On Self-Predicative Universals in Category Theory

TL;DR

This paper explores how category theory's universals serve as models for self-predicative universals in philosophy and mathematics, highlighting their foundational significance and introducing new conceptual frameworks like brain functors.

Contribution

It demonstrates the role of always-self-predicative universals in category theory as a foundation for understanding philosophical and mathematical universals, and introduces the brain functor concept.

Findings

Category theory universals model Platonic Forms and Hegelian universals.

Self-predicative universals contrast with non-self-predicative universals in set theory.

Introduction of brain functors as a new conceptual model.

Abstract

1. 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.