Ten Misconceptions from the History of Analysis and Their Debunking

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545

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

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.Comment: 52 pages, 1 figur

Active layer thermal regime in two climatically contrasted sites of the Antarctic Peninsula region

Permafrost controls geomorphic processes in ice-free areas of the Antarctic Peninsula (AP) region. Future climate trends will promote significant changes of the active layer regime and permafrost distribution, and therefore a better characterization of present-day state is needed. With this purpose, this research focuses on Ulu Peninsula (James Ross Island) and Byers Peninsula (Livingston Island), located in the area of continuous and discontinuous permafrost in the eastern and western sides of the AP, respectively. Air and ground temperatures in as low as 80 cm below surface of the ground were monitored between January and December 2014. There is a high correlation between air temperatures on both sites (r=0.74). The mean annual temperature in Ulu Peninsula was -7.9 ºC, while in Byers Peninsula was -2.6 ºC. The lower air temperatures in Ulu Peninsula are also reflected in ground temperatures, which were between 4.9 (5 cm) and 5.9 ºC (75/80 cm) lower. The maximum active layer thickness observed during the study period was 52 cm in Ulu Peninsula and 85 cm in Byers Peninsula. Besides climate, soil characteristics, topography and snow cover are the main factors controlling the ground thermal regime in both areas.info:eu-repo/semantics/publishedVersio