Platonists and nominalists disagree about whether mathematical objects exist. But they almost uniformly agree about one thing: whatever the status of the existence of mathematical objects, that status is modally necessary. Two notable dissenters from this orthodoxy are Hartry Field, who defends contingent nominalism, and Mark Colyvan, who defends contingent Platonism. The source of their dissent is their view that the indispensability argument provides our justification for believing in the existence, or not, of mathematical objects. This paper considers whether commitment to the indispensability argument gives one grounds to be a contingentist about mathematical objects.
- Azzouni, J. (1997). Applied mathematics, existential commitment and the quine-putnam indispensability thesis. Philosophia Mathemaica, 3(5), 193–209.Google Scholar
- Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press.Google Scholar
- Colyvan, M. (2007). Mathematical recreation versus mathematical knowledge. In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical knowledge (pp. 109–122). Oxford: Oxford University Press.Google Scholar
- Colyvan, M. (2008). Indispensabilty arguments in the philosophy of maths. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/mathphil-indis.
- Field, H. (1980). Science without numbers: A defence of nominalism. Oxford: Blackwell.Google Scholar
- Hale, B. (1987). Abstract objects. Oxford: OUP.Google Scholar
- Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Clarendon.Google Scholar
- Maddy, P. (1990). Realism in mathematics. Oxford: OUP.Google Scholar
- Miller, K. (2010). Minimalism and modality: the nature of mathematical objects. In A. Hazlett. (Ed.), New Waves in Metaphysics. London: Palgrave McMillan.Google Scholar
- Putnam, H. (1967). Mathematics without foundations, reprinted in Putnam (1979), Mathematics matter and method: Philosophical papers Vol. I, second edition. (pp. 43–59). Cambridge: Cambridge University Press.Google Scholar
- Quine, W. V. O. (1953). Two dogmas of empiricism, In From a logical point of view (pp. 20–46). Harvard University Press: Cambridge, MA.Google Scholar
- Schiffer, S. (1996). Language created, language independent entities. Philosophical Topics, 24, 149–167.Google Scholar
- Sider, T. (2002). The ersatz pluriverse. The Journal of Philosophy, 99, 275–315.Google Scholar
- Wright, C. (1983). Frege’s conception of numbers as objects. Aberdeen: Aberdeen University Press.Google Scholar