Abstract

Hellman [2003] raises interesting challenges to categorical structuralism. He starts citing Awodey [1996] which, as Hellman sees, is not intended as a foundation for mathematics. It offers a structuralist framework which could denned in any of many different foundations. But Hellman says Awodey's work is 'naturally viewed in the context of Mac Lane's repeated claim that category theory provides an autonomous foundation for mathematics as an alternative to set theory ' (p. 129). Most of Hellman's paper 'scrutinizes the formulation of category theory ' specifically in 'its alleged role as providing a foundation ' (p. 130). I will also focus on the foundational question. Page numbers in parentheses are page references to Hellman [2003]. 1. Categorical Set Theory Hellman correctly notes (p. 131) that Mac Lane often writes as if he used a set-theoretic foundation. Indeed Mac Lane does use a set-theoretic foundation—categorical set theory. Categorical set theory begins with the theory WPT (for well-pointed topos) which is the elementary topos axioms plus extensionality, also called well-pointedness: For any parallel functions f,g: A — • B, either / = g or there is some x € A with fx j=gx. A function is fully determined by its values on elements. 1 An element is a function from a singleton set; so x 6 A says x: 1 — » A. In ZF each element x £ A determines a unique function from any singleton 1 to A, taking the single element of 1 to x, and vice versa. In categorical set theory the function is the element.

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