Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics

TL;DR

This paper critically examines Connes' criticisms of Robinson's hyperreals, analyzing the philosophical, logical, and foundational aspects, and challenges some of Connes' claims using model theory and set theory insights.

Contribution

The paper provides a detailed analysis of Connes' arguments against hyperreals, highlighting the role of definability, model theory, and foundational assumptions, and offers counterarguments based on recent mathematical developments.

Findings

Definable models of hyperreals exist, challenging Connes' 'virtual' theory claim.

Connes' reliance on non-constructive principles like the Hahn-Banach theorem is scrutinized.

The analysis reveals potential circularity in Connes' philosophical reasoning.

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 his 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 Goedel'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.