TOWARD A SPIRITUALIZED
SCIENCE AND MATHEMATICS

by Donald Reed
Tachion Energy Research

This article was sourced from the Journal of Borderland Research Vol XVLIII, No 6 Nov-Dec 1987

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The scientist of today has penetrated deep into the mysterious interior of matter, “unearthing”, in the literal sense of the word, the abysmal powers that lie hidden there. But he is at a loss, for what he finds in stars and atom is no longer in any real sense of the word; it is a pure nexus of relations, an intellectual shadow of a subtle inner essence as yet eludes him.

Notwithstanding the impressive fruits of 20th century technology, it cannot be denied that mathematical physics, the underpinning of technological progress, is very largely groping in the dark. Mechanistic in spirit and outlook, it seeks to answer the searching questions of mankind with symbolisms ever more abstruse as time goes on. Its root ideas have grown further removed from the realms of experience, moreover, from anything that is accessible to human imagination.

Part of the reason for the undue abstruseness of such a system devolves upon its built in restrictions for modelling the full spectrum of human experience, both physical and metaphysical. Indeed, the mathematical system chosen to consciously perceive that reality. If one’s perceptual understanding and belief systems are limited to one small aspect of reality, then the mathematical expression of physical laws will become correspondingly unduly complicated so circumscribed by our perceptions. For this reason, and others voiced recently by physicist Allen J. Jackson, (1) the system of physical science has for centuries remained incompatible with the equally valid body of metaphysical knowledge.

The outcome of all this is that without abandoning the skeptical and undogmatic frame of mind which it professes, science today is in need of a deep philosophical change. A new metaphor is urgently required that will serve as the foundation for a synthesized framework of physics and metaphysics. We require a fresh paradigm that will put the mechanistic approach into its proper perspective, not merely repudiate it and leave us to wander around in the resulting emptiness. It must be a model of the universe and man around which human thought, feeling, and action can find a fresh mode of expression.

There is, side by side with the intricacy of the analytic school abstracted from Cartesian geometry and Newtonian calculus, another geometrical system. This latter system represents a novel avenue of thought and imagination. By its precepts we shall be led far more directly into that cosmic and ethereal realm of metaphysics where will be found the living links, bot in the universe and ourselves, between the spiritual heart of things, and the outer surface.

We speak here of the science of projective geometry, which is far from new. Hidden away in the general textbooks and regarded as a rather abstract branch of higher mathematics, projective geometry has not presented many immediate aspects for practical application in technical areas. this, as well as its abstract guise, has kept it in the background for centuries. The important thing about it is the quality of its forms of thought, and this is modern. Because it reflects a modern outlook, projective geometry will hereinafter be referred to in this article as the “new” geometry.

The following exposition is an expansion of the inspiring and ground-breaking work done in this field by the German mystic Rudolf Steiner, and his interpreters. (2) The interested reader is encouraged to consult their works for further treatment of the subject.

Physical science is based upon Cartesian geometry, which deals primarily with rigid and lifeless forms circumscribed by measurement. In contrast to this, the new geometry moves in a realm of light and metamorphosis, out of which the ethereal and archetypal forms are fashioned. It is characteristic of the new geometry not to refer the geometric forms and figures to an arbitrary framework, but to evolve their qualities, as it were, out of their own nature. The new geometry enables us not only to approach the forms externally, as in the Cartesian format, but to perceive them in their inherent mutual relations–relations that proceed from the forms themselves, not from extraneous coordinates. This corresponds to a transition from a discursive and analytic treatment of nature, to an imaginative perception of her living and unfolding process. In this sense the science of the new geometry approaches the spirit and practice of art, while remaining firmly grounded in the rational edifice of mathematical precision.

The basic concepts of the new geometry are not difficult, but they require a kind of thinking to which we are not yet accustomed, nurtured as we have been for so long on the one-sidedly Cartesian-Euclidian way of thinking and the analytical method. The new geometry requires a qualitative grasp of mathematical forms, and the cultivation of an activity of though which encompasses more than the momentary manifestation of form.

In the new geometry, the ideal structure of 3-dimensional space does not proceed one-sidedly from the point alone (as in Cartesian systems), but from two opposite but mutually dependent entities–point and plane, each of which plays a fully equivalent part in the fundamental structure. A simple illustration will serve to show the spatial significance and mutual polarity of point and plane. Imagine a spherical surface expanding and contracting. At one extreme it contracts into the central point; at the other, it expands into the plane that disappears into the infinite distance of space. We say a plane rather than an infinite sphere because of necessity when the radius is infinite, the curvature vanishes altogether. Although the infinitely distant plane is not perceptible through the senses, it is nevertheless a clear and exact though which finds expression in corresponding mathematical formalism. Moreover, the infinite plane stands on an equal footing with every other plane of finite space.

Thus the polarity of point and plane is really one of expansion and contraction, not only quantitative but qualitative. In this polarity the straight line plays the mediating factor, its formal relations to point and plane being mutually equivalent. It is the inclusion within the system of infinitely distant entities, which create a new geometry based solely on the interrelationships of point, line, and plane. Projective geometry is not merely a geometry of created forms, but the geometry of the relationships between form-creating entities.

Implied in such a structure exhibiting mutual polarity is the existence of a space in dual or polar relationship to our ordinary Euclidian space, not independent but interpenetrating the latter. Its nature is qualitatively different in terms of function and activity, and quantitatively different in regards to form. Euclidian space is structured from points raying outwards centrifugally from a center towards an unreachable infinite periphery (the so-called “infinite plane” of space). In contrast, the polar-Euclidian (or Counterspace, in the terminology of Rudolf Steiner) is characterized by planes, starting from an infinite periphery and taking on a centripetal or inward motion, thereby engendering a plastic or molding quality as they attempt to  extend themselves toward an all-relating point in the center (Rudolf Steiner called in the “star point”).

As we shall subsequently see, it is in the realm of Counterspace that all forces of living organic growth have their domain. Consequently, by means of this Counterspace structure, the function and expression of all living forces of nature can be properly understood.

Yet, the introduction of Counterspace into our geometrical repertoire, at the same time necessitates some shifting of our mode of thought in regards to relationship of geometrical elements. Specifically, our ideas of “part” and “whole” are profoundly changed by this new geometry. Ordinarily, we think of a plane for example as composed of infinitely many points. But now we have to think of a point as made up of infinitely many planes. Admittedly, the second concept is not as easily accessible to the imagination. But the two polar aspects interweave. In the first case, the plane is the whole–greater than any of its parts (points). Whereas, in the latter case, paradoxical as it may seem, the point is the whole and the planes which pass through it are parts, or members of the points.

Unlike current forms of geometry, which find their basis in the egocentricity of the point alone, the new geometry is a true reflection of spiritual and moral ideas. The interweaving of planar entities working towards a common goal, in conjunction with earthly point centers thrusting out, is evocative of the view of brotherhood held by the Rosicrucians and other mystical orders–of the community where each stands as an individual, yet all are interwoven into the whole, and by their cooperative actions achieve a common accord with themselves and their environment.

Let us now turn from the world of pure form to that of active forces. In addition to the centric forces described as the “field theories” of physics, the new geometry establishes a concrete basis for the existence of another type of force which can be described as “peripheral” or “etheric”. Much like the aformentioned geometrical model of Counterspace, these forces can be considered as originating from the periphery, or infinite plane of space, and tending towards the material bodies of living things–above all towards the germinating centers of fresh life. But the relative center  toward which they work is not their source, but rather their infinite receiver. Again, we must invert the accustomed notions of center and periphery to get at the right idea. A physical force, emanating from the center–such as gravitational, electromagnetic, etc., needs the surrounding space into which to expand and dissipate. The infinite periphery has to be there to receive it. Accordingly, a peripheral force needs the living center towards which it expends itself. And yet this center is radically different from the point center of physical forces. Its activity will be receptive, rather than self-centered. It will be active by virtue of what is around it, not within it. Just as the operation of centric forces are quantitatively described by Cartesian related systems, the workings of peripheral forces find their proper expression in the interplay between point and plane underlying the new geometry.

As abstract as the operation of etheric forces seems, nowhere is their activity more apparent than in the the world of nature, specifically in the morphology and physiology of living things. A pertinent example is afforded by the pupation of the caterpillar. At the stage where the metamorphosis into the butterfly is to take place, inside the caterpillar chrysalis a breakdown of organs and tissues is occurring. Yet at this stage, the shape and position of the organs of the forming butterfly are already stamped on the pupa. These marks are outside and there is nothing yet formed inside to correspond with them. Clearly, no centric-type force could produce this result, but invisible etheric (peripheral) forces outside the insect have stamped upon it the shape corresponding to that final cause which is inherent in its being.

Now, the formation of sugar, starch and cellulose in the leaves and other organs of the plant is a phenomena, not of expansion but of contraction. The chemical constituents of which these carbohydrates are formed–carbon dioxide of the air and water raised from the soil–are specifically lighter than the resulting product in the body of the plant. The chemical process itself is endothermic and therefore cooling and contracting. Even the polymerization, forming the di-and polysaccharides, the different forms of starch, cellulose and lignin, is a condensing process. Once again we see the activity of peripheral forces in ample evidence.

Even in the inorganic realm we find evidence of etheric forces working. For instance, although it has largely gone unrecognized by the orthodox science community, the mathematical laws governing the science of crystallography owe their existence to projective geometrical relationships. In addition to the centric forces of stress and strain which are operating in the crystal after formation, invisible etheric forces are chiefly responsible for sustaining the crystal as a stable entity in time and space. All that is localized at the place where the crystal appears to our senses is a kind of rigidification, as it were, as shock or inner rupture of the archetypal ethereal form which pervades all space. In this connection the esoteric metaphor, “All matter is ‘woven’ light”, takes on a fresh meaning.

The medical treatment procedure known as homeopathy is process whose effectiveness has thus far eluded an explanation from the fields of modern chemistry or medicine. Briefly, homeopathy is a medical treatment wherein through successive dilution of an original amount of medicinal preparation (usually shaken or succussed), that substance become potentized and more effective in combatting the disease or affliction. Such an outcome is indeed difficult to account for both in terms of rough and ready common sense and of prevailing scientific notions. Nevertheless, this phenomenon becomes understandable when considered from the standpoint of Counterspace. Precisely because of the polar nature of centric and peripheral forces, when we dilute a physical substance (make it smaller), we are really intensifying or concentrating the sphere of ethereal force which is invested within it. Thus by physical dissolution we make the ethereal or curative influence, stronger.

Finally, the ground-breaking work that Eric Dollard is conducting as an extension of Nikola Tesla’s research, and recently reported in the Journal of Borderland Research (May-June 1987), is the clearest replicable evidence to date of the working of living Counterspace forces by virtue of the operation of an electro-mechanical devices. The living organic forms Eric and his colleagues have observed in discharge tubes, and the planar discharge patterns mimicking the structure of morphological forms of nature incorporating the golden-mean ratio, suggest that the unique dielectric wave set in motion is resonantly coupling with and entraining the subtle but powerful Counterspace forces of light and life which pervade our everyday existence. Moreover, this research is pivotal since it holds the promise of ultimately extricating the subject of etheric forces of nature from the nebulous realms of speculation and pure abstract mystical thought, and place it on a firm rational basis.

Because of the limited space in this journal, in this introductory discussion we can only treat the subject on merely a superficial basis. Nevertheless, even in the foregoing encapsulated discussion, it has been demonstrated that such a system of Counterspace shows promise as a basis for the mathematical modelling of metaphysical operations. Moreover, it may serve as a fresh viable alternative to the lofty but lifeless theoretic abstractions attendant with the current mathematical edifice of science.

Above all, if formally integrated in scientific thought, the tenets of the new geometry like few others yet formulated, may hold profound implications for the status of our very social, political, and religious structures as well. Unlike the impersonally motivated interactions in modern sub-atomic physics which help to sustain our illusory belief in the supremacy of the ego, the philosophical basis of the forces that are engendered by the new geometry of Counterspace is an able reminder of the humility of purpose which should be the guiding forces in our lives. Like the elements of point and plane, in the ideal social structure individuals will work together independently, side by side, yet achieving a common accord. United in the circle of their activity, they will work together joyfully in freedom, each with his own task yet conscious of the common bond. With the full import of these simple truths appreciated, the new geometry will perhaps afford a rational justification of the formerly “unprovable” mystical doctrine of the omnipresent intelligent plan behind the workings of the universe at all levels of Creation. We will come to realize the profound truths of Einstein’s vision of ultimate reality couched in his heretofore cryptic statement, “God is subtle, never malicious”.

FOOTNOTES

(1) Jackson, A (1982); “Cause and Effect in Metaphysics and Science” Rosicrucian Digest (60) 10:24-27

(2)
Adams, G. (1965); Physical and Ethereal Forces, Rudolf Steiner Press, London
Adams, G. (1977); George Adams, Interpreter of Rudolf Steiner, Henry Goulden Limited, East Grinstead, Sussex
Adams, G., Whicher, O. (1949); The Living Plant, Stourbridge
Adams, G., Whicher O. (1952); The Plant Between Sun and Earth, Stourbridge
Adams, G. (1933); Space and the Light of Creation, Rudolf Steiner Press, London
Adams, G., Vreeede E. (1965); Strahlende Weltgestaltung, Dornach, 2nd Edition
Whicher, O., (1971); Projective Geometry: Creative Polarities in Space and Time, Rudolf Steiner Press, London
Locher-Ernst, L. (1937); Urphanomene Der Geometrie, Zurich
Locher-Ernst, L. (1957); Raum and Gegenrauum, Dornach
Locher-Ernst, L. (1940); Projektive Geometrie, Zurich
Gschwind, P. (1977); Der Linare Komplex, Mathematisch-Astronomische Sektion, Dornach
Gscwhind, P. (1979); Methodische Grundlagaen zu einer Projektiven Quntenphysik, Mathematisch-Astronomische Sektion, Dornach
Gscwhing, P. (1985); Raum, Zeiot, Geschwindikeit, Mathematisch-Astronomische Sektion, Dornach