Fermats «adæquare» – und kein Ende?

Barner, Klaus

doi:10.1007/s00591-010-0083-5

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…In his 1853 text [15], Björling exploits this distinction to argue against a purported counterexample, published by F. Arndt [5] in 1852, to Cauchy's 1821 "sum theorem". 7 Namely, Björling points out that in fact Arndt's counterexample only converges "for every given value", i.e., value from the narrow A-continuum. Meanwhile, it does not converge "for all values", i.e., values from the enriched B-continuum.…”

Section: Bråting's Close Readingmentioning

confidence: 99%

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Cauchy's Continuum

Katz¹,

Katz

2011

Perspectives on Science

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Cauchy's sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in 1853 by introducing uniform convergence, whose traditional formulation requires a pair of independent variables. Meanwhile, Cauchy's hypothesis is formulated in terms of a single variable x, rather than a pair of variables, and requires the error term r n = r n (x) to go to zero at all values of x, including the infinitesimal value generated by 1 n , explicitly specified by Cauchy. If one wishes to understand Cauchy's modification/clarification of the hypothesis of the sum theorem in 1853, one has to jettison the automatic translation-to-limits. Contents 1. Sifting the chaff from the grain in Lagrange 2 2. Cauchy's continuity 3 3. Bråting's close reading 5 4. Cauchy's 1853 text 8 5. Is the traditional reading, coherent? 9 6. Conclusion 12 Appendix A. Spalts Kontinuum 12 Appendix B. Fermat, Wallis, and an "amazingly reckless" use of infinity 17 Appendix C. Rival continua 21 Acknowledgments 24 References 24

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Section: Bråting's Close Readingmentioning

confidence: 99%

“…Struik notes that "Fermat uses the term to denote what we call a limiting process" [83, p. 220, footnote 5]. K. Barner [7] compiled a useful bibliography on Fermat's adequality, including many authors we have not mentioned here. 19 See main text around footnote 15 above for a mention of Barrow's role, documented by H Breger.…”

Section: Appendix a Spalts Kontinuummentioning

confidence: 99%

Cauchy's Continuum

Katz¹,

Katz

2011

Perspectives on Science

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“…We will analyze E. Bishop's implementation of Brouwer's nominalistic project. 8 It is an open secret that the much-touted success of Bishop's implementation of the intuitionistic project in his 1967 book [16] is due to philosophical compromises with a Platonist viewpoint that are resolutely rejected by the intuitionistic philosopher M. Dummett [49]. Thus, in a dramatic departure from both Kronecker 9 and Brouwer, Bishopian constructivism accepts the completed (actual) infinity of the integers Z.…”

Section: An Anti-lem Nominalistic Reconstructionmentioning

confidence: 99%

“…See also G. Kreisel [98]. 8 It has been claimed that Bishopian constructivism, unlike Brouwer's intuitionism, is compatible with classical mathematics, see e.g. Davies [44].…”

Section: An Anti-lem Nominalistic Reconstructionmentioning

confidence: 99%

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

Katz

2011

Found Sci

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

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Fermat’s Dilemma: Why Did He Keep Mum on Infinitesimals? And the European Theological Context

2017

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Abstract. The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be so treated and involve additional tools-leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat's manuscript but it is Breger who tampers with Fermat's procedure by moving all terms to the left-hand side so as to accord better with Breger's own interpretation emphasizing the double root idea. We provide modern proxies for Fermat's procedures in terms of relations of infinite proximity as well as the standard part function.

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Fermats «adæquare» – und kein Ende?

Cited by 4 publications

References 8 publications

Cauchy's Continuum

Cauchy's Continuum

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

Fermat’s Dilemma: Why Did He Keep Mum on Infinitesimals? And the European Theological Context

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