Here are my responses to your many interesting questions. They raise so many issues that it would really take a considerably longer posting to do them justice. However, I do hope you find some value in my comments.
Q. Beyond your
description of basis number forms, how far have you developed your model?
I would dispute that I am just describing
basis number forms. Though the very use of words such as "irrational",
"imaginary", "transcendental" and "transfinite" strongly suggest that there
are hidden transrational dimensions to numbers, mathematics clings to areduced
What I am attempting to explain - I will illustrate presently - is how each number type (in the complete number system) has complementary quantitative (mathematical) and qualitative (psychological) interpretations.
Whereas the former aspect is absolute and static, the latter is inherently dynamic and relative. This in itself is a major undertaking and - even within mathematics - could help to clarify many of the paradoxes surrounding numbers (esp. transfinite). However I am mainly interested in using these qualitative numbers to precisely order the various levels and stages of the psychological spectrum.
I have at this stage worked out the
spectrum in considerable detail - focusing mainly on the more advanced
levels and have completed two books on the subject. The first is more accessible
in its presentation (with the number system largely implicit). The second
involves the translation of all levels and stages explicitly in mathematical
The book I am presently working on extends the spectrum further to include physical reality. The thesis now is that physical and psychological reality are themselves complementary with an underlying structure that is mathematical (in this qualitative sense).
The scientific paradigm of the rational level of course separates physical and psychic worlds. However at all other levels a complementary relationship exists so that changing one changes the other. Thus for each "higher" level of psychological development a corresponding "lower" level of physical reality exists with the intuitive understanding of the "higher" interpreting the "lower" level.
The problem with modern physics is that so many of its finding are non intuitive (e.g. Superstring Theory). This is due to an attempt to read "lower" levels from the perspective of a rational perspective which is inappropriate. So in effect I am attempting to provide an appropriate framework to intuitively interpret reality (both physical and psychological) at all levels.
Q. Have you
worked out calculations that can be interpreted with meaning? How do you
apply your mathematics? To what?
Holistic mathematics involves a study
of the basic symmetry patterns inherent in dynamic relationships, and is
not primarily concerned with new calculations. Indirectly however it is
useful for suggesting new ways of tackling mathematical problems. (I have
found it very helpful in the study of prime numbers!) Also it has enabled
me put "the fire" into several important mathematical equations. Most importantly
it leads to a new form of mathematical "proof". When one accepts the complementarity
of quantitative and qualitative structures, then the truth of a formulation
in one mode necessarily establishes truth in the other. I have a strong
conviction that many problems in current mathematics can be "proven" through
formulating viable (complementary) qualitative interpretations.
(I have established an alternative "proof" for the infinitude of primes in this manner).
The application of my mathematics is - as I have stated - mainly in relation to clarifying the psychological spectrum. It can in applied in principle however to the integration of any dynamic system.
Q. How do you use it in your economics profession?
Economics as a scientific discipline is deeply flawed. It is based on an outdated Newtonian framework and employs many highly questionable assumptions. On two previous occasions I have attempted to develop an alternative dynamic structure. The many issues this raised led to deep involvement in other fields culminating in holistic mathematics.
I intend to return in the near future to this difficult task utilising the insights I have since acquired.
Q. How in practice
do you express the fundamental structure of each level of understanding
mathematically in more detail than your summary of forms of numbers?
This is central and will require a rather lengthy reply to indicate what is involved.
Ignoring for the moment pre-rational
states, I employ four main levels linear, circular, point and radial.
Each level is expressed in terms of directions of experience with the linear level being one directional (i.e. one dimensional). These directions as I stated in my postings are literally reduced expressions of unity derived from a dynamic interpretation of successive roots of 1.
We start with the unconscious which is dynamically neutral consisting of positive and negative charges. Consciousness then involves the separation of these complementary charges enabling the literal positing of phenomena in experience.
The task of the linear rational stages is the specialisation of conscious development. One views life in terms of just one direction, literally as a continual extension of activity adding to one's sum of knowledge, skills, money etc.
Thus the linear level emphasises the positive (conscious) direction of experience, which creates an imbalance in terms of the related unconscious.
For some - usually on the threshold
of adulthood - this neglect of the unconscious leads to an existential
crisis and a transition to the higher circular level.
Just as conscious rational development requires the activation of the positive pole,
unconscious (intuitive) development requires the activation of the complementary negative pole. This involves the dynamic negation or undoing of phenomena and is well documented in the mystical literature. It is painful and distressing involving considerable erosion of former experience. In Christian mysticism it is referred to as purgation. Terms
such as "learned ignorance" and "cloud of unknowing" have also been used. I refer to them as mirror structures, anti-structures or most simply negative structures.
In static mathematical terms when we combine a positive number with it's negative we get zero. Likewise here, when we combine the already developed positive direction (of rational experience) with its negative we get nothing. However this complementary concept of nothingness is inherently dynamic representing the pure potential for existence (i.e. intuition). St. John of the Cross who is the most prominent exponent of this type of experience literally refers to it's goal as nada (i.e. nothing).
So just as the linear level involves the explicit development of the positive, the transition from linear to circular involves the beginning of explicit development of the negative direction of experience.
The circular level (subtle) commences when one starts to combine these two directions simultaneously to give both positive and negative directions. This leads to the development of new intuitively based rational structures which I term irrational. This is not an arbitrary choice of words but rather directly complements the notion of "irrational" in mathematics.
Conventional reason is based on an
either/or logic based of separation of poles. Something is or is not absolutely.
However intuition is based on an alternative logic based on the dynamic
identity of poles.
Thus something both is and is not in this logic (i.e. has a relative existence).
Now in terms of the logic of reason intuition is deeply paradoxical. Indeed it is
irrational. This reflects the difficulty of translating a logic which is inherently dynamic and two dimensional in linear one dimensional terms.
A corresponding problem exists in
conventional mathematics when we try to obtain the square root of a number.
The result gives an irrational answer with both positive and negative values.
So one again we have an exact complementary correspondence as between (dynamic)
psychological and static mathematical operations.
The circular level involves the development therefore of irrational structures in both positive and negative directions. The positive direction (illumination) involves the outpouring of intuition and an attempt at conscious subtle translation of reality leading to inevitable rigidity and confusion. The negative direction (purgation) involves the undoing of this (limited) translation leading to the movement to progressively "higher" intuitive states.
My approach here is actually very
much in keeping with the standard Western tradition (with greater emphasis
on intellectual development). Thus Evelyn Underhill writes of
The Awakening of the Self (concrete irrational structures - positive direction), The Purgation of the Self (concrete irrational structures - negative direction), The Illumination of the Self (formal irrational structures - positive direction) and The Dark Night of the Soul (formal irrational structures - negative direction). If one successfully negotiates this level it leads close to a transcendent void - the nada, the nothingness, which is pure intuition concealed in darkness.
There is considerable misrepresentation - in my opinion - of the next level and this is where I feel I can make a valuable contribution. In the West it is normally left out entirely and treated as a continuation of the circular. By contrast in the East it is well documented (the causal realm) with it's spiritual intuitive characteristics very accurately described. However I find the treatment invariably adopted unduly passive, insufficiently existential and not truly human. There is a failure to address key psychological dynamics which ultimately reflects a faulty understanding of the role of repression in experience (a point indeed which Ken has made).
Let us go back for a moment to the earlier linear level. When one separates the two poles of understanding to posit phenomena (externally) in consciousness, a simultaneous reverse movement takes place by which the unconscious is to some degree blocked off or negated (i.e. repressed). Thus the very act of consciousness therefore necessarily entails repression. The irony is that when the positive direction is sharply differentiated there is such significant unconscious blocking that the validity of this understanding is not even questioned (as with conventional science). The unconscious then acts to provide a supporting framework of assumptions and prejudices of which one remains completely unaware.
Now the key problem of the circular level is that the movement to "higher" states of consciousness still involves subtle rational control and an inevitable build up of repression. Thus the arrival at the transcendent void, generally coincides with a severe emotional crisis due to the growing concentration of unexpressed primitive elements in the unconscious.
The transition to the point level involves a new remarkable development by which the unconscious starts to freely project itself in an involuntary manner. (We have here "dark night" radiation in a manner which remarkably parallels Stephen Hawking's view of black hole radiation). This leads to the development of what I term "imaginary" structures. Again I am using this word not in a merely literary or figurative sense but rather in a precise mathematical fashion. The repression represents the forgotten and lost negative direction of understanding. Now the unconscious is inherently two dimensional (i.e. two directional). However to express itself this repressed unconscious must become embodied in phenomenal (i. e. one directional form). In other words a two dimensional quality (which is negative) is translated in (reduced) one dimensional terms. This exactly complements the mathematical notion of an imaginary number which results from attempting to extract the square root of a negative quantity. Thus once more the "imaginary" quantity (statically understood) in mathematics is complemented by an "imaginary" quality (dynamically understood) in transpersonal psychology.
Virtual particles in physics - which represent spontaneous projections or emissions of matter from a fundamental ground - are equally "imaginary" in this dynamic mathematical sense. Thus reality now must be structured in "complex" rather than "real" terms. (Interestingly these very ideas are finding their way into quantum physics!)
The transition from circular to point
level involves a remarkable inversion. Formerly the world was understood
in conscious terms (as out there). Now the world is understood in unconscious
terms (as in here as one's own psyche) with only an indirect meaning (as
out there) through unconscious projection. This leads to the birth of "imaginary"
structures in both positive and negative directions. Because of the considerable
erosion of former conscious structures one now is increasingly exposed
to one's shadow self.
The point level now involves the attempted reconciliation of both (remaining) "real" and "imaginary" structures in both positive and negative directions. The personality increasingly splits into an even "higher" (real) self involving extremely subtle rational translations and a "lower"(imaginary) self involving increasing exposure to the primitive fantasies and projections of the repressed unconscious.
This leads to the birth - in both selves - of what I term transcendental structures. Again the dynamic psychological interpretation exactly complements the mathematical meaning. The most famous transcendental number in mathematics is pi. which reflects the simple relationship of line and circle (i.e. ratio of circumference to line diameter). These transcendental structures equally involve the relationship of line and circle. One now increasingly views reality in terms of the relationship between the rational (linear) and holographic (circular) paradigms.
These stages again go through concrete
and formal stages. At the concrete stages one interprets reality in psycho-physical
terms readily identifying parallels as between physical and psychological
reality. At the formal stages one sees reality in psycho-mathematical terms
now identifying even deeper parallels as between mathematical (rational)
and psychological (intuitive) understanding.
(My contention that integral mathematics involves complementary static and dynamic aspects is a reflection therefore of this formal transcendent stage).
So just as Piagetís concrete and
formal operational stages have universal application at the linear level,
these corresponding stages have - in my mapping of the spectrum - equal
validity at the point (causal) level.
I have gone into these concrete and formal stages in great detail in my books (identifying in each case eight distinct sub-stages). I feel that this demonstrates the power holistic mathematics in helping to clearly identify important stages, which seem to be entirely missing in other maps.
The corresponding micro "imaginary" lower level leads to ever more intimate and deep rooted projections. One gradually learns to see each all phenomenal projection as an embodiment of holistic (unconscious) desire. When this is achieved the involuntary nature of projection - which always reflects repression - ceases.
When I say that objects have both "real" and "imaginary" aspects this means that all objects serve a dual identity both as actual (individual) entities and as potential (collective) symbols of a universal order. With instinctive behaviour, unconscious desire expresses itself involuntary in direct attachment to objects making its potential aspect extremely difficult to recognise. That is why intense exposure to the primitive self takes place at the point level where one has sufficient maturity to deal with the highly intimate sensations evoked. When eventually projections cease an extremely pure form of intuition remains revealing the true "imaginary" nature of all objects.
Thus when Blake saw "a world in a grain of sand" he was understanding both "real" and "imaginary" aspects of the object. While recognising its actual individual "real" existence, he was equally recognising its potential collective "imaginary" existence.
With the erosion of all attachment
to phenomena associated with both the "lower" physical self and the "higher"
spiritual self one arrives at the point or centre which represents the
pure activity of will (which co-ordinates both). This is a new void with
two related aspects (i.e. "real" transcendent and "imaginary" immanent).
It represents pure process (completely without structure). However it has a fascinating interpretation in terms of our circular diagram. The diagonal lines (rays) in the complex plane bounded by the circle have equal "real" and "imaginary" co-ordinates and therefore a numerical value equal to zero. These diagonal lines can be used to represent physical light). Equally they can be used to represent spiritual light (i.e. pure intuition). All finite measurements are relative to the speed of light. Light itself does not travel in time and therefore - in terms of itself - eludes measurement. It can only be measured indirectly through itís interaction with matter.
It is exactly similar with spiritual light which only obtains finite meaning through interaction with phenomena.
Thus the transition from point to
radial levels involves the purely infinite in a plenum-void. Nothingness
(in terms of physical phenomena) is unity in terms of spiritual meaning.
However because the infinite is an inherently intuitive notion in terms
of mathematical quantities it can only be given a highly reduced interpretation
as transfinite numbers.
We are now into the radial level
(non-dual reality) where another remarkable translation takes place. The
infinite - as pure intuition - now starts to express itself as finite reality
where it obtains a reduced one dimensional interpretation. Now - as stated
- mathematically the co-ordinates of the diagonal lines (representing intuition)
have equal "real" and "imaginary" values and represent the additional four
complex roots (obtained from taking the eight roots of unity). Thus in
terms of expressing the interaction of spiritual light with phenomena we
can say that it has equal "real" and "imaginary" aspects. What this simply
entails psychologically is that we now have the complete harmony of cognitive
and affective structures (external and internal) which are "real" and "imaginary"
with respect to each other. This could be similarly expressed in Jungian
terms as the successful differentiation of all four functions. (Indeed
Jung was undoubtedly searching for a satisfactory psycho-mathematical explanation
of his functions and came very close to the true number definition i.e.
"complex" with his identification of two rational and two irrational functions).
Thus the radial level now involves
eight directions. We have finite "real" and "imaginary" aspects in the
interaction of cognitive and affective experience. These have additional
"positive" and "negative" aspects in the interaction of external and internal
The background spiritual light then provides the four complex directions necessary to "see" the finite phenomena in each case.
This thinking explains simply the paradox of the nature of physical light. Physical light - like spiritual light - is inherently two directional and infinite. However in (reduced) interactions with matter it necessarily expresses itself in complementary "real" and "imaginary" fashion (i.e. "particle" and "wave" aspects).
To conclude this answer let us go back to the pre-rational levels which I have not dealt with.
In my approach there is direct complementarity
as between pre-linear and post-linear levels. I sometimes think that Ken
overstates his pre-trans fallacy. Whereas it is perfectly true that transrational
states should not be confused with pre-rational, there is however a remarkable
structural complementarity as between earliest and latest levels whereby
the implicit potential of the earlier level becomes explicit at the later
My first pre-rational level is the binary which represents the total confusion of all eight directions. It is thus an undifferentiated qualitative binary state where physical unity is indistinguishable from psychological nothingness (i.e. no experience at all). Now in complementary fashion, the highest level (radial) can be described as a fully differentiated binary state, a state of pure transformation where unity and nothingness coincide (the plenum-void or dazzling darkness).
The first pre-rational transition then involves implicit separation in experience of actual reality from potential ground (i.e. finite and infinite directions)
The next pre-rational level, still
using number terminology is the prime (or primitive) where remaining four
directions are confused. A prime number - by definition - has no factors
(i.e. no dimensional characteristics). This is literally true here of psychological
behaviour where the infant is still unable to provide a dimensional framework
This level complements the point level where the four directions are fully differentiated.
The next transition involves the gradual implicit separation of "real" objects and "imaginary" dimensions in experience (sensori-motor stages).
The third pre-rational level, which I call the natural, involves a psychological multiplication process through which objects (now placed within dimensions) readily appear in experience. This again complements the natural number system which is obtained from a multiplication process involving prime numbers. We still however have remaining confusion of the two directions of experience (complementing the circular post-linear level).
The final pre-rational transition
involves the implicit differentiation of these directions of experience
which brings us up to the linear level.
Q. How do you translate indirectly the holistic aspect of integral mathematics which is directly intuitive in precise mathematical terms?
I think I have demonstrated this in the previous lengthy answer.
Q. Do you work out formulas and interpret them?
I have a deep conviction that the really fundamental questions have simple answers. My whole system can be represented by a simple equation x (raised to the power of n) = 1.
When n = 1, we have the linear level, n = 2, we have the circular level; n = 4 equates with the point level and finally n = 8 yields the radial. However the decisive difference in my approach is that I give these simple equations a qualitative dynamic interpretation which is circular rather than linear. The more holistic an approach to an overall system, the more circular the approach should become so that ultimately one is left with totally paradoxical relationships. Once again my contention is that a complete interpretation of the system requires paradoxical symmetries that are horizontal (positive and negative), vertical (real and imaginary), and diagonal (finite and transfinite). Thus I believe the Theory of Everything (TOE) - which is the great quest of physics - at its most fundamental requires nothing more than the application of these symmetries. Equally at the complementary psychological level the Theory of Everything simply involves a matching complementary set of symmetries (which unfold at the radial level).
However these symmetries - because they are totally paradoxical - really point to that pure experience of mystical union which is totally ineffable and mysterious.
you tried to translate or fully map out Ken Wilberís spectrum of consciousness
in your terms? What does it look like.
There is a fundamental similarity in levels in both spectrums (though of course I accept as Ken states that the final "level" is not really a level). Thus I believe my mathematical interpretation in terms of directions can give a new insight into Kenís spectrum.
However the translation of the stages of the levels is very different especially at the "higher" levels.
Perhaps it is an unfair criticism of Kenís spectrum but there seems to me to be a certain discontinuity in that it is very Western in emphasis on personal development and Eastern on transpersonal development. Though lacking the intuitive clarity of Eastern mysticism I personally find the Western variety - at its best - more dynamic and attractive. (This preference however could reflect insufficient knowledge on my part of the various traditions which Ken has in abundance).
The main difference is that I have been seeking all along a thoroughly contemporary approach. My approach is somewhat different to Ken. I am mainly seeking answers to problems not dealt with in existing accounts. Ken on the other hand puts large store on extensive and meticulous research leading to a highly articulate synthesis of existing traditions. Because of Kenís huge - and richly deserved - authority in the field there is a danger that his presentation of the spectrum becomes the "conventional wisdom". My reason for entering this debate is to show that radically different translations are possible (and desirable) with all sorts of useful applications never before associated with mysticism.
Q. What do
you make of more sophisticated mathematical functions?
It is an axiom of my approach that every mathematical function has in principle a qualitative dynamic interpretation applicable to both physics and psychology.
However we have to learn to walk before running. The important thing at this early stage is to translate very basic relationships (which is what I am doing).
how do you apply your model?
I think I have already answered that.
Points of Detail
Q. What do you make of the holon? How is it represented?
I have no real problem with the term (though each holon could equally be classed as a parton!) I do however have some reservations about Kenís use of the notion. When Ken states that all holons have four facets, I agree that this true at a certain level (i.e. the point level where understanding is four directional). It is not however strictly true at the other levels.
Let me illustrate with reference to a particularly interesting holon (viz. a number).
At the linear level of rational understanding , which is a solely positive direction, this has one facet. In other words a number is understood in static objective terms. It has an absolute independent existence not affected through interaction with the mind.
This is very much the conventional
interpretation of a number.
At the circular level of irrational (intuitive) understanding, understanding subtly changes. A number now has two directions (positive and negative). In other words a dynamic mind-matter interaction takes place of (objective) number to (subjective) self and (subjective) self to (objective) number. Because the interaction of both poles is involved, a number is understood as having a relative rather than absolute meaning.
(To generalise, correctly understood
all mathematical relationships should be understood in strictly relative
terms at this level).
At the point level of transcendental understanding (rational and intuitive), understanding is even more subtle and four directional. One now appreciates that a number (perception) only has meaning in the context of a background number concept. Now the number concept strictly is an infinite collective notion (applying to potentially all numbers). As we have seen when a symbol has a potential - as opposed to actual - meaning then it is correctly defined mathematically in "imaginary" terms. Thus properly speaking the relationship between the (individual) number and its (collective) concept is a relationship as between what is (relatively) "real" and "imaginary". Generalising further the relationship between any perception and corresponding concept is mathematically complex in this dynamic sense.
The clear implication for science is that - at this level - we move from the "real" to the "complex" world. The reason we believe we are in a "real" world is due to the consistent way we reduce concepts to perceptions (understanding them in actual rather than potential terms).
To sum up therefore at this level, a number has four directions of real-imaginary (interaction of number perception with concept) and positive-negative (interaction in both cases of object and subject).
At the radial level of pure spirit (what connects both reason and intuition), a number is now understood as having eight directions. The first four are the finite directions already identified. However in addition we include four purely spiritual directions which correspond to the light necessary to (literally) see the finite connections. Without this background spiritual light, no understanding - at any level would be possible. However at other levels it is not explicitly included in interpretation.
This spiritual light can be given a reduced finite interpretation as four complex directions (which complement the four finite directions). In each case the complex direction has an equal "real" and" imaginary" component. This can be interpreted simply as saying that in the dynamics of experience at the radial level, where all eight directions are involved cognitive and affective experience must be fully harmonised.
Q. In your
point level why do you equate "real" with conscious and "imaginary" with
I hope I have already explained this? Our very definitions of reality are based on conscious understanding. What is real is actual (in conscious terms).
The "imaginary" by contrast relates to what is potential (in unconscious terms). However the unconscious can only reveal itself indirectly though attaching itself to phenomenal objects which then carry a potential rather than actual meaning. So though the "imaginary" relates primarily to the unconscious it always has an indirect conscious interpretation.
In stressing that all objects have "real" and "imaginary" directions, I am simply saying that they serve a dual function in being both actual and "real" (conscious interpretation) and potential and "imaginary" (holistic symbols or archetypes indirectly expressive of unconscious meaning).
Q. How do real
and imaginary numbers equate as expressions of individual and collective?
I have answered this. The "real" relates to conscious understanding which is basically a process of differentiation leading to the creation of individual units in experience.
The "imaginary" - though given indirect conscious expression - relates directly to unconscious understanding which is basically a process of integration leading to holistic or collective understanding.
Q. Is the following summary of your analogies - regarding circular point and radial levels - correct?
You have raised a number of points here which I have already attempted to explain.
I take issue with the your use of
the word "analogies". The mathematical terms I am using are not intended
to be merely analogous. What I am saying is that every mathematical relationship
(conventionally understood) has a complementary dynamic interpretation
(which is equally valid). Put another way mathematics has both "particle"
and "wave" aspects, with at present only the conventional "particle" aspect
recognised. Of course this new and yet unfamiliar "wave" interpretation
of numbers and relationships initially will appear very strange. However
it is important to be clear on the basic point.
Q. Is there
a reference to good/bad, to irrationality being seen as a negative trait?
Again, I am using irrational and
negative in a carefully defined mathematical and not in the conventional
sense. Thus in my terminology, irrational means a logic based on complementarity
of opposites (two directional). Negative means the dynamic undoing of consciousness.
The logic of the four directional complex plane could be expressed simply as the interaction of rational logic (based on either/or) and intuitive (irrational) logic of the complementarity of opposites (based on both/and).
Alternatively it can be expressed as the interaction of linear and circular logic. (Holistic mathematics represents this form of logic).
The logic of the eight directional approach would represent the integration of linear and circular logic in an ineffable fashion. Any possessive or self conscious attempt at understanding now goes. In other words understanding becomes a spontaneous expression of the spirit.
Finite and transfinite reality could
be translated as the interpenetration in experience of the phenomenal world
(finite) with spirit (transfinite).
Q. Diagonals: in geometrical terms where do you place them on a spatial diagram?
I am not sure if a posted diagram would be recognised by the web, so I will have to try and explain.
Imagine a circular with a horizontal line (real axis) and vertical line (imaginary axis)both drawn through the centre giving the form of a cross. Now the diagonal lines are drawn threw the centre in each direction so that any point on these lines is at an equal distance from horizontal and vertical lines.
Q. In making
your analogy to Ken Wilberís "Four Corners of the Kosmos" you seem to be
dropping the diagonal third level of your analysis. Why? Is it impossible
Is Ken Wilber missing one more level in his model?
I am very impressed by your observation here. I will try and clear up the apparent inconsistency.
The complex number system is indeed
the most comprehensive in mathematics.
Now the diagonal lines that I use in the radial level represent an important special case of the complex number system (where real and imaginary co-ordinates are equal). These lead to null lines (of zero length) which have been used in physics to represent physical light (and that I use to represent spiritual light).
The best way of answering this is exactly as Ken himself has suggested that - what I call - the radial level is not a level as such but the foundation of all levels. In other words it represents the ultimate refinement of the other levels, and used to harmoniously integrate these levels. Thus the radial "level" is the linear, circular and point levels, now mutually interpenetrating. However until we reach it, the radial level will appear as another level. In that sense it think it is valid enough to refer to it in separate terms.
The eight directional approach still of course holds here.
Q. It seems that your circular level would already bring on intuition, direct knowing etc. Can you explain?
You are quite right. It does. All the "higher" levels require intuition. However the nature of this subtly changes with each level. He circular level represents a specialised development of the transcendent otherworldly variety. This is kept under refined rational control leading in its pure form to what is sometimes referred to in the Western tradition as dark contemplation (symbolised by night).
At the point level, due to spontaneous projections, the affective dimension is included more in intuitive experience, leading to dim contemplation (symbolised by dawn).
At the radial level, cognitive and affective directions are harmonised leading to the purest form of intuition what is both a a contemplation of immanence and transcendence . This simply means that one now has the intuitive capacity to deal with the world and spirit in equal manner.
Q. Your paragraph on light and null lines need some explanation.
Now the diagonal line represents the hypotenuse of a right angled triangle with the two other lines representing real and imaginary lines of equal length. Using the Pythagorean Theorem we obtain the length of this diagonal line by squaring the two other which cancel out to give a value of zero. Thus these diagonal lines have no length.
They are very important in Relativity Physics as they represent light - which in terms of itself has no measurement. These lines equally - in psychological terms - can be used to represent spiritual light.
Q. You refer
to Ken Wilberís theory of holons Ö why not call this holarchic mathematics
to avoid controversy about the word holisticÖ
I do not use the holon terminology in my own writings. I used it in my postings to provide a reference point for readers familiar with Kenís work.
I have an open mind about the terminology for my mathematics. To my great amazement when I carried out a search on the Internet, I found no one already using the term Holistic Mathematics and I personally liked it. I have used other names - holo-mathematics, holomatics, transrational mathematics etc. so I may well rename it again in the future.
The word holarchic is not in common usage and seems to me too restrictive. Most people are familiar with the word "holistic" even if it can be used in different ways.
Q. Have you
tried mathematically to formulate the meaning of Kenís tenets?
No, not as yet, though I have little
doubt that it can be done. However I would have to be careful to avoid
simply reducing his approach -which is temperamentally different - simply
to my own.
In conclusion, Stella, I very much
enjoyed answering your thoughtful queries and found them helpful in clarifying
my own thoughts.