Article

Foundations of Science

, Volume 17, Issue 1, pp 51-89

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

  • Karin Usadi KatzAffiliated withDepartment of Mathematics, Bar Ilan University
  • , Mikhail G. KatzAffiliated withDepartment of Mathematics, Bar Ilan University Email author 

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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.

Keywords

Abraham Robinson Adequality Archimedean continuum Bernoullian continuum Burgess Cantor Cauchy Completeness Constructivism Continuity Dedekind Du Bois-Reymond Epsilontics Errett Bishop Felix Klein Fermat-Robinson standard part Infinitesimal Law of excluded middle Leibniz-Łoś transfer principle Nominalistic reconstruction Nominalism Non-Archimedean Rigor Simon Stevin Weierstrass