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The Constructed Objectivity of Mathematics and the Cognitive Subject
, 2001
"... Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out ..."
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Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out the mathematical conceptualization from any other form of knowledge, and give unity to it. Yet, this very conceptualization is deeply rooted in our "acts of experience", as Weyl says, beginning with our presence in the world, first in space and time as living beings, up to the most complex attempts we make by language to give an account of it. I will try to sketch the origin of some key steps in organizing perception and knowledge by "mathematical tools", as mathematics is one of the many practical and conceptual instruments by which we categorize, organise and "give a structure" to the world. It is conceived on the "interface" between us and the world, or, to put it in husserlian terminology, it is "de
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
The Cognitive Foundations of Mathematics: human gestures in proofs and mathematical incompleteness of formalisms1.
"... The foundational analysis of mathematics has been strictly linked to, and often originated, philosophies of knowledge. Since Plato and Aristotle, to Saint Augustin and Descartes, Leibniz, Kant, Husserl and Wittgenstein, analyses of human knowledge have been largely endebted to insights into mathemat ..."
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The foundational analysis of mathematics has been strictly linked to, and often originated, philosophies of knowledge. Since Plato and Aristotle, to Saint Augustin and Descartes, Leibniz, Kant, Husserl and Wittgenstein, analyses of human knowledge have been largely endebted to insights into mathematics, its proof methods and its conceptual constructions.
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 Constructed Objectivity of Mathematics and the Cognitive Subject 1
"... ÇThe problems of Mathematics are not isolated problems in a vacuum; there pulses in them the life of ideas which realize themselves in concreto through out human endeavours in our historical existence, yet forming an indissoluble whole transcend any particular scienceÈ [Hermann Weyl, 1949]. ..."
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ÇThe problems of Mathematics are not isolated problems in a vacuum; there pulses in them the life of ideas which realize themselves in concreto through out human endeavours in our historical existence, yet forming an indissoluble whole transcend any particular scienceÈ [Hermann Weyl, 1949].
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.
Proofs and Programs 1
"... In order for machines to do Mathematics what is required first of all is a language that describes Mathematics in adequate terms for machines. This language has to be completely formalised and without any semantic ambiguity. Computers cannot make operational choices as a function of the meaning of a ..."
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In order for machines to do Mathematics what is required first of all is a language that describes Mathematics in adequate terms for machines. This language has to be completely formalised and without any semantic ambiguity. Computers cannot make operational choices as a function of the meaning of a phrase, especially if it is uncertain or depends on the context, but only on analysing its
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:
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
1ESPACE, TEMPS ET COGNITION A PARTIR DES MATHEMATIQUES ET DES SCIENCES DE LA NATURE
"... RESUME: La cognition humaine paraît étroitement liée à la structure de l’espace et du temps relativement auxquels le corps, le geste, l’intelligibilité semblent devoir se déterminer. Pourtant, ce qui, après les approches physicomathématiques de Galilée et de Newton, fut caractérisé par Kant comme f ..."
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RESUME: La cognition humaine paraît étroitement liée à la structure de l’espace et du temps relativement auxquels le corps, le geste, l’intelligibilité semblent devoir se déterminer. Pourtant, ce qui, après les approches physicomathématiques de Galilée et de Newton, fut caractérisé par Kant comme formes de l’intuition sensible, n’a cessé au cours des siècles qui suivirent de se trouver remis en cause dans leur saisie première par les développements théoriques. En mathématiques d’abord, avec les géométries noneuclidiennes, en physique ensuite, où relativité générale puis théories quantiques et critiques ont dû remanier profondément l’objectivité de ces concepts pour en faire des catégories, certes toujours aussi essentielles, mais de plus en plus contreintuitives, et maintenant en biologie où la temporalité, notamment, et la causalité se révèlent largement différentes de celles de la physique. C’est ce que nous tentons de présenter et de discuter dans ce texte en vue d’en dégager la pertinence pour la cognition ellemême. MOTSCLES: espace, temps, cognition, théories scientifiques. ABSTRACT: Human cognition seems strictly related to the structure of space and time where bodily presence, action and intelligibility are to be determined. Yet, the classical