David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Philosophia Mathematica 19 (3):227-254 (2011)
Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a category-theoretical approach will be highly appropriate. But if sets have a richer `nature' than is preserved under isomorphism, then such an approach will be inadequate.
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Solomon Feferman (2013). Foundations of Unlimited Category Theory: What Remains to Be Done. Review of Symbolic Logic 6 (1):6-15.
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