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Internal Logic: Foundations of Mathematics from Kronecker to Hilbert

Springer Verlag (2002)

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Hermann Weyl on Minkowskian Space–Time and Riemannian Geometry.Yvon Gauthier - 2005 - International Studies in the Philosophy of Science 19 (3):261 – 269.details Hermann Weyl as a founding father of field theory in relativistic physics and quantum theory always stressed the internal logic of mathematical and physical theories. In line with his stance in the foundations of mathematics, Weyl advocated a constructivist approach in physics and geometry. An attempt is made here to present a unified picture of Weyl's conception of space-time theories from Riemann to Minkowski. The emphasis is on the mathematical foundations of physics and the foundational significance of a constructivist philosophical (...) Download Export citation Bookmark 2 citations
Arithmetic without the successor axiom.Andrew Boucher - details Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition. Download Export citation Bookmark 1 citation
Historical and Foundational Details on the Method of Infinite Descent: Every Prime Number of the Form 4 n + 1 is the Sum of Two Squares.Paolo Bussotti & Raffaele Pisano - 2020 - Foundations of Science 25 (3):671-702.details Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the sum (...) Download Export citation Bookmark 5 citations
La descente infinie, l’induction transfinie et le tiers exclu.Yvon Gauthier - 2009 - Dialogue 48 (1):1.details ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does not correspond to the infinite (...) Download Export citation Bookmark
The construction of chaos theory.Yvon Gauthier - 2009 - Foundations of Science 14 (3):153-165.details 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 (...) Download Export citation Bookmark

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