Volume 9, Issue 2, pages 64-107
The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the vector (Hilbert) space version of the mathematics of partitions at the set level. Since partitions are the math tool to describe indefiniteness and definiteness, this shows how the reality so well described by QM is a non-classical reality featuring the objective indefiniteness of superposition states. The lattice of partitions gives a skeletal model of quantum reality with the partition versions of pure states, non-classical mixed states (i.e., including a superposition state), and the completely distinguished classical state that satisfies the partition logic version of the Principle of Identity of Indistinguishables. Both the key notions of quantum states and quantum observables are respectively the (density) matrix versions of partitions and vector space version of a numerical attributes. The operation of projective measurement given by the Lüders mixture operation is the vector space version of the partition join operation between the partition prefiguring the density matrix of the state being measured and the partition prefiguring the observable being measured. All this adds specific key concepts and structure to Abner Shimony’s idea of a literal understanding of QM to form what might be called the “Objective Indefiniteness Interpretation” of QM.
