Holistic Mathematics - Four Types of Complementarity

Recap on Holistic Mathematical Notions

Q Before we move on, I wonder if you could briefly recap on the holistic mathematical notions covered so far.

PC Certainly! We started from an arithmetical perspective to cover both the fundamental binary digits (1 and 0) and the mathematical operations of addition and subtraction.

We showed how - in holistic terms - 1 is implicit in the very notion of form and 0 in the notion of emptiness. As all development processes represent the interaction of form and emptiness (and emptiness and form) the holistic binary digits are inherent in all transformation processes providing the appropriate means for their scientific encoding.

We then demonstrated how these binary digits in turn are intimately associated with the dynamic operations of addition and subtraction (positing and negating).

A phenomenon - in any development context - must be (consciously) posited as unitary form for identification to take place. Likewise such form must be likewise (unconsciously) negated and voided for (spiritual) emptiness to occur.

So all development processes entail - in dynamic holistic terms - the continual positing (+) as form and corresponding negation (-) as emptiness.

We fundamentally identified differentiation (in any development context) as the positing of (unitary) form. This leads to the separation of one pole (which is thereby recognised) from the corresponding opposite pole (which is not recognised). In this manner, for example, we identify exterior phenomena as independent "objects" (without reference to the opposite interior pole).
Integration by contrast entails negation of what has been posited in experience leading to a dynamic notion of (spiritual) emptiness (as the absence of phenomenal form).
Integration - wherever it occurs - implicitly entails the complementarity of polar opposites as emptiness, though this may subsequently be confused with the identification of phenomenal form.

Next we were able to show an extremely close holistic mathematical connection as between the fundamental geometrical notions of line and circle and the binary digits and operations (of addition and subtraction).

So what is linear (in geometrical terms) can be directly identified both with the binary digit "1" (as unitary form) and the operation thereby of independent positing (+) leading to the separation of opposite poles.

What is circular (in geometrical terms) can be likewise be directly identified with the binary digit "0" (as empty void) and the operation thereby of negation (-) of what has already been posited as independent form. This leads to the complementarity (and ultimate identity) of opposite poles.

A linear approach - whereby experience with respect to opposite poles is dealt with in relative isolation from each other - leads to an unambiguous asymmetrical type treatment of development variables and is directly appropriate for the differentiated understanding of development relationships.

A circular approach - whereby opposite poles are understood as complementary (and ultimately identical) - leads to a paradoxical symmetrical treatment of variables and is directly appropriate for (intellectual) understanding of the integral aspect of development relationships.

So differentiation and integration are based on qualitatively distinct logical systems (which should not be confused with each other).

Development - wherever it occurs - necessarily entails the interaction of both its differentiated and integral aspects (as form and emptiness) respectively.
This requires a holistic binary interpretation (i.e. that combines both linear and circular modes of understanding).

By far the most common problem in intellectual translation is the attempted reduction of the integral to the differentiated aspect of understanding. Here integration is approached through a multiple composite of (partial) asymmetrical type systems of relationships. This necessarily leads to imbalance and inconsistency from an overall perspective.

We the went on to identify three distinct linear (and corresponding circular type) relationships as between opposite poles.

Firstly we have horizontal type relationships e.g. exterior and interior (and interior and exterior). This represented by the horizontal diameter line of the circle.

Secondly we have vertical type relationships e.g. whole and part (and part and whole). This is represented by the vertical line diameter of the circle.

Thirdly we have diagonal type relationships e.g. form and emptiness (and emptiness and form). This is represented by the twin diagonal line diameters of the circle.

The understanding of development with respect to opposite poles can always take place in both differentiated and integral terms.

In the differentiated approach opposite poles are separated (as positive). The understanding of development then occurs in linear asymmetrical terms with respect to one pole (which is temporarily fixed as reference frame).

In the integral approach opposite poles are viewed as complementary (and ultimately identical). Here the pole which has already been posited (through differentiation) is dynamically negated in spiritual terms (as empty void). The indirect phenomenal interpretation now takes place in circular paradoxical terms (which is the very means of eroding dualistic type distinctions). Thus, though the integration aspect is ultimately void and empty (as Spirit), indirectly it is associated with the paradoxical interpretation of form.

Basically three types of approach are possible with respect to both the differentiated and integral appreciation of development.

The first approach concentrates merely on the horizontal relationships (within a given level) as between exterior and interior (and interior and exterior) polarities.
We can define this as a two-circular or twin-circular approach (though strictly the horizontal diameter divides the circle into semi-circles).

The second - as well as dealing with the horizontal - concentrates on the vertical relationships as between whole and part (and part and whole) polarities.
We can define this - quite literally in holistic mathematical terms - as a four-quadrant approach.
(As I have stated before I have considerable reservations regarding the current four-quadrant approach associated with Ken Wilber. It is heavily based on the differentiated understanding of the four quadrants. It then misleadingly attempts to approach integration as a multiple composite of such fragmented quadrant understanding. However appropriate integral understanding is qualitatively very distinct and correspondingly requires a very distinct intellectual translation).

The third approach - as well as dealing with horizontal and vertical polarities - concentrates crucially on the twin diagonal relationships as between form and emptiness (and emptiness and form) polarities. As diagonal polarities combine the simultaneous recognition of both horizontal (within levels) and vertical (between levels) simultaneously, such diagonal understanding (with corresponding interpretation) entails the most truly integral understanding, that culminates in pure nondual awareness.
We can define this - again using holistic mathematical language - as an eight-sectoral approach.

Holistic Mathematical Interpretation of a Point

Q Have we enough at this stage to define a holistic mathematical TOE for development?

PC We will discuss the nature of various types of TOEs in considerable detail later. However in answer to your question we now have enough to define the most basic of - what I call - the integral TOEs.

Now a properly integral TOE requires that the methods of interpreting differentiation and integration be clearly distinguished using unique logical systems. Otherwise an "integral" approach entails considerable confusion - in intellectual interpretation - of the integral with the differentiated aspect of development.
In the context of my own approach I am attempting to refine the integral approach in a precise scientific manner (using as my tools the holistic appreciation of mathematical symbols).

However there is a considerable unresolved problem (which we will be addressing shortly). Indeed it took me many years to properly recognise the nature of this difficulty.

So far we have dealt with the basic issue of dealing with differentiation (as the linear separation of opposite poles) and integration (as their circular complementarity and ultimate identity).

Q Can I briefly interject again to raise an important issue. You keep mentioning the notions of circular complementarity and ultimate identity. In what sense are these distinct?

PC You must remember that integration always starts with the differentiated understanding of phenomena (where linear dualistic notions predominate).

So understanding the complementarity of opposite poles is a process by which we gradually recognise the paradoxes associated with the separated dualistic interpretation. So the actual process of developing complementary understanding always entails the mix of both dual (unambiguous) and nondual (paradoxical) understanding.

Now when this process reaches a considerable degree of refinement, dual linear notions rapidly dissolve (as soon as they are posited) so that the ultimate identity of opposite poles can be continually affirmed through a permanent state of nondual spiritual awareness.
This likewise represents the ultimate state of integration of these poles.

Carrying our holistic mathematical geometrical representation further, we can envisage a situation where the linear diameter continually shrinks leading to corresponding shrinkage of the circle until ultimately both have shrunk to the non-dimensional point at the centre (bindu). Though we must indirectly represent this point in finite dimensional terms, strictly it points to a non-dimensional reality. Likewise the ultimate state of integration (where opposite poles are perfectly reconciled) can be represented in holistic mathematical terms as a point - that literally points to a non-dimensional state of spiritual awareness.

So at the central point both the phenomenal notion of line and circle dissolve. Likewise at the central point of being (and reality), phenomenal linear and circular notions dissolve as empty non-dimensional spiritual awareness (though in dynamic experiential terms this is a state which can be approached rather than finally realised).
So the point (as well as line and circle) have very important holistic mathematical interpretations.

Distinctive Types of Complementarity

Q Thanks for the clarification! Can you continue on now with your clarification of the unresolved problem in relation to an integral TOE?

PC Yes! It is possible to recognise the three fundamental polarities of development (in horizontal, vertical and diagonal terms). It is also possible to recognise that every stage of development can be encoded in terms of the holistic binary system (defining a unique configuration of differentiation and integration for each stage). However this in itself is only sufficient for - what I call - an Integral 1 approach.

Q So what is an Integral 1 approach?

PC An Integral 1 approach is based on Type 1 complementarity as between opposite poles.

In geometrical terms, poles which can be represented in geometrical terms as at opposite ends of a line diameter are subject to Type 1 complementarity.

Thus - as we have seen - the horizontal polarities i.e. exterior and interior (and interior and exterior) can be represented as at opposite ends of the horizontal line diameter through the centre of the circle (of unit radius).

Likewise the vertical polarities i.e. whole and part (and part and whole) can be likewise represented as at opposite ends of the vertical line diameter through the centre of the circle.

Finally the diagonal polarities i.e. form and emptiness (and emptiness) can be similarly represented as at opposite ends of the twin diagonal line diameters through the centre of the circle.

In each of these cases the integral relationship as between opposite poles is defined in terms of Type 1 complementarity. So again the dynamic relationship between positive and negative polarities (in either horizontal, vertical or diagonal terms) is subject to Type 1 complementarity.

Q Is this not enough in considering the dynamic relationship as between these opposite polarities?

PC No! We have also to consider the relationship as between horizontal and vertical (and vertical and horizontal) polarities. These are represented in geometrical terms by the opposite end of line radii that are at right angles (90 degrees) from each other.
The proper understanding of these polarities - which requires very subtle appreciation - constitutes Type 2 complementarity.
The task therefore in terms of a precise scientific approach to development is to define Type 2 complementarity in appropriate holistic mathematical terms.

You may remember in our last discussion that we saw that the horizontal and vertical notions of the holistic binary digit "1" are quite distinct.

If we represent in horizontal terms "1" as representing an actual (finite) form as quantity, then - relatively - in vertical terms "1" represents a potentially (infinite) form as the quality i.e. dimension (to which the quantity is related).
Of course in dynamic terms we can always switch reference frames with the (infinite) dimension represented in horizontal terms and the related (finite) quantity in vertical terms.
And in the dynamics of understanding this switching of reference frames continually takes place!

Type 2 Complementarity

Q. You would maintain that these considerations are of the most vital significance?

PC Indeed! The fact is that - by definition - every development process necessarily entails Type 2 complementarity.

For example a perception (as finite) has no meaning without being related to its corresponding concept (as potential dimension applying to all perceptions within its class). Likewise - in reverse terms - a concept (as potentially infinite) applies to all perceptions to which it relates within its class.

So from a psychological perspective, conceptual understanding literally provides the dimensions (of space and time) thereby providing a framework for (actual) perceptual phenomena.
Because physical and psychological and physical domains are themselves complementary in integral terms, this implies that the very interpretation of objective phenomena, with respect to space and time, intimately depends on Type 2 complementarity.

As we have seen there are extremely close links as between the holistic geometrical understanding of the relationship between line and centre and the holistic arithmetical relationship as between the binary digits (which entails both holistic addition - as positing - and holistic subtraction - as negation).

So just as we can represent the horizontal, vertical and diagonal poles as line diameters of the circle in geometrical terms, equally we can represent these three poles in arithmetical terms (in terms of the binary digits of form and emptiness).

However to do this we need to define two distinct ways in which the number system can be represented in linear terms.

Q So if I attempt to summarise, Type 2 Complementarity is concerned with the manner of transformation in which - in relative terms - a quantitative perception (as an actual finite phenomenon) gives way to its corresponding qualitative concept (as a potential infinite phenomenon) or in terms of your binary approach how the holistic binary notion of 1 as representing a quantity of form (on the horizontal axis) gives way to the binary notion of 1 as a corresponding dimension of form (on the vertical axis)!

PC Alternatively of course it equally represents - in reverse manner - how the dimension of form (as concept) gives way to its related quantitative phenomenon (as perception).
(Also remember we discussed at length at an earlier stage how - with respect to all dynamic interactions - reference frames can be shifted so that perceptions equally have qualitative aspects and concepts have quantitative aspects respectively)!

We could equally say that Type 2 complementarity is concerned with the manner in which a hol-on (which is defined with reference to its whole aspect) dynamically switches to an on-hol (which is now defined in terms of its part aspect).
Likewise it explains - in reverse manner - how an on-hol switches to a hol-on.

Looking at it yet another way, Type 2 complementarity deals with the manner in which, in psychological terms, both conscious and unconscious (and unconscious and conscious) dynamically interact with respect to phenomenal experience.
From the corresponding complementary physical perspective, it explains how actual finite phenomena interact with the universal potential infinite ground of reality (and vice versa).

I define an Integral 2 approach as one that incorporates Type 2 with Type 1 complementarity in the interpretation of phenomenal relationships.

Type 3 Complementarity

Q What briefly is Type 3 complementarity?

PC This requires the most subtle kind of understanding that only properly unfolds when very refined cognition can interact with pure spiritual awareness in an extremely fluid manner.

There is in fact a close relationship as between the three types of complementarity and the "higher" stages of development.

Type 1 complementarity is associated with H1 (i.e. the subtle realm) which combines dynamic interaction of the complementarity of opposites with growing spiritual awareness so that conscious (rigid) attachment is considerably eroded.

Type 2 complementarity is associated with H2 (i.e. the causal realm) which combines the very refined appreciation of the dynamics through which the conscious and unconscious interact in phenomenal terms. This is likewise associated with a purer realisation of spiritual intuitive awareness so that now unconscious (rigid) attachment to phenomena is considerably eroded.

Type 3 complementarity is associated with H3 (i.e. nondual reality). This relates to even more intricate dynamics of the phenomena of form combining the manner in which both conscious and unconscious themselves are related in all phenomenal interactions with spiritual understanding. So at this stage there is no longer any separation as between the extremely refined understanding of phenomenal form on the one hand and pure unobstructed spiritual awareness on the other. Thus with the culmination of this type of awareness, form and emptiness (and emptiness and form) directly coincide. With this full unfolding, not alone does rigid attachment to both conscious and unconscious type phenomena largely cease but also any remaining attachment to the very process by which they interact!
In the mystical ascetical literature this would be referred to as the direct purification of the volitional aspect of the will (which serves to harmonise both conscious and unconscious).

In holistic mathematical terms Type 3 complementarity represents the manner in which the twin diagonal polarities i.e. form and emptiness (and emptiness and form) interact with both the horizontal polarities i.e. exterior and interior (and interior and exterior) and the vertical polarities i.e. whole and part (and part and whole) and - in reverse manner - how both the horizontal and vertical in turn interact with the diagonal polarities.

I thereby define an Integral 3 approach as one which is based on the extremely refined cognitive interpretation of how both conscious and unconscious phenomenal understanding interact with the purest realisation of spiritual intuitive awareness (and how in turn such spiritual awareness interacts with phenomenal understanding in both conscious and unconscious terms).

Type 0 Complementarity

Q And you believe that these three types of complementarity are extremely important!

PC Yes they are! In fact this notion of complementarity provides the most precise means for defining the structure of all the main stages of development.
However there is another type which I define as Type 0 complementarity. This really represents the absence of complementarity and is identical with linear understanding.

Putting it another way linear understanding - representing Type 0 complementarity - is complementarity with itself thereby tends to closing itself off from dynamic interaction (in horizontal, vertical and diagonal terms).

Overall Approach to Development

Q I know we will be going into his in more detail when you have clarified in holistic mathematical terms the precise nature of the various types of complementarity. However can you give us some indication of the overall nature of your approach to the stages of development?

PC  I use the term stage to refer to any sequence of development which can be defined in terms of a distinct holistic mathematical interpretation of the binary digits.

Because this is a radial approach, every stage is defined in a dynamic interactive context representing a unique configuration of both linear and circular understanding (enabling in every case proper interpretation of both the differentiated and integral aspects of the stage).
My most general classification of stages are Bands and my most specific Modes (though again in dynamic terms these necessarily interact).

We will be going into all this later in considerable detail. However just to give some indication of the comprehensive nature of scientific approach made possible through Holistic Mathematics, I will list the main categories of stages here (with in every case a binary interpretation of both the linear (1) aspect as differentiation and the circular (0) aspect as integration respectively.



Transitions between Levels


Mirror Stages


Modes - primary
            - secondary

Personality (Types)


As far as I am aware, this classification covers all the facets of other major stage approaches while doing so in a more refined manner (i.e. where both the differentiated and integral aspects are made properly distinct). It also covers other important areas necessary for a true dynamic approach such as mirror stages and dimensions which I have yet to see properly discussed in other methods.
In terms of this approach what are referred to as streams or lines of development would constitute just one aspect (i.e. differentiated) of the secondary Modes.

Q What about the levels (or planes) of reality, states and bodies - how are these handled in your approach?

Remember in this holistic mathematical TOE, not alone are all dynamic processes (e.g. stages) defined in binary fashion (i.e. linear and circular) but also with respect to the three sets of fundamental polarities (horizontal, vertical and diagonal)!

By applying the horizontal polarities i.e. exterior and interior (and interior and exterior) each stage can be viewed as a stage of reality (exterior) or - in relative terms - a stage of self.

From a (linear) differentiated perspective the overall Spectrum will manifest itself as either a Spectrum of Consciousness (stages of self) or alternatively a Spectrum of Nature (stages of reality). Of course in (circular) integral terms such divisions are rendered paradoxical. However the binary approach is equipped to deal with both interpretations.

Be applying the vertical polarities i.e. whole and part (and part and whole) each stage can be viewed either in terms of structure or in terms of an (underlying) spiritual state.
Again in (linear) differentiated terms, the overall Spectrum can be viewed either as a series of stages defined in terms of states or alternatively a series of stages defined in terms of (phenomenal) structures.
In (circular) integral terms states and structures (and structures and states) are dynamically interdependent. So once again the binary approach is designed to incorporate both perspectives.

Finally by applying the diagonal polarities i.e. form and emptiness (and emptiness and form) each stage can be viewed either in terms of (affective) bodies or alternatively - in relative manner - in terms of (cognitive) mind.
In (linear) differentiated terms we can then view the overall Spectrum as a set of stages (defined as bodies) or alternatively as a set of stages (defined as mind).
In circular integral terms, once again bodies and mind (i.e. cognitive and affective aspects) interact for all stages (ultimately coinciding as empty Spirit).

Q Can you briefly show how you use this binary approach to define your levels (which are largely synonymous in your approach with the "main stages" of development).

PC We cannot yet do this in a comprehensive manner as the mathematical encoding for the three (or four if we include Type 0) types of complementarity has not been properly defined.

However we can give some idea!

As this is a radial approach I define levels so as to lend themselves to both linear and circular interpretation. There are nine levels in all.

Linear Approach to Stages

From a linear perspective we have three lower levels - which starting from the lowest - I term L3, L2 and L1 respectively. Then we have one broad middle level (which is neither higher nor lower). So I term this level as L0 (alternatively as H0) or more commonly as L0, H0 (i.e. between lower and higher). The higher levels - in ascending order - are H1, H2 and H3. Then there are two radial levels R1 and R2

Using more conventional terminology, L3, L2 and L1 correspond to the prepersonal levels (archaic, magic and mythic respectively). L0, H0 corresponds to the personal or rational level (which includes conop, formop and vision logic). H1, H2 and H3 correspond to transpersonal levels (subtle including psychic, causal and nondual).
R1 and R2 correspond to - what is referred to in Christian mysticism as levels that include all other levels i.e. the unitive life as the more contemplative (passive) and transforming (active) varieties.
In a sophisticated linear approach we can recognise the levels alternatively as ascending stages of self (psychological) or ascending stages of reality (physical).
This arises from separation of the horizontal polarities within each level.

Equally we can recognise the levels as a series of ascending (phenomenal) structures or alternatively as a series of ascending states.
This arises from the separation of the vertical polarities between levels.

Finally we can recognise the levels as a series of ascending bodies (based on affective appreciation) or alternatively as a series of ascending minds (based on cognitive appreciation).

Circular Approach to Stages

From a circular perspective we have complementarity both within and between levels (in horizontal, vertical and diagonal terms). This complementarity is of four kinds Type 1, Type 2, Type 3 and Type 0.
Indeed the linear classification which we have just listed represents the application of Type 0 complementarity.

In horizontal terms, each level has both a physical (as stage of reality) and a psychological expression (as stage of self) which are - relatively - complementarity as the exterior and interior (and interior and exterior) aspects of the levels.

In vertical terms, each level has both a (phenomenal) structure and a (spiritual) state which are - relatively - complementarity as the higher and lower (and lower and higher) aspects of the levels. So L3 is complementary with H3 (and H3 with L3); L2 is complementary with H2 and H2 with L2; finally L1 is complementary with H1 and H1 with L1. What this implies is that prepersonal confusion at a "lower" stage (e.g. L1) is a precondition for the subsequent emergence of transpersonal maturity at the corresponding "higher" stage (i.e. H1). Likewise transpersonal maturity at the "higher" stage (i.e. H1) is a precondition for unravelling the prepersonal confusion at the corresponding "lower" stage (L1).
So in dynamic terms both the structures and states (and states and structures) of complementary levels, continually interact with each other in a vertical fashion.

In diagonal terms each level has both an (affective) body and a (cognitive) mind which are relatively both horizontally and vertically complementary to each other.
So, for example, the affective understanding of the exterior aspect of a "lower" stage such as L2 is complementary with the cognitive understanding of the interior aspect of the corresponding "higher" stage H2 (and of course vice versa).

Now such complementarity is of the Type 1 variety entailing straight line opposite polarities (in horizontal, vertical and diagonal terms).

This enables integration of H1 with L1 (and H1 with L1); L2 with H2 (and H2 with L2); L3 with H3 (and H3 with L3) and L0 with H0 and H0 with L0 (which of course entails that the middle level is complementary with itself).

Type 2 complementarity enables the integration of H1 and H2 (and L1 and L2) and then the combination of Type I and Type 2 complementarity leads to the integration of these corresponding levels i.e. "lower" and "higher" (and "higher" and "lower") with each other.

Type 3 complementarity enables the integration of H3 (with H1 and H2) and L3 (with L1 and L2). The combination then of Type 1, Type 2, and Type 3 complementarity enables the top-down integration in turn of the three "higher" (H1, H2 and H3) with the three "lower" (L1, L2 and L3) and also - in reverse fashion - the bottom-up integration of three "lower" (L1, L2 and L3) with the three "higher" (H1, H2 and H3).

The full integration of the Spectrum entails the combination in turn of Type 1, Type 2 and Type 3 with Type 0 complementarity (and - in reverse - Type 0 with Type 1, Type 2 and Type 3).

This leads to the full emergence of the Radial levels which entail the relative interdependence of all levels (integration) as continuous with the relative differentiation of each level as discrete.

Here I distinguish R1 and R2 (i.e. Radial 1 and Radial 2).

With R1, the more integral complementarities (i.e. Type 1, Type 2 and Type 3) dominate to a degree their relationship with Type 0 (which represents the most differentiated form of understanding).
This therefore entails a somewhat passive contemplative engagement with phenomenal reality.

With R2, Type 0 is brought into proper balance with Types1, 2 and 3 leading to a more dynamic transforming engagement with practical affairs that is equally fully integral (contemplative) and fully differentiated (active).

Overall Dynamics of Differentiation and Integration in Development

Q Can you describe in general terms how the differentiated and integral aspects typically unfold in development.

PC At the beginning of life, as no differentiation has yet taken place equally there can be no integration. So we have here a totally confused integral state (with the commencement of L3). (This state - where no integration has taken place is this complementary with the opposite state of total integration i.e. the completion of H3).
Of course in a dynamic context these always have a relative rather than an absolute meaning. So total integration represents a state which can be approximated but never fully attained (while one still lives a phenomenal existence)!

The first task of development is to emerge from this totally confused state of integration (with respect to the three fundamental polarities) and gradually achieve successful differentiation. This task broadly defines - what in linear terms - we can refer to as the prepersonal levels (L3, L2 and L1).
With respect to stages of self the unfolding of L3 (archaic) is largely concerned with the initial separation of the diagonal polarities i.e. emptiness and form (and form and emptiness).
So for example the infant gradually learns to differentiate the bodyself (as form) from the merely potential initial spiritual state (as emptiness).

An equivalent unfolding at L3 (archaic) with respect to the stages of reality is concerned with the gradual differentiation of (physical) forces from the an inchoate state of matter-emptiness.

Now with sufficient consolidation of this form of differentiation, the next task - which defines L2 (magic) - is the gradual differentiation of the vertical polarities i.e. the whole aspect of reality as distinct from part (and part from whole).
This relates to the differentiation of the emotional self and culminates in a distinct - though yet primitive - notion of a separate personal identity.

The equivalent unfolding with respect to the stages of reality entails the gradual separation of matter phenomena from dimensions (though they still remain to a degree intertwined). This corresponds to the level of string (or membrane reality) where it is recognised that dimensions are in a sense contained in the strings. This culminates in the first manifestly observable particles.

Again given sufficient consolidation of this differentiated form, the task defining L1 (mythic) is the differentiation of the horizontal polarities i.e. the exterior aspect of reality as distinct from the interior (and the interior from the exterior). This leads to the gradual emergence of a mental self that can think in abstract terms.

The equivalent unfolding with respect to the stages of reality is the increasing development of natural forms in sub-atomic particles which  have not yet obtained a fully independent status.

Though integration is necessarily associated with development at each of these stages it is in a context which is mainly concerned with the successful (independent) differentiation of forms.

The culmination of this differentiation with resepct to horizontal, vertical and diagonal polarities takes place at the middle personal level i.e. L0, H0 (which is neither prepersonal nor transpersonal).

In particular this leads to an independent ego identity that is considerably defined in terms of its mental ability to abstractly engage in rational manner with the environment.

This differentiation of the ego self is in turn - with respect to the corresponding middle stage of reality - is associated with a world of stable forms that likewise have now assumed an independent existence as separate "objects"

So the prepersonal levels can be viewed as a movement away from the initial state of totally confused integration towards a properly differentiated appreciation of reality.
This then culminates with the middle (personal) levels which represents the specialisation of differentiated type development.

Though integration necessarily also takes place at the earlier and middle stages, it is of a largely reduced nature that accommodates itself to the differentiation that has already taken place. Thus the integration of the centaur - entailing Spirit, mind, emotions and body (representing the culmination of the middle level) is necessarily somewhat limited .
Proper integration requires development at the higher levels.

H1 (subtle) entails the horizontal integration of this level (as a stage of self) with the corresponding level (as a stage of reality).
The implications here are that the self and the world are now viewed as dynamically complementary. Thus the self has no meaning apart from the world; likewise the world has no meaning apart from the self. So rather than being viewed in largely separate terms they are now viewed as ultimately completely interdependent (through their identity as Spirit).
Thus from one perspective our scientific appreciation of reality is inseparable from the psychological constructs (through which we view this reality). Likewise the self is inseparable from its interactions with (exterior) reality.

As horizontal and vertical polarities are themselves implicitly interdependent, a limited amount of vertical integration also takes place so that H1 is integrated with L1 (top-down) and L1 with H1 (bottom-up).
Such integration is based on Type 1 complementarity.

H2 (causal) entails from a top-down perspective the horizontal-vertical type integration of this level (as state) with the corresponding "lower" level L2 (as structure) and also - in reverse fashion - the integration of the "higher" level (as structure) with the "lower" level as "state".
Equally from a bottom-up perspective it entails the horizontal- vertical type integration of the "lower" with the corresponding "higher" level (alternating as between states and structures).
Such integration is based on Type 2 complementarity.
When combined with Type 1, top-down integration of both H1 and H2 with L1 and L2 and also bottom-up integration of both L1 and L2 with H1 and H2 is possible. Also significant integration of both H1 with H2 (and L1 with L2) can also occur.

One of the implications of this type of integration is that the structure of a "lower" level can only be properly interpreted from the perspective of the corresponding "higher" level. Thus for example a prepersonal stage of self such as L2 (magic) can only be properly understood from the corresponding perspective of the H2 (causal).
Equally the confused nature of L2 points to the corresponding "higher" level for its proper transcendence.

Likewise a prepersonal stage of reality such as L2 (which is the level of string reality) can only be properly interpreted from the corresponding transpersonal structure/state level of reality of H2 (causal). Thus the integral scientific interpretation of H2 i.e. Integral 2, is required to properly understand in a holistic fashion the corresponding physical relationships of L2.

H3 entails the most refined simultaneous integration of the complementary appreciation of form (Type 1 and Type 2) with the pure interaction of Spirit (which characterises Type 3 complementarity).

This then enables integration of the "highest" level H3 with the "lowest" level L3 (and likewise the integration of the L3 with H3) in diagonal terms.

What this entails that integration takes place with respect to opposites in both horizontal and vertical terms. Thus - for example - the exterior expression of H3 (as stage of reality) is integrated with the interior expression of L3 (as stage of self), though in truth very little separation of opposites now remains.

Also there is further complementarity, in that the mental expression of the one level (cognitive) is associated with the body expression of the other (affective).
Thus - with respect to our example - if the exterior expression of H3 as stage of reality is in mental (cognitive) terms, then the corresponding interior expression of L3 as stage of self will be - relatively - in body (affective) terms.
And of course both corresponding "higher" and "lower" levels as stages of self and stages of reality equally have both body (affective) and mental (cognitive) interpretations.

This perhaps helps to explain why the bodyself - which is first to be differentiated in development is the last to be properly integrated with both mind and Spirit.

Thus by combining Type1, Type 2 and Type 3 complementarity, substantial integration of the "higher" (H1, H2 and H3) and the "lower" (L1, L2 and L3) stages and the "lower" (L1, L2 and L3) and the "higher" (H1, H2 and H3) can now take place.

Thus integration culminates in a pure passive contemplative type awareness (where very refined appreciation of phenomena of form interacts with spiritual emptiness).

However one big problem that remains is that the middle level (i.e. L0, H0) - based on Type 0 complementarity (representing specialised differentiation of phenomena) is not yet properly incorporated in development.

Thus mature radial development relates to the manner in which L0, H0) is itself both properly integrated and also differentiated with the other levels.
This culminates at R2 where the relative interdependence of all levels with each other (as integration) co-exist with their relative independence (as differentiation).

Thus R2 in its most complete expression leads to a life that is both fully contemplative (spiritually) and also fully active (in practical terms).

R1 would characterise a state where active (dualistic) involvement is successfully incorporated with contemplative awareness in a - somewhat - secondary manner (so that contemplative orientation still - relatively - dominates).

R2 would then entail more successful balance in terms of both aspects.

Summary in terms of Binary System

Q Can you briefly summarise this in terms of your binary system?

PC Once again, we are not yet in a position to properly encode all key features of development in holistic mathematical terms.

However in general terms we can say that all development starts from a totally confused holistic binary state where (unitary) form (1) is confused with the void of emptiness (0).

The earlier stages of development are largely geared to the movement away from an initial totally confused sense of integration (0) to a properly differentiated appreciation of form (1) with respect to the three fundamental polarities (in diagonal, vertical and horizontal terms).

This culminates at the middle level (L0, H0) with successful differentiation - in one-directional terms i.e. with respect to the - merely - positive direction of form (+).

The three higher levels are then largely concerned with the recovery of the hidden negative direction (-) and the successful incorporation of this in dynamic terms with the corresponding positive direction through the complementarity of opposites.

Thus integration as pure spiritual emptiness in mature terms (0) is dynamically approached through the complementarily of the opposite directions of form i.e. + 1 - 1.
This complementarity is thus achieved with respect to the three fundamental polarities (horizontal, vertical and diagonal) which it its most advanced expression culminates in pure integration as contemplative awareness (0).

Radial development is then concerned with the mature combination of both holistic binary digits (1 and 0) leading ultimately to balanced experience that is both fully contemplative as nondual awareness (0) and fully active as refined dualistic form (1).

Q Just one final question at this stage! Presumably in the way you define your stages you do wish to convey the impression that all development proceeeds in a smooth predictable fashion in the manner outlined.

PC The key point about the radial approach is that the discrete experience of any particular stage is always - by definition - combined with the continuous experience of all stages.

Thus though one at a given point in development may well identify mainly with the understanding appropriate to a particular stage, this will necessarily interpenetrate to a degree with the understanding of all stages.

Thus I would like to keep open a wide number of possible ways in which the mature development of stages can be acquired. Indeed this would be a key feature of recognising distinctive personality types where development will tend to display unique patterns (characteristic of a certain type).

However we can say in a qualified universal manner, that when the dynamics of development are misunderstood, blockages to the proper unfolding of stages will inevitable occur.
A key feature of my own approach is that it attempts to make explicit the manner in which such misunderstanding takes place.

Like all approaches, this one can only be properly appreciated in the context in which it is written. It is deliberately designed as a scientific approach that makes clearly explicit - in holistic mathematical terms - the structural features that are associated with every stage (and their corresponding relationship with other stages).

However such a scientific approach - even in its holistic integral sense - provides but one perspective on development and in no way excludes the need for other distinctive styles that can perhaps better deal with other important aspects of experience.

However having said this the scientific approach - certainly in the manner that I define it - has been greatly neglected. Therefore I view my own efforts as a much needed gesture to bridge a very important gap in our present understanding of development.