Ellerman, David
(2022)
Follow the Math!:The mathematics of quantum mechanics as the mathematics of set partitions linearized to (Hilbert) vector spaces.
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Abstract
The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) linearized to vector spaces, particularly in Hilbert spaces. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac's Complete Sets of Commuting Operators (CSCOs). The mathematics of partitions is first developed in the context of sets and then linearized to vector spaces where it is shown to provide the mathematical framework for quantum mechanics. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM--as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being ``definite all the way down."
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