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**1 - 7**of**7**### 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

### MECHANIZED CULTURAL REASONING AS A TOOL TO ASSESS TRUST IN VIRTUAL ENTERPRISES

"... The globalized knowledge society generates virtual enterprises that are usually set up and managed on the web, and the new trend is to make the relevant technologies available on intelligent portable devices. The existence of trust is a mandatory condition to make such enterprises successful. Trust ..."

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The globalized knowledge society generates virtual enterprises that are usually set up and managed on the web, and the new trend is to make the relevant technologies available on intelligent portable devices. The existence of trust is a mandatory condition to make such enterprises successful. Trust has many facets ranging from very theoretical ones to fully heuristic features. One point is that trust can arise when one understands better the behavior of partners. In this paper we outline a new technology leading to the possibility to include inter-cultural issues among the factors having a strong impact on trust. This technology is called Abstraction-Based Information Technology. Its goal is to enable to design tools in artificial intelligence to perform so-called cultural reasoning that ensures better trust among inter-cultural communities. We outline how Abstraction-Based Information Technology becomes feasible when working with virtual knowledge communities. An argument in favor of our approach is that it relies on a bottom-up approach, particularly suitable for the web technology and for intelligent wearable devices. The solution of intercultural troubles then amounts to solve knowledge conflicts among virtual knowledge communities.

### 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.

### 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.

### 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.

### WHY ARE (MOST) LAWS OF NATURE MATHEMATICAL?

"... Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas —Einstein In a frequently quoted but scarcely read paper, the Hungarian physicist Eugene Wigner rediscovered a question that had been implicitly posed for the first time by ..."

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Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas —Einstein In a frequently quoted but scarcely read paper, the Hungarian physicist Eugene Wigner rediscovered a question that had been implicitly posed for the first time by the Transcendental Aesthetics of the “Critique of Pure Reason”. More precisely, rather than asking, in the typical style of Kant, “how is mathematics possible”, Wigner was wondering how it could be so “unreasonably effective in the natural sciences ” (Wigner, 1967). The effectiveness in question refers to the numerous cases of mathematical theories, often developed without regard to their possible applications, that later have played an important and unexpected descriptive, explanatory and predictive role in physics and other natural sciences. A frequently given example is that of the conic sections, already known by the Greeks before Christ and used by Kepler many centuries after their discovery to describe the orbits of celestial bodies. Even more striking is the case of non-Euclidean geometries, applied by Einstein to describe how heavy matter bends the structure of spacetime in his general theory of relativity: the theory of curved, non-Euclidean spaces had already been built a century earlier by Gauss, Lobacevski and Riemann. A literary quotation addressing the role of complex numbers, due to the German writer Robert Musil, will conclude my necessarily short list of

### 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 short-sided 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