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Countability and self-identity

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Abstract

The Received View of particles in quantum mechanics is that they are indistinguishable entities within their kinds and that, as a consequence, they are not individuals in the metaphysical sense and self-identity does not meaningfully apply to them. Nevertheless cardinality does apply, in that one can have n> 1 such particles. A number of authors have recently argued that this cluster of claims is internally contradictory: roughly, that having more than one such particle requires that the concepts of distinctness and identity must apply after all. A common thread here is that the notion of identity is too fundamental to forego in any metaphysical account. I argue that this argument fails. I then argue that the failure of individuality and identity applies also to macroscopic physical objects, that the problems cannot be constrained to apply only within the microscopic realm.

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Notes

  1. For fermions this means that the many-particle wave function is antisymmetric in the exchange of the spin-spatial coordinates for any pair of particles. ‘An antisymmetrical eigenfunction vanishes identically when two of the electrons are in the same orbit. This means that in the solution of the problem with antisymmetrical eigenfunctions there can be no stationary states with two or more electrons in the same orbit, which is just Pauli’s exclusion principle.’ (Dirac, 1926, 669-70).

  2. It has to be stressed that this aspect of QM is independent of interpretational issues. It was not deduced from ideas about ‘superposition’, or entanglement, which came later, contra a remark in Lowe (1998).

  3. If it is to be a property of a quantum system what Hermitean operator is there to measure it? What are its eigenstates?

  4. See also Dieks and Versteegh (2008), Dalla Chiara and Toralda Di Francia (1995) and Heathcote (2020).

  5. Strictly speaking French and Krause hold to an underdetermination thesis in (2006), but subsequent publications of Krause suggest a firmer adherence to the Received View. I don’t follow them in this underdetermination thesis. My defence here concerns the Received View.

  6. Such monomials had appeared earlier in Weyl (1931) ch IV. These monomials are such that the exponents are integers mod 2, i.e. occupation numbers are either 1 or 0. It is in this sense that, as Weyl says, they encapsulate the Pauli exclusion principle. See Weyl (1949) p. 247.

  7. Much of my argument will also apply to that in Dorato and Morganti (2013). There is a comprehensive response to Bueno (2014) in Krause and Arenhart (2018). There is a different line of argument against the Received View in the work of Simon Saunders and Fred Muller, either separately or together, but I’ve discussed their view elsewhere, as have many others, and so will only briefly touch on it here.

  8. There are additional reasons why this dispute is tangential to my present concerns. Wehmeier treats self-identity as unproblematic — his focus is on the eliminability of the binary relation of identity.

  9. A large literature has grown up on plural quantification, as can be seen from the collection Rayo and Uzquiano (2006) and the references given therein. An early criticism of the idea is Linnebo (2003). The more recent Linnebo and Rayo (2012) gives a guide to further issues. The Stanford Encyclopaedia entry on Plural Quantification was updated in 2017. My thanks to a reader of the journal for suggesting the inclusion of this issue.

  10. A referee has argued that Dorato and Morganti’s ‘can and should’ should be taken in a pragmatic sense. Since I confess I don’t know how to do this consistent with Weyl’s argument I leave it to the judgement of the reader.

  11. I’ve given this argument elsewhere (Heathcote, 2021) and applied it also to Black’s spheres. Here I focus on its application to quantum particles.

  12. It should be noted that the argument to follow does not depend on the details of the translation of the sequence into set theory. It is the pairing with ordinals that is the crux. Also the details of the satisfaction of open sentences does not change in moving to plural quantification. See Boolos (1985a, 1985b).

  13. The idea that there is a property ‘behind the veil’ of the quantum state is what we know of as a hidden variable theory; it seems that those who are trying to think of relations as lying contained in the quantum state are engaged in an extension of this same project. In QM relations get inferred from the probability statistics of the actions of Hermitian operators acting on spatially separated systems. Post-measurement descriptions of what has become a product state cannot be read back into the pre-measurement non-product states. Since all we have in QM are states of composites, it is very hard to see where this definite relation might lie. If one wants to base one’s metaphysics on relations then there are some high hurdles in QM yet to clear. The issue of entanglement is discussed again in Section 4.

  14. He says: ‘Arbitrary reference so understood, as Breckenridge and Magidor (2012, 398) highlight, is especially suitable to explain what goes on when we refer to indiscernible objects.’ (Berto, 2017, p. 18). This rather exaggerates what Breckenridge and Magidor claim.

  15. The fact that we can and do name the roots, albeit in an arbitrary fashion, as positive and negative, suggests that Ladyman’s idea in Ladyman (2007) of representing then by an unlabelled irreflexive symmetric graph must be open to doubt.

  16. If we accept this then it blocks some of the responses to Black’s case, for example that of Hacking (1975).

  17. Hesse claimed that it did in Hesse’s (1963), and this was a mistake. Unfortunately many have followed her lead. But Schrödinger had gotten the matter right in his (Schrödinger, 1950).

  18. It would not be difficult to adapt this to the antisymmetric subspace idea of Dirac and Weyl.

  19. Can a single vacancy be a vacancy for two clubs — say a maths club, that has to have one member in the chess club? In other words an intersection of members and vacancies? I see no reason why not.

  20. Berto (2017) p. 4 gives the same truncated quote but introduces five punctuation errors into the quotation, which further serve to distort the meaning.

  21. In the German the names were Hans and Karl. This was changed to Mike and Ike by the translator after a popular cartoon by Rube Goldberg ‘Mike and Ike (They look alike)’. In 1927 this cartoon was turned into a number of feature films starting with Dancing Fools — ironically starring two actors who did not look alike! This change ensured that the idea would be conveyed to audiences in 1930’s America.

  22. This application was not considered in Breckenridge and Magidor (2012). My thanks to a reader for the journal for suggesting the inclusion of this work.

  23. Unger thinks that the myriad precisifications may lead us to conclude that there are a myriad of cats but overlooks the fact that no given precisification would have it that there are even two cats on the mat, let alone many. What is plural (as noted by Geach) are the possible precisifications, not the cats. The myriad of possible precisifications creates a disjunction of possibilities that can qualify as the cat, not a conjunctive sum of many cats.

  24. ‘Compactness’ is a topological term roughly meaning a bounded, closed space. The surface of a 2-sphere is a good example.

  25. Tsirel’son’s bound is equivalent to Grothendieck’s constant used in the paper. The complexity of multipartite entanglement is well discussed in the review Horodecki et al. (2009). This discovery was initiated by Peres (1996).

  26. This notion was introduced in Horodecki et al. (1998). See also and Smolin (2001). Bounded entanglement was confirmed in the laboratory in 2010, see Barreiro et al. (2010).

  27. They continue: ‘This result astonished the physicists. It transpired that the structure of entanglement is not uniform!’

  28. For the interaction between entanglement and indistinguishability see Wiseman and Vaccaro (2003) or Tóth and Gühne (2009). This is an area of ongoing research.

  29. There is a persuasive refutation of Evans argument in Lowe (1998). The view I am suggesting is one of the possible counterarguments that Lowe considers.

  30. People are not suitable to be members of sets. As Groucho Marx (almost) said: You wouldn’t want to be a member of any set that would have you as a member.

  31. In a footnote to this Weyl cites Dirac’s (1929) paper on many electron systems.

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Acknowledgements

My thanks to the two readers of this journal who made some excellent suggestions for improvements.

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  1. Adrian Heathcote

    Present address: , Gordon, Australia

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    Heathcote, A. Countability and self-identity. Euro Jnl Phil Sci 11, 109 (2021). https://doi.org/10.1007/s13194-021-00423-z

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