Reconciling Anti-Nominalism and Anti-Platonism in Philosophy of Mathematics
Abstract
The author reviews and summarizes, in as jargon-free way as he is capable of, the form of anti-platonist anti-nominalism he has previously developed in works since the 1980s, and considers what additions and amendments are called for in the light of such recently much-discussed views on the existence and nature of mathematical objects as those known as hyperintensional metaphysics, natural language ontology, and mathematical structuralism.
References
Anderson, Alan Ross and Nuel D. Belnap, Jr (1975). Entailment: The Logic of Relevance and Necessity, Princeton: Princeton University Press, pp. 473–492.
Benacerraf, Paul (1965) “What Numbers Could Not Be”, Philosophical Vol. 74, No. 1, pp. 47-73. doi: 10.2307/2183530
Borges, Jorge Luis (1962, Spanish original 1940). “Tlön, Uqbar, Orbis Tertius”, translated by Alastair Reid. In: Ficciones: Selected Essays and Short Stories of Jorge Luis Borges, edited by Anthony Kerrigan, New York: Grove Press, pp. 17–36.
Burgess, John P. (2015). Rigor and Structure, Oxford: Oxford University Press.
Carnap, Rudolf (1950). “Empiricism, Semantics, and Ontology”, Revue Internationale de Philosophie 4, no. 11, pp. 20–40.
Chihara, Charles (1973). Ontology and the Vicious Circle Principle, Ithaca: Cornell University Press.
Chihara, Charles (1990). Constructibility and Mathematical Existence, Oxford: Oxford University Press.
Drake, Stillman (1957). Translator and editor, Discoveries and Opinions of Galileo, New York: Doubleday
Field, Hartry (1980). Science without Numbers: A Defense of Nominalism, Princeton: Princeton University Press.
Gödel, Kurt (1947). “What Is Cantors Continuum Problem?”, American Mathematical Monthly 54, no. 9, pp. 515–524.
Hart, Wilbur D. (1977). “Mathematical Knowledge by Mark Steiner”, Review of Steiner (1975), Journal of Philosophy¬ 74, no. 2, pp. 118–129.
Hellman, Geoffrey (1989). Mathematics without Numbers, Oxford: Oxford University Press.
Hennis, Boris (2018). “Lichtenberg’s Point”, Grazer Philosophische Studien 95, no. 2, pp. 265–286. doi: 10.1163/18756735-000040
Hicks, R. D. (1925). Translator and editor. Diogenes Laertius: Lives of Eminent Philosophers vol. 2, Loeb Classical Library, Cambridge: Harvard University Press, pp. 22–83.
James, William (1907). Pragmatism: A New Name for Some Old Ways of Thinking, Cambridge: Harvard University Press.
Kneale, William and Mary Kneale (1963). The Development of Logic, Oxford: Clarendon Press.
Moltmann, Friederike (2013). “Reference to Numbers in Natural Language”, Philosophical Studies 162, no. 3, pp. 499–536. doi: 10.1007/s11098-011-9779-1
Nietzsche, Friedrich (1967, German original 1887). On the Genealogy of Morals. Translated by Walter Kaufmann and R. J. Hollingdale, New York: Vintage.
Quine, W. V. O. (1960a). Word and Object, Cambridge: MIT Press.
Quine, W. V. O. (1960b). “Variables Explained Away”, Proceedings of the American Philosophical Society 104, pp. 343–347.
Steiner, Mark, (1975). Mathematical Knowledge, Ithaca: Cornell University Press.
Strawson, Peter (1959). Individuals: An Essay in Descriptive Metaphysics, London: Methuen.
Whorf, Benjamin Lee (1956). “Science and Linguistics”, In: Language, Thought, and Reality, edited by J. B. Carroll, Cambridge: MIT Press.
Copyright (c) 2022 Disputatio. Philosophical Research Bulletin
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
