David Ellerman:From Abstract Objects in Mathematics to Objectively Indefinite States in Quantum Mechanics

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Abstract: Given an equivalence relation ~ on a set U, an abstraction operator @ takes two equivalent entities to the same abstract thing: u~u′ iff @(u)=@(u′). In mathematics there two different notions of `abstraction' at work. The #1 version is just the equivalence class, e.g., a homotopy type is just the equivalence class of homotopic spaces. The #2 version is a more abstract object that is, for example, definite on what is common to all the spaces homotopic to each other but is indefinite on where they differ.

1. We show how to mathematically model the two versions starting just with a subset S⊆U and then focus on the less familiar #2 interpretation where S is viewed not as a set of distinct elements but as a more abstract entity `S-ness' that is definite on what is common between the elements of S and indefinite on how they differ.
2. The point is that the #2 case dovetails precisely with the quantum mechanics notion of a superposition of distinct eigenstates of some observable which is a state definite only on what is common to the superposed eigenstates and is objectively or onticly indefinite between them. The process in general of classifying by some attribute to make the #2 indefinite state more definite then emerges as the process of measurement in QM.