Holistic Mathematics - Type 2 Integration

Recap on Holistic Mathematical Approach

Q. Before starting on Integral 2 understanding can you briefly recap on the holistic mathematical nature of your approach?

PC The fundamental basis of this approach is in the remarkable hidden meaning of mathematical symbols.

We can distinguish as between two distinct logical systems, which in binary holistic mathematical terminology are linear (1) and circular (0) with respect to each other.

The linear (either/or) system, which is based on the clear separation of polar opposites, leads to an unambiguous asymmetric type interpretation of relationships that is suited for appreciation of the differentiated aspect of development.

The circular (both/and) system, which is based on the complementarily (and ultimate identity) of polar opposites leads to a paradoxical dynamic appreciation of relationships that is by contrast (indirectly) suited for appreciation of the integral aspect of development.

Experience necessarily requires the interaction of both systems. However in formal terms most intellectual understanding is based on an undue reliance on the linear (either/or) logical approach.

Though not always the case, mathematical understanding has become largely synonymous in our culture with analytic interpretation based solely on linear logic.

Despite its great achievements, it therefore offers very limited access to the rich philosophical meaning inherent in its symbols

However all mathematical symbols have a corresponding holistic interpretation based on the circular (both/and) system.

Because a comprehensive understanding of either aspect is only possible in the dynamic context of their mutual interaction, the radial approach then requires the incorporation of both linear and circular aspects for all symbols.

Taking the most basic notion of number I have been demonstrating the nature of the radial approach in our past few discussions.

Number provides the accepted means of our quantitative ordering of reality. This corresponds in turn with the (merely) linear interpretation of number symbols.

What is not properly recognised however is that number equally provides the most efficient means possible for the qualitative ordering - more properly the dynamic interaction of quantitative and qualitative ordering - of reality relating to all development structures. However this requires appropriate appreciation of the corresponding circular interpretation of number.

I was at pains to show that every number can be defined with respect to two distinct aspects.

We have a quantitative (linear) aspect (defined within a given qualitative dimension) and - in reverse fashion - a (circular) qualitative dimensional characteristic (to which each number quantity is raised).

Of course equally we can start with qualitative aspect as linear in which the corresponding quantitative aspect is circular. So both quantitative and qualitative have both linear and circular aspects.

The important point here is that in a comprehensive radial approach the number symbol (in this context representing quantity) and the number symbol (representing the dimensional quality) are linear and circular with respect to each other and defined therefore with respect to distinct logical systems.

However in the conventional analytic interpretation, there is no formal role for circular understanding. Therefore symbols - that properly have a circular meaning - can only be dealt with in reduced linear fashion.

Therefore - in a linear approach - all numbers are ultimately defined with respect to an invariant 1

^{st}dimension (i.e. raised to the power of 1) so that the circular aspect thereby becomes redundant.So we do not customarily represent this reduced dimensional aspect of 1 in defining numbers. Thus for example 2

^{1}is written as 2 (i.e. without dimension).However - even within an analytic approach - the circular aspect of number is still indirectly apparent.

Thus when we attempt to extract the roots of numbers, a clear circular aspect is revealed.

As we have seen if x

^{2}= 1^{2}, then x (i.e. x^{1}) = + 1 or - 1 and lies as two points on the circle of radius 1 (at the end of the line diameter bisecting the circle).However though the circular nature of mathematical roots can be easily demonstrated in analytic terms, because of its linear philosophical bias, their remarkable holistic significance remains hidden.

Therefore it takes circular understanding (using both/and) logic to properly appreciate the holistic significance of square (and higher) roots.

The radial approach to mathematical symbols can equally demonstrate another remarkable feature of dimensions (that remains hidden in an analytic interpretation).

Thus with 1

^{2}, the 2 - that here represents the (qualitative) dimension - has a complementary circular interpretation in both/and terms. Thus the dimension relates in circular terms to the situation where opposite polarities (that are + and - with respect to each other) cancel out as nothing. So strictly speaking the (qualitative) dimension is - literally - empty in quantitative terms. In this way we do not confuse what is qualitative with what is quantitative.Now in (indirect) linear terms - where in effect it is reduced in a quantitative fashion - this dimension has an explanation as 2 (representing the identification of the two opposite poles). However we can only make sense of this reduced linear explanation in the context of the circular interpretation (establishing their interdependence).

Thus by properly recognising the circular aspect we are able to establish a proper correspondence here as between the notion of the square of a number and its root.

Q I appreciate this point! It puzzled me a lot when I was doing mathematics as to the asymmetry evident in squaring a number and then obtaining its root. Thus if I square 1 and then reverse the operation by obtaining the square root, I would expect to get 1. Only the answer is + 1 or -1. However what you are saying makes a great deal of sense i.e. that there is a hidden unrecognised aspect to analytic mathematical understanding. (And I can see that it takes the inclusion of circular interpretation to show that this is the case)!

PC In the radial interpretation - which is inherently dynamic - obtaining the power of a number involves a horizontal/vertical switch as between linear (either/or) and circular (both/and) logic. Obtaining the corresponding root entails the reverse vertical/horizontal reverse switch from circular (both/and) to linear (either/or) logic.

So in radial terms the 2 representing the dimension entails the complementarity (and ultimate identity) of both polarities (+ 1 and - 1). Though in direct qualitative terms this is 0 (empty), in absolute linear terms it is written as 2.

In reverse fashion (using linear interpretation) when these polarities are separated as the roots we get either + 1 or - 1.

Q So your view is that the proper appreciation of either the linear (analytic) or circular (holistic) interpretation of mathematical symbols must take place in a radial context (which differentiates and integrates both)?

PC Yes! When we return to the analytic approach from the radial, an enriched appreciation emerges (where the hidden shadow side of analytic thinking is revealed). This greatly enhances the creative possibilities for analytic interpretation while equally providing the correct interface for holistic understanding.

Likewise we return to the holistic interpretation from the proper context (where it is properly related to analytic). This in turn enhances the application of holistic abilities within a practical context.

Summary of Integral 1 Approach

Q Can you now briefly recap for us the nature of holistic understanding in the context of Integral 1 understanding?

PC Integral understanding in holistic mathematical terms is always associated with a refined form of bi-directional understanding.

In Integral 1 appreciation, phenomenal reality is based on circular interpretation of 2 and expressed in terms of fundamental opposite "real" poles which are positive (+) and negative (-) with respect to each other. In psychological terms "real" corresponds with (direct) conscious understanding; in physical terms "real" corresponds to object phenomena that are the products of such (conscious) understanding.

This entails that with respect to any given (horizontal) level, reality always expresses itself in terms of two opposite poles i.e. exterior and interior (and interior and exterior) that are positive and negative with respect to each other. When the Integral 1 approach is extended to the other fundamental poles (vertical and diagonal) they also will be interpreted in a similar manner.

Q Putting it another way you, are saying that all holons are conditioned by exterior and interior i.e. Right-Hand and Left-Hand aspects that are dynamically complementary?

PC Indeed! This would be one way of expressing it. However we are referring here to an inherently dynamic understanding of Right-Hand and Left-Hand (and Left-Hand and Right-Hand) that renders any fixed notion of location paradoxical.

Once again we must distinguish differentiation from integration. In a linear (Integral 0) approach, the integral is somewhat confused in intellectual terms with the differentiated aspect. Until we define the direction of movement, Right-Hand and Left-Hand are ambiguous terms. From a nondual (integral) perspective, direction has no strict meaning. Therefore in dualistic terms we can always fix location in opposite ways that are equally valid within their respective reference frames.

Though interpretation is unambiguous (with respect to each separate frame of reference), when considered in dynamic interaction with each other as interdependent, they are paradoxical.

So for example scientific understanding can be given either a Right-Hand (as empirical facts) or Left-Hand interpretation (as - relatively - theoretical constructs) which are unambiguous when considered independently of each other. However when we consider both in mutual interaction they are in fact paradoxical (where Right-Hand and Left-Hand lose any fixed identity).

However in an Integral 0 approach there will be an attempt to fix science with just one frame of reference (e.g. in exterior terms with the Right-Hand quadrant). This leads therefore to unambiguous uni-directional asymmetric understanding of its relationships with subsequent integration largely reduced to differentiated interpretation.

However with Integral 1 understanding the differentiation of scientific understanding takes place in a bi-directional fashion (with interpretations in both Right-Hand and Left-Hand quadrants). Though these (differentiated) linear interpretations are unambiguous in isolation, they are deeply paradoxical in terms of each other. So the process of integration - which is now based on circular both/and interpretation - is clearly distinguished from differentiation.

However though the Integral 1 approach clearly distinguishes in this way the process of integration from differentiation, as we have seen, it is based on Type 1 complementarity which is not able to properly interpret in scientific manner the quantitative and qualitative aspects of understanding.

So transformations - which are intrinsic to all whole/part (and part/whole) interactions - require the Integral 2 approach (with associated type 2 complementarity) and it is to this that we now turn.

Integral 2 Approach

Q From your holistic mathematical perspective, the Integral 2 approach is intimately related to the circular interpretation of the number 4. I know you consider this of the utmost importance!

PC Yes! Jung realised its great importance as exemplified by his emphasis on mandalas (based on this notion) as of profound integral significance. However though holistic mathematical understanding was deeply implicit in his understanding, it was not properly formalised in an explicit manner.

In geometrical terms the circular "4" can be literally represented as a circle that is divided into four equal quadrants. It provides in fact the scientific basis for a truly dynamic appreciation of the 4 quadrants. From another perspective it provides the fundamental integral interpretation of the four dimensions (and their relationship to each other).

Most importantly it opens the way for true philosophical appreciation of the notion of an imaginary number which potentially has an enormous role to play in the subtle interpretation of all dynamic interactions.

The circular 4 which can be written 1

^{4}represents 4 as a (vertical) dimension rather than a (horizontal) quantity.Though - as we have seen from our radial approach - the true meaning in this context of dimension is circular, indirectly it has a reduced linear interpretation (where it is expressed with respect to the first dimension). We do this by obtaining its 4 roots.

So if x

^{4}= 1^{4}, then x (i.e. x^{1}) = + 1, - 1, +iand -i(wherei= square root of - 1).When we represent these 4 values in geometrical terms (in the complex plane) they lie as 4 equidistant points (representing opposite poles of the horizontal and vertical line diameters drawn through the centre of the circle). So following the convention of defining the horizontal axis as "real" the vertical axis is then "imaginary" the geometrical pattern looks something like this!

However though these roots have an important (linear) analytic interpretation, a much greater significance attaches to their inherent (circular) holistic meaning.We can express this by saying that in dynamic integral terms, all phenomena are conditioned by both horizontal and vertical poles (which are "real" and "imaginary" with respect to each other).

We have already looked at the horizontal poles in our previous discussion, where we found that at any given level, all phenomena have both exterior and interior (and interior and exterior) aspects that are dynamically complementary.

However we can now also include in a holistic mathematical manner the vertical poles (which operate between levels). These vertical poles represent the whole and part (and part and whole) aspects of all phenomena, which are also dynamically complementary (along the vertical axis). However in holistic mathematical language they are defined - relatively - in an "imaginary" rather than a "real" manner.

Holistic Interpretation of an Imaginary Number

Q How do you establish the dynamic mathematical meaning of an "imaginary" number in holistic terms?

PC As you know in analytic term

iis defined as the square root of - 1. Soiis the imaginary equivalent of the real number 1.In holistic terms we have already established that linear (one-dimensional) understanding arises from the separation of opposite poles (with in any context just one posited in an arbitrary manner).

In psychological terms this relates directly to conscious (differentiated) understanding of phenomena; in corresponding physical terms it relates to what science identifies as phenomenal reality i.e. which in this context is thereby "real".

So conscious understanding and the scientific interpretation of reality mutually support each other in a linear manner.

However to move from differentiated to integral understanding we must - in dynamic terms - negate what has already been posited (and differentiated). This then enables a switch - in psychological terms - from conscious to unconscious understanding (which is the basis for integration). Therefore - though conscious and unconscious are necessarily interdependent in the dynamics of understanding - it is initially helpful to identify the differentiated with the conscious and the integral with the unconscious aspect respectively.

So the linear differentiated aspect is based on the positing (+) of one pole in a one-dimensional manner.

By contrast the circular integral aspect is based on the negation (-) of what has been posited in two-dimensional manner (combining positive and negative poles as complementary).

However to generate phenomena in a differentiated manner we must necessarily express understanding in linear (one-dimensional) terms.

Thus as the unconscious is two-dimensional and dynamically associated with the negative direction of understanding, to indirectly express this in linear (one-dimensional) fashion, we must reduce understanding from two dimensions to one. This entails the holistic equivalent of obtaining a square root.

Therefore the indirect expression of circular understanding (in a reduced linear manner) entails in exact fashion the dynamic equivalent of the mathematical notion of an imaginary number.

Thus we now can distinguish as between two notions of (unitary) phenomenal form.

Firstly we have the direct conscious expression which corresponds with "real" interpretation (1).

Secondly we have the indirect conscious expression (pointing to what is circular unconscious meaning) which corresponds with "imaginary" interpretation (

i).From an Integral 2 perspective, in holistic mathematical terms all phenomena are "complex" (with both "real" and "imaginary" aspects). In other words such phenomena necessarily serve as the embodiment of both conscious and unconscious meaning (though the latter aspect may not be properly recognised).

Q What you are saying is that the mathematical notion of the imaginary is the indirect interpretation of the (circular) both/and logical system that is expressed in a customary linear (either/or) fashion. This is truly remarkable as it provides a means for "converting" as it were between two logical systems that are qualitatively distinct from each other.

So to convert circular interpretation in linear form we simply switch from "real" to "imaginary" linear interpretation. Thus with scientific understanding, where linear phenomena are considered "real", circular phenomena (corresponding to unconscious understanding) will be considered "imaginary".

Likewise to convert linear to circular format we switch from "real" to "imaginary".

Thus from a mystical perspective spiritual truth pertaining to the unconscious will be considered "real" with material phenomena as "imaginary" (i.e. illusionary).

First question! If what you say is true regarding the fundamental philosophical importance of an "imaginary" number, why is it not already recognised?

PC We dealt with this to a degree in our last discussion. Conventional mathematics is firmly based on the logic of (static) analytic (either/or) interpretation. For example a number can be + 1 or - 1 but it cannot, by conventional reasoning, both + 1 and - 1 simultaneously.

However in terms of dynamic holistic interpretation, positive and negative poles are combined. So in dynamic interaction both the positive and negative aspects of unitary form (+ 1 and - 1) - with a merely paradoxical identity - are necessarily equal to each other.

However once again we can only express this paradoxical two-dimensional logic in a reduced linear manner.

So the notion of an imaginary number in analytic terms points to the alternative logical (both/and) system which remains hidden within its own restricted interpretation.

Thus though imaginary numbers can be used in an analytic context their true identity necessarily remains concealed.

Q Can we say therefore that i represents the number which is both + 1 and - 1 simultaneously?

PC Yes! This would be valid. However once again we cannot give meaning to such an interpretation in merely analytic terms.

Q Does this not lead to a problem in interpreting - i. How can we distinguish it from+ i?.

PC Remember from a dynamic perspective direction is very important!.

So if +

iis expressed as both + 1 and - 1 simultaneously, then -ientails the reversal of direction so that it is both - 1 and + 1 simultaneously. So in dynamic holistic terms - until full integration is achieved - we can should distinguish for example top-down integration (where "lower" stages are integrated from the perspective of "higher") with bottom-up (where "higher" are integrated from the perspective of "lower"). Thus if the first direction of integration (i.e. top-down) is defined as positive, then the other direction (i.e. bottom-up) is thereby - relatively - negative.

Q Does the holistic mathematical notion of an "imaginary" number bear any relationship to what is imaginary in a literally sense?

PC Indeed it does! Though conventional mathematicians may try to maintain otherwise, in fact the meanings are very closely associated.

In common understanding what is imaginary in literary terms relates to the creative prompting of the unconscious. Likewise what is imaginary in holistic mathematical terms likewise relates in a more precise scientific manner to the unconscious (though it also has an important complementary meaning relating to physical reality).

Q What is this scientific counterpart? And why is such an important notion missing from our understanding?

PC Again we have the same problem as in mathematics in that conventional science is based firmly in formal terms with linear (either/or) logic.

Though so many findings of modern physics firmly contradict this paradigm, the philosophical approach has not greatly changed. So even cutting edge physics such as string theory is approached in a remarkably unambiguous fashion with physicists seeking clear-cut explanations for the fundamental workings of the universe.

Because in dynamic terms, the psychological and physical aspects of reality are complementary, this requires that we have a counterpart interaction in physical terms to that of the (psychological) conscious and unconscious perspective.

The problem with present science is that it is largely based on "real" notions of phenomena corresponding to direct conscious observation.

However as we have seen when we include the unconscious (with conscious understanding) our psychological interpretations of reality have - in holistic mathematical terms - both "real" and "imaginary" aspects. This entails that physical reality likewise has "real" and "imaginary" aspects.

So from an Integral 2 perspective we have "complex" psychological interpretations (i.e. "real" and "imaginary") complementing "complex" physical reality (i.e. "real" and "imaginary" object phenomena).

Therefore from a physical perspective, material phenomena (corresponding to "real" conscious interpretations) continually interact with an immaterial ground (corresponding to "imaginary" unconscious interpretation).

However in a linear interpretation, the immaterial ground is reduced to consciously observed (i.e. real) material phenomena.

Examples of Imaginary

Q Can you give some examples from the psychological perspective of the "imaginary" aspect.

PC One obvious manifestation would be in terms of dreams and fantasies where symbols can be deeply expressive in various ways of the unconscious desire for meaning. Symbols that appear here often have more of an "imaginary" than a "real" identity.

Another noteworthy example relates to religious and national myths, where again symbols can have a strong "imaginary" meaning.

The unconscious is also greatly involved in all sorts of projections. Though people may believe that they are being "fair" or "objective" in their views so often in fact they are deeply influenced by unconscious projections which are not properly recognised. So one again all phenomena are in dynamic terms "complex" with an "imaginary" as well as "real" interpretation.

The philosophy of materialism represents a good example of this confusion of the "imaginary" with the "real" aspect. Here material goods represent the embodiment of unconscious desires. For example one may refer to a dream car or dream home.

However though the conscious manifestation of economic goods is material, the unconscious desire for their attainment is spiritual. Because these two aspects are qualitatively distinct, therefore the (mere) conscious attainment of material goods cannot ultimately satisfy the (unconscious) desire for spiritual meaning.

One particularly interesting example is provided by scientific interpretation. Scientists might like to pride themselves on their objectivity but in strict terms the overall scientific process is far from objective. What constitutes important research, the recognition of meaningful work, the creative insight necessary for truly original contributions, and even the belief that science is a worthwhile activity are all based to a considerable degree on the unconscious aspect of understanding.

At the most fundamental level, the very recognition of phenomena and their related psychological constructs necessarily entails both a conscious (differentiated) and unconscious (integral) aspect. So once again the recognition of such phenomena - in holistic mathematical terms - is dynamically "complex" (with both "real" and "imaginary" aspects). The recognition of the (differentiated) "real" aspect of a phenomenon corresponds directly with conscious recognition; the recognition of the (integral) "imaginary" aspect corresponds directly with unconscious recognition (that is indirectly expressed in a phenomenal manner).

Q Let me put this another way! Are you saying that all phenomena have an unambiguous local identity (corresponding to conscious recognition) and an archetypal holistic identity (corresponding to unconscious recognition). Therefore though the combination and quality of understanding with respect to these two aspects may vary greatly depending on context, all phenomena therefore necessarily entail a dynamic interaction of "real" (conscious) and "imaginary" (unconscious) recognition.

PC Yes! This would be an excellent way of putting it! However remember the "imaginary" aspect does not relate directly to the unconscious but rather its indirect phenomenal expression.

Therefore by definition there is always an element of confusion in associating the unconscious indirectly with conscious symbols. Thus when "imaginary" understanding becomes very pure and unrestricted (as at the causal level) the phenomenal symbols through which the unconscious spiritual light is mediated become extremely fluid and transparent!

Q Your examples here of the importance of the "imaginary" relate largely to psychological understanding. Are there any examples of the "imaginary" notion that arise from physical manifestations?

PC In terms of physical manifestations a major problem remains in that science is still very much based on "real" notions of understanding (relating directly to conscious recognition). Therefore its very manner of interpreting phenomena tends to distort proper recognition of the "imaginary".

However I can point to the very clear example of virtual particles.

At the very dynamic "lower" level of sub-atomic activity the necessary interaction of virtual with real (in dynamic particle interactions) is well recognised.

Now virtual particles could equally be defined as imaginary particles.

Indeed with such particles opposite polarities (i.e. matter and anti-matter) are very closely related and continually fuse with each other. Therefore virtual particles - which enjoy but a transient fleeting identity - lie very close to the immaterial ground (from which they freely emerge). So in this way form (as virtual particles) continually emerges from emptiness and instantly returned to emptiness (as its immaterial ground).

We could say quite accurately therefore that virtual particles represent a temporary fleeting indirect material expression of the immaterial ground (from which they emerge)

Q What is the relationship here as between this dynamic behaviour in the physical world and psychological understanding?

PC In a dynamic integral sense both "higher" and "lower" and "lower" and "higher" are in vertical relationship to each other. Therefore the "lower" we go in probing the physical origins of all development processes, equally the "higher" we must go to obtain the spiritual inspired dynamic understanding that can properly integrate such findings. Thus - when understood appropriately - a clear complementary structural pattern is evident to both "lower" physical and "higher" psychological understanding.

Again we come back to a key paradox regarding modern physics. It seems the closer that it comes to a TOE (as with string and M- theory), the less intuitive grasp we have of the significance of its findings.

This problem is due to seeking a merely (differentiated) analytic interpretation of reality without equal emphasis on an (integral) holistic appreciation.

However this (integral) holistic appreciation requires the understanding of the "highest" levels of the Spectrum which are vertically complementary with the "lowest".

So the "higher" inherently dynamic spiritually inspired understanding is necessary to holistically interpret the "lower" equally dynamic physically inspired material manifestations.

I

llustration of Four Directional Dynamics

Q Can you know explain with reference to the understanding of a number how these Integral 2 four-directional dynamics operate?

PC It is important to point out that the dynamics that I will describe universally apply to the Integral 2 understanding of all phenomena. However in view of the holistic mathematical nature of these ideas there may be a special value in concentrating here on the notion of number.

The number quantity - say 2 - has a bi-directional interpretation in both exterior and interior (and interior and exterior) terms. We can start by looking on 2 as an exterior number object (to which the corresponding interior number perception is related). Equally from the opposite interior perspective we can look on 2 as the interior number perception (to which the corresponding exterior object is related).

So (exterior) object and (interior) perception are relatively positive and negative in "real" conscious terms (with respect to each other).

The fusion of both polarities (as complementary) then enables the switch to the concept of number which serves as the qualitative notion of number i.e. dimension.

The concept of number - based on both/and logic - does not strictly represent any actual number but rather the potential for number existence. In a direct sense it relates to unconscious (rather than conscious understanding). However this can be expressed in an indirect conscious fashion in "imaginary" terms.

Just as the "real" actual number (i.e. 2) has both positive and negative expressions (as - relatively - exterior object and interior perception), likewise the "imaginary" potential number has both positive and negative expressions (as - relatively - exterior dimension and interior concept respectively).

Thus in dynamic interactive terms when the direct recognition of the actual number is "real" the indirect recognition of the potential number concept is "imaginary".

However we can - as always - reverse these dynamics so that the general number is consciously understood as actual in "real" terms with respect to exterior dimension and interior concept (which are positive and negative with respect to each other).

In this case it is the fusion of both aspects that enables the (unconscious) switch to the corresponding number perception (which is now understood as the potential for individual number existence).

So now in dynamic interactive terms the actual collective number (as exterior dimension and interior concept) is "real" with the corresponding individual number (as exterior object and interior perception) as "imaginary".

Q. How would you describe these dynamics with respect to holons?

PC In an Integral 2 approach all holons have - in horizontal terms - exterior and interior aspects and in vertical terms whole and part aspects which are dynamically positive and negative with respect to each other. This entails Type 1 complementarity.

Furthermore, the switch from horizontal to vertical poles (and vertical to horizontal) requires the interaction of both actual and potential aspects (which are "real" and "imaginary" with respect to each other). In psychological terms this means that conscious and unconscious continually interact; in physical terms it means that matter and the (potential) immaterial ground also continually interact. Therefore the horizontal and vertical poles are - relatively - "real" and "imaginary" with respect to each other. This entails Type 2 complementarity.

Integral 2 Approach and Causal Realm

Q This would imply that Integral 2 understanding requires the attainment of a very "high" level of spiritual development?

PC In my approach each major form of integral approach is related in turn to a corresponding major level of the Spectrum.

So the Integral 0 approach entailing Type 0 complementary, understanding is largely related to the middle level of the Spectrum (comprising conop, formop and vision-logic).

The Integral 1 approach entailing Type 1 complementarity is then related to H1 i.e. the first of the "higher" levels (i.e. subtle realm).

The Integral 2 approach entailing Type 2 understanding is then related to H2 i.e. the second of the "higher" levels (i.e. causal realm).

Q I see! You are saying that associated with each of the higher spiritual (or transpersonal) levels is a refined form of bi-directional cognitive understanding. The mutual increasingly dynamic interaction of spiritual intuition with this bi-directional form of interpretation then properly constitutes "higher" integral understanding. So Integral 1 is bi-directional in "real" terms and Integral 2 is bi-directional in both "real" and "imaginary" terms. And the structure of this understanding is perfectly provided by the holistic interpretation of circular notions of number?

PC Yes! I must stress however that the understanding of these "higher" levels is likewise necessary to enable the holistic mathematical interpretation of number symbols. In other words we cannot attempt holistic mathematical interpretation from the perspective of the analytic understanding of H0 (the middle level).

Q This raises an important issue. If we accept that understanding generally does not extend beyond the middle level, is there any point in even attempting to communicate such ideas that are the expression of the "higher" levels. Surely you risk considerable misinterpretation of your ideas (i.e. based on conventional analytic understanding).

PC Good point! What you say is largely correct. During the few years when I was actively engaged in communicating these ideas I would say that the majority did not properly appreciate the nature of my approach. And perhaps this was inevitable!

However having said this, there was a significant minority who in many ways did get a proper handle on what I was attempting and often surprised and delighting me with the quality of their contributions.

The ability to understand reality in a holistic mathematical manner represents just one way - though I would maintain a potentially very important way - of interpreting reality.

The gift for this - like all talents - is largely innate. So it is quite possible that someone who would not conventionally have attained in significant manner to "higher" spiritual levels could immediately resonate with my approach. Hopefully it could then act in some manner as a catalyst for further spiritual development (which would thereby considerably enhance appreciation of its nature).

Likewise there are others who clearly have already attained to "higher" spiritual levels who - due to their own personality and gifts - would never show any interest in such a holistic mathematical approach. So the situation is very fluid and I am now much more open to the possibility of surprise in the kind of people who can genuinely respond.

However, I am confident that I am raising several important issues e.g. regarding the nature of an integral approach to development, which urgently need to be addressed (whether in the context of holistic mathematical interpretation or otherwise).

Q You say that Integral 2 understanding relates directly to the understanding of H2 (the causal realm). Can you say a little more about this?

PC I must stress that there many distinct ways of experiencing - what might be referred to as - the causal realm and likewise many distinct ways of interpreting its nature. Indeed in important respects I am deliberately reacting here to somewhat stereotypical mystical accounts (largely based on Eastern esoteric traditions) which I personally find greatly lacking from a human existential perspective.

A key problem which needs to be faced in spiritual development - and this is something on which I believe all the traditions are agreed - is that of self-centred attachment to phenomena.

In basic terms H1 is largely concerned with detachment from possessive identification with all conscious phenomena. Thus when successively attained, rigidity of experience is greatly reduced. This then enables the free unrestricted interpenetration of refined phenomenal form that is dynamically consistent with the maintenance of pure contemplative awareness.

However once the self is largely denuded of (rigid) attachments at a conscious level, its remaining (and often deep) attachments at an unconscious level can surface and are indirectly projected into consciousness in a very fluid and transient manner. This type of (indirect) consciousness is very distinct from the former direct type and during the causal stage one has ample time to watch its movements.

From a "high" spiritual level perspective the causal realm is associated with very refined virtual archetypes (where symbols of form - emerging from the unconscious - freely interpenetrate with spiritual awareness). They can be of a transcendent or immanent nature. In terms of my holistic mathematical approach, the former would initially often be associated with the pure "imaginary" appreciation of collective concepts (or dimensions). In relative terms the latter would be associated with pure "imaginary" appreciation of individual perceptions (or objects). Though typically, the latter is initially dominated by the former expression.

From a complementary "low" physical perspective, the causal level is associated with the intimate exposure to primitive desires (that are still in some manner confused with their phenomenal expressions). So the "higher" one goes in spiritual terms the "lower" one must go in physical bodily terms (as the pureness of the reception of the light at one level requires the removal of any remaining instinctive confusion at the vertical complementary level).

Put another way as long as the transcendent spiritual aspect dominates at a "higher" level, remaining confusion will remain in terms of the corresponding immanent "lower" physical level. Thus the removal of instinctive confusion at the "lower" level is a prerequisite to harmonising both transcendent and immanent aspects (where ultimately the body is integrated with Spirit).

So to summarise H2, the causal realm is necessary to properly disentangle "imaginary" unconscious elements of personality from their conscious expressions. Likewise the Integral 2 approach is designed - in terms of refined cognitive appreciation - to likewise disentangle (bi-directional) "imaginary" from "real" interpretation of relationships.

Dynamic Interaction of Whole and Part

Q Can you briefly concentrate with respect to a holon - on the dynamic relationship as between whole and part (and part and whole)?

PC When we were dealing with the horizontal polarities we saw that in order to differentiate any holon it must be posited (in conscious terms).

Thus when the exterior aspect of a phenomenon is recognised, it is thereby consciously posited; likewise when the interior aspect is recognised it is likewise posited (in conscious terms).

Thus the independent recognition (i.e. differentiation) of both exterior and interior arises from the fixing of the frame reference with just one pole.

However the important ability to switch as between exterior and interior (and interior and exterior) requires the corresponding negation of each pole. Such ability represents (empty) integral appreciation based the dynamic (and paradoxical) interdependence of both poles. Thus the degree of integration achieved sets limits on the quality of switching interaction that can take place. Thus when the integral aspect is not properly recognised, it is largely reduced to differentiation. This in turn leads to the rigid positing of phenomena (with respect to both poles) with little dynamic switching taking place.

Thus in static analytic terms (corresponding to the analytic Integral 0 approach) opposite poles are posited (and thereby differentiated) in experience. Integral understanding is then effectively reduced to differentiation as the multiple composite of these posited poles.

Thus any four-quadrant approach based on the attempted unambiguous fixing of holons with distinct quadrant locations represents the Integral 0 approach (as I define it).

With an Integral 1 approach we refine such understanding by by saying that the differentiation of any holon requires that it be posited in both "positive" and "negative" terms so that continual switching takes place as between the exterior and interior (and interior and exterior) aspects of all holons. Thus in dynamic terms to posit the exterior aspect we must negate the interior. Likewise to posit the interior we must negate the exterior. Therefore relative to each other these horizontal poles are "positive" and "negative".

So differentiation (in either/or terms) - with respect to any holon - is bi-directional entailing independent polar reference frames (that are opposite to each other).

Integration (in both/and terms) is also bi-directional now involving the interdependence of both poles (based on the mutual complementarity of opposites).

So an Integral 1 approach enables differentiation and integration to be properly distinguished from each other.With an Integral 2 approach we clarify such understanding further by saying that the differentiation of any holon requires that it be posited in both "real" and "imaginary" terms.

So in dynamic terms we now must recognise the continual switching that takes place as between the quantitative and qualitative (and qualitative and quantitative) aspects of all holons. Thus - in any context - the dynamic switch from part to whole (or alternatively) whole to part requires a corresponding switch from quantitative to qualitative recognition (or alternatively qualitative to quantitative).So from a qualitative perspective the switch from a perception to its corresponding concept (or alternatively concept to corresponding perception) requires a new type of dynamic recognition.

In complementary physical terms the switch from an object phenomenon to its dimensional recognition (or alternatively dimension to object) requires the same kind of recognition.

As we have seen this switching requires the introduction of "imaginary" consciousness as the indirect expression of the role of the unconscious (or from the complementary physical perspective the immaterial ground of reality).

So Integral 2 understanding requires the inclusion of "imaginary" as well as "real" understanding with respect to both positive and negative directions.

Thus every holon in Integral 2 terms is bi-directional with respect to both horizontal "real" and vertical "imaginary" poles.

Again differentiation is based on the relative separation of poles as regards both "real" and imaginary" (and positive and negative) aspects; integration is based on their mutual complementarity.Another way of saying this is that every holon - by definition - dynamically belongs to all four quadrants.

Q Can you illustrate this with an example?

PC Let us take our example of the atom. Now in horizontal "real" terms this has both exterior and interior aspects (that are positive and negative with respect to each other).

Thus in relative terms, we can view an atom as an exterior "object" or alternatively as an "interior" perception. The refined differentiated appreciation comes from recognising both exterior and interior interpretations (within their independent frames of reference).

In this way we have two opposite ways of differentiating recognition of the atom with respect to Right-Hand and Left-Hand quadrants (separately)

However the corresponding integral appreciation requires the simultaneous interdependent recognition of polar reference frames (recognising them as paradoxical).

Thus from an integral perspective the quadrants are empty so that Left-Hand and Right-Hand quadrants have no strict meaning.

The next stage is to recognise that the atom has both an individual and collective identity. In psychological terms we cannot understand the individual perception of an atom without relating it to the corresponding collective concept of atom.

From the corresponding physical perspective, we cannot recognise an atom without placing it a dimensional context (of space and time).

So when the focus of a holon is on one aspect (in "real" terms) the other aspect dynamically intervenes as "imaginary".

In other words when the "real" focus is on the perception of an atom (corresponding to a specific object), the unrecognised background recognition (i.e. of its collective dimensions) is "imaginary". However if we now focus on the collective notion of "atom" this holon is now "real" with the background context (i.e. specific atoms) "imaginary".

So in dynamic terms with every holon, what is "real" and "imaginary" keeps switching depending on context.

Once again however when we fail to recognise the hidden "imaginary" aspect, it sets severe limits on the quality of dynamic switching that can take place as between whole and part (and part and whole). In other words our recognition of holons then becomes somewhat rigid.

Q How does this inclusion of the "imaginary" relate to the holarchical approach to development?

PC As we have seen holarchy - as conventionally defined - is based on a one-sided interpretation of the relationship as between whole and part.

So with holarchy, every holon is a defined as a whole/part so that the (lower) whole is part of a (higher) more collective whole. However this requires - in any context - predefining the meaning of the whole (though in dynamic terms it is inseparable from the part)

So we equally can have partarchy. Here every holon is defined as a part/whole so that the (lower) part is whole in the context of a (higher) more unique part. However this entails the opposite problem of predefining the meaning of the part (though again it is dynamically inseparable from the whole).

So in dynamic terms hol-ons (whole/parts) continually keep switching to on-hols (part/wholes) and in reverse fashion on-hols (part/wholes) keep switching to hol-ons (whole/parts). So to deal with the dynamics of development, interpretation likewise needs to switch as between holarchical and partarchical (and partarchical and holarchical interpretation).

Again in refined differentiated terms, we have two "real" interpretations (which are unambiguous within their own separate frames of reference).

However for integral appreciation we must combine both interpretations as interdependent (based on paradoxical both/and reference frames).

So this ability to switch as between hol-ons and on-hols (holarchical and partarchical interpretation) is intimately dependent on the intervention of "imaginary" understanding.

In this context, the "imaginary" aspect is the dynamic basis of the transformation of all phenomena.

"Real" interpretation aims at the stable fixing of identifiable form.

The "imaginary" aspect - by contrast - always aims at transformation and the dissolution of unambiguous form.

So in "real" differentiated terms, development can be characterised in either holarchical or partarchical fashion (as form).

The "imaginary" aspect is the basis of the dynamic transformation in either holarchical or partarchical terms (leading to their ultimate dissolution)

So in dynamic integral terms where "real" and imaginary" are fully reconciled, development is spiritually empty (so that holarchy and partarchy no longer strictly apply).

Space and Time and Integral 2 Approach

Q Can you know apply the Integral 2 approach to the understanding of space and time?

PC We saw that with Integral 1 terms that object phenomena (with respect to both space and time) have - relatively - positive and negative directions.

Likewise these dimensions (with respect to corresponding objects) have - relatively - both positive and negative directions.

So space and time are symmetrical with both positive and negative directions (with respect to objects and dimensions).

If we identify the positive direction with the physical aspect, then - relatively the negative direction applies to the corresponding psychological aspect (the perceptions and concepts embodying experience of space and time).

However if we identify the positive direction with the psychological aspect then -relatively - the negative direction will imply the physical aspect.

The Integral 2 approach now allows us to properly relate objects and dimensions, which are "real" and "imaginary" with respect to each other.

Thus if an object in space is "real" then - in dynamic integral terms - the dimension to which it relates is - relatively - imaginary.

Likewise if an object in time is "real" then the dimension to which it relates is - relatively - "imaginary".

However if we initially identify the dimensions as "real" then the objects to which they relate are - relatively - "imaginary".

Thus space and time are symmetrical this time continually alternating as between "real" and "imaginary" states with respect to positive (physical) and negative (psychological) polarities.

So again in (conscious) differentiated terms, the dimensions of space and time (and the object phenomena to which they relate) are both interpreted in "real" terms.

However the important dynamic switch, enabling interaction as between objects and dimensions (and dimensions and objects) requires the "imaginary" aspect.

Without this "imaginary" aspect, interaction becomes limited with rigid objects forming in rigidly understood dimensions.

Now we not quite ready yet to express space in terms of time (and time in terms of space).

This requires the inclusion of twin diagonal polarities in an eight-sector approach, which is the basis of the Integral 3 approach.

Q What are the limitations of the Integral 2 approach?

PC There are different contexts for the understanding of the Integral 2 approach.

In terms of the understanding of the level to which it relates i.e. H2 (causal) it can tend to be unduly contemplative (which indeed is true of all the integral approaches).

However - even in terms of integration - it is not yet able to deal successfully with a key remaining issue. This relates to the successful reconciliation of both affective and cognitive aspects of understanding (which are personal and impersonal in relation to each other). It requires the Integral 3 approach (associated with Type 3 complementarity) to deal with this problem.

From the perspective of either the Radial 1 or Radial 2 approaches, a fuller appreciation of both the strengths and limitations of Integral 2 emerges.

Indeed this applies to the appreciation of all stages of development. To understand any stage fully it must be placed in the context of all stages. Thus while particular stages are still emerging in development this proper context is not fully available.

This in an important sense, the understanding associated with each stage remains in continual transition throughout development. In a qualified sense this is even true of radial development which can only approach rather than ever fully attain completion.

Q To conclude why do you believe so strongly in the value of the holistic mathematical approach?

PC I find it truly remarkable that all the key structural dynamics of reality are to be found encoded in the holistic mathematical explanation of what is - literally - a reduced expression of unity.

This fits in with my conviction that the most profound explanations are inherently very simple yet also capable of the most complex interpretations.

Thus the holistic interpretation of the two roots of unity provide the horizontal polarities i.e. exterior and interior (and interior and exterior) which are dynamically positive and negative with respect to each other.

The four roots of unity (i.e. the four quadrants) provide both horizontal i.e. exterior and interior (and interior and exterior) and vertical i.e. whole and part (and part and whole) polarities, which are relatively "real" and "imaginary" with respect to each other (in positive and negative terms).

The eight roots of unity (i.e. the eight sectors) provide diagonal, vertical and horizontal polarities the clarification of which dynamics will be the subject of our next discussion.

Suffice it to say that we will be then in a position to encode in scientific manner all

the key stages of development - to a remarkable level of detail - in a precise holistic mathematical fashion.From this perspective the holistic interpretation of number is essential for the proper scientific ordering of reality (in qualitative terms) just as the analytic interpretation is equally essential for its scientific ordering (in quantitative terms).

.