Foundations of Science 18 (2):259-296 (2012)
|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|
|Through your library||Configure|
Similar books and articles
S. Shanker (1998). Review of J. Changeux and A. Connes, Conversations on Mind, Matter, and Mathematics. Edited and Translated by M.B. DeBevoise. [REVIEW] Philosophia Mathematica 6 (2):241-245.
Alexandre Borovik, Renling Jin & Mikhail G. Katz (2012). An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals. Notre Dame Journal of Formal Logic 53 (4):557-570.
Raymond M. Smullyan (1992). Gödel's Incompleteness Theorems. Oxford University Press.
Ali Enayat (2001). Power-Like Models of Set Theory. Journal of Symbolic Logic 66 (4):1766-1782.
Vasco Brattka & Guido Gherardi (2011). Effective Choice and Boundedness Principles in Computable Analysis. Bulletin of Symbolic Logic 17 (1):73-117.
Ali Enayat (2004). Leibnizian Models of Set Theory. Journal of Symbolic Logic 69 (3):775-789.
Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
Zofia Adamowicz & Teresa Bigorajska (2001). Existentially Closed Structures and Gödel's Second Incompleteness Theorem. Journal of Symbolic Logic 66 (1):349-356.
Paul E. Howard (1973). Limitations on the Fraenkel-Mostowski Method of Independence Proofs. Journal of Symbolic Logic 38 (3):416-422.
Timothy Bays (2009). Skolem's Paradox. In Edward N. Zalta (ed.), Stanford Encyclopedia of Philosophy.
Added to index2012-11-10
Total downloads14 ( #83,218 of 550,802 )
Recent downloads (6 months)11 ( #6,133 of 550,802 )
How can I increase my downloads?