Abstract: Numbers and other mathematical objects are exceptional in having no locations in space and time and no causes or effects in the physical world. This makes it difficult to account for the possibility of mathematical knowledge, leading many philosophers to embrace nominalism, the doctrine that there are no abstract entities. It has also led some of them to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects, eliminating so-called ontological commitment to numbers, sets, and the like. These projects differ considerably in the apparatus they employ, and the spirit in which they are put forward. Some employ synthetic geometry, others modal logic. Some are put forward as revolutionary replacements for existing mathematics and science, others hermeneutic hypotheses about what they have meant all along. We attempt to cut through technicalities that have obscured previous discussions of these projects, and to present concise accounts with minimal prerequisites of a dozen strategies for nominalistic interpretation of mathematics. We also examine critically the aims and claims of such interpretations, suggesting that what they really achieve is something quite different from what the authors of such projects usually assume.

Keywords: abstract entities, Burgess, mathematical knowledge, mathematical objects, modal logic, nominalism, ontological commitment, philosophy, philosophy of mathematics, Rosen, synthetic geometry