Enhanced ‘If-Thenism’, Fictionalism, and Realist Anti-Platonism

‘If-thenism’ sees mathematical practice as a matter of working out what follows from our mathematical hypotheses without regard to whether those hypotheses are true. Enhanced if-thenism, as Maddy envisages it, adds a further component to mathematical practice: the identification of mathematically valuable hypotheses to investigate (where axioms as hypotheses are “chosen with an eye to facilitating important mathematical goals”, and only some amongst all the logically possible collections of axioms are supported as tracking “underlying contours of mathematical depth”). While accepting that Maddy is right that the identification of mathematically interesting axioms is a sufficiently important component of mathematical practice to warrant a departure from ‘if-thenism’ in its purest form (according to which any consistent axioms would be fair game for mathematical investigation), this chapter argues that empirical science can also provide support for mathematical axioms. Indeed, a plausible reading of Hilary Putnam’s rather elusive indispensability argument is as an argument against pure ‘if-thenism’ and in favour of an enhanced if-thenism where axioms are chosen for scientific reasons.