If (or when) mathematics is dispensable for solving empirical problems, why can the detour through mathematics be so useful? Typical discussions of applicability only approach this problem indirectly, saying just enough about it to argue that whatever gain mathematics offers (e.g., a gain in inferential power) can be enjoyed without ontological costs. In this paper, I first clarify this problem and distinguish it from other, related ones; in particular, I make it clear that the problem has nothing specifically to do with the application of mathematics to empirical sciences, and arises in the same way for many so-called applications of mathematics to mathematics (which I argue are best defined as deployments of new mathematical resources to make progress on a mathematical problem). I then survey possible solutions, including a particularly promising one, inspired by Ken Manders and in line with recent work on reformulations: mathematics offers tools for systematic expressive restriction, allowing for attentional management in problem solving.
David Waszek (Mon,) studied this question.