On Cauchy's Notion of Infinitesimal

Cutland, Nigel J.; Keßler, Christoph; Kopp, Ekkehard; Ross, David A.

doi:10.1093/bjps/39.3.375

Cited by 25 publications

(39 citation statements)

References 2 publications

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“…Namely, assume for the sake of simplicity that the target is the unit interval [0, 1]. A more satisfactory answer than the one above is provided in terms of a hyperfinite grid Grid H = 0, A nonstandard number is some sort of chimera 43 which is impossible to grasp and certainly not a concrete object.…”

Section: Of Darts Infinitesimals and Chimerasmentioning

confidence: 99%

Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

2012

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We examine some of Connes' criticisms of Robinson's infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes' own earlier work in functional analysis. Connes described the hyperreals as both a "virtual theory" and a "chimera", yet acknowledged that his argument relies on the transfer principle. We analyze Connes' "dart-throwing" thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being "virtual" if it is not definable in a suitable model of ZFC. If so, Connes' claim that a theory of the hyperreals is "virtual" is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren't definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes' criticism of virtuality. We analyze the philosophical underpinnings of Connes' argument based on Gödel's incompleteness theorem, and detect an apparent circularity in Connes' logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes' magnum opus) and the Hahn-Banach theorem, in Connes' own framework. We also note an inaccuracy in Machover's critique of infinitesimal-based pedagogy.

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Section: Of Darts Infinitesimals and Chimerasmentioning

confidence: 99%

Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

2012

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“…Rather, Lakatos, Laugwitz, Cutland et al [15], Bråting (2007, [9]) and others have argued that infinitesimals as employed by Cauchy have found set-theoretic implementations in the framework of modern theories of infinitesimals, just as Kanovei had done for Euler in 1988 [36]. The existence of such implementations indicates that the historical infinitesimals were less prone to contradiction than has been routinely maintained by triumvirate historians, who invariably cite Berkeley's flawed empiricist critique.…”

Section: Triumvirate Historymentioning

confidence: 99%

A Cauchy-Dirac Delta Function

Katz¹,

Tall²

2012

Found Sci

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“…The tenor of Hobson's remarks, 39 as indeed of a majority of historians of mathematics, is that Weierstrass's fundamental accomplishment was the elimination of infinitesimals from foundational discourse in analysis. Infinitesimals were replaced by arguments relying on real inequalities and multiple-quantifier logical formulas.…”

Section: The Triumvirate Nominalistic Reconstructionmentioning

confidence: 99%

“…Modulo suitable foundational material, one can ensure that every infinitesimal is represented by a null sequence; an appropriate ultrafilter (called a P-point ) will exist if one assumes the continuum hypothesis, or even the weaker Martin's axiom. See Cutland et al [39] for details.…”

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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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On Cauchy's Notion of Infinitesimal

Cited by 25 publications

References 2 publications

Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

A Cauchy-Dirac Delta Function

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

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