Response to Colyvan

Abstract

In 'Weaseling Away the Indispensability Argument' (Melia 2000) I argued that although quantification over mathematical objects may indeed be indispensable to our scientific theories, it is not always intellectually dishonest to deny the existence of some of the objects over which we quantify. I argued also that, though the postulation of mathematical objects may increase the attractiveness or utility of our scientific theories, the way in which they do so is unlike the way in which the postulation of theoretical physical entities increases the utility of our scientific theories. Because of this, we can be justified in adopting different attitudes to the different entities postulated by our best scientific theories. In particular, we can have good reason for accepting the existence of theoretical physical entities, but little reason for accepting mathematical ones. Accordingly, I urged that the mere fact that we may have to quantify over mathematical objects in our best physical theory is not enough to establish that we ought to believe that there are such things as mathematical objects. Colyvan claims that, even if we accept my distinction between the different kinds of utility the postulation of physical and mathematical entities may afford, the indispensability argument still works (Colyvan 2002). For there are clear cases in which postulating mathematical entities has exactly the right kind of theoretical virtues. For instance, unificatory power is a clear virtue of a physical theory and postulating complex numbers can increase the unificatory power of a theory. Similarly, explanatory power of a theory is also a virtue of a theory and the Minkowski geometric explanation of certain relativistic effects gives us a clear example of an explanation that employs mathematical entities. In my view, Colyvan's strategy is the best way for those who want to defend the indispensability argument. Were there clear examples where the postulation of mathematical objects results in an increase in the same kind of utility as that provided by the postulation of theoretical entities, then it would seem that the same kind of considerations that