Holistic Mathematics 3 - Part B

Q.  We have now reached the radial level. Can you explain what this entails in holistic mathematical terms?

PC  Yes, this is perhaps the key question of all. How does (finite) phenomenal reality emerge from the (transfinite) fundamental void?

Remarkably, our simple mathematical approach can give a compelling explanation.
We have already seen that the diagonal or null lines, which in mathematical terms have a magnitude of zero, precisely represent - in holistic terms - the (transfinite) void.
However - as these diagonal lines lie in the two dimensional complex plane - their intersection with the circumference of the circle can be represented as the sum of two coordinates, representing the distance of the points from both horizontal (real) and vertical (imaginary) axes. These complex coordinates i.e. 1+ i, 1- i, -1 + i, -1 - i (in each case multiplied by the reciprocal of the square root of 2) in fact represent the four complex solutions (in addition to the two real and two imaginary) obtained from the eight roots of unity.

Thus our transfinite diagonal lines (initially involving a two dimensional transformation using the Pythagorean Theorem), alternatively can be explained in one dimensional finite terms as the sum of equal real and imaginary components.

In psychological terms the transfinite void (representing the purely intuitive two dimensional unconscious), can be alternatively explained in reduced one dimensional conscious rational terms as the equal combination of real (quantitative) and imaginary (qualitative) experience. Thus this void - representing the supersymmetry of spiritual "forces" as pure potential, is continually actualised - in reduced asymmetrical terms - as real (cognitive) and imaginary (affective) experience.

In complementary physical terms, the transfinite void (representing the empty implicit ground of reality) can be expressed in reduced finite terms as the equal combination of real (quantitative) and imaginary (qualitative) aspects. Thus again this void - representing the supersymmetry of physical forces as pure potential, is continually actualised - in a world of broken symmetry - as (real) matter and (imaginary) dimensions.

Thus both physically and psychologically the transfinite empty void (representing pure potential) is continually actualised in reduced finite terms (quantitative and qualitative).

This reduced actuality is likewise continually transformed in some measure (through both the processes of evolution and involution) back to that empty potential void.

Q.  So one can explain both physical and spiritual "forces" alternatively in simple or (reduced) "complex" terms. Is this the case?

PC  Yes, this is precisely the case. We saw in our earlier discussion of (physical) light
its utterly mysterious true nature. Though providing the standard by which space and time measurements take place, in terms of itself it has no finite meaning in space and time (In other words its true nature is transfinite).

However light (as with all of the forces) has an alternative (reduced) finite interpretation (through its interaction with phenomena). Now this reduced finite translation involves
equal "real" and "imaginary" components. This at once provides a key to the understanding of the wave-particle duality of light.

The "real" component of light is most easily understood in terms of its particle aspect
(where light can be identified at a specific location in space as photons).

The "imaginary" component of light is by contrast represented by its wave aspect (whereby light travels in a generalised fashion through time).

Put alternatively in reduced finite terms, light exhibits both quantitative (particle) and qualitative (wave) aspects. In terms of my holistic mathematics, this is exactly what we would expect to find.

Now I treat all four forces in the same fashion. In holistic mathematical terms the electromagnetic and weak forces serve as (horizontal) opposites (positive and negative with respect to each other). The gravitational and strong forces likewise serve as horizontal opposites. These related pairings in turn serve as (vertical) opposites (with respect to each other) and can be converted into each other through a "real" to "imaginary" (or "imaginary" to "real" transformation) Thus in the supersymmetry of the void all the forces are diagonally complementary. Each of the four forces can be explained in finite terms by one of the complex solutions representing the coordinates of the diagonal lines in each quadrant. Each therefore will manifest both "real" (particle) and "imaginary" (wave) aspects. (The fact that it has not been yet experimentally possible to validate this contention for all four forces, in no way weakens my assertion).

Of course this thinking equally applies to the "spiritual" forces.

Once again these are represented in mystical terms as the contemplation of immanence and transcendence both of which can be given - in horizontal terms - an external (physical) or internal (psychological) interpretation.

Now these spiritual "forces" in their own terms are transfinite and absolute (providing the standard through which relative phenomena are measured). However they again have a reduced finite interpretation with both "real" and "imaginary" components.

Thus spiritual light has a "real" (quantitative) immanent aspect and an "imaginary" (qualitative) transcendent aspect. The same logically applies to spiritual "gravity".

As we have seen in relation to the dark night there is a very close relationship as between spiritual "gravity" and spiritual light. Thus what is internalised as gravity is released in "imaginary" form as radiation (light).

Thus in holistic mathematical terms if x(1 - i) (where x is the reciprocal of the square root of 2) represents the spiritual electromagnetic force then to obtain gravity we transform "real" to "imaginary" ("imaginary" to "real") and obtain negative signs. This gives us x(-1 + i). Thus "spiritual" gravity can be represented as diagonally opposite to the electromagnetic force. Likewise the other two forces (strong and weak) are diagonally opposite.

Of course if this applies to the "spiritual" forces it equally applies - in complementary fashion - to "physical" forces. In holistic mathematical terms therefore the electromagnetic and the gravitational force can be represented as physically symmetric in diagonal "complex" terms. Likewise the weak and strong forces are diagonally symmetric.

The electromagnetic and weak, and gravitational and strong are alternatively both horizontally and vertically symmetric with respect to each other.

To conclude this section the "forces" in their ultimate symmetrical state - physical and spiritual - can be represented as both absolutely simple (i.e. as transfinite null lines) and absolutely "complex" (with equal finite "real" and "imaginary" components).

So here in holistic mathematical terms simplicity and complexity are understood quite clearly as two expressions of the same reality. This interpenetration of simplicity and complexity characterises the radial level.

Q.  Can you now outline the overall structure of the radial level in holistic mathematical terms?

PC  As we have seen earlier, each of the transitions between levels involves a major development which has a precise holistic mathematical interpretation.

The linear level involves the specialisation of the "real" positive (conscious) direction of experience.

The transition from linear to circular levels involves the specialised development of the complementary negative (unconscious) direction.

The circular level is then defined in terms of the development of structures with both positive and negative aspects.

The transition from circular to point levels involves the specialised development of the "imaginary" (projected) conscious direction of experience.

The point level then involves the (additional) development of structures with both real and imaginary aspects.

The transition from point to radial levels involves the specialised development of the (simple) transfinite aspect of experience. Sometimes misleadingly - esp. in Eastern spirituality - the arrival at this simple non-dual experience of reality is represented as the completion of the spiritual journey. True reality is identified here solely with the hidden underlying symmetry implicit in all (manifest) phenomena.

However a complete expression of reality - more typical of Western spirituality - involves the journey into radial level with the (additional) development of structures having both finite (actual) and transfinite (potential) aspects. This return to finite reality arises from the realisation that what is absolutely simple (in intuitive transfinite terms) is equally absolutely "complex" (in rational finite terms). Thus the recognition of the complementary rational pole of experience entails the rebirth of finite reality and the world of broken symmetry.

Overemphasis on the merely rational expression of experience, confines one unduly to the finite asymmetrical world (consequently blotting out the hidden transfinite domain of perfect symmetry). This is the position which characterises Western science.

However overemphasis on the merely intuitive expression of reality, leads to over absorption in this hidden transfinite ground in an unduly passive experience (blotting out the world of finite phenomena). This is the position which too often characterises the emphasis of Eastern spirituality.

Clearly truly comprehensive experience of reality implies equal attention to both rational and intuitive poles at the radial level. Remarkably holistic mathematics can precisely demonstrate the simultaneous simple and "complex" nature of this level.

So in many ways the radial level - far from representing the final goal of spiritual experience represents a brand new beginning. In the light of spiritual realisation one is now able to return - in varying capacities - to the world of broken symmetry offering a healing presence.

Thus the radial level is properly characterised by the increasing dynamic interaction as between the transfinite and finite, emptiness and form, the simple and the "complex", symmetry and asymmetry.

Q.  Can you now outline the stages of the radial level.

PC  I divide the radial level into two sub-levels.

The first sub-level involves the slow emergence from the transfinite void and the gradual rebirth of "transformed" finite reality. However for some considerable time experience may suffer from what I call "process split" (where finite and transfinite remain to some degree separated). In other words involvement in actual phenomenal affairs does not yet serve as an adequate expression of one’s spiritual vision.

The second sub-level is more dynamic and active where finally one finds a satisfactory
worldly expression of deep spiritual convictions.
Though secondary personality characteristics may vary greatly, people living at the radial level are essentially centroverts (i.e. primarily centred on God).

Of course the best known exponents of this final phase are the great spiritual leaders who with superhuman energy and endurance transform the face of history.
Others - though less known - exercise a profound influence for good on those whom they encounter. Finally - as at all levels - there are many who are largely anonymous, with their intense commitment remaining a secret to others (and perhaps even to themselves).

However it is important to avoid any elitist notions in terms of the radial level. All ego comparisons with others in terms of "better" and "worse", "higher" and "lower" are only true in a relative sense. The predominant attitude of the radial level - which reflects a cosmic based personality - is one of radical equality (in horizontal, vertical and diagonal terms).

In holistic mathematical terminology the radial level is eight directional. Thus we have the successful differentiation (with consequent integration) of positive (external) and negative (internal) experience; of real (quantitative) and imaginary (qualitative) experience, of finite (actual) and transfinite (potential) experience.,

We can deal with both radial sub-levels in terms of spiritual, intellectual and emotional experience which are very closely interrelated.

Spiritual involvement involves both transcendent (masculine) and immanent (feminine) poles. On the one hand there is the discovery that divine identity (inherent in every person). Also and equally important one is at last free to be human without masks or pretensions. So human and divine identity now coincide.

Cognitive intellectual development is of course closely related with the spirit.
We have now the unfolding of a dynamic "complex" paradigm in understanding. A diagonal form of complementarity is at work. At a local immediate level the "real" analytical aspect of understanding is restored; at the global level the "imaginary" holistic aspect predominates. The task of the radial level is to progressively integrate both aspects
until they are no longer distinguishable.

Affective emotional development initially remains polarised to some degree. Again at an immediate personal level one feels intimately loved - perhaps for the first time - in an unconditional fashion; at a global level a more detached platonic awareness initially predominates.

It may take many years of ongoing development before these diagonal poles are properly harmonised.

Thus the first sub-level of the radial level involves increasing immersion in phenomenal reality (which is now reborn in transformed fashion). At both spiritual, cognitive and affective levels, however, initial difficulty will be experienced in properly harmonising finite (partial) with transfinite (holistic) experience.

The second sub-level involves is highly dynamic representing the supreme manifestation
of integrated experience. Here transfinite and finite aspects approach equal balance (the marriage of contemplation with activity). Though the manner of expression is unique to each individual, a satisfactory worldly expression of one’s deepest spiritual desires is at last obtained. Though this brings joy and fulfilment, equally it entails much suffering through growing compassionate involvement with the problems of the world.

Q.  What is the relevance of the radial level for mathematics.

PC  Once again each level is characterised by a distinctive mathematical approach.

The linear level is characterised by the rational quantitative approach (Standard Mathematics) and is analytical in nature.

The circular level is characterised by the intuitive qualitative approach (Holistic Mathematics 1) and is synthetic in nature.

The point level is characterised more subtly by the relationship as between the rational quantitative and intuitive holistic approaches (Holistic Mathematics 2).

Thus each level involves the specialisation of a particular mathematical system.
The radial level now involves the gradual integration of these mathematical approaches
(already successfully differentiated at the other levels). It therefore involves the mutual interaction of all three systems with the relative independence of each system.

I refer to this as Holistic Mathematics 3. Now this terminology might seem misleading as increasing attention is likewise paid now to analytical mathematics. However full integration is not obtained immediately and the qualitative holistic element is likely to remain prominent for some time.

I do recognise a final idealised dynamic state of mathematical understanding where analytical and holistic aspects are highly integrated. This can be referred to as Comprehensive Mathematics.

One of the earlier mathematical tasks of the radial level is the formulation of "A Theory of Everything" which represents an extreme in terms of a purely holistic understanding of the fundamental ground of reality (non-dual reality). (I will deal with this in detail in my next posting!)

Subsequent mathematical development involves the study of the relationship as between the underlying fundamental ground and the now (reborn) world of finite phenomena. In other words it involves the mathematical study of the relationship between symmetry and asymmetry.

The proper investigation of this symmetrical-asymmetrical reality involves therefore a "complex" rational paradigm. This comprises a "real" analytical and an "imaginary" holistic aspect. Thus Holistic Mathematics is now seen in its appropriate context as the "imaginary" component of the "complex" rational paradigm.

Q.  Can you say a little more about the nature of this "complex" paradigm.

PC  One of the great weaknesses of the standard "real" approach is that because it lacks an appropriate holistic framework, it can lead to much naivety where fundamental deep questions regarding the nature of reality are involved. Indeed most of the key issues in mathematics and physics are really philosophical. Holistic Mathematics provides the ideal vehicle for dealing with such philosophical issues. It also provides the appropriate intuition for interpreting mathematical hypotheses. Superstring Theory is an interesting example of where Holistic Mathematics can make a valuable contribution by giving it a meaningful intuitive explanation. (As this illustrates so well Holistic Mathematics in action, I may return to it in a future posting).

One of the interesting philosophical findings of Holistic Mathematics is that a consistent Theory of Everything (TOE) must be an expression of the empty transfinite ground of reality, and can only be adequately expressed in the language of Holistic as opposed to Standard Mathematics.

Also, when one deals with phenomenal reality, a comprehensive approach requires a "complex" paradigm comprising both "real" analytical and "imaginary" holistic aspects.

This approach is subject to the uncertainty principle. Any attempt to maximise the "real" analytical approach leads to increased fuzziness in terms of the complementary "imaginary" holistic aspect. In other words we run into paradoxes and inconsistencies which cannot be explained in "real" analytical terms. (This very problem lies at the root of the continuing difficulty in terms of reconciling Quantum Mechanics with the Theory of Relativity).

By contrast any attempt to maximise the "imaginary" holistic approach leads to a purely circular qualitative approach (thus eliminating the scope for quantitative analysis).

Thus we have two extremes. The "real" analytical component is suited to the study of the quantitative "parts" (in isolation from the qualitative whole). The "imaginary" holistic component is geared to the study of the qualitative whole (in isolation from the quantitative "parts").

Thus any truly overall understanding i.e. (a TOE) must be couched in holistic mathematical terms (and expressed in purely circular paradoxical rational fashion).
Any truly partial limited understanding i.e. (a TOS or Theory of Something) must be couched in analytical mathematical terms (in standard linear non-paradoxical fashion).

Thus the use of standard mathematics by definition will always lead to partial explanations of reality. Therefore when we get into the difficult area of mathematically interpreting the dynamic interpenetration of "parts" and "wholes" we require a "complex" paradigm.

Again the "real" analytical component will be used to formulate quantitative theories. The "imaginary" holistic component will provide a background intuitive framework to interpret such theories and establish their scope and limitations. Thus standard mathematical and enlightened philosophical understanding of reality now go hand in hand.

Q.  Can you go into this relationship between the "real" and "imaginary" aspects of mathematics in more detail.

PC  Remember that the linear level (which provides the paradigm for "real" mathematics) is strictly one directional.

Now as we have seen reality is more subtle. At the circular level it becomes dynamically two- directional with the complementarity of positive (external) and negative (internal) directions. At the point level reality now becomes four directional with the additional complementarity of "real" (quantitative) and "imaginary" (qualitative) aspects. At the radial level it becomes eight directional with additional transfinite as well as finite directions.

For the purposes of exposition we will restrict ourselves here to four rather than eight directions.

Thus the one directional approach of the linear level always requires the positing of one of these directions absolutely (to the exclusion of three other related aspects).

Now this in fact is all very relevant to understanding Ken Wilber’s four quadrant approach. As Ken indicates we can have theories which emphasise the external and internal aspects of experience (separately). Also we can have theories which emphasise the individual and collective aspects of experience (separately).

By the very nature of the linear level there will be a tendency to emphasise one partial explanation as the whole truth (thus excluding equally valid explanations).

Now Ken of course quite rightly emphasises the need to concentrate on all four facets to achieve a more balanced understanding. However by definition - as it is not suited to the synthesis of opposing directions of experience - this can only be done in a very limited fashion at the linear level. At best one will form a differentiated understanding of four (rather than one) alternative facets. But these will not be properly integrated in experience.

At the circular level one forms the understanding that positive (external) and negative (internal) aspects of experience are dynamically complementary (in horizontal terms). In other words they are ultimately integrated as the same unified experience. This leads to heightened intuitive awareness. One is now enabled to form one directional understanding (separately) of each facet at the linear (rational) level while dynamically unifying both aspects at the circular (intuitive) level.

At the point level one forms the additional understanding that "real" (individual) and "imaginary" (collective) understanding are dynamically complementary (in vertical terms).
Again they are ultimately unified in dynamic intuitive fashion at this level. One is now enabled to form one directional understanding (separately) of all four facets at the linear (rational) level while dynamically integrating all aspects at the ("higher" intuitive) point level.

Thus the appropriate rational differentiation of each facet of experience is properly the task of the linear level. However the appropriate integration of these facets is the task of the "higher" intuitive levels. That is why I maintain that my holistic mathematical approach (which is designed to integrate reason with intuition) provides the appropriate context to interpret Ken’s four quadrant approach. Properly understood - in such holistic mathematical terms - the four quadrants relate to the complex number axes (interpreted in qualitative terms).

(Of course a full interpretation of holons involves the relationship as between finite (phenomenal) and transfinite (formless) aspects and is eight directional requiring the specialised understanding of the radial level).

Q.  Have you anything more to say here before we leave this topic?

PC  Using a four directional approach, the standard scientific paradigm is built on the asymmetrical interpretation of dimensions (3 and 1) i.e. three dimensions of space and one of time.

The holistic scientific paradigm is based on the - more mathematically correct - symmetric interpretation of dimensions (2 and 2) i.e. two dimensions of space and two of time each having positive and negative directions. Indeed we can alternatively define reality here in terms of 4 space dimensions (two real, two imaginary) or alternatively 4 time dimensions (two imaginary, two real).

Now experience can be seen as a continual correction process by which the asymmetrical nature of conventional space and time ceaselessly switches to its underlying symmetrical nature (and the symmetrical in turn switches to the asymmetrical). Of course when these dynamics are limited - as at the linear level - the correction process in turn is very restricted so that the finite phenomena of broken symmetry assume an absolute rigid identity.

Now conventional science - even in linear terms - greatly misrepresents the nature of experience. We can represent actual phenomena from four different aspects.

• We can experience (perceptual) objects - in relation to the self. (This gives the common sense observation of objects being three dimensional in space. Now in this interaction the time dimension directly results from the mental contribution. So in this formulation we have 3 dimensions of space (positive) and one of time (positive).
• We can experience the self - in relation to (perceptual) objects. However whereas before, we experienced the object externally in space and time, now it has an internal direction. However conventional science makes no distinction as between these two aspects.
• We can experience (conceptual) objects - in relation to the self. Now this experience is abstract yet still misleadingly identified as three dimensional (in spatial terms). For example the new bulky printer I have purchased - as (quantitative) perception - is certainly three dimensional (in spatial terms). However it is highly misleading to refer to the (qualitative) holistic concept of "printer" as three dimensional in space. Thus there is an immediate confusion in conventional science as between the partial quantitative and holistic qualitative interpretations of objects.
• We can experience (conceptual) objects - in relation to the self. Again though these holistic qualitative objects have internal as well as external directions, this distinction is not made in conventional science.

So we can have here four distinctive interpretations of the same observation. Thus any equation - even at the linear level - has (at a minimum) four interpretations.

Thus if a represents the hypotenuse of a right angled triangle and b and c the other two sides, then a2 = b2 + c2.

Now this can be given (a) a partial quantitative (external) interpretation as objective truth (i.e. that the result will apply for an actual triangle), (b) a partial quantitative (internal) interpretation as subjective truth )i.e. that the result has a psychological interpretation), (c) a holistic qualitative (external) interpretation as objective truth (i.e. that the result applies abstractly to "all" triangles) and (d) and finally a holistic qualitative (internal) interpretation as subjective truth (i.e. again that the result has a general interpretation in psychological terms).

However though we can identify these four possible interpretations of the same equation, there is no way of reconciling them at the linear level. This is why the linear level is asymmetrical.

Now considerable progress is made at the circular level.

When we take the same four aspects of the experience of an object, the external and internal directions are clearly differentiated. If we take the objective direction as positive, then - in relative terms - the subjective direction is negative. Also the perceptual and conceptual experiences are differentiated in "real" terms. Quite simply if the perception is identified in spatial terms, then - relatively speaking - the corresponding concept is identified in temporal terms (and vice versa).

So lets now look again at the interpretation of the four aspects.

• When we experience the external perception of the object (in relation to the self), it is given a positive interpretation. Thus we identify the object as three dimensional in (positive) space and one dimensional in (positive) time.
• When we experience the internal perception (in relation to the world), it is given - in relative terms - a negative interpretation. Thus we now identify the object as three dimensional in (negative) space and one dimensional in (negative) time.
• When we experience the external concept (in relation to the self), space and time dimensions are exchanged. Thus we now identify the object as three dimensional in (positive) time and one dimensional in (positive) space.
• Finally when we experience the internal concept (in relation to the world) the positive direction turns to negative. So we now experience the object as three dimensional in (negative) time and one dimensional in (negative) space.

Though each aspect of phenomenal reality is experienced in asymmetrical terms, overall in terms of the differing four aspects there is symmetry. We have the same number of negative as well as positive directions (8 in all). Also we have the same number of temporal as well as spatial dimensions.

Thus the very ability to switch as between the differing asymmetrical aspects, implicitly depends on an underlying ground where space and time are symmetrical (two dimensions each with both positive and negative directions).

If scientists are to understand that the "same" equation can have several different interpretations (as complementary directions), then rational analytical understanding must be underlined by intuitive holistic understanding.

When we move into the point level, there is now the further appreciation that space and time in dynamic terms are "real" and "imaginary" with respect to each other. Thus we can describe an object in "complex" terms using solely spatial or temporal dimensions.

Thus using this "complex" perspective we can describe an observation of an object in the following four ways.

• The external perception of the object (in relation to mind and concept) is in "real" (positive) space. "Real" time is now "imaginary" space. Thus the observation is four dimensional (three of "real" space and one of "imaginary" space).
• The internal perception (in relation to object and concept) is in "real" (negative) space (three dimensions) and "imaginary" (negative) space (one dimension).
• The external concept of the object (in relation to mind and perception) is in "imaginary" (positive) space (three dimensions) and "real" (positive) space (one dimension).
• Finally the internal concept (in relation to object and perception) is in "imaginary" (negative) space (three dimensional) and "real" (negative) space (one dimensional).

Thus we have here described the four aspects of the object in terms of "complex" space. (We could equally describe them in "complex" time).
(An eight directional interpretation would of course be even more subtle).

Again the ability to switch between these four asymmetrical interpretations of the observation involves implicitly an underlying "complex" space symmetry (i.e. where space has two "real" and two "imaginary" dimensions (with positive and negative directions).

Thus when combined with holistic mathematics, standard mathematical equations require a far more refined multifaceted interpretation.

Q.  Can you demonstrate how the "arrow of time" is eight directional at the radial level?

PC  In Holistic Mathematics 1, I dealt with the two-directional arrow of time.
As all events in experience involve the dynamic relationship of (internal) observer and (external) event, relativity is always involved with the event both past and future in time.

Now the four directional interpretation involves a distinction as between the quantitative and qualitative interpretations of phenomena. The quantitative (partial) interpretation is given by the perception of "the event"; the qualitative (holistic) interpretation is given by the corresponding concept of "event". In holistic mathematical terms if the (partial) quantitative definition of a phenomenon is "real", then - in relative terms - the (holistic) qualitative definition is "imaginary".

Experience of time is now polarised in both horizontal and vertical fashion.
Thus in terms of the relationship of (internal) observer and (external) event as defined in quantitative (partial) terms, time has two directions - positive and negative (relatively past and future) - in "real" time.

In terms of the relationship of (internal) observer and (external) event as defined in qualitative (holistic) terms, time has again two directions - positive and negative (relatively past and future) - in "imaginary" time.

Now these four directions of time are measured in finite terms.
However each of these four finite directions can be given a transfinite grounding. Whereas the polarised finite approach relates to rational understanding of the experience, immediate transfinite understanding relates to corresponding intuitive understanding.
This intuitive understanding is absolute (not relative) and best described simply as the present moment.

Thus each of the four finite directions in time are grounded in a transfinite experience of the present moment. In this sense we have eight directions to the experience of time.

What is fascinating is that the transfinite (null) diagonal directions can be given a reduced finite explanation as the combination of equal "real" and "imaginary" components. And as both "real" and "imaginary" components can have positive or negative signs, this allows for four "complex" interpretations.

Thus an eight directional experience of an event - which characterises the radial level - involves two horizontal opposites (the "partial" quantitative aspect) that are - relatively - positive and negative in "real" time.

It also involves two vertical opposites (the "holistic" qualitative aspect) that are - relatively - positive and negative in "imaginary" time.

It finally involves four diagonal opposites (the essential aspect) - that are transfinite representing experience of the present moment (which underlines each of the four finite relative directions).

These four diagonal opposites can be given a reduced finite explanation (as the intersection of "partial" quantitative and "holistic" qualitative aspects) that are relative in "complex" time with equal "real" and "imaginary" components. (These are "real" + and
"imaginary" +, "real" - and "imaginary" +, "real" + and "imaginary -, "real" - and "imaginary" -).

These distinctions are far from academic. For example a properly integrated explanation of evolution would have to maintain a balance as between horizontal, vertical and diagonal complementarity.

This complementarity is often missing in practice. Indeed I would characterise my biggest criticism of Ken Wilber’s work as a lack of sufficient vertical complementarity. This leads - among other things - to an elitist view of evolution with a continued tendency to elevate "higher" above "lower" life forms. Thus the statement that the noosphere represents a "higher" holarchy than the physiosphere is only true in a relative limited sense. (What Ken seems to often forget is that our very interpretations of this "lower" physioshere are derived from the "higher" noosphere). Thus when not balanced by a recognition of the opposite polarity (i.e. that in a certain equally valid sense that the physiospere is "higher" than the noosphere) we can obtain a distorted perspective and fail to realise the ultimate radical equality of all creation.

When one considers that conventional science is still based on just a single direction of time, one realises how limited is its perspective.

Q.  What do you precisely mean when you say that the radial level is multidimensional?

PC  Firstly I want to clear up a possible confusion in relation to terms.

In holistic mathematics directions relate directly to dimensions. Thus to say that the radial level is eight directional, implies that it is eight dimensional.

However the correct way to interpret these "dimensions" - as I have already stated - is in terms of three binary systems (horizontal, vertical and diagonal) relating to information, transformation and essential states respectively. As the final diagonal binary system can be represented as the equal intersection of both information and transformation states, at its simplest we can represent reality dynamically in terms of a double binary system involving the processing of information and transformation respectively.

So just as the application of the horizontal binary system leads to the generation of information as multiple quantities (objects), the application of the vertical binary system leads to the generation of transformations as multiple qualities (dimensions).

Thus properly understood, corresponding to each object as quantity is a corresponding dimension as quality. Thus there are as many dimensions as objects. So just as reality consists of multiple objects, it is also multidimensional.

I already showed in Holistic Mathematics 2, how each "real" object can be viewed alternatively as an "imaginary" dimension. Likewise each "real" dimension can be viewed alternatively as an "imaginary" object.

When we identify objects solely as quantities in conventional scientific terms (and thereby in abstraction from dimensions) there is much confusion.

Properly understand the "object" represents a dynamic interaction pattern that is both quantitative and qualitative. (In psychological terms the perception of the object represents the "partial" quantitative aspect; the concept of the object represents the "holistic" qualitative aspect). Thus in dynamic terms - at all levels of reality - quantitative and qualitative aspects interpenetrate. Objects and dimensions are truly complementary (in vertical terms).

Now for a static analysis of reality we have to keep one of these poles fixed. Of course the usual convention is to view multiple objects (quantitative) against a fixed dimensional background (qualitative).

However it is equally valid to view multiple dimensions (qualities) against a fixed object background (quantitative). This latter approach leads - in very much complementary fashion to psychology to the prediction of a "shadow" world of matter. Interestingly this result is predicted by Superstring Theory (which is not surprising given the highly dynamic interaction as between "objects" and "dimensions" at this level of reality).

Of course the radial level involves the additional realisation that all "objects" and "dimensions" - which are "real" and "imaginary" with respect to each other have both finite and transfinite aspects. In other words the uniqueness of all quantitative "objects" and qualitative "dimensions" (reflected psychologically through perceptions and concepts respectively) are continually mediated through a radiant spiritual light.

Though of necessity we must make polar distinctions in communication, these have only a relative limited use with no essential value. Thus reality which appears polarised in terms of objective-subjective aspects (matter and mind) and quantitative-qualitative aspects (wholes and parts) ultimately comprises a seamless invisible web (without distinctions).

Q.  Can you now deal with the important area of mathematical proof. Does this have any meaning in the context of holistic mathematics?

PC  Yes, this is a very important and interesting area. I have already dealt with this in part in Holistic Mathematics 1 where I strongly questioned the accepted meaning of mathematical proof. Standard mathematics expresses "proofs" formally in explicit rational terms. However intuition is always essential to literally "see" what is implied by the logical connections established through reason. Thus in dynamic terms two elements are involved in standard analytical "proofs"; a rational explicit element (which is solely recognised in formal presentation) and an intuitive implicit element (which enables one to "see" what is implied by starting axioms). All "proofs" in fact are thus subject to the uncertainty principle and in dynamic terms strictly relative. Indeed they represent no more than an especially important form of social consensus. There are thus two types of understanding.

The rational type responds readily to logical linear connections. However when carried to extremes this leads to a lack of creative insight and a considerable narrowing of vision.

The intuitive type responds readily to holistic circular connections often enabling instant insight (without the need for formal rational connections) into what is implied by initial axioms. However once again when carried to extremes this can lead to undue vagueness and a lack of sufficient rigour.

However a significant imbalance remains. Whereas most important mathematical "proofs" initially depend essentially on creative intuition, formally they are presented in merely rational terms.

Surprisingly Holistic Mathematics has its own type of distinctive "proof". However the verification of such "proof" - in contrast to Standard Mathematics - is more personal and intuitive.

We can identify three distinctive levels of proof corresponding to each of the three levels of Holistic Mathematics.

Holistic Mathematics 1 is concerned with establishing complementary connections of the horizontal type (i.e. positive external and negative internal aspects).

Such complementary connections are obtained directly by intuition. Now the role of reason here is in providing an appropriate indirect rational translation of such relationships. Holistic Mathematics itself has been designed to provide such reduced translations.

Thus for example when I say that numbers are strictly dynamic entities with positive and negative polarities, this requires - in direct terms - the appropriate intuition to verify this insight, and - indirectly - the appropriate reason to appreciate the mathematical translation employed.

Whereas standard mathematical "proofs" require linear understanding,
proofs relating to Holistic Mathematics 1 require the understanding of the "higher" circular level. Thus there is a great danger of reductionism in relation to Holistic Mathematics. If people are unable to understand in terms of the "higher" levels, then inevitably they will try and reduce it to the "lower" linear level and thereby misrepresent its findings.

Thus "proof" in relation to Holistic Mathematics 1 is essentially circular whereby one successfully establishes horizontal complementarity (at the required level of investigation), indirectly expressed in appropriate reduced rational fashion. In the precise qualitative language of Holistic Mathematics this entails (algebraic) irrational - rather than rational - "proof".

Holistic Mathematics 2 involves vertical as well as horizontal complementarity.
This implies a far closer relationship being maintained as between rational and intuitive modes of expression.

A good example of this type of "proof" is my assertion that the entire Spectrum of Consciousness constitutes the qualitative number system.
In this approach there is both horizontal and vertical complementarity. The qualitative definition of each number type referring to a particular level or stage of development requires horizontal complementarity in the manner I have described. Vertical complementarity is then provided through establishing direct correspondence as between
pre-linear and trans-linear stages of development.

Acceptance of my number spectrum then depends on the validity of the complementary number relationships established (both horizontal and vertical).
However once again this requires the appropriate understanding of the point level.

Holistic Mathematics 3 is especially interesting as it involves the three types of "proof" thus established.
One of the practical implications of this that properly understood a comprehensive "proof" requires all three types.

Firstly we need to provide a quantitative analytical proof; then we need a qualitative holistic proof; finally we need to establish a satisfactory correspondence as between both quantitative and qualitative proofs.

One example of where I believe this has been done is in relation to the irrational nature of the square root of 2.

A quantitative analytical "proof" has been long established that the square root of 2 is irrational.

A qualitative holistic "proof" (i.e. as to why the square root of 2 is irrational) was provided in my posting on Holistic Mathematics 1).

Finally I was able to establish a direct dynamic correspondence in this posting as between both quantitative and qualitative behaviour.

The implications of this comprehensive approach are extremely interesting.

No proposition is fully proven until quantitative and qualitative "proofs" exist with a satisfactory correspondence between both demonstrated. For example a quantitative proof for Fermat’s Last Theorem now exists. However a comprehensive solution requires that a holistic qualitative proof be also found and that a satisfactory correspondence as between both quantitative and qualitative "proofs" established. My own view is that the qualitative "proof" of Fermat’s Last Theorem can be expressed very simply. Direct correspondence as between quantitative and qualitative aspects would then require the establishment of an equally simple quantitative "proof".

It could well be that since Fermat some essential key insight has been missing preventing such a simple approach. Thus I believe that it is entirely possible that Fermat did prove his theorem in an ingeniously simple (quantitative) fashion.

One highly creative approach to many of the remaining unsolved problems in quantitative mathematics (many of which can be simply stated) would be to approach them from a holistic perspective attempting to "solve" them in qualitative terms. If such a qualitative proof for a proposition could be successfully found, then this would suggest that a quantitative "proof" also exists. A proposition could therefore be deemed provisionally true pending the establishment of a satisfactory rational "proof".

Indeed there is a very important conjecture in relation to prime numbers for which I have established a qualitative "proof". This would therefore suggest that - though still lacking a rational "proof" - that the conjecture is in fact true. (As this has played an important role in much of my thinking I may return to this later).

To sum up, mathematics has an analytical quantitative aspect (which is rational) and a holistic qualitative aspect (which is intuitive). A creative approach requires combining both aspects.

Q.  Can you now sum up your main findings in relation to Holistic Mathematics 3.

PC  Once again each of the major levels has a distinctive mathematical approach associated with it.

The linear level is based on a rational logic and involves the specialisation of the analytical (quantitative) aspect. (Standard Mathematics).

The circular level is based on an (algebraic) irrational logic and involves the specialisation of the synthetic (qualitative) aspect (Holistic mathematics 1).

The point level is based on a transcendental logic and involves the specialisation of the relationship as between analytical and synthetic approaches. (Holistic Mathematics 2).

The radial level is based on a "complex" logic and involves the progressive integration of all three approaches.
Thus we have a "real" component which represents the rational analytical aspect.
We have an "imaginary" component which represents the intuitive holistic aspect.
Finally these are combined as a "complex" component representing the integration of analytical and holistic aspects.

A truly comprehensive paradigm must necessarily be "complex" (in a holistic mathematical sense) recognising both quantitative and qualitative approaches to understanding.

Holistic Mathematics 1 involves horizontal complementarity; Holistic Mathematics 2 involves in addition vertical complementarity; Holistic Mathematics 3 involves diagonal complementarity. This level relates directly to the radical equality of the transfinite void; in reduced terms it involves a "complex" relative hierarchy (involving "real" quantitative and "imaginary" qualitative aspects).

Ultimately both the analytical and holistic aspects of mathematics are fully integrated. This is what I term Comprehensive Mathematics. My hope is that one day that this will be understood as simply mathematics.