Categorical Foundations and Mathematical Practice

Philosophia Mathematica 20 (1):111-113 (2012)
Abstract
Linnebo and Pettigrew's critique in this journal of categorical foundations well emphasizes that the particulars of various categorical foundations matter, and that mathematical practice must be a major consideration. But several categorists named by the authors as proposing categorical foundations do not propose foundations, notably Awodey, and the article's description of current textbook practice seems inaccurate. They say that categorical foundations have justificatory autonomy if and only if mathematics can be justified simply by its practice. Do they seriously believe philosophers may be able to justify some currently unpractised mathematics better than current practice justifies itself?
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