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

Abstract

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 an ontologically limitative disposition on historiography, teaching, and research.

Rubik’s for Cryptographers

“…Rather, his approach to continuity was via what is known today as microcontinuity (see the subsection "Continuity"). Several recent articles, (Błaszczyk et al [14]; Borovik and Katz [16]; Bråting [20]; Katz and Katz [62], [64]; Katz and Tall [69]), have argued that a proto-Weierstrassian view of Cauchy is one-sided and obscures Cauchy's important contributions, including not only his infinitesimal definition of continuity but also such innovations as his infinitesimally defined ("Dirac") delta function, with applications in Fourier analysis and evaluation of singular integrals, and his study of orders of growth of infinitesimals that anticipated the work of Paul du Bois-Reymond, 8 Cantor's dubious claim that the infinitesimal leads to contradictions was endorsed by no less an authority than B. Russell; see footnote 15 in the subsection "Mathematical Rigor".…”

“…The philosophical underpinnings of such a belief were analyzed in (Katz and Katz 2012a [64]), where it was pointed out that in mathematics, as in other sciences, former errors are eliminated through a process of improved conceptual understanding, evolving over time, of the key issues involved in that science.…”

“…Moreover, no such claim for a single scientific development is made either by the practitioners or by the historians of the natural sciences. It was further pointed out in [64] that the term mathematical rigor itself is ambiguous, its meaning varying according to context. Four possible meanings for the term were proposed in [64]:…”

“…It was further pointed out in [64] that the term mathematical rigor itself is ambiguous, its meaning varying according to context. Four possible meanings for the term were proposed in [64]:…”

Leibniz’s Laws of Continuity and Homogeneity

“…The existence of such implementations indicates that the historical infinitesimals were less prone to contradiction than has been routinely maintained by triumvirate historians. 10 The issue is dealt with in more detail by Katz and Katz [22], [23], [24], [25]; Błaszczyk et al [2]; Borovik et al [3]; Katz and Leichtnam [26]; Katz, Schaps, and Shnider [27].…”