A Fundamental Dualism in the Mathematical and Natural Sciences: From Logic to Biology
This is an essay in what might be called “mathematical metaphysics.” There is a fundamental dualism that runs through mathematics and the natural sciences. The dualism starts at 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 dualism 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 dualism 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 common-sense 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 dualism.
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