Meaning and duality: from categorical logic to quantum physics - Maruyama, Y (2016)
Duality abounds in science, both pure and applied: in mathematics there is duality between space and algebra; in physics there is duality between states and observables; in information science there is duality between systems and properties/behaviours; in logic there is duality between models and theories; and even in philosophy there is duality between realism and antirealism (Dummett), or substance metaphysics and process/function metaphysics (Whitehead/Cassirer). They look akin at a level of abstraction, and yet, taking a closer look at subtleties involved, one may find a parting of the ways. The present thesis builds upon category theory and universal algebra to explicate and articulate the dynamics of duality, the mechanism of how duality emerges, changes, and breaks. A generic duality theory is first pursued in the abstract via categorical topology and algebra, and subtler duality theories are then explored to the end of shedding light on more nuanced facets of concrete dualities such as coalgebraic dualities. Each duality theory thus developed is cashed out to uncover formerly unknown dualities; e.g., the first duality theory tells us a dual equivalence between domains and convex structures, which remained an open problem before. Dualities are often induced by Janusian objects, which sometimes form truth value objects as in topos theory. To elucidate this link between duality and categorical logic, categorical universal logic is developed on the basis of Lawvere's hyperdoctrine and Hyland-Johnstone-Pitts' tripos, thereby expanding the realm of (first-order/higher-order) categorical logic so as to encompass, inter alia, classical, intuitionistic, quantum, fuzzy, relevant, and linear logics. Finally, duality meets symmetry in quantum physics, yielding a categorical understanding of the Wigner theorem, i.e., a purely coalgebraic characterisation of the quantum symmetry groupoid. In passing, operational quantum duality is systematised via Chu space theory. In light of these, duality would arguably serve as a unifying principle to overcome the fragmentation of science, and to soothe the dualistic divide of philosophy between realist and antirealist worldviews.