Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena

ISBN-13: 9781402018688 / Angielski / Twarda / 2004 / 235 str.

A. Iu Khrennikov; Andrei Khrennikov; Andrei Khrennikov

Opis

In this book we develop various mathematical models of information dynamics, I -dynamics (including the process of thinking), based on methods of classical and quantum physics. The main aim of our investigations is to describe mathematically the phenomenon of consciousness. We would like to realize a kind of Newton-Descartes program (corrected by the lessons of statistical and quantum mechanics) for information processes. Starting from the ideas of Newton and Descartes, in physics there was developed an adequate description of the dynamics of material systems. We would like to develop an analogous mathematical formalism for information and, in particular, mental processes. At the beginning of the 21st century it is clear that it would be impossible to create a deterministic model for general information processes. A deterministic model has to be completed by a corresponding statistical model of information flows and, in particular, flows of minds. It might be that such an information statistical model should have a quantum-like structure.

Spis treści

List of Figures. Introduction. 1: Processing Information on p-Adic Trees. 1.1. Ultrametric Spaces. 1.2. m-adic Geometry. 1.3. Geometry of Information Spaces. 1.4. Dynamical Processing of Information. 1.5. Role of Hierarchical Structure. 1.6. Role of Chance in Processing of Cognitive Information. 1.7. Information Reductionism. 2: Hierarchy of Information. 2.1. Hierarchical Coding of Information. .2.2. Flows of Associations and Ideas. 2.3. How Can the Brain Play Dice? 2.4. Constraints on Information Spaces. 3: p-Adic Dynamical Systems. 3.1. p-Adic Numbers. 3.2. Roots of Unity. 3.3. Dynamical Systems in Non-Archimedian Fields. 3.4. Dynamical Systems in the Field of Complex p-adic Numbers. 3.5. Dynamical Systems in the Fields of p-adic Numbers. 3.6. p-adic Ergodicity. 3.7. Newton's Method (Hensel's Lemma). 3.8. Computer Calculations for Fuzzy Cycles. 4: Random Processing of Information. 4.1. Random Dynamical Systems. 4.2. Long-term Behaviour, Dynamics on the Attractor, Examples. 4.3. Consequences for Cognitive Sciences. 5: Information Quantum Mechanics. 5.1. Quantum-Like Formalism for a One-Layer Brain. 5.2. Motivation Observable. 5.3. Neuron Activation Observable. 5.4. Complex Cognitive Systems: Evolution. 6: Bohmian Mechanics on Information Spaces. 6.1. Newton Laws for Information Processes. 6.2. Bohmian Mechanics for Hierarchical Information. 6.3. Interactions between Information Systems. 6.4. Hamiltonian Equations and Active Information. 6.5. Information Mass. 6.6. Wave Functions Taking Values in p-adic Fields. 6.7. Information Waves on p-adic Trees. 6.8. p-adic Bohmian Mechanics and Waves of Brain Activation. 6.9. Conservation Laws. 6.10. Mechanics of a System of Information Transformers, Constraints on Information Spaces. 6.11. Social and Anomalous Phenomena. 7: Abstract Ultrametric Information Spaces. 7.1. Abstract Ultrametric Spaces. 7.2. Hierarchy of Associations. 7.3. Topology and Materialism. 7.4. Existence of Universal Mental Space. 7.5. Towers of Associations. 7.6. Infinite Information Towers. 8: Pathway Representation of Cognitive Information. 8.1. Model: Thinking on a Cognitive Tree. 8.2. Dynamics in the Information Space. 8.3. Diffusion Model for Dynamics of a Mental State. 8.4. Information Phase Space. 8.5. Mental State as the Distribution of a p-adic Random Walk. 8.6. Discussion of the Neural Pathways Thinking Model. 9: Contextual Approach to Quantum Theory. 9.1. The Växjö Interpretation of Quantum Mechanics. 9.2. Contextual Viewpoint of Quantum Stochastics. 9.3. Law of Statistical Balance in Nature. 9.4. Experiments on Quantum-Like Behaviour of the Mind. 9.5. Experimental Confirmation. 10: Frequency Analysis of Foundations of Quantum Mechanics. 10.1. Classification of Transforms of Probability. 10.2. Classical, Quantum and