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A 4x4 Mathematical General Systems Matrix

by Stewart C. Dodd, Univ. Wash. Seattle

For those who can understand the mathematics, I present below (in 3 jpeg images) the General Systems model of the late Stewart C. Dodd (deceased 1975). The complete version of this model can be found in my father's book: Trance, Art, Creativity: An Investigation of the Numinous Element and the Metaphysical Realm. A book by Prof. J. C. Gowan Sr. (see the Appendix) on his memorial website.

The following is an excerpt from "Trance, Art, Creativity" concerning General Systems theory and Stewart Dodd in particular:

4.38 Creative Organization: General Systems Theory

"In the lower levels of creative production, the individual engages in creative problem-solving. In the higher levels, the mind becomes an organizer of the knowledge which wells up in it from creative openings in the preconscious. Organization is anti-entropy; it is order in place of disorder. Consequently, it validates our general theory of creativity to find that it introduces "a new and higher order" into experience. One would expect this emergent property to occur if creativity is a stepping stone on the pathway to self-actualization.

The essence of this order or organization is to find unity in diversity, the same process in different products, a universe filled with isomorphisms. Metaphor, analogy, and homology are primitive aspects of this process, but there are higher considerations to which we need to turn.

There are several examples of this emerging order in man's understanding of nature. Mathematics, especially set theory, is one; cybernetics, based on the feedback principle another; and information theory a third. Systems and human engineering theories are a fourth, decision theory a fifth, and general semantics a sixth. Since each of these areas has its own extensive literature, we shall turn to a seventh, that of general systems theory, which is far younger, less organized, and much less well known.

It is generally accepted that while some earlier writers had glimpsed the outlines of the subject, general systems theory was founded by von Bertalanffy in his classic of the same title (1968).

Bertalanffy (1968:vii) defines his subject as follows:

Systems theory is a broad view which far transcends techno- logical problems and demands, a reorientation that has become necessary in science in general . . . It is operative with varying degrees of success, in various realms, and heralds a new world view of considerable impact.

Bertalanffy (1968:38) states the purposes and aims of general systems theory as: "a tendency toward integration, centered in a general theory of systems, aiming at exact theory in nonphysical fields, which develop universal principles toward a goal of unity in science, which can lead to integration in scientific education."

Although Bertalanffy had published before then, general systems theory got its formal start in an informal meeting in 1954 at Palo Alto of K. Boulding, the economist; A. Rapoport, the biomathematician; R. Gerard, the physiologist; and Bertalanffy. They founded the society for General Systems Research, which later became a division of AAAS. The yearbooks General Systems edited by A. Rapoport have served as the house organ.

The genius of Bertalanffy, the founder of General Systems Theory, was according to Laszlo (1972:4-8) that he was the first to recognize that the process of organization of scientific knowledge might be as important as the product. This concept involved holism rather than analysis, integration rather than differentiation of scientific knowledge, the unity of nature in a diversity of forms, and the emphasis on scientific humanism rather than mechanical technology. It has come, concludes Laszlo (1972:11) "to represent a new paradigm of contemporary scientific thought," and it provides science with a new and very powerful tool.

Buckley (1967:39), a sociologist, points out that systems theory concentrates on organization and involves the following advantages: (1) a common vocabulary across several disciplines, (2) a technique for treating organized complexity, (3) a synthetic approach where a holistic analysis must be made, (4) a study of relations not entities, and (5) an operational study of purposeful, goal-seeking behavior.

Bertalanffy (1968:81ff) also notes that general systems theory depends on isomorphisms. These in turn rest on cognition, reality, and the organization of the universe in mathematical terms. He points out analogy (superficial similarities), homologies (identical basic laws in different disciplines), and the explanation of specific laws as special cases. These general notions "acquire exact expression ... only in mathematical language."

Others who have made efforts in the direction of general systems, but whose work is too demanding for our summary treatment are the physicist lberall (1972), the economist Boulding, the linguist Watzlawick (1967), the physiologist Gerard, the educator Clark (Laszlo, 1972) and the mathematician Rapoport (Laszlo, 1972).

Of all the ways of expressing the basic concepts of general systems theory, the most useful is that of set theory in mathematics. It is none other than Laszlo (1972b:19) who says: "Looking at the world in terms of such sets of integrated relations constitutes the systems view."

The individual who has contributed most to the application of mathematical set theory to general systems is Stuart Dodd, a retired professor of sociology at the University of Washington. (A summary of his Epicosm Model of the Universe will be found in the Appendix.) Briefly (but incompletely) stated, actants (the set of all names) interact in all possible ways to organize the cosmos (the set of all things namable) in all of its parts. Nature works in the cosmos to organize creation in terms of exponents (logarithms) to the base two (bit-logs).

Bertalanffy (1968:42) points out that the bit-log of N equals the amount of information from N questions. "This measure of information happens to be similar to negative entropy, since entropy is also defined as a logarithm of probability. But entropy is a measure of disorder; hence negative entropy is a measure of order or organization . . . "

So Dodd's system works in bit-logs, with four fundamental operations: pairings (twice x), squaring (x squared) , norming (2 raised to the x), and fulfilling (x raised to the x). These operations are special cases of the enumerative generator (1 + 1 / n) (raised to the n) which give rise to basic constants (such as sq. root 3 ) whose four fundamental function values constantly recur. Since general systems is viewed as the only science of which all other sciences are but applications, these basic sets and constants are related to all physical laws and constants (such as E=mcc and the speed of light), all of which may be derived from them."

Below Dodd explains his matrix (from the Appendix of "Trance, Art, Creativity":


Exhibit D is entitled "The reiteratings matrix, Rrc, for cosmists". It tells broadly how scientists, starting from scratch without even a language, can build models for the past, present, and future cosmos. Row A of this 4 x 4 reiteratings matrix tells how every symbol that man ever uses is built by the trio of acts called "reitering". Reitering is operationally defined in Row A as the intersect or product of three acts, namely combining a name and a thing named; repeating that naming among people, occasions, and contexts; without permuting or changing it.This trio of combinatoric speech acts seems to create every standardized and socialized and recurring symbol known or knowable to man.

Then Row B tells how four successive and cumulative rounds of reitering produce the four "key operators" and operations of all mathematics, logic, and language. Compounding these operations builds all syntax. Syntax in turn relates all symbols together producing all language and the communicated knowledge of man.

Row C, next, tells how the simplest and most probable self-reiterings of the four Key operations form the four "key processes". These, when mixed and blent as sums and products, etc., seem to produce the regular entities and their formulas of whatever exists or happens around us and within us.

Finally, Row D organizes cosmic on-going or phenomena in four submodels by tenses. These try to help explain the past causation of the cosmos; to describe its present contents; to predict its future consequences; and to control more and more of its compounding at any time. The synthesis of these submodels produces the epicosm model. This tells how the n cosmic actants, interacting in their nnways that tend to fulfil all their possibilities, continually organize the cosmos. Its summarizing epicosm equation

U0= 1 = a/ct

asserts for testing that the unitary cosmos, if defined as the universal set, U0, of all things namable, is organized in eight levels or subsets of random and reiterant actants (at=8 )which are ceaselessly and constantly formed and unformed by the creation rate (c) and the time taken (t).



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