This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy of mathematics. It belongs to anti-realism, but can meet those challenges and can perhaps convince some realists, at least those who are also naturalists.
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Baker A. (2001) Mathematics, indispensability and scientific progress. Erkenntnis 55: 85–116
Baker A. (2005) Are there genuine mathematical explanations of physical phenomena? Mind 114: 223–238
Benacerraf P. (1973) Mathematical truth. Journal of Philosophy 70: 661–679
Burgess J.P. (2004) Mathematics and bleak house. Philosophia Mathematica 12(3): 18–36
Burgess J.P., Rosen G. (1997) A Subject with no object. Clarendon Press, Oxford
Chihara C. (2005) Nominalism. In: Shapiro S. (eds) The oxford handbook of philosophy of mathematics and logic. Oxford University Press, Oxford, pp 483–514
Colyvan M. (1999) Confirmation theory and indispensability. Philosophical Studies 96: 1–19
Colyvan M. (2002) Mathematics and aesthetic considerations in science. Mind 111: 69–74
Dummett M. (1973) The philosophical basis of intuitionistic logic. In: Benacerraf P., Putnam H. (eds) Philosophy of mathematics: Selected readings (Reprinted, 1983). Cambridge University Press, Cambridge, pp 97–129
Field H. (1980) Science without numbers. Basil Blackwell, Oxford
Field H. (1998) Which undecidable mathematical sentences have determinate truth values? In: Dales H.G., Oliveri G. (eds) Truth in mathematics. Oxford University Press, Oxford
Goodman N., Quine W.V. (1947) Steps toward a constructive nominalism. Journal of Symbolic Logic 12: 105–122
Hellman G. (2005) Structuralism. In: Shapiro S. (eds) The oxford handbook of philosophy of mathematics and logic. Oxford University Press, Oxford, pp 536–562
Hoffman S. (2004) Kitcher, ideal agents, and fictionalism. Philosophia Mathematica 12(3): 3–17
Lakoff G., Núñez R. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books, New York
Leng M. (2002) What’s wrong with indispensability? Synthese 131: 395–417
Leng M. (2005) Revolutionary fictionalism: A call to arms. Philosophia Mathematica (III) 13: 277–293
Maddy P. (1997) Naturalism in mathematics. Clarendon Press, Oxford
Maddy P. (2005a) Three forms of naturalism. In: Shapiro S. (eds) The oxford handbook of philosophy of mathematics and logic. Oxford University Press, Oxford, pp 437–514
Maddy P. (2005b) Mathematical existence. Bulletin of Symbolic Logic 11: 351–376
Melia J. (2000) Weaseling away the indispensability argument. Mind 109: 455–479
Papineau D. (1993) Philosophical naturalism. Basil Blackwell, Oxford
Renyi A. (1967) Dialogues on mathematics. Holden-Day, San Francisco
Rosen G., Burgess J.P. (2005) Nominalism reconsidered. In: Shapiro S. (eds) The oxford handbook of philosophy of mathematics and logic. Oxford University Press, Oxford, pp 515–535
Sober E. (1993) Mathematics and indispensability. The Philosophical Review 102: 35–57
Yablo S. (2001) Go figure: A path through fictionalism. Midwest Studies in Philosophy 25(1): 72–102
Yablo S. (2002) Abstract objects: A case study. Noûs 36: 220–240
Ye, F. (2008a). Naturalism and abstract entities. Available online at http://sites.google.com/site/fengye63/.
Ye, F. (2008b). On what really exist in mathematics. ibid.
Ye, F. (2008c). Naturalism and objectivity in mathematics. ibid.
Ye, F. (2008d). Naturalism and the apriority of logic and arithmetic. ibid.
Ye, F. (2008e). The applicability of mathematics as a scientific and a logical problem. ibid.
Ye, F. (2008f). Strict finitism and the logic of mathematical applications. ibid.
Ye, F. (2008g). A strictly finitistic system for applied mathematics. ibid.
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Ye, F. What anti-realism in philosophy of mathematics must offer. Synthese 175, 13–31 (2010). https://doi.org/10.1007/s11229-009-9535-7
- Philosophy of mathematics