Platonism and anti‐Platonism: Why worry?

Abstract

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti‐Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the ‘no miracles’ argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific ‘realists’ should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.

Acknowledgements

Thanks to Mark Colyvan, Jeffrey Ketland, Alexander Paseau, Michael Potter, Peter Smith, and two anonymous referees for helpful comments on this paper. This paper was originally written for the 31st Annual Philosophy of Science Conference in Dubrovnik (April 2004), and I am grateful to audience members there for their comments.

Notes

Mary Leng is a research fellow in philosophy at St John's College, Cambridge.

Correspondence: Mary Leng, St John's College, Cambridge, CB2 1TP, UK. Email: [email protected]

This goes against the grain of many approaches to the philosophy of mathematics. Thus, many philosophers of mathematics follow Paul Benacerraf (1973) in taking the central problem for the philosophy of mathematics to be to account for the sense in which our mathematical theories can be said to be true. The approach suggested here is to postpone the issue of accounting for the truth of our theories until it has been ascertained whether we have any reason to believe those theories to be true.

And this is so, even if the particular ontological beliefs of practitioners can be used to explain features of their practice. What is being denied is not that beliefs about ontology will not feature in our account of how and why unapplied mathematics is done in the way that it is done, but rather, that the ‘facts’ concerning mathematical ontology will not matter here.

More formally, and avoiding all talk of models of ZFC, for every axiom of ZFC, the statement that results from relativising all the quantifiers in that axiom to constructible sets is provable in ZFC (i.e. provable from the unrestricted axioms). Furthermore, the relativisation of the continuum hypothesis to constructible sets is provable in ZFC (See Maddy 1997, 65–6 for more details).

Since I am sketching a nominalist analogue of van Fraassen's empirical adequacy, I am using a slightly non‐standard definition of empirical adequacy here. Van Fraassen makes use of the semantic view of theories, and defines empirical adequacy with reference to the models of a theory. This definition would clearly be unacceptable from a nominalist's perspective. However, the spirit of van Fraassen's notion of empirical adequacy (that a theory is empirically adequate exactly if what it says about the observable things and events in this world, is true (van Fraassen 1980, 12)) can be captured by the claim that the theory is correct in only some of its consequences (in van Fraassen's case, its observable consequences). As I noted earlier, the fictionalist refuses to accept the model theoretic defintion of consequence, and instead takes ‘P is a logical consequence of T’ to be a primitive modal notion.

Of course, van Fraassen himself does not accept such a response, since he rejects the demand for explanation on which it is based (van Fraassen 1980, 25).

Whether such explanations are genuinely explanatory is somewhat controversial. Mark Colyvan (2002) and Alan Baker (forthcoming) have, however, both presented quite compelling considerations in favour of the existence of mathematical explanations of physical phenomena.

See Leng (forthcoming) for a defence of this view.

The thought experiment in this form is due to Mark Balaguer (1998, 132). It is, of course, contentious. For example, Alan Baker (2003) criticises it for illicitly relying on spatiotemporal intuitions (if mathematical objects are not spatiotemporal, then it makes little sense to think of them suddenly disappearing). However, the force of the thought experiment doesn't derive from the temporal presentation, so much as the thought that the hypothesis of the existence of acausal mathematical entities should not lead us to expect any different observable phenomena than the hypothesis that they do not exist. It is this intuition that suggests that ‘cosmic coincidence’ considerations should not apply here.

Additional information

Notes on contributors

Mary Leng

Mary Leng is a research fellow in philosophy at St John's College, Cambridge. Correspondence: Mary Leng, St John's College, Cambridge, CB2 1TP, UK. Email: [email protected]