Rev. 1.0 of 15 April 2000, begun 22 Mar 2000

Vibrational Resonance and Cognitive Internalization

Dedicated to Erich Jantsch (1929-1980) and Terence McKenna (1946-2000)

Abstract. Continuing in the spirit of earlier works, we propose a mathematical model for the process of internalization of ideas. This entire concept presupposes a paradigm of mind with internal and external regions, which we accept provisionally for the sake of discussion. In short, we envision a physical model comprising several excitable, continuous media in parallel planes, interconnected by a process of resonance of vibrations. The mathematical model for this physical analog is then discretised, and proposed verbatim as a computational model for the mental system. This model is typical of complex dynamical systems, as they have evolved during the last twenty years or so.

CONTENTS

1. Introduction

EINSTEIN: The problem begins, whether truth is independent of our con- sciousness.

TAGORE: What we call truth lies in the rational harmony between the sub- jective and objective aspects of reality, both of which belong to the super- personal man.

To Einstein, the external aspect was true reality. His use of the word consciousness above seems to mean individual (internal) consciousness or awareness, as opposed to the more literal meaning of shared (external) understanding, also known as conscience. To Tagore, the two aspects together comprised true reality: awareness and conscience in harmony. This fundamental disagreement was the topic of the conversation from which the quotation above is taken.

Naturally the question arises: Why, in this postmodern world, would we wish to add new mathe- matical models to a verbal discussion ongoing for thousands of years? The new mathematics we are now injecting into this long-ongoing dialogue P complex dynamics, chaos theory, fractals, and the like P belongs to both the internal and external aspects of reality. And being new, or newly discovered, it partakes of the evolutionary aspect of reality. It is because of this mathematical evolution, or more properly, the coevolution of mathematics and conscience, that new models are appropriate. New math, new shared understanding, new individual awareness, new math, etc. This hermeneutical cycle is basic to the evolution of conscience. In science, it is manifest in the concepts which are purely mathematical rather than actually physical, such as the gravitational field of Newton and the electromagnetic field of Maxwell. [Kline, 1985] The wave and heat equations of mathematical physics are derived from these field concepts, and these in turn are the tools of the modern theories of morphogenesis, such as the reaction-diffusion equations of biological morphogenesis, which is such a large part of mathematical biology, following from the works of Rashevsky, Kolmogorov-Piscunin-Pontriagin, and Turing.

Using these tools of mathematical morphogenesis, we go on now to develop a model, for the internalization of a concept from the objective (external) aspect or field of conscience to the individual (internal) field of awareness, or vice versa, by a process of vibrational resonance or harmony. In other words, a model for Tagore's reality. The outline of our presentation is shown in Figure 1.1.

2. An exemplary complex dynamical system from classical physics

Controversy regarding the mechanism producing these patterns led to experiments in a vacuum, and revealed that grex vortices in the air over the plate, excited by its vibrations, moved the sand. A grex is a toroidal motion, in which the fluid moves downward through the center of the tori, and upwards around the outside. Further experiments using a light powder in place of sand gave further information regarding the vortices in the air. The heavier sand particles are moved by the bottom of the vortices, collecting at the nodes. The lighter powder particles are lifted over the top of the vortices and deposited at the antinodes. All this is shown in Figure 2.1,

Fig. 2.1. Section of a grex.

More recent experiments by Hans Jenny using a mixture of black sand and white powder dramatically demonstrated this filtration by density in the vortices, with the sand moving to the nodes, and the powder to the antinodes. The pattern so produced is highly suggestive of biochemical pattern formations in embryogenesis, and other morphogenetic processes. [Jenny, 1967] An experimental arrangement for the study of Faraday crispations is shown in Figure 2.2.

Fig. 2.2. The Jenny macroscope.

An electrical wave delivered to a transducer (loudspeaker) creates a sound wave in the air beneath a plastic plate, inducing and maintaining a pattern of vibration. This in turn activates a related pattern of vibration in the thin layer of liquid resting on the plate, and thus a pattern of grex vortices in the layer of air over the plate. Air, plastic, liquid, air: four layers of coupled pattern formation processes.

3. An exemplary complex dynamical model from classical mathematics

We next consider two layers of vibrating media. For example, a thin, horizontal layer of air, lying upon a thin layer of water. For each layer, we imagine constructed a mathematical model, a dynamical scheme. Visualizing each model as a black box, we thus have two boxes, one above the air. But we further suppose that the two layers are in contact. Thus the motion of the water influences the motion of the air, and vice versa. To represent these mutual influences in our mathematical models we must connect the two black boxes. First, the state of the water affects the control parameters of the air. In the theory of complex dynamical systems, this is called a serial coupling. [Abraham, 1993b] Next, the state of the air affects the control parameters of the water, another serial coupling. Taken together, the two links represent a serial bicoupling, in the language of complex dynamical systems theory.

Our physical experience of vibratory systems in contact, strings of a musical instrument for example, includes the phenomenon of resonance. In this phenomenon, the vibrations of each string encourages the vibration of the other, if the vibrations are harmoniously related. More generally, we may say that a space-time pattern in one vibratory medium induces a harmonious space-time pattern in another medium. This is, in fact, the situation with the experiments of Chladni and Faraday, in the 19th century, as described above. Specific dynamical schemes have been elaborated for the crispations of Faraday, in particular, and there is a growing literature regarding them in mathematical and scientific journals today. But we will be discussing only the computational form of these models, to which we now turn.

4. The modern computational form of the complex dynamical model

A dramatic example of the computational approach, before the advent of computing machines, was provided by Richard Southwell. As part of the war effort in Great Britain during the Second World War, Southwell commanded a troop of arithmetically gifted youth, drafted from universities for this purpose. Following a computational method of his own devising, still used today, Southwell directed the clever kids, scampering in stocking feet over huge paper carpets, from atop a tall step ladder. Doing arithmetic upon numbers they read from the paper at their feet on small scratch pads in their hands, they would stoop from time to time to write a new result on the large paper carpet, crossing out the previous result.

The computer revolution tipped the scales, giving great advantage to computational methods over the analytical techniques. Not only faster and more general, these new methods could be applied by any computer-literate person. This is the dominant style of applied mathematics today. Applied to our problem of the spaced-time pattern of vibration maintained by an external force acting on an excitable medium, such as an elastic membrane, the Southwell method reduces the membrane to a two-dimensional lattice of identical cells, each described by a small oscillator P for example, a simple pendulum, or a weight hanging from a coil spring. Each such cell is coupled somehow to its four nearest neighbors. Such a mathematical object is called a cellular dynamical system, or cellular dynamaton, or CD for short.

The CD for one membrane, or layer, is visualized on the computer monitor as an apparently continuous image of changing colors. Each pixel (point on the screen) is given a color which is a code for a number. For example, red might denote the displacement of a pendulum to the left by one inch, while blue indicates displacement to the right by one inch. Colors in between may be coded for intermediate positions of the pendulum. One starts with any convenient pattern of colors on the screen. The equation of evolution for the vibrating state of the membrane, interpreted by the Southwell method, then determines, from this initial pattern, a new one. And repeated application of this method with the new pattern yields still another, and so on. Thus, a moving image is seen on the screen, and after a few iterations of the computational method of Southwell (if all is going according to Hoyle) we are viewing the space-time pattern determined by the equation of evolution chosen at the start of the process.

Finally, this strategy may be applied to the complex system of two coupled membranes. Each membrane is modeled by a CD, and the two CDs are coupled to each other, cell-by-cell, as shown in Figure 4.1.

5. Application to conscience

Just as the early cartographers made maps of the world by gluing together individual charts brought back by pioneering navigators, shared understanding may have been constructed by convergence, or emergence, of individual understandings. This map of the intellectual world, constantly evolving, becomes stored in libraries and perhaps in an immaterial or mental field of living, collective memory. New individuals P infants, children, young adults, immigrants -- are brought into this collective field, or acquire conscience, through an educational process commenc- ing perhaps in conversation with family members or neighbors. There is also evidence that some sort of telepathy, such as morphic resonance, contributes to this induction process. [Sheldrake, 1981; Radin, 1997] Such a resonance process of mental fields, individual and collective, is consistent with the Sanskrit heritage which informed Tagore and Sheldrake. In the Kashmiri Shaivite tradition, a divine oscillation (Spanda) is the driving force which maintains an excited state of vibration in the collective field. [Dyczkowski, 1992] The existence of such fields is, of course, controversial. Even assuming their existence, the idea that information is encoded into the field as a vibratory space-time pattern -- like a radio broadcast into the electromagnetic field -- is a further flight of fantasy. And yet, this vibration image has been a part of our conscience for thousands of years, due to prophesy, sacred texts, gnostic experiences, and the like. Finally, accepting all this for the sake of discussion, there is the further question: Is the vibrating field evolving, or eternal? We summarize all views in one image in Figure 5.1.

And in Figure 5.2 we see, side-by-side,

• the physical experiment of Chladni, Faraday, Jenny, myself, and others, and
• the vibratory model of conscience

6. Conclusion

References

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