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

Foundations of Science 18 (2):259-296 (2013)

Authors
Thomas Mormann
University of the Basque Country
Abstract
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.
Keywords Axiom of choice  Dixmier trace  Hahn–Banach theorem  Inaccessible cardinal  Gödel’s incompleteness theorem  Klein–Fraenkel criterion  Noncommutative geometry  Platonism  Skolem’s non-standard integers  Solovay models
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Reprint years 2013
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DOI 10.1007/s10699-012-9316-5
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References found in this work BETA

Real Patterns.Daniel C. Dennett - 1991 - Journal of Philosophy 88 (1):27-51.
Mathematics as a Science of Patterns.Michael D. Resnik - 1997 - New York ;Oxford University Press.
Fair Infinite Lotteries.Sylvia Wenmackers & Leon Horsten - 2013 - Synthese 190 (1):37-61.

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Citations of this work BETA

Infinitesimal Probabilities.Sylvia Wenmackers - 2019 - In Richard Pettigrew & Jonathan Weisberg (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 199-265.

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