The indispensability thesis maintains both that using mathematical terms and assertions is an indispensable part of scientific practice and that this practice commits science to mathematical objects and truths. Anti-realists have used several methods for attacking this thesis: Hartry Field has tried to show how science can do without mathematics by showing that it is possible to replace analytic mathematical scientific theories with synthetic versions that make no reference to mathematical objects. Phillip Kitcher and Charles Chichara have tried, instead, to maintain the mathematical formalism in science without being committed to mathematical realism, by giving a non-realist account of mathematical objects. Finally, Geoffrey Hellman devised a modal structuralism that uses modal operators to translate standard mathematical language into a ‘structuralist’ language. In this chapter I discuss these positions and claim that, on the one hand, they have failed to show that science can do without mathematical objects, and on the other, that these approaches to mathematics do not represent an ontic and epistemic gain over standard realism.
Keywords: Chichara, Field, Hellman, indispensability thesis, Kitcher, mathematical formalism, mathematical realism, mathematical structuralism, modal operators, structuralism