ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish (...) between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$. (shrink)

A response is given here to Benacerraf's (1965) non-uniqueness (or multiple-reductions) objection to mathematical platonism. It is argued that non-uniqueness is simply not a problem for platonism; more specifically, it is argued that platonists can simply embrace non-uniqueness—i.e., that one can endorse the thesis that our mathematical theories truly describe collections of abstract mathematical objects while rejecting the thesis that such theories truly describe unique collections of such objects. I also argue that part of the motivation for this stance is (...) that it dovetails with the correct response to Benacerraf's other objection to platonism, i.e., his (1973) epistemological objection. (shrink)

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics (...) as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)