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

Thomas Mormann
Ludwig Maximilians Universität, München (PhD)
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
Categories (categorize this paper)
Reprint years 2013
DOI 10.1007/s10699-012-9316-5
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

 PhilArchive page | Other versions
Through your library

References found in this work BETA

The Principles of Mathematics.Bertrand Russell - 1903 - Cambridge, England: Allen & Unwin.
Real Patterns.Daniel C. Dennett - 1991 - Journal of Philosophy 88 (1):27-51.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press.
Mathematics as a Science of Patterns.Michael David Resnik - 1997 - Oxford, England: New York ;Oxford University Press.
The Principles of Mathematics.Bertrand Russell - 1903 - Revue de Métaphysique et de Morale 11 (4):11-12.

View all 83 references / Add more references

Citations of this work BETA

Infinitesimal Probabilities.Sylvia Wenmackers - 2016 - In Richard Pettigrew & Jonathan Weisberg (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 199-265.
Reverse Formalism 16.Sam Sanders - 2020 - Synthese 197 (2):497-544.

View all 14 citations / Add more citations

Similar books and articles

Power-Like Models of Set Theory.Ali Enayat - 2001 - Journal of Symbolic Logic 66 (4):1766-1782.
Leibnizian Models of Set Theory.Ali Enayat - 2004 - Journal of Symbolic Logic 69 (3):775-789.
On the Philosophical Relevance of Gödel's Incompleteness Theorems.Panu Raatikainen - 2005 - Revue Internationale de Philosophie 59 (4):513-534.


Added to PP index

Total views
922 ( #6,940 of 2,498,492 )

Recent downloads (6 months)
44 ( #19,314 of 2,498,492 )

How can I increase my downloads?


My notes