Version: v1, Published online: 1998
Retrieved May 31, 2020, from https://www.rep.routledge.com/articles/thematic/category-theory-applications-to-the-foundations-of-mathematics/v-1
Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue.
Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics.
Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics.
Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical.
This article assumes knowledge of category theory on the level of Category theory, introduction to §1.
McLarty, Colin. Category theory, applications to the foundations of mathematics, 1998, doi:10.4324/9780415249126-Y071-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/category-theory-applications-to-the-foundations-of-mathematics/v-1.
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