Abstract
We discuss differences in mathematical representations of the physical and mental worlds. Following Aristotle, we present the mental space as discrete, hierarchic, and totally disconnected topological space. One of the basic models of such spaces is given by ultrametric spaces and more specially by m-adic trees. We use dynamical systems in such spaces to model flows of unconscious information at different level of mental representation hierarchy, for “mental points”, categories, and ideas. Our model can be interpreted as an unconventional computational model: non-algorithmic hierarchic “computations” (identified with the process of thinking at the unconscious level).
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References
The Complete Works of Aristotle, ed. J. Barnes, 2 vols. (Princeton Univ. Press, Princeton, 1984).
S. Albeverio, A. Yu. Khrennikov and P. Kloeden, “Memory retrieval as a p-adic dynamical system,” Biosystems 49, 105–115 (1999).
D. Dubischar, V. M. Gundlach, O. Steinkamp and A. Yu. Khrennikov, “A p-adic model for the process of thinking disturbed by physiological and information noise,” J. Theor. Biology 197, 451–467 (1999).
S. Freud, The Ego and the Id (W.W. Norton and Company, New York, 1923).
S. Freud, “The interpretation of dreams,” In Standard edition 1954–1974 (Vols. 4, 5, pp. 1627) (London: Hogarth Press).
A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordrecht, 1997).
A. Yu. Khrennikov, “Human subconscious as the p-adic dynamical system,” J. Theor. Biology 193, 179–196 (1998).
A. Yu. Khrennikov, Classical and Quantum Mental Models and FreudTs Theory of Unconscious Mind Series Math. Modelling in Phys., Engineering and Cognitive Sciences, 1 (Växjö Univ. Press, Växjö, 2002).
A. Yu. Khrennikov, “Toward an adequate mathematical model of mental space: Conscious/unconscious dynamics on m-adic trees,” Biosystems 90(3), 656–675 (2007).
R. Lauro-Grotto, “The unconscious as an ultrametric set,” Amer. Imago 64, 535–543 (2007).
F. Murtagh, “On ultrametricity, data coding, and computation,” J. Classification 21, 167–184 (2004).
F. Murtagh, “Ultrametric model of mind, I: Review,” p-Adic Numbers Ultr. Anal. Appl. 4, 193–206 (2012).
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Sci. Publ., Singapore, 1994).
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adic mathematical physics,” p-Adic Numbers Ultr. Anal. Appl. 1(1), 1–17 (2009).
A. Khrennikov, “Representation of cognitive information with the aid of probability distributions on the space of neuronal trajectories,” Proc. Steklov Inst. Math. 245, 117–134 (2004).
A. Khrennikov, “Probabilistic pathway representation of cognitive information,” J. Theor. Biology 231, 597–613 (2004).
A. Khrennikov and M. Nilsson, p-Adic Deterministic and RandomDynamical Systems (Kluwer-Springer, Dordreht, 2004).
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Khrennikov, A.Y. Cognitive processes of the brain: An ultrametric model of information dynamics in unconsciousness. P-Adic Num Ultrametr Anal Appl 6, 293–302 (2014). https://doi.org/10.1134/S2070046614040049
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DOI: https://doi.org/10.1134/S2070046614040049