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 f (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.
|Number of pages||38|
|Journal||Foundations of Science|
|State||Published - Jun 2013|
Bibliographical noteFunding Information:
Acknowledgments V. Kanovei is grateful to the Fields Institute for its support during a visit in 2012. The research of Mikhail Katz was partially funded by the Israel Science Foundation grant 1517/12. The research of Thomas Mormann for this work is part of the research project FFI 2009–12882 funded by the Spanish Ministry of Science and Innovation. We are grateful to the referees for numerous insightful suggestions that helped improve an earlier version of the article. We thank Piotr Błaszczyk, Brian Davies, Martin Davis, Ili-jas Farah, Jens Erik Fenstad, Ian Hacking, Reuben Hersh, Yoram Hirshfeld, Karel Hrbácˇek, Jerome Keisler, Semen Kutateladze, Jean-Pierre Marquis, Colin McLarty, Elemer Rosinger, David Sherry, Javier Thayer, Alas-dair Urquhart, Lou van den Dries, and Pavol Zlatoš for helpful historical and mathematical comments. The influence of Hilton Kramer (1928–2012) is obvious.
- Axiom of choice
- Dixmier trace
- Gödel's incompleteness theorem
- Hahn-Banach theorem
- Inaccessible cardinal
- Klein-Fraenkel criterion
- Noncommutative geometry
- Skolem's non-standard integers
- Solovay mode