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**1 - 10**of**10**### THE LAWS OF NATURE AND THE EFFECTIVENESS OF MATHEMATICS

"... In this paper I try to evaluate what I regard as the main attempts at explaining the effectiveness of mathematics in the natural sciences, namely (1) Antinaturalism, (2) Kantism, (3) Semanticism, (4) Algorithmic Complexity Theory. The first position has been defended by Mark Steiner, who claims that ..."

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In this paper I try to evaluate what I regard as the main attempts at explaining the effectiveness of mathematics in the natural sciences, namely (1) Antinaturalism, (2) Kantism, (3) Semanticism, (4) Algorithmic Complexity Theory. The first position has been defended by Mark Steiner, who claims that the “user friendliness ” of nature for the applied mathematician is the best argument against a naturalistic explanation of the origin of the universe. The second is naturalistic and mixes the Kantian tradition with evolutionary studies about our innate mathematical abilities. The third turns to the Fregean tradition and considers mathematics a particular kind of language, thus treating the effectiveness of mathematics as a particular instance of the effectiveness of natural languages. The fourth hypothesis, building on formal results by Kolmogorov, Solomonov and Chaitin, claims that mathematics is so useful in describing the natural world because it is the science of the abbreviation of sequences, and mathematically formulated laws of nature enable us to compress the information contained in the sequence of numbers in which we code our observations. In this tradition, laws are equivalent to the shortest algorithms capable of generating the lists of zeros and ones representing the empirical data. Along the way, I present and reject the “deflationary explanation”, which claims that in wondering about the applicability of so many mathematical structures to nature, we tend to forget the many cases in which no application is possible. mathematics; laws of nature; algorithmic complexity theory; evolution; semantics. 2 MAURO DORATO Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas. (Einstein,1933) 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.

### CAUSES AND SYMMETRIES IN NATURAL SCIENCES. THE CONTINUUM AND THE DISCRETE IN MATHEMATICAL MODELLING1.

"... How do we make sense of physical phenomena? The answer is far from being univocal, particularly because the whole history of Physics has set, at the center of the intelligibility of phenomena, changing notions of cause, from Aristotle’s rich classification, to which we will return, to Galileo’s (too ..."

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How do we make sense of physical phenomena? The answer is far from being univocal, particularly because the whole history of Physics has set, at the center of the intelligibility of phenomena, changing notions of cause, from Aristotle’s rich classification, to which we will return, to Galileo’s (too strong?) simplification and their modern understanding in terms of “structural

### Laplace, Turing and the "imitation game " impossible geometry: randomness, determinism and programs in Turing's test 1.

"... Abstract�: From the physico-mathematical 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 discrete-state machine, yet he tries to show the undetectabilit ..."

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Abstract�: From the physico-mathematical 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 discrete-state 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 discrete-state 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 physico-mathematical nature, but the analysis is purely conceptual.

### CNRS et Dépt. d'Informatique.

"... Abstract: From the physico-mathematical 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 discrete-state machine, yet he ..."

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Abstract: From the physico-mathematical 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 discrete-state 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 discrete-state 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 non-linear 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 physico-mathematical nature, but the analysis is purely conceptual.

### 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 physico-mathématiques 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 physico-mathématiques 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 non-euclidiennes, 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 contre-intuitives, 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 elle-même. MOTS-CLES: 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

### Reflections on Concrete Incompleteness?

"... Abstract. How do we prove true, but unprovable propositions? Gödel produced a statement whose undecidability derives from its “ad hoc” construction. Concrete or mathematical incompleteness results, instead, are interesting unprovable statements of Formal Arithmetic. We point out where exactly lays ..."

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Abstract. How do we prove true, but unprovable propositions? Gödel produced a statement whose undecidability derives from its “ad hoc” construction. Concrete or mathematical incompleteness results, instead, are interesting unprovable statements of Formal Arithmetic. We point out where exactly lays the unprovability along the ordinary mathemat-ical proofs of two (very) interesting formally unprovable propositions, Kruskal-Friedman theorem on trees and Girard’s Normalization Theo-rem in Type Theory. Their validity is based on robust cognitive perfor-mances, which ground mathematics on our relation to space and time, such as symmetries and order, or on the generality of Herbrands notion of prototype proof. Introduction: some history, some philosophy Suppose that you were asked to give the result of the sum of the first n integers. There exist many proofs of this simple fact (see [Nelsen93] for this and more examples), an immediate one (allegedly (re-)invented by Gauss at the age of 7

### 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.

### I rapporti tra i fondamenti della matematica e della fisica: dialogo1.

"... Lo scopo è anche quello di analizzare i metodi della fisica da un punto di vista simile e, ..."

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Lo scopo è anche quello di analizzare i metodi della fisica da un punto di vista simile e,