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Peter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.

Clarifying Perspectives 1

Primary Perspectives and The Holistic Binary System

Peter Collins

In holistic mathematical terms, all perspectives can be scientifically interpreted in an interactive binary digital manner, arising from the continual switching as between on/off states with respect to three fundamental sets of polarities. These polarities - which quite literally have a reduced expression in terms of unity - are geometrically represented by “The Eight Sectors” (arising from the division of a circle - of unit radius - by horizontal, vertical and diagonal lines).

As this holistic mathematical approach clearly distinguishes linear from circular notions of interpretation, it is ideally suited to clarify:

1. The (linear) differentiation of distinct perspectives through alternate on/off switching as between sector locations.

2. The corresponding (circular) integration of an overall holistic perspective through the simultaneous switching (on/off and off/on) with respect to complementary locations (in horizontal, vertical and diagonal terms).

3. The precise (binary) manner in which both the linear (1) and circular (0) aspects of perspectives are related for every stage of development.

Perspective on Development

As it is very relevant for what follows, I will start by saying a little about my overall perspective on development.

Development consists of the twin processes of both differentiation and integration. Though both are necessarily involved in all experience they are in fact qualitatively distinct requiring in turn distinctive methods of intellectual interpretation.

I have maintained for many years that in terms of cognitive translation, Ken Wilber substantially reduces the integral aspect of development to differentiated interpretation.

Therefore to avoid this confusion regarding the treatment of these two aspects, I define three approaches.

1. The (linear) asymmetric approach suited for interpretation of the differentiated aspect. This is based on the treatment of development variables as relatively independent.

2. The (circular) complementary approach suited for interpretation of the integral aspect. This is based on the bi-directional treatment of development variables as relatively interdependent.

3. The (binary) radial approach suited for dynamic interpretation of the combined interaction of both differentiated and integral aspects. [1]

Thus it is in the context of the radial that we can provide the appropriate perspective from which to appraise the limitations of the other two methods.

I will now briefly illustrate the nature of these approaches with respect to interpretation of the stages of development.

From the (linear) asymmetric perspective, stages are defined in a somewhat discrete hierarchical fashion i.e. as the movement from unambiguously defined lower to higher levels. For example in this context the lowest stages could be defined as prepersonal, the middle as personal and the highest as transpersonal. Then overall development is viewed as broadly moving from prepersonal through personal to transpersonal stages. This characterises very much Ken Wilber's general approach to development. Here integration is typically understood in a reduced (merely top-down) manner i.e. where lower stages are eventually subsumed in the higher.

However from the (circular) complementary approach all stages are defined in a continuous manner i.e. as simultaneously existing in some measure throughout development. In this context, the "lower" stages are complementary with the "higher" (and the "higher with the "lower"). The middle stages which are neither "higher" nor "lower" are thereby complementary with themselves. However what is neither "higher" nor "lower" is ultimately complementary with what is both "higher" and "lower". Thus full integration of the middle stages with both "higher" and "lower" can only take place after "higher" and "lower" have themselves been substantially integrated.

This explains why the radial stages (entailing both contemplation and activity) properly unfold following the attainment of specialised contemplative awareness (representing the integration of "higher" and "lower" stages).

Thus when development commences - what we identify in discrete terms as - the "lowest" is directly complementary with the corresponding "highest" stage. And because what is both "highest" and "lowest" is likewise complementary with neither "highest" nor "lowest" (i.e. the middle stages) this entails that we have here the greatly confused continuum of all stages (as mere potential for eventual actualisation). Thus starting development properly represents a state of greatly confused (circular) integration (due to corresponding lack of proper differentiation).

Because with earliest development, appropriate differentiation of structures has not yet commenced, therefore prepersonal and transpersonal aspects are necessarily intertwined (in an undifferentiated manner).

Likewise when spiritual development reaches its fullest expression - what we again would identify in discrete terms as - the "highest", is now directly complementary with the corresponding "lowest" stage (this time in a mature fully integrated fashion).

Because both types of structures have now been properly differentiated, they can thereby be likewise fully integrated with each other this time in a completely unattached manner. Put another way, Spirit (as transpersonal) can be fully integrated with the Bodyself (prepersonal) in top-down fashion; in like manner the Bodyself can be fully integrated with Spirit in - relative - bottom-up manner. So prepersonal and transpersonal once again have a purely relative meaning (in mature integral terms).

Then with radial development - where the refined multifaceted differentiation of phenomena can take place without secondary attachment - prepersonal and transpersonal can be given both a continuous (integral) and discrete (differentiated) interpretation that is appropriate for any given stage of development.

Circular relationships (suited for interpretation of the integral aspect) are bi-directional and paradoxical from a linear perspective.

Thus when we attempt to translate a bi-directional circular relationship in (reduced) linear terms, two equally valid interpretations can always be given. This means in effect that both transpersonal and prepersonal have twin meanings (which in dynamic terms unfold in opposite directions to each other).

From a spiritual perspective overall development can be defined in terms of two directions i.e. transcendent and immanent respectively. In terms of the transcendent one keeps going beyond - as it were - the more limited perspectives of matter, body, emotion, mind and soul to ultimately attain pure Spirit (as emptiness). Thus in this context pure Spirit is - literally without - created phenomena (as their final goal). However from the corresponding immanent perspective one equally keeps going within - in relative reverse manner - soul, mind, emotion, body and nature to ultimately experience Spirit (as essential form). Thus, in this context pure Spirit is inherent in all created phenomena (as their very source). (And in dynamic circular terms, both immanent and transcendent as source and goal have ultimately merely relative meanings which are interchangeable!)

Thus when we attempt to view development in hierarchical asymmetrical manner, we can give it two distinct spiritual directions (i.e. transcendent and immanent). From the transcendent perspective, development is generally characterised in holarchical terms as the movement towards ever more collective notions of wholeness ultimately culminating in pure nondual awareness i.e. from (confused) Bodyself to Spirit. [2]

From this perspective earliest prepersonal development is directly identified - in discrete terms - with the instinctive life of the body. Then with the (middle) personal stages, specialised mental capacities unfold. Finally with the (higher) transpersonal stages full spiritual awareness is attained. From this perspective therefore overall development is interpreted as the movement from confused notions of form (early prepersonal) to mature experience of emptiness (late transpersonal).

However from the corresponding immanent perspective it all looks somewhat different. By contrast development - in relative terms - is interpreted as the movement towards ever more unique notions of partness ultimately culminating in pure transparent matter (i.e. where the very essence of all created phenomena is revealed as Spirit). So here the direction is from (confused) Spirit to mind to mature Bodyself (i.e. where the Spirit can be made fully immanent in bodily form).

From this perspective therefore overall development is characterised as the movement from confused notions of emptiness (early prepersonal) to mature experience of form (late transpersonal).

Thus from a (linear) asymmetrical perspective, therefore we can give two equally valid interpretations of the spiritual direction of development (which are consistent within their respective independently defined polar reference frames). So once again the first interpretation is defined in terms of the transcendent whereas the second is defined - relatively - in terms of the immanent direction.

However when we view these reference frames as interdependent, then deep paradox results. So what moves in a positive (forward) with respect to one direction moves - relatively - in a negative (backward) direction with respect to its complementary opposite and vice versa. In other words in the dynamics of development, transcendent and immanent directions unfold in an opposite manner with respect to each other.

Thus an appropriate integral interpretation of development - dealing with the relative interdependence of variables - requires a bi-directional circular approach (based on the dynamic complementarity of polar opposites). This in turn requires a more subtle differential interpretation - dealing with the relative independence of development variables - where two equally valid (opposite) asymmetrical interpretations can be given in any context.

Ken Wilber from an Integral Perspective

When I look at Ken Wilber's overall intellectual approach I find it deeply flawed from this integral perspective.

For all his refinements and nuances - which quite frequently are inconsistent in terms of each other - he typically attempts to define concepts in a linear asymmetrical manner where the direction of development is unambiguously defined.

So in spiritual terms his forward direction is defined (solely) with respect to its transcendent aspect. Therefore when looked at from this (independent) context, development notions such as prepersonal and transpersonal appear to have unambiguously defined discrete meanings (i.e. lower stages are prepersonal; higher stages are transpersonal).

However as we have seen we can - and should - equally define the forward direction of development in terms of its corresponding immanent aspect. Once again in this (independent) context prepersonal and transpersonal appear to have unambiguously defined discrete meanings.

However when we relate both interpretations simultaneously with each other (as dynamically interdependent) deep paradox results. Then we realise that both prepersonal and transpersonal have in fact twin opposite meanings that are subtly contained in each other (through such interaction).

Thus the movement from linear asymmetrical understanding of a one-directional nature (suited for the limited differentiation of development) to an appropriate integral appreciation (where relationships are properly understood in an interdependent fashion) requires initially, bi-directional linear interpretation with respect to every developmental relationship.

Therefore an appropriate integral interpretation - as I define it - not only entails the bi-directional paradoxical treatment of all development variables (as interdependent) equally it entails a more subtle bi-directional approach to differentiation. This again requires that two equally valid asymmetrical interpretations be provided for every relationship conditioned by polar opposite aspects (e.g. prepersonal and transpersonal). Once again, despite his more nuanced interpretation, the basic nature of Ken Wilber's holarchical model can be accurately characterised. It is a one-directional asymmetrical approach greatly lacking in the dynamic treatment of interactive relationships and thereby quite unsuited for proper integral interpretation (based on the complementarity of opposite poles).

The issue I am raising goes to the very roots of what constitutes an appropriate integral approach (in intellectual terms) and therefore is of fundamental importance. Quite frankly the reduced cognitive nature of interpretation that is currently widely accepted in "Integral Studies" by its very nature cannot yield a consistent overall view of development.

For example though development entails two equally important spiritual (i.e. diagonal) directions, Ken Wilber defines the forward direction of his model exclusively in terms of transcendence, i.e. each higher stage transcends and includes the lower stage. [3] However as we have just seen the immanent direction (as forward) is equally valid.

Likewise though development entails two equally important vertical directions (holarchy and onarchy), Ken Wilber characteristically defines his model in terms of the holarchical i.e. where each (lower) whole is also part of a higher whole. However development can be equally defined in terms of its onarchical direction i.e. where each (higher) part is also whole (in the context) of lower parts. Emphasis on this other onarchical direction would lead to similar emphasis on both exclusion and inclusion with respect to the unfolding of stages and a more balanced approach to integration (that is both top-down and bottom-up).

Finally in heterarchical terms again all events entail both psychological and physical asymmetrical explanations, which - in interactive terms - unfold in opposite directions to each other. However once again Ken typically views such relationships from just one perspective e.g. in the manner he attempts to deal in physical terms with the atom, molecule, cell and organism as if somehow independent of mind.

So once again to sum up, the linear approach suited for differentiation is based on unambiguous asymmetrical connections as between development variables. From an overall integral perspective however this approach is very one-sided and unbalanced leading to many inconsistencies.

The circular approach - properly suited for the dynamic interpretation of integration - by contrast is based on the paradoxical bi-directional treatment of complementary opposite variables. In turn this approach is supported by a more subtle approach to differentiation, where two equally valid opposite interpretations are available in any developmental context.

The radial (or binary) approach, - which is the most comprehensive - combines both of these more limited perspectives (i.e. linear and circular) on development. Moreover the proper scientific version of this approach - using the tools of Holistic Mathematics - provides the precise configuration for the interaction of both linear and circular aspects for every stage of development. In other words in this scientific radial approach the configuration of linear and circular aspects (i.e. the holistic binary digits) precisely defines the very nature of each stage.

Perhaps it would be helpful to outline very briefly here the overall nature of development from this radial perspective.

Here we can distinguish 4 main bands, lower, middle, higher and radial (each with three main stages).

Life commences from a state of totally confused integration (before differentiation of structures has yet commenced). The lower band - which in dynamic terms is complementary with the higher - is then interpreted as the gradual movement away from this confused state of integration through appropriate differentiation of structures. Such development then culminates during the middle band with specialised development of linear (i.e. differentiated) understanding. Of course appropriate integration also takes place during these stages. However it is necessarily of a somewhat reduced nature and basically designed as a supportive framework for the increased specialisation of (dualistic) differentiated activity.

So therefore from a radial perspective the lower band is characterised by the movement away from confused circular structures of integration towards mature linear structures of differentiation.

The higher band is then formally characterised by the gradual movement away from mature (one-directional) linear stages of differentiation towards refined mature (bi-directional) circular stages of integration culminating in pure spiritual awareness. In this way "higher" development can be seen as complementing - in a mature fashion - the confused nature of "lower" development. [4]

The radial band is then finally concerned with the mature incorporation of the middle with both "higher" and "lower" stages (representing simultaneously the interaction of both the differentiated and integrated aspects of experience).

In turn the three levels of each band can be given a precise radial configuration (indicating the appropriate nature of both linear and circular aspects that characterises each stage). [5] One of the important implications here is that a unique interpretation of development is associated with each of the 12 major levels (as defined by this approach) which then applies to every concept in development. [6] Therefore in this context it makes little sense for example to talk of the pre/trans fallacy (implying that one definitive explanation applies for all development). In fact we now have 12 distinct types of pre/trans fallacies (representing the corresponding cognitive interpretation of each level).

Primary Perspectives

Now having outlined the basic nature of the linear (differentiated), circular (integral) and binary (radial) approaches it is time to say a little about - what Ken refers to as -primordial or indigenous perspectives. (I will refer to them here simply as primary perspectives!)

What I will do here is to first outline how this problem is dealt with through my holistic mathematical approach (which I have been refining now for over 30 years).

Again it is important to carefully distinguish the differentiation of distinct perspectives (as discrete) from their corresponding overall integration (where they are continuous with each other), before then incorporating both aspects as radial perspectives. Here the interaction of both aspects is preserved (in a manner appropriate for the stage of development in question).

The starting point for this treatment of perspectives is the dynamic nature in which quadrant locations keep switching in experience. For simplicity we will begin with just two locations interior and exterior respectively and - as conventionally understood - identify the interior (in subjective personal terms) and the exterior (in an objective impersonal manner).

Now given these two quadrant locations (Left-Hand and Right-Hand) we can initially generate two perspectives. Thus - using holistic mathematical language - we can only posit the interior (in a personal manner) with reference to the corresponding exterior (impersonal) perspective (which is thereby negated). So the interior is dynamically posited in the Left-Hand (+) through dynamically negating the corresponding exterior perspective in the Right-Hand quadrant (-).

Then in like manner we posit the exterior (impersonal) in the Right-Hand quadrant (+), by dynamically negating the interior (personal) perspective in the Left-Hand (-).

When we concentrate on what have been thereby posited (without reference to the dynamic interactive context), we have the differentiation of two distinct perspectives that can be unambiguously identified with each quadrant (i.e. personal interior with the Left-Hand and impersonal exterior with the Right-Hand respectively). This (merely) differentiated treatment typifies Ken Wilber's standard exposition of the quadrants. [7]

Thus - because of the lack of a proper interactive context - differentiation is based on interpreting each perspective as merely positive (i.e. with respect to its on state). However integration is based on mutually relating both perspectives as positive and negative in terms of each other (i.e. with respect to on/off states). In a dynamic context something remarkable then happens when we switch back to the differentiation of perspectives in each quadrant (from this overall integral perspective).

Remember we started by identifying the interior with personal experience (in subjective terms). However when we switch back to this quadrant (from the exterior), its location changes so that we now view such experience in an impersonal objective manner. Likewise through we initially identified the exterior with the impersonal (in objective terms) when we next switch back to this quadrant (from the interior), again its location switches so that it is now viewed in a personal subjective manner. Thus when reference frames switch - due to interaction with integral (nondual) understanding - the perspectives we associate with these frames likewise switch so that we can no longer unambiguously locate them or indeed quadrants in dynamic terms. (Truly from this interactive perspective what is Left-Hand is also Right-Hand and what is Right-Hand is also Left-Hand).

In other words in dynamic terms the perspectives keep switching in an on/off fashion between both locations. [8] If we take a static freeze frame as it were at any time, a perspective will appear to be unambiguously associated with either Right-Hand or Left-Hand. In other words if the Right-Hand is on (with respect to the perspective), the Left-Hand will be off. Alternatively if the Right-Hand is off, the Left-Hand will be on. However in dynamic interactive terms both Right-Hand and Left-Hand locations are necessarily involved for every perspective. [9] The importance of this is truly far reaching for these dynamic on/off states clearly suggest that a fundamental holistic binary system can be used to encode all interactions as between quadrants and thereby all primary (and indeed by extension composite) perspectives. So this holistic binary digital system that can potentially encode all transformation processes, complements (in dynamic terms) the well-known analytic system that can potentially encode information.

The holistic mathematical interpretation of the four quadrants is fundamentally based on this binary system. As it has the undoubted capacity to provide the most coherent scientific interpretation possible of development relationships, I will say a little about it here.

Remarkably all relationships - and thereby perspectives - can literally be expressed as a reduced form of 1 (in holistic terms). 1 in holistic terms is closely associated with form (as oneness is implicit in the very recognition of form) and 0 with nothingness (i.e. emptiness). With very minor modifications the very symbols we use to represent one and nothing (zero) are the line and circle respectively. So linear logic (based on unambiguous asymmetrical relationships) relates directly to the differentiation of form (1). Circular logic (based on paradoxical complementary relationships) relates - indirectly - to integration as emptiness (0). In other words true integration (i.e. interdependence) is nondual (where separate dualistic distinctions cease).

Two Sector Model

When we obtain the square root of 1 in analytical terms two results arise i.e. + 1 and - 1 respectively.

These are represented in geometrical terms by a circle bisected by its line diameter - with conventionally + 1 in the Right-Hand and - 1 in the Left-Hand sector. So this representation combines the notion of both line and circle. Equally the holistic interpretation combines the notion of both circular and linear interpretation. When we separate (i.e. differentiate) the lines from the centre we get two opposite linear interpretations of form. However when we combine both lines (i.e. integrate), they are enclosed by the circle. So the circular understanding is bi-directional and ultimately empty combining both positive and negative polarities (i.e. + 1 – 1 = 0).

However we can switch the frame of reference in both analytic and holistic terms. We could equally represent the two roots of 1 with + 1 in the Left-Hand and - 1 in the Right-Hand sectors respectively.

Now this is vital for appropriate dynamic appreciation where frames of reference continually switch.

So let us look briefly at the two unique sets of perspectives thereby generated.

Though we conventionally view the (interior) self as personal (Left) and the (exterior) world as impersonal (Right), in dynamic interactive terms both the self and the world have personal and impersonal aspects.

Certainly the (interior) self can feel in a personal subjective manner; equally the self can view itself in a detached impersonal way (as the object of thought).

Likewise though (exterior) phenomena can be scientifically dealt with in an objective impersonal fashion, equally they can be viewed in more personal subjective terms through affective type appreciation. So from a cognitive perspective we may relate to an "object" in 3rd person terms as "it". However - strictly - from an affective perspective - we relate directly to it in 2nd person terms as "you" [10] Poets for example typically reflect this more personal aspect of objects in nature!

Thus starting with our two quadrant positions Left-Hand and Right-Hand we find that we can generate 4 distinct primary differentiated perspectives (i.e. 2 X 2). In other words Left-Hand and Right-Hand quadrants can be associated with both personal and impersonal perspectives relating to the self and the world respectively.

And the switching in quadrant locations (creating distinct new perspectives) is brought about through the interaction with overall integral understanding (where opposite perspectives are understood as mutually complementary). So we can see that it makes little sense in dynamic interactive terms to try and unambiguously link Left-Hand or Right-Hand with distinct perspectives.

Thus to differentiate perspectives we must arbitrarily fix the frame of reference by unambiguously assigning polar directions. However with two poles this can always be fixed in two - equally valid - ways. So once again we thereby generate 4 distinct perspectives (from two starting poles).

Four Sector Model (The Four Quadrants)

(In experiential terms continual switching of opposite quadrant locations - horizontal and vertical - takes place. Therefore in a static freeze frame fashion we can give four distinct mappings for quadrant locations).

When we start with four quadrant locations (UL, UR, LL and LR) we can differentiate 16 (i.e. 4 X 4) primary perspectives.

In geometrical terms the four sectors (i.e. four quadrants) can be literally represented as a reduced expression of 1 (in both analytic and holistic terms).

Thus when we take the four roots of 1, we get two real + 1 and - 1 and two imaginary results + i and - i (where i is the square root of - 1).

Conventionally these roots are represented with the horizontal diameter representing the real and the vertical diameter the imaginary poles respectively. This suggests that the holistic notion of the imaginary number is vitally important from an integral four-quadrant perspective. As we shall see shortly it is fundamentally involved in all holon (and onhol) interactions. In other words we cannot properly explain in scientific terms the interaction between wholes and parts (and parts and wholes) without this holistic mathematical notion of an imaginary number. Conventionally in terms of the quadrants we start by fixing the UL with the individual expression of (interior) subjective personal experience. However we have seen that such experience can also be given an objective impersonal meaning. Thus we have the subjective "I" and the objective "I". However equally these (horizontal) aspects can be given both individual and collective meanings. In other words we can distinguish as between the (individual) private "I" and the (collective) social "I" (both of which can be given personal and impersonal interpretations). So the "I" that initially had one unambiguous meaning in the UL quadrant (i.e. as the (individual) personal "I" now can be seen to incorporate meanings associated with all four quadrants. Thus with four quadrants based on both horizontal (interior, exterior) and vertical (individual, collective) poles, we have four arbitrary ways of fixing locations. So depending on which designation is used, we can differentiate each of the four perspectives (as listed above) associated with the "I".

Likewise we could conventionally start in the LL quadrant with plural "We" as a collection of "Is". However once again this notion of "We" can be given both personal and impersonal interpretations. Likewise "We" can be viewed in (quantitative) individual terms (as the aggregate of separate "Is"). However equally it can be viewed in (qualitative) social terms (as what is collectively shared by a group of "Is"). So once again we can see that four distinct perspectives - representative of each of the four quadrants - are associated with "We".

If we start with the UR quadrant, in conventional terms it will be associated with the (individual) object as an "it". However as we have seen any such "object" can be given both personal and impersonal interpretations. This means in effect that strictly every "object" can be related to in 2nd person terms as "you". (In this sense a human being represents the most dynamic animated form of "object")

Likewise whereas the quantitative view of an "object" leads directly to its individual notion, its qualitative view leads to the alternative collective notion. In other words I can only identify an individual rock (in quantitative terms), if the rock likewise shares the collective quality of "rockness" (with all other rocks in its class). So once again we can see that four differentiated perspectives (entailing the properties of all quadrants) are associated with this quadrant (as arbitrarily defined through fixing of polar reference frames).

Finally we could conventionally start by fixing the LR quadrant with (collective) "its" (as the quantitative aggregate of individual "its"). However once again such "its" can be given both personal and impersonal interpretations. Likewise they have an individual meaning (as the quantitative aggregate of separate "its"). In fact this conventional view of "its" (as a collection of individual units) is not the appropriate collective interpretation (but rather a reduced version of it). The true collective meaning of "its" relates to the shared qualitative property which all "its" of a certain class must possess (in order to be identified).

Therefore to differentiate primary perspectives we must arbitrarily fix the polar frames of reference (with respect to both horizontal Left-Right and vertical Upper-Lower poles). Thus with each arbitrary designation we can differentiate 4 distinct quadrant perspectives.

However because these poles can be fixed in 4 different ways this means that overall we can differentiate 16 (i.e. 4 X 4) primary perspectives.

Thus it can thereby be seen - in this relative static type of approach - that all quadrant locations are in fact associated with each (arbitrarily fixed) location.

Then when perspectives have been properly differentiated in each quadrant (with respect to each arbitrary fixing of reference frames), appropriate integration can take place though the mutual complementary understanding of perspectives in opposite quadrants (horizontal and vertical).

I will just briefly comment on the significance of this in terms of my overall approach.

Initial spiritual development at H1 (i.e. psychic/subtle realm) is largely concerned with the integration of polarities (and thereby perspectives) in heterarchical terms (interior and exterior). Put another way, this equally relates to the mutual interpenetration of stages of self with stages of reality.

Spiritual development at H2 (causal realm) relates increasingly to the additional integration of polarities at a hierarchical level (individual and collective). It is at this stage the appropriate holistic mathematical interpretation of the four quadrants properly unfolds (where the imaginary notion is used). This requires the very subtle refinement of both the quantitative and qualitative interpretation of phenomena, which entails that both (analytic) conscious and (holistic) unconscious notions of holons interpenetrate in a very refined manner. From a holistic mathematical perspective, this requires that we now define holons in dynamic "complex" terms. So each "complex" holon is seen to represent the dynamic interaction of both a "real" conscious aspect (as holon) and an "imaginary" unconscious aspect (as corresponding onhol). Put another way this requires that the quantitative and qualitative aspects of holons - which are "real" and "imaginary" with respect to each other - be incorporated in scientific interpretation.

For example, from this new "complex" perspective, we can see how Ken Wilber's failure to properly recognise the true nature of "heaps" is due to a common form of reductionism. Ken unfortunately attempts to view a heap in "real" conscious terms as the quantitative aggregate of individual sentient holons. However a "heap" equally has an "imaginary" holistic qualitative meaning (as the indirect exterior counterpart of unconscious understanding) that cannot be reduced to its individual components. In fact the "individual" sentient holons require this shared collective meaning for their own identity. So in this context, if we fix the frame of reference initially with the individual holons as "real" then the corresponding heap is "imaginary" (as an onhol). However if we shift the frame of reference to the collective "heap" it now becomes the "real" holon (with the individual components as "imaginary" onhols).

In other words in dynamic interactive terms - from the perspective of H2 - all phenomena are defined (without restriction) in terms of "complex" holons, which keep switching between "real" (holon) and "imaginary" (onhol) status (both of which have a purely arbitrary meaning depending on context).

So we see here the extraordinarily important role of the holistic mathematical interpretation of the "imaginary" number (i.e. the square root of - 1) as necessary to satisfactorily explain the dynamic nature of the relationship between wholes and parts (and parts and wholes). Put another way, if we are to understand the dynamic interaction of both the quantitative and qualitative aspects of phenomena in any development context (without gross reductionism) we must incorporate both "real" and "imaginary" interpretation in scientific terms. From the corresponding psychological perspective this means that proper understanding of the relationship as between quantitative and qualitative aspects explicitly requires both (analytic) conscious and (holistic) unconscious appreciation of reality.

However once again, the precise nature of such "complex" understanding (i.e. "real" and "imaginary") can only properly unfold at H2, where the bi-directional nature of all hierarchies becomes fully apparent.

Once again - what is truly remarkable - is that all such appreciation is deeply implicit in the holistic geometrical representation of the four quadrants (as the four roots of unity).

In the analytic interpretation the real roots + 1 and -1 are represented on the horizontal axis (with conventionally the Right-Hand associated with the positive and the Left-Hand with the negative result respectively). Then the imaginary roots + i and - i are represented on the vertical axis (with the Upper positive and the Lower negative respectively). However this fixing of roots is just conventional as I say, and they can be validly represented in four different ways.

Now in the holistic mathematical interpretation of the four quadrants the real roots + 1 and - 1 represent opposite polarities of form conforming directly to conscious understanding. The imaginary roots + i and - i represent opposite polarities of form relating directly to holistic unconscious understanding (though indirectly expressed in conscious terms).

Thus in the dynamic interpretation of the quadrants we have the continual interaction of both "real" and "imaginary" aspects of form (with respect to both positive and negative directions) where quadrant locations continually switch as between their opposites (in horizontal and vertical terms). And as we have seen, associated with each arbitrary fixing of the four quadrants are 4 distinct differentiated perspectives. As there are 4 such arbitrary fixing arrangements possible, this leads to 16 primary perspectives.

The great value of the holistic mathematical approach is that it provides the precise rationale for both (linear) differentiation and (circular) integration of perspectives (with the quadrants literally representing the four-part division of the circle by horizontal and vertical lines). Put yet another way we can refer to this four-quadrant approach as the means of differentiating and integrating perspectives with respect to both (horizontal) stages of self and reality and (vertical) stages as state and structure. [11]

The Eight Sector Approach to Perspectives

(In the diagram the two diagonal lines - representing polarities of form and emptiness - combine both "real" and "imaginary" aspects as form in equal manner. However they also have an alternative interpretation in terms of emptiness as null lines = 0. From a dynamic interactive perspective the switching of opposite sector locations - horizontal, vertical and diagonal - continually takes place yielding eight unique freeze frame sector mappings).

However an even more comprehensive level of both differentiation and integration that can take place (relating to H3 understanding). Here we must adopt an eight-sector model (that goes significantly beyond the four-quadrant approach). In my opinion this is both necessary and sufficient (using both its linear and circular interpretations) to provide the fundamental modelling for all stages of development (and by extension all perspectives appropriate for every stage).

In the eight-sector model we define three sets of polarities.

Once again the horizontal (interior and exterior) operate heterarchically (within a given stage).

The vertical (individual and collective) operate hierarchically (between different stages).

The (fundamental) diagonal polarities (form and emptiness) operate simultaneously within and between all stages of development.

These polarities can once again be literally defined as a reduced expression of 1 (in analytic and holistic terms). Remarkably the diagonal lines can be given both a coherent interpretation as form and emptiness respectively (with that of emptiness symbolised by the diagonals representing null lines = 0).

Once more associated with each arbitrary arrangement of the eight sectors are eight distinct perspectives. However because these sectors can be arranged in eight different ways this means that we can differentiate 64 (8 X 8) distinct perspectives in all.

I will now briefly illustrate the nature of these 64 perspectives.

Each of the four quadrants (interior, exterior, individual and collective) which - as we have seen - continually switch locations as perspectives change, can now be further subdivided into two sectors by the diagonal lines bisecting the quadrants (representing both form and emptiness).

So if we start again with the conventional location of the "I" with the UL quadrant this now is subdivided in two sectors with additional interpretations in all sector locations as emptiness and form.

So starting with emptiness - which conventionally is associated with transcendent Spirit - each of the four meanings of "I" - that we already defined - reflecting the four quadrants, now have a empty (spiritual) interpretation as transcendent i.e. the pure (empty) spiritual archetype of the self. So we have the transcendent interpretation of the (individual) private "I" (as subject). We also have the transcendent interpretation of the (collective) social "I" (as subject). Then we have the transcendent interpretation of the (individual) private "I" (as impersonal object of thought). Finally we have the transcendent interpretation of the (collective) social "I" (again as object of thought).

However associated with each of these four spiritual interpretations (as emptiness) are also four phenomenal interpretations of form as the self viewed according to the masculine principle. There are two basic tendencies here. One can seek to move beyond the present notion of the phenomenal self through active control (representing the masculine) or alternatively relax back into acceptance of this present self through passive response (representing the corresponding feminine principle). So the masculine principle in the desire to go beyond the present notion of self would in this context be more closely associated with the transcendent aspect of Spirit. So therefore we also have here four phenomenal interpretations of self (as form) representing the masculine principle. So again the (individual) private personal "I", the (collective) social "I", the (individual) impersonal objective "I" and (collective) impersonal "I" (as social object) can all be experienced as phenomenal form (according to the masculine principle).

Now in the other sector of "I" we have eight additional perspectives, initially defined according to the aspect of form. So we have here the phenomenal "I" experienced according to the feminine principle again with respect to the four aspects i.e. the (individual) private personal "I", the (collective) social personal "I", the (individual) impersonal private "I" and the (collective) impersonal social "I". However associated with each of these phenomenal interpretations as form are four corresponding spiritual meanings (as immanence) So we have the immanent meaning of the individual personal "I" (i.e. where Spirit is made manifest through acceptance of one's physical bodyself), the immanent meaning of the (collective) social "I" (as personal), the immanent meaning of the (individual) private "I" (as impersonal) and finally the immanent meaning of the (collective) social "I" (as impersonal).

So we have now defined 8 perspectives for each of the two sectors (initially identified with the individual personal "I")

We can do the same with respect to the two sectors (initially identified in the LL quadrant with the collective "We") So again we have the four spiritual transcendent interpretations as applying to the (collective) "We" as qualitative personal identity, the (individual) "We" (i.e. as quantitative collective of individual personal "Is"), the impersonal objective "We" as applying in both the qualitative (collective) and aggregate (individual) sense. Associated with these are then the corresponding four interpretations of form (according to the masculine principle). In the other sector of "We" the feminine principle applies to the same four categories (as form) associated with four corresponding spiritual immanent meanings.

Moving to individual "it" which was conventionally defined with respect to the UR quadrant again we now subdivide into two sectors (of emptiness and form respectively). Here we had the (individual) impersonal "it" and the (collective) social "it" (both as impersonal). However these also had meanings as personal "you" and social "you" (through switching reference frames). So now these all have transcendent interpretations (as emptiness) and associated phenomenal meanings as form (according to the masculine principle). Then in the other sector all four categories are defined with respect to the feminine principle (as form) with associated spiritual (empty) interpretations as immanent.

Finally moving to the (collective) "its" conventionally defined with the LR quadrant again this is now subdivided in two sectors of emptiness and form. Again in four quadrant terms, this has four meanings as the (collective) qualitative property defining all "its", as the aggregate quantitative interpretation of (individual) "its", as the (collective) personal qualitative property defining all "its" (of an appropriate class) and finally as the (individual) aggregate of "its" (as defined as the sum of personal individual units). These four categories now have both a transcendent meaning (as emptiness) and a phenomenal interpretation of form (according to the masculine principle).

Then in the other sector the same four categories can be defined as form with respect to the feminine principle and in spiritual terms with respect to immanent interpretation.

From one extreme, the diagonal perspectives (in the context of both approaches) relate to basic psychophysical interactions (representing the simultaneous instinctive interaction of mind and body aspects). From the other extreme where body and mind are successfully integrated they represent the purest expression of both the transcendent and immanent aspects of Spirit.

In other words the purest attainment of Spirit - with respect to both its transcendent and immanent aspects as emptiness - requires that both the masculine and feminine principles of form be fully integrated (as represented by the diagonally opposite lines in each of the four quadrants). In earliest development (L3) body and mind remain greatly fused together. Insofar as experience takes place it is of an immediate (fleeting) instinctive nature (which cannot be held in memory due to the lack of differentiation entailed).

However finally - in complementary terms in my approach - at H3 we have the successful differentiation and corresponding integration of all primary perspectives with respect to (horizontal) stages of self and reality, (vertical) stages as states and structures and finally these more complex (diagonal) stages. From the perspective of form - in diagonal terms - we simultaneously now have psychophysical stages of (instinctive) body and mind and equally from the perspective of emptiness, Spirit (as simultaneously transcendent and immanent).

So once again the purest expression of Spirit (as transcendent and immanent) requires the most refined degree of instinctive behaviour (where body and mind become very largely differentiated from each other). In this manner Spirit can then be properly integrated with the Body (strictly Bodymind). Likewise the Bodymind can be integrated with Spirit.

So the key to successful integration at the higher levels is that a dynamic balance be maintained as between complementary opposite perspectives.

Firstly at H1 (psychic/subtle) this is largely achieved with respect to "real" opposite interior and exterior poles (i.e. where dynamic balance is maintained as between the personal and impersonal aspects - viewed in conscious terms - with respect to all primary perspectives).

Secondly at H2 (causal) this is largely achieved with respect to additional individual and collective poles which are now "real" and "imaginary" relative to each other. Here dynamic balance is maintained as between individual and collective aspects in terms of all primary perspectives. As we have seen this requires that unconscious (as "imaginary") be fully incorporated with conscious interpretation (as "real"). In this way states and structures are properly integrated. Quite simply when one attempts to understand development with respect to merely conscious interpretation then inevitably it leads to a reduction of the qualitative aspect (in terms of the quantitative) or alternatively the quantitative aspect (in terms of the qualitative). [12]

Finally at H3 (nondual) this is substantially achieved with respect to the fundamental poles of form and emptiness i.e. where dynamic balance is now maintained as between the instinctive psychophysical interactions of the Bodymind - combining conscious and unconscious (as both simultaneously "real" and "imaginary") - and the pure nondual experience of Spirit as void (with respect to both transcendent and immanent expressions).

So to briefly summarise:

1. The actual experience of differentiating perspectives entails both positing and negating. In other words we posit a perspective with respect to one quadrant location by negating its opposite pole (in horizontal, vertical and diagonal terms). This means in effect that for every perspective that is revealed a corresponding (mirror) perspective remains hidden which can only be revealed when the polar reference frame switches.

Thus if we fix - as in conventional terms - the UL quadrant with the individual expression of personal (interior), the LR quadrant with the individual expression of impersonal (exterior), the LL with collective expression of personal (interior) and the LR with the collective expression of impersonal (exterior) we dynamically posit each perspective in turn through negating the other perspectives (in horizontal and vertical terms). Though in static reduced manner we thereby recognise the differentiation of four distinct perspectives in this way (i.e. as posited), in dynamic terms all the quadrants are necessarily involved in each case (through corresponding negation). However by its very nature the very act of differentiation always entails a lack of the full dynamic context of experience. Put another full understanding with respect to form in its true dynamic context, entails the purest experience of (integral) nondual awareness.

2. Because the four quadrants can be arranged in four distinct ways this entails that four sets of distinct perspectives can be differentiated for each arrangement, giving 16 in all. However in a fully comprehensive eight-sector approach which includes diagonal polarities we can differentiate 64 (i.e. 8 X 8) primary perspectives.

3. The integration of primary perspectives entails combining positive and negative aspects in a dynamically balanced manner. This entails clear recognition that opposite perspectives (in horizontal, vertical and diagonal terms) are ultimately fully complementary. Therefore to fully experience any (single) perspective properly we must ultimately experience all fully. And as one begins to experientially realise the true interdependence of all perspectives (as form) one likewise attains to the true nondual experience of Spirit as pure emptiness.

Thus the complete integration of perspectives (based on the mutual complementary interaction of the differentiated primary perspectives) culminates in pure contemplative awareness of reality.

The integral aspect i.e. where opposite perspectives in horizontal, vertical or diagonal terms are viewed as complementary and thereby interdependent, is vital in terms of the switching of such perspectives. And the greater the integration the more perspectives can flexibly arise in a balanced manner in experience (without undue attachment). Where the emphasis is mainly on differentiation however, experience is likely to (a) become very rigid and (b) be largely confined to a composite mix based on a limited set of the primary perspectives.

4. All perspectives entail the mutual interaction as between on and off states (with respect to the fundamental horizontal, vertical and diagonal polarities). These can all be encoded using the holistic version of the digital binary system combining the linear logic of form (1) for differentiation and the circular logic of emptiness (0) - indirectly - for integration. The horizontal polarities entail "real" (conscious) interpretation in a linear (1) and circular (0) manner. The vertical polarities entail - relatively - "imaginary" (unconscious) interpretation in a linear (1) and circular (0) manner. The diagonal polarities entail simultaneously both "real" (conscious) and "imaginary" (unconscious) interpretation in full freedom of Spirit. Here dynamic interaction of opposite polarities of form approaches as close as possible to pure emptiness. So what is posited as form is successfully negated as emptiness the moment it arises in experience i.e. 1 - 1 = 0 (in holistic terms).

Composite Perspectives

Though the primary are indeed fundamental we can generate - from these primary - composite perspectives (without limit) of an increasingly indirect nature.

We can distinguish at least two types of composite perspectives.

The first is based on a combination of two or more primary perspectives (that are themselves experienced 1st hand).

Take for example phenomenology or existential philosophy. Fruitful work here would require authentic interior experience (of a personal subjective nature). However equally it would require the ability to objectively reflect on such experience. So we have - at a minimum - the combination of two 1st hand primary perspectives.

The second is based on the combination of primary experiences that can be 2nd, 3rd, 4th, 5th, …. nth hand.

For example I might give my objective impersonal view (3rd person) on some matter.

However I could equally attempt to give someone else's view e.g. Ken Wilber's. So I am now relaying a 2nd hand view (i.e. my interpretation of his interpretation of reality). Furthermore I could attempt to mix perspectives by giving my interpretation of his interpretation in the light of my own interpretation. However I could go further by attempting say to relay someone else's interpretation of Ken Wilber's viewpoint again mixing perspectives with respect to three different people. And this can go on without limit. Also a variety of individual and collective perspectives (of both a personal and impersonal nature) and a complex mix of such perspectives can be used to comprise the composite versions.

However composite perspectives - by their very nature - can create a considerable problem in terms of subsequent integration. In other words once we become attached to these perspectives it becomes increasingly difficult to unravel them for integral purposes. Therefore I would suggest that the task of true contemplative integration must necessarily be based initially on the reconciliation of primary perspectives through experiential recognition of their complementary opposite features. This would for example explain my own strong preference for basing views with respect to development mainly on 1st hand primary experience (rather than academic research which is necessarily of a secondary nature).

Anyone who is seriously striving to attain to pure contemplative awareness needs to base experience - initially at any rate - on direct primary perspectives.

Indeed such direct understanding of reality is what largely defines authentic spiritual development. However with the unfolding of radial reality it would then become possible to combine increasingly complex composite perspectives (as differentiated in terms of form) with the continual attainment of nondual (integral) spiritual capacity. Thus the full integration of the radial level would be more complete relating to the reconciliation of composite as well as primary perspectives.

So when we look at development in radial terms we can see how each stage is characterised by a certain unique way of achieving the differentiation of separate perspectives with an overall manner of then seeking holistic integration of these same perspectives. Thus each level is characterised by a set of differentiated perspectives and an overall holistic (integral) perspective (that typifies the stage)

Briefly the earlier prepersonal stages of the lower band (L3, L2 and L1) represent the movement away from a confused overall perspective (resulting from lack of sufficient differentiation with respect to primary elements). The personal stages of the middle band (L0, L0 H0 and H0) represent the mature differentiation of primary (and indeed composite) perspectives. The transpersonal stages of the higher band (H1, H2 and H3) then entail the movement to the mature integration of primary perspectives culminating in the pure overall perspective (without any particular perspective) of contemplative nondual awareness. The comprehensive stages of the radial band (R1, R2 and R3) finally represent the further flexible differentiation of both primary and composite perspectives, together with the continuing integration of these in a comprehensive overall holistic perspective through the permanent attainment of nondual spiritual awareness.

Notes

1. The conventional binary system is based on an analytic logic of clear separation i.e. 1 is not equal to 0; 0 is not equal to 1 The holistic binary system is based on a dynamic logic of complementarity and ultimate identity of these same digits: i.e. 1 - 1 (i.e. the dynamic interactive notion of oneness) = 0; 0 = 1 - 1. However the holistic system - unlike the analytic - requires both real and imaginary interpretations of the binary digits.

2. The transcendent direction can equally be associated with onarchical interpretation (as from a nondual perspective all such designations are merely arbitrary). However if we designate the forward direction with one aspect (e.g. transcendence) then - in relative terms - the other aspect will thereby have the opposite direction.

3. Though the holarchical (and onarchical) directions are vertical, the transcendent (and immanent) are diagonal. This is due to the fact that successful development in both cases entails both heterarchical and hierarchical aspects.

4. Development equally relates to states (as well as structures). Remarkably however it is possible to express states indirectly as structures through use of the holistic mathematical notion of the "imaginary". Thus states comprise "imaginary" - as opposed to "real" - structures. Equally, structures comprise "imaginary" - as opposed to "real" - states. So states and structures in development are dynamically "real" and "imaginary" with respect to each other. Though reality is "complex" in interactive terms, science properly recognises only the "real" aspect, which it literally identifies as "reality" (because of sole concentration on structure).

5. Briefly the 12 (main) levels are configured in the following manner. We start with totally confused circular integration in terms of the three sets of polarities (i.e. diagonal, vertical and horizontal). The three “lower” levels in the first band entail gradual linear differentiation (one by one) with respect to these three sets. The middle levels then entail the mature specialised (one-directional) differentiation with respect to the polarities. The “higher” levels then entail the gradual movement away from differentiation, with mature circular integration - in bi-directional terms - taking place (with respect to horizontal, vertical and diagonal polarities). This in turn entails the integration of “higher” and “lower” (and “lower” and “higher”) levels. Finally the radial entail the gradual mature integration and continuing differentiation of all “higher” and “lower” with the middle levels.

6. In fact it is even more multifaceted than this. Because from an integral perspective the structures associated with each stage are in continual transition and their nature thereby transformed (with reference to the understanding of a higher level) this entails that a whole series of enhanced versions are available for the cognitive understanding of lower stages. Thus as well as the interpretations associated with each new level we can likewise give enhanced interpretations of development from the understanding associated with lower levels (that are now transformed in the context of the enlarged perspective of the higher stage).

Thus for example with respect to formop, we have the formop interpretation of the formop stage. However equally we have enhanced interpretations of formop from the perspective of "higher" stages. So we also have enhanced vision-logic, subtle, causal, nondual and radial interpretations of formop.

7. I would not deny that Ken Wilber has an integral view of the quadrants e.g. where they are seen as mutually coarising with each other in experience. The clear problem however - certainly as viewed from my perspective - is that he fails to consistently demonstrate how this integral view (of mutually coexisting quadrants) can be successfully reconciled with the differentiated view (where each quadrant is seen in a partial - relatively independent - manner). This requires an approach that is dynamically interactive by its very nature (which Ken Wilber fails to adopt). So I would see here - as generally in his work - a considerable discontinuity as between his integral view (associated with nondual states) and his differentiated interpretation (associated with dualistic structures). A proper dynamic approach however requires the mutual interpenetration of both aspects.

8. Heraclitus was perhaps the first recorded Western thinker to properly grasp the nature of dynamic interactive activity “The way up is the way down; the way down is the way up”

From a nondual perspective all directional systems (through separation of polar opposites) are merely arbitrary. For example if we are lost in a desert and encounter a straight road we could arbitrarily assign one direction as “up” in which case the opposite direction would be unambiguously defined as “down”. However we could initially have used the opposite labelling system to define direction. Thus in terms of the second - equally valid system - what is “up” is “down” in terms of the first. Likewise what is “down” in terms of the second is “up” in terms of the first. Therefore when we view all arbitrarily fixed dualistic reference frames from a nondual (integral) perspective, they increasingly keep switching in a dynamic interactive manner in experience.

9. There are strong similarities here with quantum mechanics (e.g. with respect to the two-slit light experiment). Therefore though each individual photon of light appears to unambiguously go through one slit (when viewed in a freeze frame particle fashion), it appears subsequently to have travelled through both slits (when viewed from an overall wave perspective). This of course clearly demonstrates that the “higher” understanding of spiritual levels is necessary to interpret the dynamic interactive behaviour of corresponding “lower” physical levels.

10 As I hope to make clear in a later contribution, I do not favour the conventional 1st, 2nd and 3rd person terminology in the context of a scientific type approach to perspectives. Firstly on close examination it is full of ambiguities (reflecting the corresponding ambiguities of conventional understanding). Secondly, it does not directly lend itself to complementary type appreciation (which is the proper basis for integration).

11 In a direct sense the conscious aspect of understanding is expressed as structure and the unconscious aspect as the corresponding state (with respect to the "higher" spiritual stages of understanding). And of course there is continual dynamic interpretation of both conscious and unconscious aspects (as structure and state respectively).

However indirectly the "real" state can be interpreted as an "imaginary" structure; likewise the "real" structure can be interpreted as an "imaginary" state. So at the level of "complex" reality - where "real" and "imaginary" aspects interpenetrate in a relative manner - we thereby have the same interpenetration of both state and structure aspects.

12 Once again the interaction of both the quantitative and qualitative aspects of experience requires the corresponding incorporation of both state and structure aspects of understanding (or alternatively "real" and "imaginary" aspects with a merely relative meaning depending on context).

References

Collins, P. Genesis of Holistic Mathematics - Appendix to "Looking for Perspective" (Chap. 15 of ongoing work "Development - the Radial Approach") http://indigo.ie/~peter/radial15a.html

Miller, A.V. (Translator) Hegel's Science of Logic Humanity Books - 1998

Mc Dermott, M. (1998) Knowledge and the Knower; Complexity and the Self. This monograph is available for download at the following web address. http://www.lightmind.com/Impermanence/Library/texts/mikem-00.html

Nalimov, V.V. (1982) Realms of the unconscious: the enchanted frontier. (Robert G. Colodny, Ed.). Philadelphia: ISI Press.

Von Franz, M-L. Number and Time : Reflections leading Toward a Unification of Depth Psychology and Physics .Northwestern University Press; (August 1974)

Wilber, K. (2003) Appendix B (An Integral Mathematics of Primordial Perspectives) to Excerpt C "The Ways we are in this Together: Intersubjectivity and Interobjectivity in the Holonic Kosmos" from Volume2 of Kosmos Trilogy available at Ken Wilber online with web address. http://wilber.shambhala.com/html/books/kosmos/excerptC/appendix-B.cfm