Article

Foundations of Science

, Volume 18, Issue 2, pp 259-296

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

  • Vladimir KanoveiAffiliated withIPPIMIIT
  • , Mikhail G. KatzAffiliated withDepartment of Mathematics, Bar Ilan University Email author 
  • , Thomas MormannAffiliated withDepartment of Logic and Philosophy of Science, University of the Basque Country UPV/EHU

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Abstract

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model \({\mathcal{S}}\) 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 \({\mathcal{S}}\), 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 \({-\hskip-9pt\int}\) (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 Hyperreal Inaccessible cardinal Gödel’s incompleteness theorem Infinitesimal Klein–Fraenkel criterion Leibniz Noncommutative geometry P-point Platonism Skolem’s non-standard integers Solovay models Ultrafilter

Mathematics Subject Classification (2000)

Primary 26E35 Secondary 03A05