On Self-Predicative Universals in Category Theory

This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato’s Theory of Forms as well as for the idea of a “concrete universal” in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. 2. The paper also shows how the always-self-predicative universals of category theory provide the “opposite bookend” to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having one theory (e.g., Frege’s Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or the other). 3. Moreover the paper considers one of the most important examples of self-predicative universals in pure mathematics, namely adjoint functors or adjunctions. It gives a parsing of adjunctions into two halves (left and right semi-adjunctions) using the heterodox notion of heteromorphisms, and then shows that the parts can be recombined in a new way to define the cognate-to-adjoints notion of a brain functor that provides an abstract conceptual model of a brain. 4. Finally the paper argues that at least one way category theory has foundational relevance is that it isolates the universal concepts and structures that are important throughout mathematics.

This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato’s Theory of Forms as well as for the idea of a “concrete universal” in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. 2. The paper also shows how the always-self-predicative universals of category theory provide the “opposite bookend” to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having one theory (e.g., Frege’s Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or the other). 3. Moreover the paper considers one of the most important examples of self-predicative universals in pure mathematics, namely adjoint functors or adjunctions. It gives a parsing of adjunctions into two halves (left and right semi-adjunctions) using the heterodox notion of heteromorphisms, and then shows that the parts can be recombined in a new way to define the cognate-to-adjoints notion of a brain functor that provides an abstract conceptual model of a brain. 4. Finally the paper argues that at least one way category theory has foundational relevance is that it isolates the universal concepts and structures that are important throughout mathematics.

文献摘要