Nominalism and the application of mathematics

A significant feature of contemporary science is the widespread use of mathematics in several of its subfields. In many instances, the content of scientific theories cannot be formulated without reference to mathematical objects (such as functions, numbers, or sets). In the hands of W. V. Quine and Hilary Putnam, this feature of scientific practice was invoked in support of platonism (the view according to which mathematical objects exist). Quine and Putnam insisted that one ought to be ontologically committed to mathematical entities since they are indispensable to our best theories of the world. This is the indispensability argument. This argument posed a formidable challenge to nominalists, who now needed to show either (i) that mathematical entities are not indispensable to mathematics or (ii) that quantification over these entities does not require ontological commitment. Since Hartry Field’s Science without Numbers (Princeton, NJ: Princeton University Press, 1980), most nominalisation strategies have attempted to show that mathematics is ultimately dispensable, thus taking up (i): Geoffrey Hellman’s modal structuralism and Charles Chihara’s constructibility approach offer two examples. Unfortunately, for a variety of technical and philosophical reasons, none of these strategies have succeeded (see John Burgess and Gideon Rosen, A Subject with No Object, Oxford: Oxford University Press, 1997). In his book, Jody Azzouni provides the most thorough and detailed attempt to explore (ii). On his view, (a) mathematical theories are in fact indispensable to science. It is often not even possible to express the content of a scientific theory without invoking mathematical objects. Moreover, (b) mathematical and scientific theories are true, and they need to be taken to be true, since often one needs to draw consequences from such theories without being able to specify exactly what their content is. Despite (a) and (b), however, Azzouni denies that mathematical objects exist. How?