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- This essay proposes a historical reconstruction in three phases of the relationship between sociology and mathematics: (a) P.F. Lazarfeld's choices regarding theory, methodology and mathematics within the objectivity paradigm; (b) the relationship between mathematics and sociology, from statistic models and artificial societies to social simulation; (c) the new possibilities that Artificial Neural Networks offer sociology.

The book shows a very original organization addressing in a non traditional way, but with a systematic approach, to who has an interest in using mathematics in the social sciences. The book is divided in four parts: (a) a historical part, written by Vittorio Capecchi which helps us understand the changes in the relationship between mathematics and sociology by analyzing the mathematical models of Paul F. Lazarsfeld, the model of simulation and artificial societies, models of artificial neural network and considering all the changes in scientific paradigms considered, (b) a part coordinated by Pier Luigi Contucci on mathematical models that consider the relationship between the mathematical models that come from physics and linguistics to arrive at the study of society and those which are born within sociology and economics, (c) a part coordinated by Massimo Buscema analyzing models of artificial neural networks, (d) a part coordinated by Bruno D`Amore which considers the relationship between mathematics and art. The title of the book "Mathematics and Society" was chosen because the mathematical applications exposed in the book allow you to address two major issues: (a) the general theme of technological innovation and quality of life (among the essays are on display mathematical applications to the problems of combating pollution and crime, applications to mathematical problems of immigration, mathematical applications to the problems of medical diagnosis, etc.) (b) the general theme of technical innovation and creativity, for example the art and mathematics section which connects to the theme of creative cities. The book is very original because it is not addressed only to those who are passionate about mathematical applications in social science but also to those who, in different societies, are: (a) involved in technological innovation to improve the quality of life, (b) involved in the wider distribution of technological innovation in different areas of creativity (as in the project "Creative Cities Network" of UNESCO). TOC: Mathematics and Society Preface Vittorio Capecchi, Massimo Buscema, Pierluigi Contucci, Bruno D`Amore 1- Vittorio Capecchi: Historical Introduction Section I (mathematics and models) 2- Pierluigi Contucci, Ignacio Gallo, Stefano Ghirlanda: "Equilibria of culture contact derived from ingroup and outgroup attitudes" 3- Oscar Bolina: "Society from the Statistical Mechanics Perspective" 4- Anna M. Borghi, Daniele Caligiore, Claudia Scorolli: "Objects, words, and actions. Some reasons why embodied models are badly needed in cognitive psychology." 5- Luca Desanctis, Stefano Ghirlanda: "Shared culture needs large social networks" 6- C. Gallo: "Mathematical models of financial markets" 7- F. Gallo, Pierluigi Contucci, A. Coutts, Ignacio Gallo: "Tackling climate change through energy efficiency: mathematical models to offer evidence-based recommendations for public policy" 8- Simone Sarti, Marco Terraneo: "An application of the multilevel regression technique to validate a social stratification scale" 9- Robert B. Smith: "The Academic Mind Revisited: Contextual Analysis via Multilevel Modeling" Section III (mathematics and neural networks) 10 - Massimo Buscema: "The General Philosophy of the Artificial Adaptive Systems" 11 - Massimo Buscema, Pier Luigi Sacco: "Auto-Contractive Maps, the H Function and the Maximally Regular Graph (MRG): a new methodology for data mining" 12- Massimo Buscema, Pier Luigi Sacco: "An Artificial Intelligent Systems Approach to - Unscrambling Power Networks in Italy`s Business Environment" 13- Giulia Massini: "Multi - Meta SOM" 14- Massimo Buscema: "How to make data mining: The Persons Arrested Dataset" 15- Enzo Grossi: "Medicine and Mathematics of Complex Systems" 16- Massimo Buscema, Enzo Grossi: "J-Net System: a new paradigm for Artificial Neural Networks applied to diagnostic imaging" 17 - S. Tangaro, R. Bellotti, F. De Carlo, G. Gargano: "Digital Image Processing in Medical Applications" Section III (mathematics and art) 18 - Giorgio T. Bagni: "Mathematics, Art, and Interpretation: an Hermeneutic Perspective" 19- Giorgio Bolondi: "Point, line and surface, following Hilbert and Kandinsky" 20- Bruno D`Amore: Figurative arts and mathematics: pipes, horses, triangles and meanings A contribution to a problematic theory of conceptual meaning, from Frege and Magritte up to the present time" 21- Michele Emmer: The idea of space in art, technology and mathematics" 22- Raffaele Mascella, Franco Eugeni,e Ezio Sciarpa: "Mathematical structures and sense of beauty" 23- Monica Idà: "Visual impact and mathematical learning" 24- Marco Pierini: "Art by Numbers.Mel Bochner, Roman Opalka and other Philarhythmics" 25-Aldo Spizzichino: "My way of playing with the computer. Suggestions for a personal experience in vector graphics" 26- Gian Marco Todesco: "Four dimensional ideas" 27- Igino Aschieri, Paola Vighi: "From Art to Mathematics in the paintings of Theo van Doesburg" Editors`s biographies