Yvon Gauthier

This paper aims at a logico-mathematical analysis of the concept of chaos from the point of view of a constructivist philosophy of physics. The idea of an internal logic of chaos theory is meant as an alternative to a realist conception of chaos. A brief historical overview of the theory of dynamical systems is provided in order to situate the philosophical problem in the context of probability theory. A finitary probabilistic account of chaos amounts to the theory of measurement in the line of a quantum-theoretical foundational perspective and the paper concludes on the non-classical internal logic of chaos theory. Finally, deterministic chaos points to a philosophy which asserts that chaotic systems are no less measurable than other physical systems where predictable and non–predictable phenomena intermingle in a constructive theory of measurement.

Le réalisme se porte mal: des philosophes comme Putnam ou van Fraassen qui naguère encore défendaient des points de vue à tout le moins apparentés au réalisme proposent des thèses anti-réalistes ou qui sont aux antipodes du réalisme métaphysique. Et ceux qui tiennent encore au réalisme le voilent prudemment. Quant au réalisme critique de Popper et des poppériens, il est relégué aujourd'hui au rôle d'une thèse historique sur le développement rationnel de la science que contredit la doctrine kuhnienne. Reste le réalisme naïf.

My discussion centers around Ulrich Kohlenbach's Applied Proof Theory: Proof Interpretations and their Use in Mathematics [21 ] which appears as a major work in the "proof mining project" of recent proof theory. The emphasis is on the tradition of proof theory originating with Hilbert and his motivations. I examine also Hilbert's Kroneckerian inspiration and I maintain that Hilbert's programme is the continuation of Kronecker's programme by logical and metamathematical means. I conclude that despite Kreisel's proposal of a shift of emphasis in traditional proof theory, Hilbert's original proof-theoretical ideal is still alive and proof mining is in the purview of Hilbert programme. © 2011 Elsevier B.V., All rights reserved.