Relationalists can be inflationists just like substantivalists

Given that relationalism is often just stated as the denial of substantivalism,76 it would be rather odd if there weren't some position opposed to sophisticated sub-stantivalism. There is: I call it unsophisticated relationalism. The idea is this. We can accept that the relationalist will classify each nominally distinct embedding of qi in space as merely different ways of describing the same single configuration (i.e. the same spatial structure): when the relations are the same the configurations are the same. However, there is nothing in relationalism that means we cannot permute the identities of the qi themselves. Thus, we can consider q1 and q2 as having masses m1 and m2 respectively in W, but as having m2 and m1 respectively in some other world WP12(m1m2) (where Pij(mimj) is a permutation operator shuffling the particles' masses). Applying this permutation procedure in our three body'- three state' Leibnizian (i.e. relational) world will give us six distinct, yet qualitatively identical possible worlds (corresponding to each possible permutation). But if we allow this, then we lose out on the parsimonious account that lead Leibniz to choose it in the first place in a bid to avoid indistinguishable possibilities violating PSR. The obvious response is to outlaw haecceitism (i.e. to enforce PII so that the action of Pij(mimj) is factored out of possibility space), so that permuting identities just gives us the original world back.77 Hence, relationalism avoids indistinguishable possibilities only if haecceitism is ruled out, just as substantivalism entails them only if it is accepted. However, neither anti-haecceitism nor haecceitism are necessary corollaries of either substantivalism or relationalism.

Strictly speaking there is no reason why there should not exist a profusion of worlds that have the same relational structure and even the same objects, only differently distributed under the relations—God may have liked a particular configuration so much that he made lots of copies (perhaps infinitely many). Suppose we begin with two worlds that differ with respect to just one relation, say R(x, y) in W1 but -R(x, y) in W2. Are we to suppose that this single difference is all that prevents W2 from being snuffed out of existence on pain of violating PII? Anti-haecceitism is not a necessary component of relationalism; it is an additional assumption that can be denied. The reason it can be denied is that relationalism involves a primitive set of (material) objects, which allow us to permute them while keeping the relational structure itself intact. If the existence of this set of objects is denied, or else if we adopt the view that they are individuated by the relational

75 Note that there is another type of anti-inflationist substantivalism according to which neither D nor I represents possibility space. Instead, one configuration is selected from each equivalence class (we might call this space S, for selective: see §3.3 for a characterization of this and 'direct' and 'indirect' interpretations in the context of gauge theory, where I show that these options carve out an adequate taxonomy for interpretations of gauge freedom). Butterfield gets this interpretation by coupling substantivalism to counterpart theory (see Butterfield [1989]). Thus, contra Hoefer ([1996], p. 15), denying PII (i.e. deflation or Leibniz equivalence) does not entail haecceitism (inflation); there's more than one way to deny it. Therefore, even if the sophisticated substantivalist argument just presented is wrong, it still stands that the substantivalist is not committed to inflation.

76 See, for example, Hoefer [1998].

77 See Wilson [1959] for an identical response to a similar permutation argument.

structure then this move won't work: permuting the individuals is either outlawed or else it is idle. Such a view is generally part of structuralist positions. But even on such a structuralist position which appears to definitively rule out haecceitistic differences there is still room to fit such differences in. Given two objects x and y, individuated by the relational structure they are part of, we can simply say that x represents de re of y that (i.e. y) might have been x (cf. Lewis [1983a], p. 395). What we have here is a profusion of possibilities without a profusion of worlds. Naturally, this position—"cheap quasi-Haecceitism"—is open to both substantivalists and relationalists too. Therefore, modal metaphysics cannot dictate spacetime ontology and spacetime ontology cannot dictate modal metaphysics.

More simply, the above argument can be spelt out in terms of the definition of relationalism I gave in §2.1:

1. Relationalism is just that view of spacetime that takes the structure to be determined by the relations holding between a primitive set of material objects.

2. The above definition does not include an analysis of the modal behaviour of this set of objects; in particular is does not involve any claim about whether the objects have haecceities or not.

3. Therefore, on the assumption that the objects do have haecceities (and ruling out e.g. essentialism, counterparts, etc.) we can clearly build up an enlarged possibility space comprising qualitatively indistinguishable worlds that differ solely in terms of which individuals get which properties.78

Of course, the argument I have just presented is similar to Clarke's tu quoque to Leibniz in the Correspondence: granted that space is a relational entity, it would be absolutely indifferent, and there could be no other reason but mere will, why three equal particles should be placed or ranged in the order a, b, c, rather than in the contrary order. ([Alexander, 1956] C III-2)79

Leibniz simply took this to be a reductio of atoms, in much the same way that he took the shift arguments to be a reductio of the substantivalist's conception space and time: both violate PSR:

'Tis a thing indifferent, to place three bodies, equal and perfectly alike, in any order whatsoever; and consequently they will never be placed in any order, by Him who does nothing without wisdom. But then He being the author of all things, no such things will be produced in nature. (L-IV.3, [Alexander, 1956], p. 36)

However, the notion of order in a relationally defined space is rather a weak base on which to construct a permutation argument of the kind Clarke wants. Moreover, there is no reason for Clarke to invoke particles that are equal. Both factors simply serve to supply Leibniz with yet more cannon fodder for PSR to target and blow away with PII. But the shuffling of properties over individuals gets to the anti-haecceitistic core of Leibniz's relationalism. The trouble is, it is nowhere spelt

78 In fact, there might even be cases where the label "unsophisticated" is a misnomer. For example, the relationalist might require truthmakers for certain modal claims that demand such worlds.

out that anti-haecceitism is a necessary part of relationalism. There is nothing internal to relationalism per se that implies the identification of those possibilities that differ solely in the redistribution of (non-spatiotemporal) properties over individuals. Relationalism concerns the ontological priority of material objects (the relata) over space and time (the relational structure); this does not obviously involve any commitment to any additional modal semantics of either ontological category beyond the basic priority claim: relationalists can be inflationists too! There is no reason why one's views about spacetime should determine one's modal commitments (and vice versa).

There is an obvious objection to the argument I have just outlined. Namely, acting on configurations of matter by Euclidean isometries was a part of the Newtonian theory, and therefore is physically well motivated, as are the reductive methods. The shifts are simply elements of the group of symmetries of Newtonian mechanics. Permuting identities is not included in this way. It is true that although we managed to allow the substantivalist to set up camp in what was believed to be the relationalist's site, D, we cannot construct a converse situation wherein the relationalist is allowed to set up camp in I. The reason for this is that the sophisticated substantivalist's occupation of D can be explicated using resources to be found within the theory and within the mathematical representation. On the other hand, the relationalist cannot similarly occupy I because the possibility that inflation is compatible with relationalism takes place outside of the physical theory and outside of the mathematical representation. Therefore, unless some operation corresponding to Pij(mimj) can be found within the theory and the geometric spaces used to represent the relevant system, I will be out of bounds, even though inflation per se is an option. The objection has some force, but recall that the whole point of going to the reduced space was to enforce the outlawing of indistinguishable possibilities. The fact that such possibilities are compatible with relationalism is enough to show that the enterprise of associating relationalism with deflation-ism is seriously misguided: possibility counting simply isn't relevant to spacetime ontology.80

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