The Way from Rota to Quantum Mechanics

Abstract

This paper traces an intellectual journey or \textit{Way} (in the sense of a Tao) that starts with some unfinished work of Gian-Carlo Rota on making a logic of equivalence relations or partitions. Rota understood the category-theoretic duality between subsets and partitions which implied there should be a logic of partitions dual to the usual Boolean logic of subsets.And just as probability starts quantitatively with the size of a subset, so he saw that information should start with some notion of size of a partition. After developing the logic of partitions and its quantitative version as logical entropy, it became clear that there is a fundamental duality, fully developed only in category theory, that runs through the exact sciences. Classical physics lies on the subset side and quantum physics on the partition side of the duality. The rest of the paper develops the treatment of quantum mechanics seen through the lens of partitions as the logic of definiteness and indefiniteness. The lattices of partitions allows the treatment of quantum phenomena in highly simplified but essential terms. Since Feynman saw the``only mystery'' of quantum mechanics in the two-slit experiment, this new approach is developed to show how to resolve that mystery. Finally, quantum statistics is treated using Rota-style enumerative combinatorics.