The Emergence of Orientation and the Geometry of Logic
Tóm tắt
In this paper it is said that cultural evolution consists of four phases which are denoted by the words nature, mythology I, mythology II and history. Pictograms, ideograms and logograms signify two phases fundamental for the later emergence of the linear writing and time as history and thought. The nowaday operational structures of cognition appear as rooted in prehistoric thought meaning vision, ideas – images seen with the inner eye – and logos as orientation of logograms. Thus it is assumed that logic in the form of the elementary first order propositional calculus should be derived from the orientation symmetry of space. The author gives an exact proof in terms of the Clifford algebra that this is indeed the case. Four basic representations in Cl2, Cl3, Cl3 and Cl3,1 of the 16 expressions of second order logic relations are given. The proposition of transformational self-reference is defined. The most basic self-referent representation of Peirce's box-X icons (the 16 binary logic connectives) is derivated from the spin-representation δD2d of the spatial congruence group of the square or the mandala of the sunwheel. Finally it is shown that the representation in terms of the idempotents of the Clifford algebra Cl3,1 is isomorphic with Zellweger's logic garnet.
Tài liệu tham khảo
Baldwin, J. M. (1906). Thought and Things. London, 3 vols.
Baylis, W. E. (1993). Electrons, photons, and spinors in the Pauli algebra. in: Z. Oziewicz et al. (eds), Spinors, Twistors, Clifford Algebras and Quantum Deformations, Dordrecht, pp. 97–106.
Belger, M. & Ehrenberg, L. (1981). Theorie und Anwendung der Symmetriegruppen. Frankfurt am Main.
Clark, G. (1989). New light on Peirce's iconic notation for the sixteen binary connectives. Paper presented at the Sesquicentennial Congress at Harvard University.
Clark, G. & Zellweger, Sh. (1993). Let the mirrors do the thinking. Mount Union Magazine, Spring: 2–5.
Flusser, V. (1992). Die Schrift. Frankfurt am Main.
Fuller, R. Buckminster. (1975). Synergetics: Explorations in the Geometry of Thinking. New York.
Kiss, S. A. (1947). Transformations on Lattices and Structures of Logic. New York.
Laing, R. D. (1976). The Facts of Life. New York.
Lounesto, P., Mikkola, R., & Vierros, V. (1987). CLICAL User Manual — Complex Number, Vector Space and Clifford Algebra Calculator for MS-DOS Personal Computers. Helsinki.
Menger, K. (1962). A group in the substitutive algebra of the calculus of propositions. Archiv der Mathematik 13: 471–478.
Peirce, Ch. S. (1902). The simplest mathematics. in: C. Hartshorne, & P. Weiss (eds), Collected Papers of Charles Sanders Peirce. Cambridge, vol 4, pp. 227–323.
Petraschen, M. I. & Trifonow, E. D. (1969). Anwendung der Gruppentheorie in der Quantenmechanik. Leipzig.
Piaget, J. (1957). Logic and Psychology. New York.
Schmeikal-Schuh, B. (1989). Soziogene der Orientierung — Ikonen der Raumzeit. WISDOM. Jg. III, 3/4: 1–18.
Schmeikal-Schuh, B. (1993). Logic from space. Quality & Quantity 27: 117–137.
Schmeikal, B. (1993). Space-time Sociology. Vienna, Research-Memorandum, Nr. 313.
Schmeikal, B. (1995). The implicate order of the void. Internal paper of the Biofield Laboratory, Vienna.
Schmeikal, B. (1996). The generative process of space-time and strong interaction, quantum numbers of orientation, in: R. Ablamowicz, P. Lounesto, & J. M. Parra (eds), Clifford Algebras with Numeric and Symbolic Computations. Boston, pp. 83–100.
Zellweger, Sh. (1982). Sign-creation and man-sign engineering, Semiotica 38,1/2: 17–54.
Zellweger, Sh. (1987). Notation, Relational Iconicity, and Rethinking the Propositional Calculus. 8th International Congress of Logic, Methodology and Philosophy of Science. Moscow.
Zellweger, Sh. (1992). Cards, Mirrors and Hand Held Models that Lead into Elementary Logic, Sixteenth Annual Conference of the International Group for the Psychology of Mathematics Education. University of New Hampshire.