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Entities Without Identity: A Semantical Dilemma

Abstract

It has been suggested that puzzles in the interpretation of quantum mechanics motivate consideration of entities that are numerically distinct but do not stand in a relation of identity with themselves or non-identity with others. Quite apart from metaphysical concerns, I argue that talk about such entities is either meaningless or not about such entities. It is meaningless insofar as we attempt to take the foregoing characterization literally. It is meaningful, however, if talk about entities without identity is taken as elliptical for either nominal or predicative use of a special class of mass terms.

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Notes

  1. For a thorough discussion of the role of quantum mechanics in provoking consideration of entities without identity, see (French and Krause 2006). For the specific argument discussed here, see also (Post 1963; Reichenbach 1999; French 2000).

  2. Roughly speaking, there are actually two possibilities corresponding to two basic kinds of particle. For bosons, the states in which one particle possesses \(M_1\) and the other \(M_2\) would receive a combined probability of 1/3 as would each of the states \(M_1(p_1) \wedge M_1(p_2)\) and \(M_2(p_1) \wedge M_2(p_2)\). For fermions, the only possible state is that in which one particle possesses \(M_1\) and the other \(M_2\).

  3. The cardinality of ‘uncountable’ sets—such as the set of real numbers—obviously cannot be expressed via an ordinal. Nonetheless, the notion of equinumerosity or equal cardinality that sustains talk of such extended notions of size is still dependent upon the notion of a bijection and thus of the identity of the elements of the set. The same arguments about the connection between cardinality and identity thus apply. But as long as we’re talking about particles in non-relativistic quantum mechanics, there are at most countably many, and so we needn’t complicate the discussion by considering collections or sets that are uncountable.

  4. Of course, some other relation must function as identity if we are to count fruit across multiple scenes.

  5. The proposal of (Domenech and Holik 2007) is spelled out in terms of the quasi-set theory discussed below. Thus, the technical definitions involve quasi-sets. However, to efficiently convey the gist of the proposal, I am using the neutral term “collection” instead.

  6. Bueno (2014) makes a related point. Proponents of anonymerity (e.g., (Domenech and Holik 2007)) often point to a physical phenomenon that ostensibly offers a clear example of counting without identity. Imagine stripping electrons off of a large atom, perhaps by blasting them with electromagnetic radiation of sufficiently short wavelength. As the electrons fly off, they can be detected by, say, their tracks in a cloud chamber. In that way, we could count how many electrons were removed from the atom, even though QM tells us we cannot say which is which. As Bueno points out, however, we cannot say that each track corresponds to a different electron unless there is a way to identify them.

  7. See, e.g., (Saunders 2003).

  8. Lowe uses the term “countability” rather than cardinality. This invites confusion in his discussions of continuously divisible substances, given the technical sense of countability.

  9. Lowe also claims, perhaps more surprisingly, that it’s possible for there to exist entities with identity but no determinate cardinality. I have been arguing that a notion of identity is conceptually prior to a notion of cardinality, and so only the first of these possibilities is logically pertinent to my argument.

  10. In fact, I have argued (in Jantzen 2011a) that treating, e.g., electrons, as sometimes entangled and sometimes not is empirically adequate under certain conditions, but is simply inconsistent as a matter of interpretation.

  11. Lowe specifically cites (French and Redhead 1988; Redhead and Teller 1992), which defend the conceptual coherence of entities without identity in the interpretation of quantum physics.

  12. The theory was later revised in collaboration with Steven French (French and Krause 2010, 2006).

  13. That is, by Zermelo–Frankel set theory without urelemente, only sets.

  14. According to Gillon (1992), it was first noted as a grammatical phenomenon in English by Jespersen (1909, vol. 2, ch. 5.2).

  15. For a finer-grained classification, see (Pelletier 2009).

  16. I agree with (Koslicki 1999) that the mass/count distinction properly applies to occurrences of a term, not the term itself. For ease of exposition, however, I will elide this distinction.

  17. Note that this use of “divisive” is subtly distinct from what Quine (1960) has in mind when he speaks of division of reference. For Quine, mass terms before the copula of a sentence act as singular terms in that they refer to a single collective rather than a multiplicity of things. In this sense, they fail to “divide their reference”. On the other hand, Quine notes that, at least to a degree, when predicative use of a mass term is true for a thing (e.g., “The liquid in the glass is water.”) it’s true for parts of the thing (e.g., “The liquid at the bottom of the glass is water.”).

  18. How could this maneuver fail? How could QM force (or at least encourage) us to reject the semantics of mass predication and embrace references to particles as canonical count terms? If the theory entailed that, in certain circumstances one could say there are exactly n particles and, furthermore, one could determine which is which—by, for instance, a procedure for re-identifying particles—this would make a mass predication reading difficult. That is, if the theory implied that the things in question behave like classical individuals, this would be a problem for the mass term view. But the theory doesn’t say this. It is the very fact that QM seems to deny this possibility that motivates the attempt to introduce anonymeres in the first place.

  19. For an overview of the relevant formalism, see (Messiah and Greenberg 1964) and (Hartle and Taylor 1969).

  20. Note the distinction between counting water and counting particular volumes, samples, or regions containing water.

  21. I am assuming that electron-stuff may be said to be present in any region of space where the wavefunction has support. The entire region in which the square of the wavefunction is non-zero is special in that one can say with certainty that it contains one unit of electron-stuff.

  22. E.g., a pure state which, when expressed as a density operator, yields only reduced density operators that are not pure states.

  23. See (Cartwright 1970) for a cogent discussion of the notion of “quantity” as I use it here.

  24. Bennett (1984) argues that Spinoza had such a metaphysic in mind. That is, he saw ordinary objects as modes of regions of space. I’m grateful to Walter Ott for pointing out this connection to me.

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Correspondence to Benjamin C. Jantzen.

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I am grateful to Walter Ott, Kelly Trogdon, and two anonymous referees for their insightful criticisms of earlier versions of this paper.

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Jantzen, B.C. Entities Without Identity: A Semantical Dilemma. Erkenn 84, 283–308 (2019). https://doi.org/10.1007/s10670-017-9958-3

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