Abstract
The significance of Max Black’s indistinguishable spheres for the nature of particles in quantum mechanics is discussed, focusing in particular on the use of the idea of weak indiscernibility. It is argued that there can be four such Black spheres but that five are impossible. It follows from this that Black’s example cannot serve as a model for indistinguishability in physics. But Black’s discussion of his spheres gave rise to the idea of weak discernibility and it is argued that such predicates are unsatisfiable in the way intended. The underlying problem with weak discernibility spreads out to also undermine the whole notion that indistinguishability rests on a notion of the permutation invariance of particles. A better foundation is indicated.
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Notes
There is a small logical lacuna in Bigaj’s argument: it doesn’t follow from the fact that a relation can be formulated by cases that it is not possible for it to be formulated in a different (case-free) manner. This is obvious once one notes that any relation can be formulated by cases even if trivially and arbitrarily. But proving the impossibility of a case-free definition would be difficult to say the least. This would throw the onus back onto Muller. The second point is that Black’s spheres can clearly be formulated by cases, simply by taking the function f(x, y) in (1) as the distance function. Thus it is unclear why Bigaj thinks it a genuine case of weak discernibility. My argument in this paper shortcircuits this dialectical to-ing and fro-ing.
Relative indiscernibility plays no part in Saunders account or in relation to quantum particles. We ignore it here.
The last sentence seems a strange piece of logic. Since, by the third sentence, all named objects are absolutely discernible, they have only one classification — that of being absolutely discernible. Named objects cannot ever be classified as ‘only weakly discernible’.
The underlying point here is not new. Problems of applying an extensional set theory to indistinguishable objects have been noted as long ago as Toraldo di Francia (1978) and Dalla Chiara (1985), Dalla Chiara and Toralda Di Francia G (1995). It was emphasised with a great deal of technical development in French and Krause (2006) and Arenhart et al. (2019) also van Fraassen and Peschard (2008) p. 23. This is merely a sample of the many publications on the subject, mostly by these authors. Of course, it probably goes without saying that there are disagreements between these authors as to how to apply this idea. See also Dieks (2020) and Dieks and Versteegh (2008).
Thus we are also meant to understand that the space in which these spheres are situated is a metric space, indeed it is a Euclidean 3-space. We are meant to take such a setting as basic and intuitive—but we should be on our guard against this appeal to a naive metaphysics, particularly when we pass on to QM. (This assumption was questioned in Hacking (1975).)
This is simply a matter of definition: a regular tetrahedron is four vertex points with a fixed distance d between them. Coxeter (1973) ch. 1.
This does not contradict the idea presented in Della Rocca (2005), whose spheres can occupy the same space.
Bas van Fraassen has reached a similar conclusion:‘If an asymmetric relational predicate, or a symmetric irreflexive one, is instantiated at all then there are at least two things. Saunders’ term “weakly discernible” does not seem apt: we have here a way to show that logically, things can be distinct without being discernible, by an argument that does not appeal to PII but to its innocuous converse. Hence I do not consider the PII saved or satisfied even in the case of fermions by reflections on “weak discernibility”, contrary to Muller and Saunders (2008)...’ Howard (2011), 234
This therefore may support the, so-called, non-reflexive logics (for which see Arenhart (2017)): the name of which I regard as very unfortunate. However, this has a very significant corollary: the universal quantifier is now only over all individuals, it can no longer be paraphrased as everything is self-identical which would require the existence of a universal set. This removes one powerful objection to the traditional “law of identity”.
It may be thought that the dyadic character of weak discernibility is merely a first-attempt simplification and that it could be adjusted so as to be polyadic. This does not seems to be so. As we have seen, the problems with the formal notion of satisfaction and the dyadic predicate are made worse as one moves to more indiscernible objects. Nor does Saunders attempt any such generalisation, nor did Muller later; nor of course did Quine himself.
In general, for n objects there are n! permutations of these objects. Stanley (2012) ch 1.
Black’s skill in making his model universe seem so present to the mind’s eye, can be seen in an innocuous detail: making the spheres not too far apart, so that one feels one can “see” them both at once. Thus one can “see” that there are two, not one. But we easily forget that there is no light at all in Black’s universe to see anything. And if we could see them the spheres would be absolutely discernible in virtue of the light to the eye constituting a bijection to our visual image. From this it follows that indistinguishable entities cannot be discernibly seen.
Pauli’s exclusion principle is often invoked in an unclear way to try to cover up the problems with the underlying ideas here. But Pauli’s principle is a consequence of the antisymmetric state space for fermions, which, mathematically speaking, is essentially an exterior product space—the Pauli principle follows from the character of this exterior product. This antisymmetric space is an irreducible representation of the symmetric group in a tensor product space, and this concept does not involve permuting particles. For \(S_{4}\), for example, there are five such irreducible representations but only two have physical significance, the two one-dimensional representations—and to the extent that permutations of the vectors (note: not particles) are significant it is only through the way that they are encoded in these representations. These representations are homomorphisms to the group of invertible linear transformations of a vector space, for one needs sums and multiplication by scalars to be defined (along with scalar products). Groups are said to act on sets, but to be represented in vector spaces. This was set out carefully by Weyl in his (1931). A modern account is Goodman and Wallach (2009). See also Heathcote (2014) and Heathcote (2020).
Schrödinger also has an analogy for Einstein-Bose statistics: three boys and two shillings to be allocated as rewards. In this case there are 6 ways of distributing the shillings. The shillings represent the particles. Mary Hesse developed this analogy in Hesse (1963), 49–50. Importantly, these analogies show that indistinguishable particles make metaphysical and semantic sense, contra the claims in Jantzen (2017).
E. J. Lowe, in Lowe (2016), has also given an argument against a weak discernibility interpretation of the Black spheres, through an entirely different argument to the one given in the present paper. He disambiguates the relations between the spheres using \(\lambda -\)abstraction and argues that under no interpretation does it justify the claim of weak discernibility. He adds much of value in his discussion of ways that entities may be non-individuals. (I thank a reader of this journal for bringing Lowe’s article to my attention.)
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My thanks to a reader of this journal for many suggestions that led to improvements.
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Heathcote, A. Five Indistinguishable Spheres. Axiomathes 32, 367–383 (2022). https://doi.org/10.1007/s10516-020-09531-6
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DOI: https://doi.org/10.1007/s10516-020-09531-6