Ellerman, David P. (2016) Category theory and set theory as theories about complementary types of universals. Logic and Logical Philosophy (Online First), 2016 (1). pp. 1-18.
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Abstract
Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought.
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| Item Type: | Published Article or Volume | ||||||
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| Additional Information: | I checked on Romeo sherpa and this journal is RoMeo Green. | ||||||
| Keywords: | universals, category theory, Plato's Theory of Forms, set theoretic antinomies, universal mapping properties | ||||||
| Subjects: | General Issues > History of Philosophy of Science Specific Sciences > Mathematics General Issues > Realism/Anti-realism |
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| Depositing User: | David Ellerman | ||||||
| Date Deposited: | 13 Aug 2016 19:02 | ||||||
| Last Modified: | 13 Aug 2016 19:02 | ||||||
| Item ID: | 12353 | ||||||
| Journal or Publication Title: | Logic and Logical Philosophy (Online First) | ||||||
| Publisher: | Nicolaus Copernicus University | ||||||
| Official URL: | http://apcz.pl/czasopisma/index.php/LLP/article/vi... | ||||||
| DOI or Unique Handle: | 10.12775/LLP.2016.022 | ||||||
| Subjects: | General Issues > History of Philosophy of Science Specific Sciences > Mathematics General Issues > Realism/Anti-realism |
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| Date: | 5 August 2016 | ||||||
| Page Range: | pp. 1-18 | ||||||
| Volume: | 2016 | ||||||
| Number: | 1 | ||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/12353 |
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