me simply hint at what such a presentation might look like. Newtonian mechanics can be derived from an action that picks out physically possible paths (i.e. trajectories y in the configuration space S). The (inhomogeneous) Galilean group Gn is a variational symmetry of this action. This implies the conservation of energy, angular momentum, and linear momentum corresponding to invariance under time translations, rotations, and boosts—this is due to Noether's first theorem, on which see Brading & Brown [2003] and §3.2.3. Given that there are 10 such conservation laws, we know that the action admits a 10-parameter Lie group as its group of variational symmetries (cf. Earman [2003b], p. 142). These symmetries, as

82 As I mentioned above, Belot [2001] defends this view whereas I deny it: PSR can be upheld in symmetrical cases without invoking PII. My defense of this claim comes in Chapters 3 and 8. Saunders too offers a way of upholding PSR whilst denying Leibniz's PII; however, he does so by putting another version of PII in its place—this version is non-reductive when it comes to interpreting objects related by non-trivial symmetries; that is, it renders distinct objects that would be deemed identical (since indiscernible) by Leibniz (see Saunders [2003b; 2003a]).

I showed above, inevitably lead to inflation. Note, however, that Newton's theory is not a gauge theory (in the usual sense), for that requires that the action admits an infinite-dimensional Lie group as a variational symmetry. However, with the tools of this framework, we can see 'up close' the source of inflation: it stems from the variational symmetries of the action. This inflation also arises for 'true' gauge theories whose actions admit as a variational symmetry, where the symmetries are 'local' (acting at points) rather than 'global' (acting at all points at once). In such cases we must again ask once about the nature of the representation relation. It arises too in the case of constrained Hamiltonian systems, of which general relativity is an example, and of which gauge theories are a subspecies. In both of these cases a one-to-one interpretation of the representation relation leads to a form of indeterminism that can always be eradicated by shifting to a many-to-one interpretation. We are now armed with the knowledge that these choices are not necessarily allied to either of substantivalism or relationalism. Rather, the choice reflects a deeper commitment to a particular understanding of identity and modal semantics.

When I introduce the concept of gauge in the next chapter, I show how the shift argument is a fairly natural consequence of interpreting the freedom to transform states that follows from the existence of variational symmetries. In brief, Newton's theory is underdetermined in the sense that absolute position and velocity are arbitrary functions of time. This underdetermination can be best seen in the Hamiltonian framework—in which the configuration space is replaced by the phase space of positions and momenta—as opposed to the Lagrangian one I implicitly assumed above. I show, within this framework, that there are standard methods for dealing with this underdetermination that are generally seen as corresponding, more or less, to 'relationalist' and 'substantivalist' positions. We know better of course!

Belot clings to the old 'Leibniz-Clarke-style' division of substantivalism and relationalism in terms of inflation and deflation. He has his reasons. He wants a clear distinction to show that the correct interpretation of spacetime as drawn from physics is still a live issue—e.g. in answer to Rynasiewicz's jibes [1996]. He has his sights set on quantum gravity, and hopes to show, by using an analogous division, that these interpretations underwrite very different approaches to quantum gravity.83 The idea is that, so long as we can get a nice division set up in this way, quantum gravity might vindicate one or the other position.84 I strongly disagree with Belot and, indeed, much of this book can be read as charting my disagreement with Belot on this point.85 Quantum gravity will almost certainly prove to

83 For this reason he is particularly scathing of sophisticated substantivalist positions. He admits that sophisticated sub-stantivalism "is very like relationalism" but "can think of no relevant difference between the two doctrines which would lead to any interesting interplay between serious physics and the (increasingly metaphysical) issue between relationalism and Lockean substantivalism" ([Belot, 1996], p. 183—here, Belot uses the sobriquet "Lockean substantivalism" to denote sophisticated substantivalism).

84 The reason being that the extended and reduced spaces that Belot sees as being underwritten by substantivalism and relationalism respectively, though classically equivalent, lead to inequivalent quantizations in general (f. Gotay [1984]). If one of these methods were successful, and if substantivalism and relationalism could be seen to be linked up to them, then quantum gravity physics would support the underwriting interpretation of spacetime.

85 More importantly, however, is the impact the source of this disagreement has on ontology and interpretation. If I am right, then there is a severe underdetermination running through physics (infecting our best (fundamental) physical be an enormous advancement in our conception of the structure of spacetime, but I doubt it can tell us anything about the correct interpretation of this structure vis-à-vis relationalism vs substantivalism.86

theories): distinct (conceptually incompatible; empirically equivalent) interpretations can occupy the same formal frameworks that are usually taken to provide a unique habitat for one or the other interpretation. By analogy with Steven French's arguments on the quantum statistics side (see French [1989; 1998; 2001]; and French & Rickles [2003]), I argue that a structuralist position is shown to be particularly well motivated by this underdetermination. However, the scope of the underdetermination envelops the French & Ladyman 'ontic' version of structural realism [Ladyman, 1998; French and Ladyman, 2003]; therefore I argue for a 'minimalist' version instead.

86 I defend the preceding claim further in what follows and also in [Rickles, 2005], of which §4 is a technically souped up version of the present chapter, focusing on general relativity and quantum gravity. See also French [2001] and §10.2 of French and Rickles [2003] for a defense of a structuralist conception of spacetime drawn from an analogy of spacetime symmetries with the permutation symmetry of quantum statistical mechanics.

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