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

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 inNoncommutative Geometry, raising the questionwhether 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) V. Kanovei IPPI, Moscow, Russia e-mail: [email protected] V. Kanovei MIIT, Moscow, Russia M. G. Katz (B) Department of Mathematics, Bar Ilan University, 52900 Ramat, Gan, Israel e-mail: [email protected] T. Mormann Department of Logic and Philosophy of Science, University of the Basque Country UPV/EHU, 20080 Donostia San Sebastian, Spain e-mail: [email protected] 123 Author's personal copy