A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

TL;DR

This paper critically examines how nominalistic tendencies have influenced the development, interpretation, and historiography of mathematics, especially in foundational analysis, through a Burgessian philosophical lens.

Contribution

It offers a Burgessian critique of nominalist reconstructions in mathematical rigor and historiography, highlighting their philosophical and pedagogical implications.

Findings

Nominalist reconstructions impact mathematical historiography and teaching.

Burgessian critique reveals philosophical issues in nominalist approaches.

Analysis of foundational work by Cantor, Dedekind, Weierstrass, Cauchy, and Bishop.

Abstract

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.