Holistic Mathematics - Differentiation and Integration

Differentiation

Can you now explain the fundamental nature of both differentiation and integration respectively?

PC Differentiation is always based on the dualistic separation of opposite poles in development.
Though in dynamic experiential terms we can only posit one pole through the simultaneous negation of its opposite, when differentiation takes place we interpret this relationship solely in terms of the positive aspect.
In this way the pole becomes separated (from its opposite) and thereby differentiated in experience.

With respect once again to the horizontal polarities i.e. exterior and interior (and interior and exterior) we will explain - using holistic mathematical language - how differentiation takes place.

At the moment while I type, there is a cup on my computer table. The very identification of this cup as an exterior phenomenon requires that I consciously posit it as separate from its corresponding interior pole. I thereby give it an "objective" identity as if existing independent of self.
Then when I become aware of self (in opposition to this cup) this is now consciously posited as an interior phenomenon (that is thereby separated from the corresponding exterior pole).

Q Can I briefly interject here! Presumably the separation of opposite poles here equally entails the separation of conscious and unconscious aspects of understanding. What happens therefore to corresponding unconscious understanding when we unambiguously posit phenomena in conscious terms?

PC Excellent point! The rigid positing of phenomena (in conscious terms) leads to their corresponding rigid negation (in unconscious terms).
In fact a basic rule applies here. The more rigid - and thereby unbalanced - the conscious positing of phenomena that takes place, equally the more rigid and unbalanced their corresponding unconscious negation. (Rigid unconscious negation in this context represents the failure to recognise the complementary role of the unconscious!)

Quite simply when the complementary role of the unconscious (as negation) is not recognised, repression occurs (in unconscious terms).
This repression thereby restricts the freedom of the unconscious to properly complement the conscious as regards phenomenal understanding.

Now indirectly the repressed - and thereby restricted unconscious - attempts to achieve balance through projections which, though veiled in conscious form, really speak of deeper unconscious needs.
However initially at any rate such projections are likely to be misunderstood and can only be addressed when they become so troublesome as to seriously threaten one's conscious identity.

We will have a lot more later to say regarding the holistic mathematical nature of unconscious projection.

So the refining of conscious understanding thereby equally requires the refining of unconscious understanding.
With such refined understanding - which necessarily interacts with spiritual contemplative awareness - differentiation still takes place (in an enhanced manner) but the temporary imbalance involved, due to separation of polar opposites, is quickly corrected thereby restoring harmony as between polar opposites (and also thereby as between conscious and unconscious).

So once again when the exterior is differentiated in experience it is posited and separated from its corresponding interior aspect.

Likewise when the interior aspect is differentiated it is posited and separated from the corresponding exterior aspect. Thus the phenomenal experience of a somewhat rigid object, gives way to the phenomenal experience of the rigid self.

In the very dynamics of experience, rigid experience with respect to one pole tends to be associated with similar rigidity with respect to the other pole.
In this way the capacity for creative insight is greatly limited as switching between poles takes place in a predictable fashion.

Again by extension differentiation with respect to all fundamental poles takes place in the same manner where each is (consciously) posited without due recognition of corresponding (unconscious) negation of its opposite.

Thus in holistic mathematical terms (+) 1 + 1 = 2

So with differentiation each phenomenal pole (as unitary form) is separately posited in a relatively independent manner thereby leading to "twoness" or duality.

Integration

Q This is very interesting. You are using a very simple holistic mathematical relationship (comprising the most basic symbols with respect to numbers and operations) to define the universal nature of differentiation - and thereby duality - everywhere in development. Can you do the same now with respect to integration?

PC Yes! I think you might have already guessed how this is done.

Whereas differentiation is based on the relative separation of opposite poles in development, integration is based on their corresponding complementarity (and ultimate identity).

So integration represents the attempt to relate both the positive and negative directions of the phenomenon.

Once again in dynamic terms the positing of one pole requires the corresponding negation of its opposite. So the very recognition of this, leads to their complementary recognition (where their mutual interdependence is recognised).

So for example, we attempt to posit an exterior phenomenon and realise that this can only be done with respect to the corresponding negation of the interior self.
In reverse fashion we try to posit the interior phenomenal self and realise that this can only be done with respect to corresponding negation of the exterior phenomenon.
So rather than both poles remaining separate (and merely positive) we thereby recognise their necessary interdependence (as both positive and negative).

For example if we go back to our example of the cup, from an integral perspective the very positing of this phenomenon in (conscious) exterior terms implies the corresponding negation of the opposite pole (self) in (unconscious) interior terms.
Likewise from the other integral perspective the very positing of the phenomenal self in (conscious) interior terms implies the corresponding negation of the opposite pole in (unconscious) exterior terms.

Now this fusion of positive and negative (which equally represents a fusion of conscious and unconscious) leads to cancellation of both polarities (as separate) leading to a certain realisation of intuition or spiritual emptiness (which is nothing in phenomenal terms).
Thus integration - wherever it occurs - always represents a spiritual intervention in development leading to the transformation of phenomenal form.

The process by which integration takes place is however never perfectly realised and generally remains largely implicit in development processes. Also because it essentially remains hidden (as the realisation of spiritual emptiness) it is generally confused with differentiated phenomena.

However the key point I am making is that integration involves a distinct logic which should not be confused with differentiation.

So in holistic mathematical terms 1 - 1 = 0
Thus the proper (unconscious) negation of what has been (consciously) posited as (unitary) form leads to the realisation of nothing (as spiritual emptiness).

Likewise in reverse form 0 = 1 - 1
So the realisation of nothing (as spiritual emptiness) leads to the (conscious) positing and corresponding (unconscious) negation of unitary form.

So in this manner, in interactive terms, (unitary) form continually gives way to spiritual emptiness (which is nothing in phenomenal terms); Likewise spiritual emptiness (which is nothing in phenomenal terms) gives way to unitary form.

In the dynamics of experience both differentiation and integration necessarily take place in both a relatively independent and also (relatively) interdependent manner.

Thus before a phenomenal pole can be successfully negated by its opposite in an integral manner (i.e. as interdependent) it must first be successfully posited in a differentiated manner (as independent).

Likewise before a phenomenal pole can be successfully differentiated as positive, its necessary complementary integral interdependence with its opposite must be recognised.

Thus differentiation is necessary for successful integration; however equally integration is necessary for successful differentiation.

However considerable imbalance as between these two aspects may be evident in practice with typically the (merely) differentiated aspect greatly dominating.
When this is the case integration insofar as it occurs, acts to accommodate the unbalanced differentiated perspective.
This accurately describes the nature of adult development in our modern culture where differentiated notions of phenomenal meaning greatly hold sway.

Q So differentiation and integration are both necessary and as you have shown dynamically interdependent (though also to a degree which greatly varies also independent of each other).
Differentiation always leads to duality (which results from the phenomenal "illusion" that arises when opposite poles are separated).
However integration leads to nonduality i.e. spiritual emptiness where the illusory phenomenal nature of phenomena gives way to nothingness.
Actual experience always entails a unique configuration of differentiation and integration i.e. duality and nonduality (though the quality of this interaction can vary greatly).

Again I can see that by starting with the two most basic numbers 1 and 0 and the two most fundamental operations + and -, by giving them their dynamic holistic mathematical interpretation, you have been able to derive the fundamental nature of differentiation and integration in a precise manner which universally applies to all development processes!

PC And we have also generated the important number 2 (with its corresponding holistic mathematical interpretation) in the process!

Q Before we go on I am still a bit confused on one point. You claim you are dealing with Holistic Mathematics. However in every case you are pairing this holistic interpretation of symbols with the corresponding analytic interpretation. So does this not represent Radial Mathematics as you define it?

PC Strictly speaking you are correct. The very basis of the comprehensive approach I define as Radial is that every mathematical symbol and relationship dual interpretations can be provided (in both analytic and holistic terms).

However the main purpose of the exercise at this stage is to derive the holistic mathematical tools required so as to successfully encode the overall development process.

Q I know you have been greatly influenced by the writing of St. John of the Cross. Can you show any parallels from your own approach?

PC In his works St John deals - perhaps more intensively than any other writer - with the process of dynamic negation i.e. nada in experience.

All of St. John's various "nights" vividly describe the erosion of the successive layers of phenomenal self (and the associated world) through dynamic negation before finally realising the fulfilment of spiritual emptiness.

Not surprisingly St. John lays strong emphasis on the need for detachment from phenomena. Though the problem of possessive ego attachment ultimately resides in the will it is always associated with a limited identification with the merely posited (differentiated) nature of phenomena which - by definition - is necessarily unbalanced.

As we have integration requires the corresponding negation of such independently posited phenomena and thus for its fullest development entails the intensive purgation that accompanies the authentic mystical quest.

So once again the problem arising from rigid ego attachment is primarily of a volitional nature, it is always associated with a distortion in cognitive understanding. Thus the unravelling of this distortion at a cognitive level can greatly assist volitional assent to the pure ultimate good (which is spiritual).

Geometrical Connections: Line and Circle

Q We have now dealt with fundamental arithmetic notions in a holistic mathematical context. Let's move on some fundamental geometrical concepts such as the line and circle. You believe that you can draw intimate connections here - in analytic and holistic terms - with both the binary digits and the operations of addition and subtraction!

PC The first clue as to their intimate connections comes from examining the very symbols we use for the binary digits. Now 1, with only the most minor modifications is a straight line; 0 - again with minor modification - is a circle.
The process through which certain symbols are chosen to commonly represent numbers or operations is itself a very interesting topic. However suffice it to say that very often symbols that are universally adopted are very rich in the holistic meaning inherent in the mathematical notions they represent.
So in this instance the (straight) line is very closely associated with the binary digit 1.
Likewise the circle is very closely associated with the binary digit 0.

Now let us look at the figure of the circle (with radius = 1) bisected by its (horizontal) straight
line diameter.

Now this line we find has two directions.
So starting from the centre point (which is represented by the o (the origin which is 0), movement to the right is defined as positive. So the radius drawn from the centre to the rights is + 1.
However movement in the opposite direction is defined as negative.
So the radius drawn from the centre to the left is - 1.

However it is important to understand that the identification of sign is merely an arbitrary convention.
We could equally define movement of the line to the left as positive, in which case corresponding movement to the right would be negative.

Once again the very symbol, which is used to represent one, is in fact a straight line. Likewise the symbol used to represent zero (nothing) is - with very minor modification - a circle.

As we have seen there are intimate connections as between 1 and 0; likewise there are intimate connections as between the line (diameter) and the circle (circumference).

Just as numerically 0 is connected to 1 through combining the positive and negative signs of 1 (i.e. 0 = 1 - 1), likewise the circle (circumference) is connected to the line through combining both the positive and negative directions of the line diameter.

So both directions of the line (drawn from the centre) are enclosed by the circumference of the circle. Likewise this bi-directional line defines the nature of the corresponding circular circumference.

Q This is very interesting! I see how you are showing how fundamental arithmetic and geometric notions are intimately related. So how does apply in holistic terms?

PC Once again there is an extremely important dynamic holistic interpretation of these same symbols, the proper appreciation of which greatly clarifies the very nature of both differentiation and integration in development.

Corresponding to the notion of the (straight) line we have in holistic terms a linear approach to development.

Corresponding to the notion of a circle we have a circular approach to development.

Positive and negative equally define opposite poles of development.

Quite simply when we attempt to differentiate development merely in terms of positive poles we thereby adopt a linear approach.

Q Just to recap! The positive pole means that it is understood as separate from its opposite. When I recognise an object it is thereby separated and made independent of the (observing) self. So the exterior pole is (consciously) posited and differentiated in experience.
So the relationship between variables that are understood in this differentiated manner will thereby be linear?

PC Yes! And just as one (1) relates to form, likewise linear (in holistic geometrical terms) likewise relates to form. So when we look at the relationship between variables that are interpreted merely in terms of (differentiated) form it will appear to be linear.

In other words a linear relationship - in this context - is synonymous with an asymmetric sequential connection as between variables.

This does not mean of course that a person necessarily believes that development takes place in an unbroken straight-line fashion. Rather it means that one will attempt to approximate the dynamics of development as a series of broken lines that can perhaps zig-zag in various fashions. In other words one will attempt to explain overall development through a variety of linear asymmetrical type connections (each of which is strictly only valid in a local context).

Q So this is why you persistently make the assertion that in intellectual terms most integral approaches are in fact very reduced as they tend to approach integration through the (sequential) asymmetric interpretation of development relationships.

PC That's it in a nutshell! Integration and differentiation are both qualitatively distinct processes - with unique logical systems - and must be preserved as such in any accurate interpretation of development processes.
Ultimately nondual are qualitatively distinct from dual notions. We cannot therefore - in intellectual terms - approach the nondual through the continual extension of (linear) dual notions. So if we attempt to do so our approach will reveal considerable imbalance and inconsistency from an overall integral perspective.

So what appears unambiguously true in any given local context is rendered paradoxical from a global perspective.

Q Can you briefly relate this holistic mathematical interpretation of linear understanding back to your earlier discussion on holarchical development and the relationship between the atom, the molecule, the cell and the organism.

PC Linear understanding always entails an arbitrary fixing of the frame of reference with just one pole (whereby it is treated as largely separate from its opposite pole).

So for example when we say that the atom is transcended and included in a molecule which is transcended and included in a cell which in turn is transcended and included in an organism and so on in an ascending sequence of more collective wholes, this represents - as I define it - the linear approach.

In dynamic terms our experience of the world entails the self in relation to the environment so that interior and exterior poles necessarily interact.

However with (linear) asymmetrical understanding there is always an attempt to define experience unambiguously with respect to one pole (as reference frame).

So in our example, the reference frame is fixed with the exterior pole and relationships then viewed as if somehow independent of the interacting mind.

Thus relationships here are defined with respect to the positive pole (in this case the exterior) and viewed as moving solely in one linear direction leading to the typical asymmetric interpretation e.g. atoms are included in molecules; thereby molecules are not included in atoms.

Now I am not denying the role of such linear interpretation. It plays an extremely important role in understanding and is vitally necessary for the proper differentiation of experience.
Rather - with the help of fundamental holistic mathematical concepts - I am defining its true nature.

So (linear) asymmetric understanding results from the attempt to understand in terms of just one arbitrarily fixed reference pole which is always positive (i.e. consciously posited) though understanding in terms of the opposite pole is equally valid.

Q So from this holistic geometrical perspective how do we properly deal with the integral aspect of development?

PC As we have seen with our circle - because the fixing of poles as positive or negative is purely arbitrary - we could equally define understanding in terms of the other pole as reference frame (in this case the interior) leading to an opposite (linear) asymmetric interpretation of development which is equally valid.

By definition we cannot hope to reconcile these paradoxes within a linear approach which is necessarily partial. Therefore when we come to translate integration in experience (as opposed to differentiation) a very distinctive approach is required.
In simple terms proper integral translation requires a balanced circular approach (which fully recognises the true complementarily of opposite poles of development).

Now this is beautifully symbolised in the diagram of the line and circle.
So in holistic terms the circular approach includes two linear directions of interpretation that are always opposite in terms of each other.
In other words circular understanding results from linear understanding that is dynamically bi-directional.

It leads in turn to the deep realisation - through clear recognition of the arbitrary fixing of polar reference frames - that all dualistic understanding can be rendered paradoxical.

Now the actual recognition of such paradox is nondual and arises through deepening spiritual awareness.

However form and emptiness are themselves dynamically complementary. Therefore the appreciation of paradox at a dualistic level acts as a catalyst towards its reconciliation through spiritual intuitive awareness. Equally the deepening of this awareness sharpens recognition of the inevitable paradox at a dualistic level. So bi-directional appreciation of form and spiritual intuition, increasingly interpenetrate mutually enhancing each other.

Q So let me summarise. Linear understanding is based on an unambiguous asymmetrical interpretation of variables properly suited for the differentiation of development. With linear understanding one pole is always posited as independent (without recognition of the corresponding negative pole).

Likewise because experience keeps switching as between opposite poles each pole can be separately posited. So for example we can posit and thereby differentiate in linear terms with respect to both exterior and interior poles (as considered relatively independent).

Circular understanding - by contrast - is based on the complementary (paradoxical) interpretation of variables properly suited for the integration of development.
Though circular understanding necessarily starts with linear understanding i.e. we must differentiate before we can integrate, crucially it is based on recognition of the corresponding negative pole. In other words with integration we realise that in any context we can only posit in conscious terms through corresponding (unconscious) negation of the opposite unrecognised pole.

Thus as integral capacity grows experience necessarily becomes more dynamic with switching between opposite poles becoming more flexible.

Differentiation necessarily takes place but now in a much more refined manner.

So as positive and negative are harmonised, experience becomes increasingly transparent as empty Spirit (resulting in a void in terms of rigid differentiated form).
So in geometrical terms the circle which literally represents a void or zero, beautifully equates with the (arithmetic) binary digit 0.

Thus the very purpose of circular understanding is to facilitate the voiding of all attachment to rigid phenomena.

Is the implication of all this that proper integral understanding in your opinion can only come through the considerable development of contemplative awareness?

PC That is my opinion. The crucial requirement for true integral awareness is this development of dynamic negation (so emphasised as we have seen by St. John of the Cross).

In Christian mystical terms this is referred to as purgation or the "dark night of the soul". This can involve prolonged intense exposure to the "shadow" unconscious which is always extremely painful.

So I never intend holistic mathematical understanding as a means of short cutting necessary spiritual development. Rather I use it to demonstrate the kind of intellectual understanding that can be properly consistent (in cognitive terms) with such development.

Q Does this not raise a big question? Surely if what you say is correct i.e. that proper holistic mathematical understanding requires contemplative type intuitive awareness, then it will be very difficult indeed to communicate its relevance. One might even suggest that there is perhaps no point of trying to communicate along these lines especially as contemplative development is generally not associated with mathematical understanding. Is there a problem?

PC Certainly there is and I am afraid that the majority of - admittedly - intelligent people who have been exposed to these ideas have failed to literally "see" what they entail. By far the most common misunderstanding is the utterly mistaken belief that I am attempting to reduce spiritual understanding to analytic mathematical notions. From my standpoint however I recognise such responses as a convenient way of avoiding serious engagement with an original approach which significantly challenges widely held conventional notions.

However there have been notable exceptions and during the several years of my active involvement in dialogue I can recall many wonderful discussions on the Forums and in e-mails with people who combined considerable intelligence and generosity of Spirit in equal measure.

For example I remember well when I took my first faltering step on to a Wilber Forum and introduced the holistic mathematical approach. My rather lengthy post remained unanswered by anyone for several days. Then rather unexpectedly I received an in-depth response from a woman in Australia which was simply brilliant. And this kind of response was repeated subsequently on many occasions.
So given a sufficient level of openness to what is admittedly a novel approach, very fruitful communication is possible.

Q Can you now illustrate how you use the holistic geometrical interpretations notion of line and circle to define the three fundamental sets of polar relationships?

We can draw the (bi-directional) line diameter of the circle as a horizontal line.
However equally we can draw it as a vertical line (through the centre at right angles to the horizontal).
Finally we can draw it as two diagonal lines (though the centre at an equal distance from both horizontal and vertical).
These lines have immense significance in dynamic holistic terms relating to the most fundamental designation of polar opposites.

Horizontal Poles

The horizontal line can be used to represent polarities that condition experience within a given level of experience.
As we have seen in earlier discussion these relate to the important objective/subjective (and subjective/objective) polarities which can be referred to in various ways e.g. exterior/interior, external/internal, outer/inner etc. In holistic mathematical terms these polarities are positive and negative with respect to each other.

In other words - whatever the level of understanding - phenomenal experience necessarily entails exterior and interior (and interior and exterior) poles that are in dynamic relationship with each other.

Again the relationship between poles can be interpreted in two ways

1) Differentiation: here opposite poles are relatively separated leading to (asymmetrical) dualistic understanding.
In precise terms this means that both poles are understood in terms of their (merely) positive aspect (i.e. 1 + 1 = 2). Understanding is here linear and unambiguous in one-directional (asymmetric) terms.

2) Integration: here opposite poles are understood as relatively complementary (and ultimately identical) leading to bi-directional dualistic understanding. This is inherently paradoxical (in dual terms) and is reconciled through a dynamic transformation to nondual (spiritual) awareness (i.e. 1 - 1 = 0). So in dynamic terms understanding here is both linear and paradoxical - in bi-directional (symmetric) terms - and circular and nondual.

So just as the bi-directional line diameter and the circular circumference are intimately related in analytic geometric terms, likewise bi-directional linear understanding and nondual circular understanding are intimately related in a holistic mathematical fashion.

In dynamic terms therefore - for all transformation processes in development - we have the ceaseless interaction of the two aspects of differentiation and integration.
Once again in the first case opposite poles separate in development (though of course never completely separate).
In the second case these opposite poles to a degree unite through recognition of their complementary nature (though again never completely).
So development - within any level - is characterised by the relationship between opposite poles that is always to a degree dual (where they are separated) and nondual (where they are complementary) though the balance can vary greatly.

Vertical Poles

Q So what about the vertical poles?

The vertical line can be used to represent poles that condition experience between various levels of experience.
These relate to the extremely important whole/part (and part/whole) polarities that can again be expressed in various ways e.g. individual/collective, particular/general, quantitative/qualitative etc. Once again in holistic mathematical terms these are positive and negative with respect to each other.

Strictly speaking whole/part transformations in development always entail changes in the level of development. So for example from the perspective of holism, evolution involves the movement of reality to more complex whole levels of (collective) organisation. Equally evolution here entails the movement from lower quantitative (physical) to higher (spiritual) structures.

As with the horizontal poles (within a given level), the relationship between vertical poles (between levels) can be given a (linear) differentiated interpretation and a (circular) integrated interpretation respectively.

Q I want to raise an important question here. In terms of your binary approach both the horizontal and vertical lines will be represented as 1 (in either direction from the centre) carrying opposite signs. So in conventional terms, the direction of the horizontal line to the right is positive and to the left negative (though this merely represents an arbitrary convention). Likewise the direction of the vertical line upwards is denoted as positive and the direction downwards is negative (though again this is merely conventional).

Again when we differentiate either pole (horizontal or vertical) only the positive direction of form is emphasised (which keeps switching alternately between poles).
This is represented by the unambiguous direction of the straight line which in holistic binary terms is represented as 1

Likewise when we integrate both poles (horizontal or vertical) both positive and negative directions are understood as complementary thereby cancelling out.
This in turn is represented by the both directions considered simultaneously (i.e. the circle) which in holistic binary terms is 0.

However surely both horizontal and vertical poles are qualitatively distinct. So how do you preserve this qualitative distinction in your binary approach?

PC This indeed is a crucial question. The full answer requires distinguishing Type 1 complementarity (where the relationships between polar opposites in either horizontal or vertical terms is considered in relative isolation) from Type 2 complementarity (where they are considered simultaneously requiring us to establish the precise relationship as between horizontal and vertical co-ordinates in holistic mathematical terms).

However for the moment it may be helpful to go back to our earlier discussion on the relationship as between perceptions and concepts.
Now a perception strictly represents an actual finite phenomenon (with respect to a potentially infinite concept).
The concept - in relative terms - then represents the potential infinite concept class (to which the actual finite phenomenon is related).

So strictly speaking - and this has far-reaching implications (in both physical and psychological terms) if the horizontal polarities represent (actual) phenomenal quantities, then the vertical polarities represent corresponding (potential) dimensional qualities.

However because in dynamic terms horizontal and vertical polarities continually interact the horizontal and vertical locations are purely arbitrary and keep switching (in dynamic terms).
Thus equally if we represent the horizontal polarities in terms of dimensions, then the vertical polarities - in relative terms - represent phenomenal quantities.

Q Let me see if I have got this correctly! Though both horizontal and vertical lines are defined in terms of similar straight lines (which are +1 and -1 with respect to each other), these lines in fact represent distinctive representations of unitary form.
So when the horizontal lines represent (actual) quantitative phenomena of form, the vertical polarities represent (potential) qualitative phenomena of form (i.e. as dimensions). So in this way we can proceed as long as we recognise this important distinction.

PC That's right! Another way of looking at this in psychological terms is to see one axis as representing conscious and the other unconscious phenomena respectively.

However there is a more satisfactory manner of dealing with this issue in mathematical terms whereby we can literally show how to convert in holistic mathematical terms from quantitative objects to qualitative dimensions (and vice versa).

Q Does what you are saying have any implication for our customary interpretation of space and time dimensions?

PC You bet! You know the remarkable thing - which has not yet been properly recognised - is that proper dynamic understanding of space-time dimensions directly conforms to the mathematical explanation of dimensions (i.e. when interpreted in a holistic fashion).
This is fully in keeping with my basic starting-point that the fundamental structure of reality is mathematical (in this dynamic holistic sense). So in this way I am unravelling an understanding that is already deeply inherent in reality

Q  Can you say something now about the Four Quadrants from a holistic mathematical perspctive?

PC   When we combine both horizontal and vertical lines through the mid-point of the circle we literally create four quadrants in geometrical terms.
In analytic terms the first important thing to observe is that these quadrants - by definition - combine both lines and circle.
Therefore in corresponding holistic terms the interpretation of the four quadrants combines both linear and circular notions.

So the Four Quadrants - which indeed is deeply relevant for structuring the fundamental nature of reality is quite literally a holistic geometrical notion (and therefore best interpreted from this standpoint).

Though in dynamic terms the signs in each quadrant, horizontally and vertically, keep switching, in a rigid linear interpretation the positive direction is solely emphasised (in each of the four quadrants). In other words understanding in each quadrant is based on fixing interpretation with one arbitrary pole (though interpretation from the opposite pole is equally valid). Thus the direction of development appears similar for each quadrant (moving from lower to higher levels in each case).

So if we for example we apply an unambiguous holarchical interpretation so that with respect to each of the four quadrants each lower holon is transcended and included in a higher holon, then we are simply giving a (rigid) linear interpretation.

However this position is untenable from a proper dynamic perspective.
Again in dynamic holistic terms, linear understanding is positive and negative (i.e. complementary) with respect to opposite quadrants.
Thus though the direction of understanding - in horizontal and vertical terms - is unambiguous for each quadrant taken separately, when opposite quadrants are related to each other the directions are complementary and paradoxical.

So once again the task of differentiating and integrating quadrants has to be clearly distinguished. When differentiating, the direction of movement (for each isolated quadrant) appears unambiguous; however when integrating the directions for opposite quadrants are complementary and paradoxical.

Thus from a dynamic perspective if we define in horizontal terms Right-Hand Quadrants - in terms of inclusion, the relatively - the Left-Hand must be defined in terms of exclusion.
Likewise if we define in vertical terms the Upper Quadrants in terms of holarchy, then again - relatively - the Lower Quadrants must be defined in terms of partarchy.

Diagonal Poles

Q Can you know deal briefly with the diagonal poles?

PC The two diagonal lines can be used to represent poles that condition experience (simultaneously) both within and between the various levels of experience.

These relate to the most fundamental form/emptiness (and emptiness/form) polarities that can be also expressed in various ways i.e. actual/potential, finite/infinite, immanent/transcendent etc.
In holistic mathematical terms these are positive and negative with respect to each other. However as they combine both horizontal and vertical aspects (in balanced fashion) the sign designation is more complex.

Thus in the UR quadrant the designation is +, +, with the corresponding designation in the diagonally opposite LL quadrant -, -.
Also in the UL quadrant the sign designation is -, +, with the corresponding designation in the diagonally opposite LR quadrant +, -.
Of course the very implication of combining polarities both within and between levels simultaneously is that the very notion of a level loses any distinct meaning.
So as pure nondual awareness is approached in mystical terms, these fundamental diagonal polarities are the last to be successfully reconciled.

Q I take it that this double sign designation means that we are balancing both actual (finite) understanding and potential (infinite) understanding in experience!

PC Correct! Alternatively from a psychological perspective we are bringing both conscious and unconscious into harmony (through the Spirit).

Q The inclusion of diagonal polarities leads to an eight-sectoral approach. Can you briefly elaborate?

PC For a comprehensive interpretation of development, reality must be defined in terms of horizontal polarities (within levels), vertical polarities (between levels) and diagonal polarities (simultaneously within and between levels).
A differentiated understanding of reality results from the relative separation of these poles yielding a (linear) asymmetrical interpretation.
However though such interpretations appear unambiguous within each sector considered in isolation, when related to each other, dynamic complementarity exists in horizontal, vertical and diagonal terms.
This in fact implies that we are dealing with eight different holarchies.

However once again, a fuller holistic mathematical expression can be given of the diagonal polarities. This in turn enables us to precisely express the nature of Type 3 Complementarity (i.e. the relationship between the diagonal and corresponding horizontal (and vertical) poles.