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**1 - 5**of**5**### 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.

### 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.