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The reasonable effectiveness of Mathematics and its Cognitive roots
, 2001
"... this paper, Mathematics is viewed as a "three dimensional manifold" grounded on logic, formalisms and invariants of space; we will appreciate by 1 ..."
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this paper, Mathematics is viewed as a "three dimensional manifold" grounded on logic, formalisms and invariants of space; we will appreciate by 1
CHAPTER 7 MATHEMATICAL CONCEPTS AND PHYSICAL OBJECTS
"... Abstract. The notions of “construction principles ” is proposed as a complementary notion w.r. to the familiar “proof principles ” of Proof Theory. The aim is to develop a parallel analysis of these principles in Mathematics and Physics: common construction principles, in spite of different proof pr ..."
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Abstract. The notions of “construction principles ” is proposed as a complementary notion w.r. to the familiar “proof principles ” of Proof Theory. The aim is to develop a parallel analysis of these principles in Mathematics and Physics: common construction principles, in spite of different proof principles, justify the effectiveness of Mathematics in Physics. The very “objects ” of these disciplines are grounded on commun genealogies of concepts: there is no trascendence of concepts nor of objects without their contingent and shared constitution. A comparative analysis of Husserl’s and Gödel’s philosophy is hinted, with many references to H. Weyl’s reflections on Mathematics and Physics. Introduction (with F. Bailly) With this text, we will first of all discuss a distinction, internal to mathematics, between “construction principles ” and “proof principles ” (see [Longo, 1999; 2002]). In short, it will be a question of grasping the difference between the construction of mathematical concepts and structures
Laplace, Turing and the "imitation game " impossible geometry: randomness, determinism and programs in Turing's test 1.
"... Abstract�: From the physicomathematical view point, the imitation game between man and machine, proposed by Turing in 1950, is a game between a discrete and a continuous system. Turing stresses several times the laplacian nature of his discretestate machine, yet he tries to show the undetectabilit ..."
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Abstract�: From the physicomathematical view point, the imitation game between man and machine, proposed by Turing in 1950, is a game between a discrete and a continuous system. Turing stresses several times the laplacian nature of his discretestate machine, yet he tries to show the undetectability of a functional imitation, by his machine, of a system (the brain) that, in his words, is not a discretestate machine, as it is sensitive to limit conditions. We shortly compare this tentative imitation with Turing’s mathematical modeling of morphogenesis (his 1952 paper, focusing on continuous systems which are sensitive to initial conditions). On the grounds of recent knowledge about dynamical systems, we show the detectability of a Turing Machine from many dynamical processes. Turing’s hinted distinction between imitation and modeling is developed, jointly to a discussion on the repeatability of computational processes. The main references are of a physicomathematical nature, but the analysis is purely conceptual.
The reasonable effectiveness of Mathematics and its Cognitive roots 1
"... “At the beginning, Nature set up matters its own way and, later, it constructed human intelligence in such a way that [this intelligence] could understand it” [Galileo Galilei, 1632 (Opere, p. 298)]. “The applicability of our science [mathematics] seems then as a symptom of its rooting, not as a mea ..."
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“At the beginning, Nature set up matters its own way and, later, it constructed human intelligence in such a way that [this intelligence] could understand it” [Galileo Galilei, 1632 (Opere, p. 298)]. “The applicability of our science [mathematics] seems then as a symptom of its rooting, not as a measure of its value. Mathematics, as a tree which freely develops his top, draws its strength by the thousands roots in a ground of intuitions of real representations; it would be disastrous to cut them off, in view of a shortsided utilitarism, or to uproot them from the ground from which they rose ” [H. Weyl, 1910]. Summary. Mathematics stems out from our ways of making the world intelligible by its peculiar conceptual stability and unity; we invented it and used it to single out key regularities of space and language. This is exemplified and summarised below in references to the main foundational approaches to Mathematics, as proposed in the last 150 years. Its unity is also stressed: in this paper, Mathematics is viewed as a "three dimensional
Space and Time in the Foundations of Mathematics, or some challenges in the interactions with other sciences 1
"... Summary: Our relation to phenomenal space has been largely disregarded, and with good motivations, in the prevailing foundational analysis of Mathematics. The collapse of Euclidean certitudes, more than a century ago, excluded ‘’geometric judgments’ ’ from certainty and contributed, by this, to isol ..."
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Summary: Our relation to phenomenal space has been largely disregarded, and with good motivations, in the prevailing foundational analysis of Mathematics. The collapse of Euclidean certitudes, more than a century ago, excluded ‘’geometric judgments’ ’ from certainty and contributed, by this, to isolate the foundation of Mathematics from other disciplines. After the success of the logical approach, it is time to broaden our foundational tools and reconstruct, also in that respect, the interactions with other sciences. The way space (and time) organize knowledge is a crossdisciplinary issue that will be briefly examined in Mathematical Physics, Computer Science and Biology. This programmatic paper focuses on an epistemological approach to foundations, at the core of which is the analysis of the ‘’knowledge process’’, as a constitutive path from cognitive experiences to mathematical concepts and structures. Contents:
CNRS et Dépt. d'Informatique.
"... Abstract: From the physicomathematical view point, the imitation game between man and machine, proposed by Turing in his 1950 paper for the journal “Mind”, is a game between a discrete and a continuous system. Turing stresses several times the laplacian nature of his discretestate machine, yet he ..."
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Abstract: From the physicomathematical view point, the imitation game between man and machine, proposed by Turing in his 1950 paper for the journal “Mind”, is a game between a discrete and a continuous system. Turing stresses several times the laplacian nature of his discretestate machine, yet he tries to show the undetectability of a functional imitation, by his machine, of a system (the brain) that, in his words, is not a discretestate machine, as it is sensitive to limit conditions. We shortly compare this tentative imitation with Turing’s mathematical modelling of morphogenesis (his 1952 paper, focusing on continuous systems, as he calls nonlinear dynamics, which are sensitive to initial conditions). On the grounds of recent knowledge about dynamical systems, we show the detectability of a Turing Machine from many dynamical processes. Turing’s hinted distinction between imitation and modelling is developed, jointly to a discussion on the repeatability of computational processes in relation to physical systems. The main references are of a physicomathematical nature, but the analysis is purely conceptual.