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

Karin Usadi Katz; Mikhail G. Katz

doi:10.1007/s10699-011-9223-1

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

Karin Usadi Katz, Mikhail G. Katz

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	51-89
Number of pages	39
Journal	Foundations of Science
Volume	17
Issue number	1
DOIs	https://doi.org/10.1007/s10699-011-9223-1
State	Published - Mar 2012

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
Nominalism
Nominalistic reconstruction
Non-Archimedean
Rigor
Simon Stevin
Weierstrass

All Science Journal Classification (ASJC) codes

General
History and Philosophy of Science

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10.1007/s10699-011-9223-1

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title = "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography",

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-{\L}o{\'s} transfer principle, Nominalism, Nominalistic reconstruction, Non-Archimedean, Rigor, Simon Stevin, Weierstrass",

author = "Katz, \{Karin Usadi\} and Katz, \{Mikhail G.\}",

note = "Funding Information: Mikhail G. Katz: Supported by the Israel Science Foundation grant 1294/06.",

year = "2012",

month = mar,

doi = "10.1007/s10699-011-9223-1",

language = "الإنجليزيّة",

volume = "17",

pages = "51--89",

journal = "Foundations of Science",

issn = "1233-1821",

publisher = "Springer Netherlands",

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T1 - A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

AU - Katz, Karin Usadi

AU - Katz, Mikhail G.

N1 - Funding Information: Mikhail G. Katz: Supported by the Israel Science Foundation grant 1294/06.

PY - 2012/3

Y1 - 2012/3

N2 - 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.

AB - 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.

KW - Abraham Robinson

KW - Adequality

KW - Archimedean continuum

KW - Bernoullian continuum

KW - Burgess

KW - Cantor

KW - Cauchy

KW - Completeness

KW - Constructivism

KW - Continuity

KW - Dedekind

KW - Du Bois-Reymond

KW - Epsilontics

KW - Errett Bishop

KW - Felix Klein

KW - Fermat-Robinson standard part

KW - Infinitesimal

KW - Law of excluded middle

KW - Leibniz-Łoś transfer principle

KW - Nominalism

KW - Nominalistic reconstruction

KW - Non-Archimedean

KW - Rigor

KW - Simon Stevin

KW - Weierstrass

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U2 - 10.1007/s10699-011-9223-1

DO - 10.1007/s10699-011-9223-1

M3 - مقالة

SN - 1233-1821

VL - 17

SP - 51

EP - 89

JO - Foundations of Science

JF - Foundations of Science

IS - 1

ER -

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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