Quality & Quantity 32: 119–154, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
The Emergence of Orientation and the
Geometry of Logic
Bioﬁeld Laboratory, Kundmanngasse 26/8, 1030 Vienna, Austria
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 Cl
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
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 Cl
is isomorphic with Zellweger’s logic garnet.
Key words: logic, geometric logic, iconic logic, Clifford algebra, orientation, spin representation,
ﬁnite groups, logical garnet.
The 1993 paper “Logic from space” (Schmeikal-Schuh 1993) released some inter-
ested responses. So I have decided to go deeper into the subject matter and add
another investigation where questions of both content and method are well posed
and answered in a deﬁnite way. The procedure is now joined with Clifford algebra,
a powerful tool that allows for a clear cut research design. As you know
hypothesis had a sociological and psychological meaning. It afﬁrmed the statement
that the operational structure of thought has emerged in human cognition during
paleolithic worship activity. The appearance of logical structures, in this view, is
based upon the discovery of orientation in space. How this happened we do not
know. But we know of some elementary activities involved such as cutting and en-
Principal of the Bioﬁeld Laboratory of the Interdisciplinary Centre for Biosynergetics and
Holistic Research (ICBHR), Vienna, lector and member of the research staff of the Institute for
Sociology, University of Vienna, Austria. This paper was made to serve as a groundwork for a
research project titled “construction and deconstruction of time”.