The Emergence of Orientation and the Geometry of Logic

The Emergence of Orientation and the Geometry of Logic 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quality & Quantity Springer Journals

The Emergence of Orientation and the Geometry of Logic

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 1998 by Kluwer Academic Publishers
Subject
Social Sciences; Methodology of the Social Sciences; Social Sciences, general
ISSN
0033-5177
eISSN
1573-7845
D.O.I.
10.1023/A:1004281109017
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Quality & QuantitySpringer Journals

Published: Oct 6, 2004

References

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