Source
Federico Holik, Juan Pablo Jorge, Décio Krause and Olimpia Lombardi (2021), LIT-107, read in full in NOTE-127 from the authors’ preprint “Quasi-set theory for a quantum ontology of properties” (PhilSci-Archive 19725, 26 pp.). The Synthese version (200(5), art. 401, 2022) was not read and may differ.
What was actually shown
The ontology (§§2–3). General magnitudes are universal U-type-properties. Observables are their instances, I-type-properties. Eigenvalues are possible case-properties, and the single realized value is the actual one. A system is a bundle of I-type-properties, that is, of determinables, with no principle of individuality. The instances are universals’ instances, “only numerically different”, and tropes are rejected because they would reintroduce distinguishability (p. 5).
Why standard logic does not suffice (§4). “An ontological domain populated exclusively by properties and non-individual bundles … cannot be adequately apprehended by any language that includes individual constants and variables” (p. 11). NOTE-127 rates this a weak-to-moderate philosophical argument.
The construction (§6, pp. 19–22). For each projector-property [Π] there is a qset U([Π]) of indiscernible instances. R(x) says that x takes exactly one instance (qc = 1) from each U([M]), U([W]), U([S]) and each U([Π]). The bundle’s kind is Q(B) := {x ∈ P(Q) | R(x)}, and Q’s Axiom of Choice gives Q(B) ≠ ∅. Any two members of Q(B) are indiscernible. N indistinguishable particles are a subqset N(B) ⊂ Q(B) with qc(N(B)) = N: a definite number with no labels. Kinds are labelled by (m, w, s) from irreducible representations of the Galilean group.
What Q supplies (§5). Indiscernibility is an equivalence relation but not a congruence, and Q does not prove substitutivity of indiscernibles: Q ⊬ a ≡ b → ∀z(a ∈ z ↔ b ∈ z) (p. 18). Swapping indiscernibles yields an indiscernible qset (Theorem 7, p. 19). Both are known results about Q (French & Krause 2006).
What could have come out otherwise. The bundles could have needed a labelling to be counted, or Choice could have had to identify which instance it picked. The paper’s answer is that Choice picks one element from each U “no matter which one” and that all members of Q(B) are indiscernible (p. 22).
What this does not say
- That the ontology is right. The paper supplies a meta-language for it. It presupposes that the algebraic formalism’s order of priority, observables before states, is a guide to ontology.
- That the ontology is formalized. States, the actualization of case-properties and dynamics are central to it (propensities, actual case-properties) and are not represented in Q. Only kinds and counting are.
- That the ontology escapes Kochen–Specker or explains entanglement. The “immunity” holds because a system is defined as a bundle of determinables, which contextuality never touches. The “mystery vanishes” for EPR because the correlations are relocated to properties of one item, not explained.
- That the Indistinguishability Principle is derived. It follows only if observables are restricted to symmetric ones, which is itself a postulate (Messiah & Greenberg). It is moved, not removed. Which symmetry sector is then selected is not discussed.
- That the construction is checked. Def. 6, Theorem 7’s proof and the notation on p. 18 contain errors, and no independent consistency check of the bundle construction is given. R(x) also quantifies “∀Π” over the collection of projector-properties as if it were classical.
- That it reaches quantum field theory. The treatment is nonrelativistic, and particle number is fixed.
Connections
- Leitgeb & Ladyman (LIT-154). NOTE-127 calls the two papers opposite answers to one puzzle. Leitgeb and Ladyman keep standard identity and accept primitive, non-discernible nodes. Holik et al. drop identity for the items and use indiscernibility instead.
- Moderate structural realism (LIT-895, THEORY-453). Esfeld and Lam keep classical identity talk and give entangled systems of one kind a primitive numerical diversity. This paper gives them a quasi-cardinal and no identity. Both count without identifying.
- Structural realism. French (2020) notes the view is close to ontic structural realism. The record’s structural-realism accounts are THEORY-032, THEORY-034 and THEORY-038.
- Wallace (LIT-151) takes up the same complaint, that non-individual ontologies cannot be stated in standard logic, by a math-first route.