Putnam and the indispensability of mathematics

In this paper, I examine Putnam's nuanced views in the philosophy of mathematics, distinguishing three proposals: modalism (an interpretation of mathematics in terms of modal logic), quasi-empirical realism (that emphasizes the role and use of quasi-empirical methods in mathematics), and an indispensability view (that highlights the indispensable role of quantification over mathematical objects and the support such quantification provides for a realist interpretation of mathematics). I argue that, as he shifted through these views, Putnam aimed to preserve a semantic realist account of mathematics that avoids platonism. In the end, however, each of the proposals faces significant difficulties. A form of skepticism then emerges.