Article ID: CBB001320860

Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics (2013)

unapi

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.

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Authors & Contributors
Zellini, Paolo
McCoy, C. D.
Bair, Jacques
Ely, Robert
Elena Gil Clemente
Kuhlemann, Karl
Concepts
Mathematics
Philosophy of mathematics
Geometry
Infinitesimals
Numbers
Philosophy
Time Periods
17th century
20th century, early
19th century
16th century
Medieval
20th century, late
Places
Italy
Germany
Europe
Mesopotamia
Great Britain
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