David Ellerman: A Fundamental Duality in the Exact Sciences – an Introduction to Mathematical Metaphysics

Abstract: There is a fundamental duality that runs through the exact sciences from logic to biology. At the logical level, it is the duality between (Boolean) logic of subsets and the logic of partitions. The quantitative versions of the dual logics are logical probability theory and logical information theory. The duality explains the duality in the category of $\mathsf{Sets}$ and its opposite $\mathsf{Sets}^{\mathrm{op}}$. The partial order in the two dual logics gives the two fundamental canonical functions and the claim is that all canonical morphisms in Sets arise (using the given information) from those two morphisms. In physics, there is the notion of “definiteness all the way down” which arises in classical physics (Boolean logic of subsets) and dually there is the notion of definiteness only down to a certain level and then objective indefiniteness that arises in quantum physics (logic of partitions).