Geometry of Meaning
Examples of Integrated, Multi-set Concept Schemes (Annex 1)
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See other Examples of Integrated, Multi-set Concept Schemes. The concept scheme described here is discussed in the paper on: Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation. This was prepared for a sub-project meeting of the Forms of Presentation group of the Goals, Processes and Indicators of Development (GPID) project of the United Nations University (UNU). The annexes were published in Patterns of Conceptual Integration. Brussels, UIA, 1984, pp. 161-204
The following points are mainly extracted from the first of two books by Arthur M Young. The books are in fact complementary:
The author is a mathematician who invented and developed the Bell helicopter
0.1 "In the pursuit of meaning, the measure formulae (of physics] are invaluable because they replace words. Words depend upon definitions, which in turn depend upon other words, making a closed circle. The measure formulae, as we have seen in the case of length (L) and its derivatives, are more explicit than words in that they define operations...Thus they anticipate the aspects or categories that we may expect a situation to have....The measure formulae are the outcome of centuries of struggle on the part of physical science to arrive at a set of terms that can be formulated in such a way as to leave no vagueness, no ambiguity." [Geometry, p.32)
0.2 "A true scientific inquiry must be based upon some paradigm, and if this is not recogninized the paradigm remains unconscious. Such is the fourfold paradigm which structures "objective" science and leads to determinism. It is false because it overlooks first cause and makes rationality the final arbiter. Its emphasis on objectivity omits the projective aspect and leads to a cul-de-sac It fails to discover the reflexive quality of the universe. The sevenfold paradigm corrects these errors and opens the closed system that the fourfold would dictate. It frees us from a misinterpretation of the constraint of law. Approaching a subject with the sevenfold paradigm can be compared to approaching a machine with a knowledge that, in addition to its structure, it has a purpose and can be turned on, whereas the objective paradigm would permit only a study of its structure." (Reflexive, :p.279)
1.1 "Let us first note that it is much easier to talk about the divided than the undivided....Of the initial unity, we can say nothing, except to regard it as a dynamic potency [we cannot predict where such potency will lead)." [Geometry, P.79)
2.1 "What then is the two-operator ? It is the operator which is not mediated by an independent duality. It differs from the dualities in fourfold analysis [quadratic analysis) in that it requires three dimensions to express it (of the chirality of structures)....Furthermore, this operator cannot be described objectively. It requires participation, which is projective....besides corresponding to the directions of time, the chirality correlates to the two directions in which one can move in the cycle of action." (Geometry, p.104)
2.2 "It is important to realize at the outset that this operator is not to be confused with "pseudo-dichotomies" such as positive and negative, true and false. Such apparent dichotomies, as I explained, actually involve four distinctions. In the case of positive and negative, the two hidden distinctions are positive and negative imaginaries; in the case of true and false, the distinctions are consistent and inconsistent, without which no use could be made of the true-false distinction." (Geometry, p.79)
2.3 "While we found that we could use the same (circular) diagram to describe both the [fourfold) learning cycle and the controlled motion of a body, recall that the order in which the steps were taken had to be reversed. While the analysis of motion used clockwise rotation on the diagram, learning was counterclockwise.. Generalizing, we may say that clockwise rotation is informed or premeditated (toward goals), whereas counterclockwise motion is natural or naive (giving rise to effects)....these considerations are so basic that for their formal expression we need a new operator - a twofold operator - to accommodate left- and right-hand rotation in our system." (Geometry, p.78-79)
2.4 "I prefer to think of the reversal of time not as negative time (-T) but as inverse time (1/T). This is, of course, conjectural, but it has some interesting implications. If one thinks of normal time as being very long (even if not infinite), then inverse time (l/T) would be very short -- eternity in an instant. In the photon, it has long been known that the energy is inversely proportional to time (h = ET). This implies that in the "anti" (matter) world there might be an unlimited amount of energy in an instant of time reversing our normal relationship between size and importance. This compaction of time would give it the character of omnipresence -- not going "backward" in time, away from the present, but instead going more deeply into the present. This interpretation has the merit of conforming to references in countless religions and mythologies to the super-sensible, nonphysical celestial world." (Geometry, p.61-82)
3.1 "In discussing the (fourfold) learning cycle, I implied that fourfold analysis is insufficient as a general description of process....the cycle includes not only actions but other factors as well. This is the threefold cycle of stimulus, response, and result. The three aspects in this case are again different categories, for which it is quite difficult to find sufficiently general names." (Geometry, p.25)
3.2 "So the generalized description of the threefold cycle is...relationship, act, state. In returning to a state, which it must do to be a cycle, the process does not necessarily return to the same state that initiated it; it only returns to a state of some kind....The threefold is not limited to this cycle. In fact, the cycle, as an analytic concept, does not fully describe the threefold. In the more general sense, this is a. way by which wholes divide into three inter related factors, often, but not always, in the form of two elements plus that which is between them....Since it is basically nonconceptual, it cannot be defined..." (Geometry, p.26-27)
3.3 "The threefold operator, represented analytically as equidistant points on a circle, is actually a three-dimensional activity, whose measure gives only its analytic aspect. The analytic aspect, which is in two dimensions, does not convey the full meaning of the cube root; it is like the shadow of a solid figure. The threefold nature of the cube root is nonanalytic. It involves categories which differ from one another more profoundly than those of the fourfold." (Geometry, p.55)
3.4 "We may think of the threefold as an endless line (curving though space in an arbitrary fashion). Between any two points on this line, we can say which is before, but we cannot look at a piece of it without getting off the line....It is a reality which is always with us, yet perpetually eludes analytic description ....If we try to analyze what it is that the threefold describes, we are in a bind, for it is just that element of participation in life that analysis cannot, and does not even pretend to, cope with." (Geometry, p.57]
3.5 "The threefold is an entirely different way of cutting the cake. It is much more fundamental (than the fourfold), and it cannot be analyzed....In fact, the threefold is the natural way we move in life. We see something, buy it, and enjoy it; food, eating, satisfaction; stimulus, response, result. There are three categories of terms: relations, acts, and states. The reader will recall the difficulty of finding sufficiently general words." (Geometry, p.99)
3.6 "But when the stimulus causes wrong action and the result is not achieved, the (fourfold) learning cycle becomes necessary. Thus the learning cycle occurs only when there is an obstacle in the larger, threefold cycle." (Geometry, p.24)
3.7 "In sum, we have shown that the threefold, despite the seamless quality which makes it difficult to get "hold of", is still in the measure formulae of science. These expressions, which constitute a basic vocabulary of science, fall into three groups of four: actions, states, relations." (Geometry, p.1023
4.1 "We have shown that the motion of a body, even one controlled by an operator, may be formally described by the four categories of measure...Postion (L), Velocity (L/T), Acceleration (L/T2 ). Control (L/T3 ). This is the form used in so-called measure formulae, the then formulae used in physics to describe the motion of a body....In describing these measures -- position and its derivatives -- we have sought to establish:
4.2 "The four types of action in the learning cycle correspond to the four measure formulae:
Spontaneous act (impulse) = acceleration Reaction (also spontaneous) = velocity (i.e. change) Observation = position (the observable factor) Control = control
The same (circular) diagram that we used to depict the cycle of action (position, velocity, acceleration, control) may be used to represent the learning cycle (spontaneous act, reaction, observation, control), but it is necessary to reverse the order..." (Geometry, p.23)
4.3 "The fourfold operator applies to analysis, concepts, forms. Consisting of two dichotomies that are mutually orthogonal, or independent, it tells us that in any situation that can be dichotomized, or divided into two opposite aspects, there must exist a second dichotomy which mediates or measures the first." (Geometry, p.94)
4.4 "While there is no assurance that this approach will solve all problems, it has the merit of including all possible permutations of the relations between knower and object....
The four states Because they are not conceptual, the human states must be apprehended through feelings. While each physical state is the rate of change of the one before, it would appear that the human states do not change into one another except through the appropriate relationships and acts; and we are reminded that the fourfold is essentially secondary to the threefold.
Both the spontaneous act and acceleration are unconscious action.
7.1 "The development of process occurs in stages. There are seven stages. Each stage develops a new power. Powers are cumulative; each one retains the powers developed in the previous stages-
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