I have been reading your recent book "A Brief History of Everything" with much interest. In terms of its vision and scope it reminded me of Hegelís "The Phenomenology of Mind", which had a considerable influence on my earlier intellectual development. I was particularly fascinated with your chapter "The Four Corners of the Kosmos". What you were describing there in terms of the four facets of every holon, can I believe, be expressed by a simple mathematical relationship which has immense significance.
Though an economist by profession, in some important respects our interests converge and I have often felt in reading your books a certain kinship of spirit. I became disillusioned with conventional scientific method at college some 25 years ago. This led me in search of spiritual truth and a growing desire to map out in contemporary fashion the various stages of mystical development. It gradually dawned on me that a remarkable and precise complementarity - at the dynamic holistic level - exists as between mathematics and (transpersonal) psychology. In one way this perhaps is not surprising. Mathematics involves the most specialised development of reason and (transpersonal) psychology in turn the most specialised development of intuition. Thus a proper synthesis of reason and intuition implies a corresponding marriage of mathematics and (transpersonal) psychology. For the past six years I have been developing a "new" holistic (or transrational) mathematics which explicitly recognises both rational and intuitive type understanding.
In this approach mathematics can be likened to a scissors. One blade corresponds to the conventional quantitative approach. The other blade corresponds to a complementary qualitative approach which in essence relates to psychological reality. This latter approach though implicit for example in the Pythagorean outlook and more recently Jungian psychology has not yet been fully articulated. The quantitative (mathematical) aspect is in direct terms rational (and indirectly intuitive). The qualitative (psychological) aspect in direct terms is intuitive (and indirectly rational). It is the dynamic interaction of both aspects that constitutes - in this integrated approach - true mathematical understanding.
Surprisingly, this qualitative approach to mathematics can be given an exact - though indirect - rational form of expression. Let me briefly illustrate as follows. In the conventional quantitative system, we have one set of natural numbers 1, 2, 3, 4, ........ Implicit here is that these numbers are all expressed with relation to the 1st dimension (i.e. raised to the power of 1).
Now the number expressing the dimension or power (i.e. 1) in direct terms represents a quality (rather than a quantity). However in conventional mathematics it is given a reduced quantitative interpretation. So when we initially express a number to a another power - e.g. the square of 3 (where the dimension is 2), its value is given in reduced quantitative format in terms of an invariant 1st dimension (i.e. 9). Indeed this one dimensional or linear approach dominates conventional mathematical activity. The qualitative dimension of understanding (intuition) is simply reduced to the quantitative (rational). Thus mathematical understanding - which in dynamic terms involves the interaction of rational and intuitive processes - is expressed in reduced static terms as merely rational.
A complementary qualitative natural system also exists. Here an invariant unit quantity (1) is raised to differing dimensional qualities or powers. 1, 2, 3, and 4 thus - in direct terms - represent in this complementary system powers or dimensions (which are qualitative). From a merely quantitative perspective this alternative (vertical) system appears trivial as the reduced value of each number is 1. However its importance becomes clearer when we engage in what might be termed (holistic) qualitative synthesis. Quite briefly, each (qualitative) "higher" order unity, has a corresponding (quantitative) "lower" order translation, which is obtained by taking the corresponding root of unity.
For example, the square of 1 (where the dimension is 2) represents the psychological (qualitative) "higher" order unity of pure intuition where polar opposites - at the linear one dimensional level - are reconciled in undivided experience. Then, the two roots of unity +1 and -1 represent the corresponding mathematical (quantitative) "lower" order translation of this union. In reduced terms, this experience has both positive and negative aspects (which are clearly separated). Furthermore all such "lower" order mathematical translations give rise to coherent circular number systems (i.e. all roots of unity lie as equidistant points on the circle of unit radius in the complex plane). Thus whereas the (quantitative) linear system has immense value in terms of static partial analysis, this reduced (quantitative) circular system has a complementary equal value - though largely unrecognised - in terms of the dynamic holistic synthesis of all systems. Putting it simply, the fundamental holistic structures of reality (e.g. physical and psychological) at all levels can be remarkably expressed in terms of appropriate roots of unity.
As you portray so well in your books, there is general agreement in mystical traditions on the major levels of understanding. For convenience, we will refer to them as the gross (inc. pre-rational), subtle (inc. psychic), causal and non-dual respectively. Using my own geometrical terminology, these are the linear, circular, point and radial levels. What is fascinating is that the fundamental structure of each of these levels can be mathematically expressed simply in terms of corresponding roots of one. Quite literally therefore, these levels can be translated - in mathematical terms - as reduced forms of unity!
The linear level is clearly a one directional approach where truth is sought in absolute positive terms. As the dimensional here is one, both the qualitative number and its quantitative translation coincide (i.e. the one root of unity is 1). This is why the very separation of quantitative and qualitative aspects of number - and indeed all phenomena - appears unnecessary at the linear level. Here, physical phenomena (quantitative) are directly represented by corresponding mental concepts (qualitative).
The circular level is a two dimensional approach where truth is sought in dynamic relative terms. This is the approach of the complementarity of opposites with both positive and negative poles. Qualitatively this can be represented by the square of 1. The corresponding quantitative translation is given by the square root of unity resulting in two answers with matching complementary poles (+1 and -1). Thus at this level all holons have both positive and negative (i.e. exterior and interior) aspects. At the level of psychological (qualitative) reality these complementary "real" aspects are perfectly fused in experience leading to a "higher" unity. At the "lower" reduced level of mathematical (quantitative) reality they are clearly separated as distinct aspects.
The point level can be represented as a four directional approach where truth is sought in even more subtle terms as the complementarity of both "real" (conscious) and "imaginary" (unconscious) opposites. This can be qualitatively represented by 1 (raised to the power of 4). The corresponding quantitative translation is given by the four roots of unity which again have matching results with complementary "real" and "imaginary" poles (+1, -1, + i and - i). What this means is that each holon is now experienced as having four facets. It has both a "real" and "imaginary" definition with positive and negative directions. In other words each holon has both an individual and collective identity with internal and external aspects. Thus if the individual or part identity is "real" then the corresponding collective or whole identity - in relative terms - is "imaginary". In correct mathematical terms each holon is now understood in "complex" rather than "real" terms.
This is of vital importance. In any system the whole is now qualitatively different from the parts. Indeed this is fundamental to the difficulties in reconciling Relativity Theory (whole) with Quantum Mechanics (parts). Therefore we should structure such systems - at this level - mathematically in "complex" terms. Once again at the "higher" psychological level of qualitative reality, these four aspects of experience are fused in a raised or transformed unity. At the corresponding "lower" mathematical level of quantitative translation, this experience is accurately expressed - in reduced terms - as involving four separate aspects (both "real" and "imaginary" with positive and negative directions).
The importance of this level is that it accurately mirrors the comprehensive "complex" number system in mathematics which now can be given a dynamic as opposed to a static interpretation. At this level reality is understood in such dynamic "complex" terms. This also throws light on the significance of the mandala (highlighted by Jung) involving the circle divided by a cross into four equal quadrants. Such a mandala is literally a pictorial geometrical representation of the four roots of unity which so precisely represents this level.
The radial level finally can be represented as an eight directional approach involving (horizontal) "real", (vertical) "imaginary" and (diagonal) "complex" poles. This even higher order psychological unity can be represented by 1 (raised to the power of 8). The corresponding quantitative representation is given by the eight roots of unity yielding two (horizontal) "real", two (vertical) "imaginary" and four (diagonal) "complex" results. The diagonal lines - representing the complex roots - can be easily shown (using the Pythagoraen Theorem) to have a finite value = 0.
In physics, it is recognised that these diagonal or null lines - of zero length represent the fundamental nature of light. In complementary manner in psychological terms, these diagonal lines represent the fundamental nature of spiritual light or intuition. Just as natural light is necessary to see objects, this spiritual light is likewise necessary to mentally "see". Thus all finite phenomenal understanding is against a background which is transfinite (i.e. spiritually intuitive).
Each holon has now eight directions or aspects. This could be expressed as saying that it has both an individual (real) and collective (imaginary) identity, with both exterior (positive) and interior (negative) directions. These four poles in turn have both natural (finite) and spiritual (transfinite) aspects. Once again at the appropriate "high" level of qualitative understanding (i.e. the radial level) all these aspects are perfectly fused. At the corresponding "low" level of quantitative mathematical translation these aspects are separate and distinct.
Indeed this eight fold psychological structure (supersymmetry) equally represents the most fundamental structure of physical reality. However it truly requires the appropriate "high" level implicit intuition of the radial level to properly interpret the corresponding "low" level explicit translation (representing the fundamental structure of physical reality). Ultimately information about reality cannot be divorced from transformation of reality.
This represents Ken, just one small - though highly important - illustration of holistic mathematics. Here, abstract mathematics no longer has any meaning, for by definition every quantitative (rational) mathematical relationship, is balanced by a corresponding qualitative (intuitive) psychological interpretation. However it will take a revolution in our customary world view to be able to properly see these connections. For example it leads to the truly multidimensional realisation that in physical terms every object phenomenon (quantitative) is in dynamic interaction with a unique corresponding dimension (qualitative). In this "complex" world-view, "real" dimensions are simply "imaginary" objects (and "real" objects "imaginary" dimensions). Ultimately therefore there are as many dimensions as objects!
I have been refining and elaborating these ideas for some time, have completed two books and am presently working on a third.
The first is a detailed treatment of the various levels and stages of the psychological spectrum. (Transforming Voyage - A Contemporary Account of Mystical Personality Development).
The second stems from my interest in holistic mathematics leading to the translation of all these levels and stages in precise number terms. Again in mathematics we have positive and negative, real and imaginary, finite and transfinite number quantities. The various psychological levels - and transitions between levels - can be accurately explained in terms of the corresponding number qualities (leading to a whole range of fascinating paradigms). We also have rational (binary, prime, natural, integers, fractions) and irrational (algebraic and transcendental) number quantities. Stages of psychological development can be exactly matched with these corresponding number qualities. Seen in this light the qualitative number system provides an amazingly precise map for ordering the entire spectrum. (The Number Paradigms: the Remarkable Complementarity of Mathematics and Psychology).
I am presently broadening my interests somewhat to look at fundamental symmetries in physical as well as psychological reality.
I realise that I have gone on at some length, but I believe that I have come up with an original approach which could well be of value in complementing your more wide ranging research.
By all accounts, Ken you are back writing more prolifically than before and pursuing ever more ambitious projects. Good luck with such endeavours in the future!