, Volume 77, Issue 3, pp 335–359 | Cite as

Mathematical Contingentism

Original Article


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.


  1. Azzouni, J. (1997). Applied mathematics, existential commitment and the quine-putnam indispensability thesis. Philosophia Mathemaica, 3(5), 193–209.Google Scholar
  2. Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press.Google Scholar
  3. Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70, 661–680.CrossRefGoogle Scholar
  4. Bird, A. (2005). The dispositionalist conception of laws. Foundations of Science, 10, 353–370.CrossRefGoogle Scholar
  5. Colyvan, M. (1998a). Is Platonism a bad bet? Australasian Journal of Philosophy, 76(1), 115–119.CrossRefGoogle Scholar
  6. Colyvan, M. (1998b). In defence of indispensability. Philosophia Mathematica, 6(1), 39–62.CrossRefGoogle Scholar
  7. Colyvan, M. (1999). Contrastive empiricism and indispensability. Erkenntnis, 51(2–3), 323–332.CrossRefGoogle Scholar
  8. Colyvan, M. (2000). Conceptual contingency and abstract existence. Philosophical Quarterly, 50, 87–91.CrossRefGoogle Scholar
  9. Colyvan, M. (2001). The indispensability of mathematics. New York: OUP.CrossRefGoogle Scholar
  10. 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
  11. Colyvan, M. (2008). Indispensabilty arguments in the philosophy of maths. Stanford Encyclopedia of Philosophy.
  12. Field, H. (1980). Science without numbers: A defence of nominalism. Oxford: Blackwell.Google Scholar
  13. Hale, B. (1987). Abstract objects. Oxford: OUP.Google Scholar
  14. Hale, B., & Wright, C. (1992). Nominalism and the contingency of abstract objects. The Journal of Philosophy, 89(3), 111–135.CrossRefGoogle Scholar
  15. Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Clarendon.Google Scholar
  16. Maddy, P. (1990). Realism in mathematics. Oxford: OUP.Google Scholar
  17. Maddy, P. (1992). Indispensability and practice. Journal of Philosophy, 89(6), 275–289.CrossRefGoogle Scholar
  18. Miller, K. (2010). Minimalism and modality: the nature of mathematical objects. In A. Hazlett. (Ed.), New Waves in Metaphysics. London: Palgrave McMillan.Google Scholar
  19. 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
  20. 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
  21. Schiffer, S. (1996). Language created, language independent entities. Philosophical Topics, 24, 149–167.Google Scholar
  22. Shoemaker, S. (1998). Causal and metaphysical necessity. Pacific Philosophical Quarterly, 79, 59–77.CrossRefGoogle Scholar
  23. Sider, T. (2002). The ersatz pluriverse. The Journal of Philosophy, 99, 275–315.Google Scholar
  24. Swoyer, C. (1982). The nature of natural laws. Australasian Journal of Philosophy, 60, 203–223.CrossRefGoogle Scholar
  25. Wright, C. (1983). Frege’s conception of numbers as objects. Aberdeen: Aberdeen University Press.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of PhilosophyThe University of SydneySydneyAustralia

Personalised recommendations