Introduction: Partitions and Quantum Reality

ThisGalileo new approach to understanding and interpreting quantum mechanics (QM) is based on the development of the notion of a partition on a set (or, equivalently, an equivalence relation on a set or a quotient set). The notion of a partition is not some ad hoc notion designed to make yet another interpretation of quantum mechanics. It is as fundamental a notion as that of a subset. But the math (and logic) of subsets was developed long before the corresponding partition math. There is a recent sequence of developments that constitute a revival or renaissance of partitions outside of the standard treatment in enumerative combinatorics. Standard logic is the Boolean logic of subsets (propositional logic being a special case). Subsets and partitions are dual concepts in category theory. The first stage in the revival was the development of the logic of partitions dual to the logic of subsets. Then just as finite Boole-Laplace probability theory starts as the quantitative version of the logic of subsets, so the quantitative version of the logic of partitions was the second stage resulting in the new foundations for information theory based on the notion of logical entropy. The third stage was based on the realization that the distinctive mathematics of QM is the vector (particularly Hilbert) space version of the mathematics of partitions. That is the subject of this book.