Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Category theory

Developed in the 1940s as a way to organize constructs in algebraic topology, category theory works at the level of whole mathematical objects rather than their elements. In the basic axioms given here the variables represent morphisms that correspond to mappings between objects. (Often morphisms are shown as arrows in diagrams, and objects as nodes.) The axioms specify that when morphisms are composed their domains and codomains must have appropriately matching types. Some of the methodology of category theory has become widely used in mathematics, but until recently the basic theory itself was not extensively studied—and its axiomatic status remains unclear. Category theory can be viewed as a formalization of operations on abstract data types in computer languages—though unlike in Mathematica it normally requires that functions take a single bundle of data as an argument.

From Stephen Wolfram: A New Kind of Science [citation]