## Exploring Categorical Structuralism (2004)

Venue: | Philosophia Mathematica |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Mclarty04exploringcategorical,

author = {Colin Mclarty},

title = {Exploring Categorical Structuralism},

journal = {Philosophia Mathematica},

year = {2004},

pages = {37--53}

}

### OpenURL

### 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.

### Citations

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(Show Context)
Citation Context ...istence is understood in physicalist empirical terms. Similarly, I doubt that any ZF set theorist takes set-theoretic reality as empirical or physical reality in anything like a Popperian way. (Maddy =-=[1990]-=- goes the farthest of anyone I know to make sets physical or empirical, but she is no Popperian.) In Mac Lane's words, with his emphasis, 'Mathematics is a formal network, but the concepts and axioms ... |

1 | California: Center for the Study of Language and Information. ARISTOTLE [1984]: The Complete Works of Aristotle - Stanford |

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(Show Context)
Citation Context ...eferred foundation is 'a well pointed topos with Axiom of Choice and a Natural Number Object' which, he notes, are the axioms of Lawvere's Elementary Theory of the Category of Sets, or ETCS (Mac Lane =-=[1998]-=-, p. 291; citing Lawvere [1964]). Hellman wants to explore whether, and how, this can serve as a foundation. He raises five main issues: (1) He asks whether it is too weak, saying: 'a powerful set the... |

1 | Frege to Godel - DEDEKIND - 1967 |

1 |
an ontologist
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(Show Context)
Citation Context ...ent. * Department of Philosophy, Case Western Reserve University, Cleveland, Ohio 44106 U. S. A. cxm7@po.cwru.edu 1 Extensionality makes the internal logic classical as noted in Mac Lane and Moerdijk =-=[1992]-=-, p. 331. Hellman stresses that 'the first-order topos axioms must be supplemented so that Boolean, classical, logic will be available. (Without supplementation, which Mac Lane and Moerdijk [1992] spe... |

1 |
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Citation Context ...inal Hu. But Mac Lane has been clear on many occasions that the easy kinds of transcendence do not interest him. See notably his debate with Adrian Mathias in Mathias [1992], Mac Lane [1992], Mathias =-=[2000]-=-, Mac Lane [2000]. He happily foregoes Nw as unused in mainstream mathematics. What interests him are new forms that could fundamentally change our daily approach to mathematics, fundamentally new mat... |