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Ontic Structural Realism and the Principle of the Identity of Indiscernibles

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Abstract

Recently, there has been a debate as to whether or not the principle of the identity of indiscernibles (the PII) is compatible with quantum physics. It is also sometimes argued that the answer to this question has implications for the debate over the tenability of ontic structural realism (OSR). The central aim of this paper is to establish what relationship there is (if any) between the PII and OSR. It is argued that one common interpretation of OSR is undermined if the PII turns out to be false, since it is committed to a version of the bundle theory of objects, which implies the PII. However, if OSR is understood as the physical analogue of (sophisticated) mathematical structuralism then OSR does not imply the PII. It is further noted that it is (arguably) a virtue of OSR that it is compatible with a version of the PII for possible worlds.

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Notes

  1. See Quine (1976, pp. 113–114), Saunders (2003a, pp. 4–5; 2006, p. 57), Ketland (2006, p. 306), Muller and Saunders (2008, p. 528), Muller and Seevinck (2009, p. 182), Ladyman and Bigaj (2010, pp. 125–127) and Muller (forthcoming, Section 3). However, not all of these authors distinguish all four grades of discernibility, and not all use the same terminology: Quine distinguishes strong, moderate and weak discriminability (corresponding to absolute, relative and weak discernibility); Saunders distinguishes absolute, relative and weak discernibility; Ketland distinguishes monadic, polyadic and weak discernibility (corresponding to absolute, relational and weak discernibility).

  2. The analogue does hold in the case of ϕ-discernibility, since relative ϕ-discernibility implies relational ϕ-discernibility and relational ϕ-discernibility is equivalent to weak ϕ-discernibility.

  3. Another intuitively plausible definition of indiscernibility, would classify a and b as indiscernibile (in a structure) if and only if there is an automorphism of the structure that maps them onto each other. If two objects are not relationally discernible then they are indiscernible in this sense:

    1. (i)

      Suppose a and b are not relationally p-discernible.

    2. (ii)

      So there is no n-ary predicate, R, such that \( \exists x_{1} \ldots \exists x_{n - 1} \neg (Ra,\, x_{1} , \ldots , \,x_{n - 1} \leftrightarrow Rb, \,x_{1} , \ldots , \, x_{n - 1} ) \) or…

    3. (iii)

      So for every n-ary predicate, R, ∀x 1…∀x n−1(Ra,x 1 ,…,x n−1 ↔ Rb,x 1 ,…,x n−1) and…

    4. (iv)

      So for every n-ary relation, R*, and every (n−1)-tuple of elements 〈a 1 ,…, a n−1〉, the n-tuple 〈a, a 1 ,…, a n−1〉 ∈ R* if and only if 〈b, a 1 ,…, a n−1〉 ∈ R* and the n-tuple 〈a 1 , a, a 2 ,…, a n−1〉 ∈ R* if and only if 〈a 1 , b, a 2 …, a n−1〉 ∈ R* and…

    5. (v)

      So the function f (f(a) = b, f(b) = a and f(x) = x for all other x) is an automorphism of the structure.

    However, the converse does not hold. Consider the structure:

    ({a, b}; {〈a, b〉, 〈b, a〉})

    The function f (f(a) = b, f(b) = a) is an automorphism of this structure, but a and b are relationally discernible (in fact, they are even weakly p-discernible).

  4. In fact, as James Ladyman has pointed out to me, this is probably not even an accurate characterisation of the version of the PII that Leibniz had in mind. Leibniz (arguably) thought that no two distinct objects share all the same intrinsic properties. But whether or not there are monadic predicates for all and only the intrinsic properties depends on the language in question. In English, for example, there are monadic predicates for what appear to be relational properties (e.g. “uncle”) and it seems likely that there exist intrinsic properties for which there are currently no predicates in English (even within the last 100 years physicists have discovered new intrinsic properties and have had to invent new predicates for them (charm, strangeness, etc.): it would be rash to suppose that this will not happen again).

  5. If one is not familiar with this debate then one might think that our scientific theories could not possibly suggest that the weakest version of the PII was false, because, on the face of it, it seems there could be no empirical grounds to postulate the existence of two distinct objects that share all the same properties and relations. But this is not correct. We would have empirical grounds to postulate the existence of two distinct but indiscernible objects if (for example) the properties of a system consisting of two such objects were not identical to the properties of a system consisting of just one of the objects (for example, the former system might have twice the mass of the latter). So even though the objects are individually indiscernible, there is a discernible difference between a system that consists of one such object and a system that consists of two such objects.

  6. In the light of the arguments of Muller and Seevinck (2009), it is no longer so clear that free photons are a counterexample to all forms of the PII.

  7. However, Rodriguez-Pereyra (2004) has argued that the bundle theory does not imply the PII.

  8. As structures themselves contain objects one might think that this does no more than assert that “elementary” particles are themselves composed of more fundamental objects, and this does not appear to be a form of OSR at all. Possibly, Muller has in mind the view that there is no lowest level of objects: any objects we find will themselves turn out to be structures on further analysis. Cf. Saunders (2003b), Stachel (2006, p. 54), Ladyman (2007a) and Ladyman and Ross (2007).

  9. This is the version of OSR that French (2010) calls “eliminativist” OSR. French characterises this as the claim that, “the very constitution (or essence) of the putative objects is dependent on the relations of the structure” (French 2010, p. 106). See also Ainsworth (2010).

  10. Cf. Dieks and Versteegh (2008) who state that, “The form taken by the ‘bundle idea’ in the case of relational properties is actually much discussed in present-day philosophy of science, and known as a from of structuralism. […] This position fits PII very well” (Dieks and Versteegh 2008, p. 927). The connection between OSR and the bundle theory of objects has also been noted by French (“in the absence of further metaphysical explication of the notion of structure itself, it is not yet clear whether or not such an approach [i.e. OSR] collapses into another form of the well-known conception of objects as bundles of properties” (French 2006, pp. 10–11)), Pooley (“the idea of the independent existence of structures suggests an obvious comparison, viz. with the view that physical objects are nothing but bundles of collocated properties (‘bundle theory’)” (Pooley, 2006, p. 93)) and Ainsworth (“[the OSRist’s] claim that relata are secondary to relations is not as revolutionary as it might at first appear: a similar claim is put forward in the so-called ‘bundle theory’ of objects” (Ainsworth 2010, p. 52)).

  11. We need to restrict ourselves to the context of a particular structure here. In the context of the structure we get if we remove the less than relation from the natural number structure 5 is not terminologically dependent on the less than relation, since we would still call an object 5 even though the less than relation does not exist (in that structure). Likewise, in the context of the structure we get if we remove 5 from the natural number structure the less than relation is not terminologically dependent on 5.

  12. There are other forms of OSR in the literature. For example, Esfeld (2004) and Esfeld and Lam (2008) develop a form of OSR according to which objects and (multi-place) relations are interdependent.

  13. It is thus the denial of the claim that there can be two distinct but indiscernible possible worlds, a doctrine sometimes known as haecceitism.

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Acknowledgments

I gratefully acknowledge the financial support of the Leverhulme foundation. This paper was inspired by discussions in the Bristol structuralism group and I owe a huge academic debt to everyone there. I’m especially indebted to James Ladyman, who made some helpful comments on an early draft of this paper. I’m also indebted to the anonymous referees of the journal, both of whom made a number of useful comments.

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    Peter Ainsworth

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Ainsworth, P. Ontic Structural Realism and the Principle of the Identity of Indiscernibles. Erkenn 75, 67–84 (2011). https://doi.org/10.1007/s10670-011-9279-x

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