Abstract: The philosophy of mathematics articulated and defended in this book goes by the name of “structuralism”, and its slogan is that mathematics is the science of structure. The subject matter of arithmetic, for example, is the natural number structure, the pattern common to any countably infinite system of objects with a distinguished initial object and a successor relation that satisfies the induction principle. The essence of each natural number is its relation to the other natural numbers. One way to understand structuralism is to reify structures as ante rem universals. This would be a platonism concerning mathematical objects, which are the places within such structures. Alternatively, one can take an eliminative, in re approach, and understand talk of structures as shorthand for talk of systems of objects or, invoking modality, talk of possible systems of objects. Shapiro argues that although the realist, ante rem approach is the most perspicuous, in a sense, the various accounts are equivalent. Along the way, the ontological and epistemological aspects of the structuralist philosophies are assessed. One key aspect is to show how each philosophy deals with reference to mathematical objects. The view is tentatively extended to objects generally: to science and ordinary discourse.

Keywords: ante rem, mathematics, modality, ontology, philosophy of mathematics, Platonism, realism, reference, reification, structuralism, universal