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.
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Mikhail G. Katz: Supported by the Israel Science Foundation grant 1294/06.
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Katz, K.U., Katz, M.G. A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Found Sci 17, 51–89 (2012). https://doi.org/10.1007/s10699-011-9223-1
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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