A Fundamental Duality in the Mathematical and Natural Sciences: From Logic to Biology

TL;DR

This essay explores a fundamental duality across mathematics, physics, and biology, linking logical structures, quantum indefiniteness, and biological mechanisms through category theory and information theory.

Contribution

It introduces a unifying duality framework connecting logic, physics, and biology, highlighting the role of category theory and logical entropy in understanding these fields.

Findings

Duality between elements and distinctions in logic and partitions

Quantum physics embodies objective indefiniteness

Biological mechanisms reflect fundamental duality

Abstract

This is an essay in what might be called ``mathematical metaphysics.'' There is a fundamental duality that run through mathematics and the natural sciences. The duality starts as the logical level; it is represented by the Boolean logic of subsets and the logic of partitions since subsets and partitions are category-theoretic dual concepts. In more basic terms, it starts with the duality between the elements (Its) of subsets and the distinctions (Dits, i.e., ordered pairs of elements in different blocks) of a partition. Mathematically, the Its $\&$ Dits duality is fully developed in category theory as the reverse-the-arrows duality. The quantitative versions of subsets and partitions are developed as probability theory and information theory (based on logical entropy). Classical physics was based on a view of reality as definite all the way down. In contrast, quantum physics embodies (objective) indefiniteness. And finally, there are the two fundamental dual mechanisms at work in biology, the selectionist mechanism and the generative mechanism, two mechanisms that embody the fundamental duality.