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

RELATIVE DIRECTED HOMOTOPY THEORY OF PARTIALLY ORDERED SPACES

Algebraic topological methods have been used successfully in concurrency theory, the domain of theoretical computer science that deals with parallel computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially ordered spaces (pospaces) as a model for concurrent systems. In this paper it is shown that the category of pospaces under a fixed pospace is both a fibration and a cofibration category in the sense of H. Baues. The homotopy notion in this fibration and cofibration category is relative directed homotopy. It is als

Topological abstraction of higher-dimensional automata

Higher-dimensional automata constitute a very expressive model for concurrent systems. In this paper, we discuss ``topological abstraction" of higher-dimensional automata, i.e., the replacement of HDAs by smaller ones that can be considered equivalent from the point of view of both computer science and topology. By definition, topological abstraction preserves the homotopy type, the trace category, and the homology graph of an HDA. We establish conditions under which cube collapses yield topological abstractions of HDAs

Pilotbericht zum Monitoring der deutschen Bioökonomie

Der Pilotbericht umfasst die Ergebnisse des Forschungsprojekts SYMOBIO. Er wurde vom Center for Environmental Systems Research (CESR) der Universität Kassel und dem Johann Heinrich von Thünen-Institut (TI), Bundesforschungsinstitut für Ländliche Räume, Wald und Fischerei mit den Fachinstituten für Marktanalyse (TI-MA), für Internationale Waldwirtschaft und Forstökonomie (TI-WF) und für Seefischerei (TI-SF) zusammen mit Kooperationspartnern des SYMOBIO-Projekts erstellt. Gesamtkoordination: Prof. Dr. Stefan Bringezu (CESR) in Kooperation mit Prof. Dr. Martin Banse (TI)Gesamtkoordination: Prof. Dr. Stefan Bringezu (CESR) in Kooperation mit Prof. Dr. Martin Banse (TI)BMBF (Förderkennzeichen 031B0281A

Pilot report on the monitoring of the German bioeconomy

This report was prepared by the Center for Environmental Systems Research (CESR) of the University of Kassel and the Johann Heinrich von Thünen Institute (TI), Federal Research Institute for Rural Areas, Forests and Fisheries with the Specialist Institutes for Market Analysis (TI-MA) in Braunschweig, for International Forestry and Forest Economics (TI-WF) in Hamburg and for Sea Fisheries (TI-SF), together with partners of the SYMOBIO project.Overall coordination: Prof. Dr. Stefan Bringezu (CESR) in cooperation with Prof. Dr. Martin Banse (TI)German Federal Ministry of Education and Research (Grant number: 031B0281A