Category Theory and Set Theory as Theories about Complementary Types of Universals

David Ellerman



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 uF ∈ uF . 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.


universals; category theory; Plato’s Theory of Forms; set theoretic antinomies; universal mapping properties

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Allen, R.E., “Participation and predication in Plato’s Middle Dialogues”, The Philosophical Review, 69, 2 (1960): 147–164. DOI: 10.2307/2183501

Allen, R.E. (ed.), Studies in Plato’s Metaphysics, London: Routledge & Kegan Paul, 1965.

Awodey, S., Category Theory (Oxford Logic Guides), Oxford University Press, Oxford, 2006. DOI: 10.1093/acprof:oso/9780198568612.001.0001

Boolos, G., “The iterative conception of set”, The Journal of Philosophy, 68, 8 (1971): 215–231. DOI: 10.2307/2025204

Desmond, W., Art and the Absolute: A Study Of Hegel’s Aesthetics, Albany: State University of New York Press, 1986.

Ellerman, D., “Category theory and concrete universals”, Erkenntnis, 28 (1988): 409–429.

Fujisawa, N., ‘Eχειν, Mετεχειν, and idioms of ‘paradeigmatism’ in Plato’s theory of forms”, Phronesis, 19, 1 (1974): 30–58.

Galluzzo, G., and M.J. Loux (eds.) The Problem of Universals in Contemporary Philosophy, Cambridge UK: Cambridge University Press, 2015. DOI: 10.1017/CBO9781316181539

Geach, P.T., “The Third Man Again”, The Philosophical Review, 65, 1 (1956): 72–82. DOI: 10.2307/2182189

Geach, P.T., Logic Matters, Berkeley: University of California Press, 1980.

Hatcher, W., The Logical Foundations of Mathematics. Oxford: Pergamon Press, 1982.

Honderich, T. (ed.) The Oxford Companion to Philosophy. New Edition, Oxford UK: Oxford University Press, 2005.

Hungerford, T. W., Algebra, New York: Springer-Verlag, 1974.

Kneale, W., and M. Kneale, The Development of Logic, Oxford: Oxford University Press, 1962.

Lawvere, F.W., “Adjointness in foundations”, Dialectica, 23, 3–4 (1969): 281–295. DOI: 10.1111/j.1746-8361.1969.tb01194.x

Lawvere, F.W., and S. Schanuel, Conceptual Mathematics: A First Introduction to Categories, New York: Cambridge University Press, 1997. DOI: 10.1017/CBO9780511804199

Mac Lane, S., “Groups, categories, and duality”, Proc. Nat. Acad. Sci. U.S.A., 34, 6 (1948): 263–267.

Mac Lane, S., Categories for the Working Mathematician. New York: Springer-Verlag, 1971. DOI: 10.1007/978-1-4757-4721-8

Mac Lane, S., and G. Birkhoff, Algebra (First Edition), New York: Macmillan, 1967.

Makkai, M., “Structuralism in Mathematics”, pages 43–66 in R. Jackendoff, P. Bloom, and K. Wynn (eds.), Language, Logic, and Concepts: Essays in Memory of John Macnamara, Cambridge: MIT Press (A Bradford Book), 1999.

Magnan, F., and G.E. Reyes, “Category theory as a conceptual tool in the study of cognition”, pages 57–90 in J. Macnamara and G.E. Reyes (eds.), The Logical Foundations of Cognition, New York: Oxford University Press, 1994.

Malcolm, J., Plato on the Self-Predication of Forms, Oxford: Clarendon Press, 1991. DOI: 10.1093/acprof:oso/9780198239062.001.0001

Marquis, J.-P., “Three kinds of universals in mathematics”, pages 191–212 in B. Brown and J. Woods (eds.), Logical Consequence: Rival Approaches and New Studies in Exact Philosophy: Logic, Mathematics and Science, Vol. II, Oxford: Hermes, 2000.

Miles, M.R., The Word Made Flesh: A History of Christian Thought, Malden MA: Blackwell, 2005.

Nehamas, A., “Self-predication and Plato’s theory of forms”, American Philosophical Quarterly, 16, 2 (1979): 93–103.

Quine, W.V.O., On Frege’s Way Out”, Mind, 64, 254 (1955): 145–159. DOI: 10.1093/mind/LXIV.254.145

Quine, W.V.O., Mathematical Logic, Cambridge, MA: Harvard University Press, 1955.

Russell, B., Principles of Mathematics, London: Routledge Classics, 2010 (1903).

Samuel, P., “On universal mappings and free topological groups”, Bull. Am. Math. Soc., 54, 6 (1948): 591–598. DOI: 10.1090/S0002-9904-1948-09052-8

Sayre, K., Metaphysics and Method in Plato’s Statesman, New York: Cambridge University Press, 2006.

Shoenfield, J., Mathematical Logic, Reading MA: Addison-Wesley, 1967.

Stern, R., “Hegel, British idealism, and the curious case of the concrete universal”, British Journal for the History of Philosophy, 15, 1 (2007): 115–153. DOI: 10.1080/09608780601088002

Vlastos, G., . Platonic Studies, Princeton NJ: Princeton University Press, 1981.

Vlastos, G., Studies in Greek Philosophy. Volume II: Socrates, Plato, and Their Tradition, edited by D.W. Graham, Princeton NJ: Princeton University Press, 1995.

Vlastos, G. (ed.) Plato: A Collection of Critical Essays I: Metaphysics and Epistemology, Notre Dame: University of Notre Dame Press, 1978. DOI: 10.1007/978-1-349-86203-0

Whitehead, A.N., and B. Russell, Principia Mathematica to *56, Cambridge UK: Cambridge University Press, 1997 (1910).

Wimsatt Jr., W.K., “The structure of the ‘concrete universal’ in literature”, PMLA, 62 (1947): 262–280.

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