Hilbertian Structuralism and the Frege-Hilbert Controversy. Issue 3 (16th October 2019)

• Record Type: Journal Article
• Authors: Doherty, Fiona T
• Is Part Of: Philosophia Mathematica. Volume 27:Issue 3(2019)
• Page Start: 335
• Page End: 361
• Publication Date: 2019-10-16
• Subjects: Mathematics -- Philosophy -- Periodicals; Wiskunde; Filosofische aspecten; Mathematics -- Philosophy; Periodicals; 510.105
• DOI: 10.1093/philmat/nkz016
• Languages: English
• ISSNs: 0031-8019

Abstract

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 $-1$ .