Introduction
 

The purpose of this brief article is to show how holistic mathematical notions can be fruitfully applied to the interpretation of the four quadrants.

Ken Wilber uses the four quadrants in his later work.
These are based on the valid belief that all holons have both (horizontal) exterior and interior aspects and (vertical) individual and collective aspects.

Now the notion of a quadrant is essentially mathematical. In the most common definition, a quadrant results when a circle is divided into four equal sections by the horizontal and vertical lines drawn through its center.
What is important to observe here is that we are dealing with the notion of both line and circle.

Each quadrant comprises a horizontal and vertical line (at right angles) bounded by a curved arc (representing a quarter of the circle’s circumference).
The implication is that in holistic mathematical terms the interpretation of quadrants involves the combination of both linear and circular notions of understanding.

Secondly the directions of the lines vary so that those in opposite quadrants have different signs. Thus if the horizontal line in the Right-Hand quadrant is positive, then the corresponding line in the Left-Hand quadrant is negative.
Likewise if the vertical line in the Upper Quadrant is positive, the line in the corresponding Lower Quadrant is negative.

The implications in holistic mathematical terms are extremely important and indeed essential for a proper dynamic relative understanding (involving the interaction of linear and circular notions).
It means in effect that if movement in development in one quadrant takes place in a positive (forward) direction then corresponding movement in the opposite quadrant takes place - relatively - in a negative (backward) direction.

I will briefly illustrate this very important difference as between absolute and (dynamic) relative notions of movement.
Imagine two drivers A and B setting out in opposite along a motorway.
Now as both interpret direction from an individual perspective, movement takes place in a (solely) positive direction. So movement for A is forward; likewise movement for B is forward.
This represents a linear asymmetric notion of movement (i.e. unambiguously in one direction).

However in dynamic relative terms the situation is different. Thus as A moves forward relative to B, B moves backward relative to A; likewise as B moves forward relative to A, A moves backward relative to B.
This represents by contrast a circular symmetric notion of movement (i.e. simultaneously in both directions).
This circular simultaneity of movement in two directions can perhaps be best appreciated by imagining a journey around the world. So as one moves away from one’s starting point - by definition - one is arriving closer to one’s destination (which is the same starting point). So one moves in two directions simultaneously - which in linear terms are - relatively positive and negative with respect to each other.

This is beautifully represented in the mathematical notion of the circle (with horizontal and vertical line diameters). Movement from the center can take place in either a positive or negative direction.
Thus when we convert this interpretation to the holistic appreciation of complementary opposites movement can equally take place (bi-directionally) in both positive and negative directions.
 
 

The Four Quadrants
 

When we examine Ken's treatment of the quadrants they clearly reveal a linear asymmetric treatment, which is untenable from a dynamic perspective.
Ken rightly recognizes that all holons can be defined in terms of four quadrants. However instead of looking at this relationship in a proper dynamically interactive sense, he tries to fix the quadrants unambiguously, which leads to considerable inconsistency.

So often he attempts to analyze reality as if it were somehow independent of the interpreting mind. For example in describing his Right-Hand quadrants he says in "The Marriage of Sense and Soul", P. 117;

"All Right-Hand events – all sensorimotor objects and empirical processes and ITs – can be seen with the mononological gaze, with they eye of flesh. You simply look at the rock, the town, the clouds, the mountains, the railroad tracks, the airplane, the flower, the car, the tree. All these Right-Hand objects and "ITs" can be seen by the senses or their extensions (microscopes to telescopes). They all have simple location, you can actually point to most of them".

This is a very emphatic statement of the "myth of the given".
However this description of Right-Hand events is not valid from an experiential perspective.
Objects do not just exist "out there" but always in relationship to the observer.
Thus in seeing a rock a bi-directional interaction is involved, where the rock is in relation to self (and the self in relation to the rock). The actual perception of the (individual) rock has both exterior and interior aspects (which mutually interact).
Thus identifying the object solely with the Right-Hand is very one-sided.
Likewise the (individual) perception of "a rock" has no meaning in the absence of the corresponding (collective) concept of "rock". Thus Upper and Lower quadrants are likewise necessarily involved in the experience.
So in dynamic terms all four quadrants are involved in the recognition of an object.

We could equally start with Ken's other rigidly defined quadrants and likewise show that in dynamic terms all four quadrants are involved.
There is in fact a basic confusion with his approach. He starts with the valid insight that every holon has exterior and interior aspects (in horizontal terms), and individual and collective aspects (in vertical terms). So by definition a holon belongs to all four quadrants.
However he then attempts to compartmentalize these same holons in terms of (horizontal) Right-Hand and Left-Hand, and (vertical) Upper and Lower quadrants.

A rock for example is clearly a holon that dynamically belongs to all quadrants. So it makes little sense to attempt to identify this holon in static terms with just one quadrant i.e. the Upper-Right.
A true integral approach requires much greater subtlety. First of all, we accept that a holon does indeed belong to all four quadrants. Therefore in reduced linear terms, we can only identify locations by arbitrarily fixing our frame of reference. We can then consistently define quadrant locations (in this relative sense).
By switching our frame of reference, we can give four equally valid quadrant explanations for any experiential event.
These explanations are paradoxical in terms of each other. However, they provide the very basis for an integrated approach.
In other words, through balanced paradox, we move from an either/or logic (where quadrants are differentiated) to a both/and logic (where they are integrated).
Thus when we differentiate the quadrants in horizontal terms, an event is either Right-Hand or Left-Hand. Depending on how we fix our frame of reference, we can give two equally valid asymmetrical interpretations of the event.
However when we integrate these same quadrants (simultaneously using both frames of reference), the event is understood as both Right-Hand and Left-Hand.
In a direct sense, these complementary opposites are reconciled through intuitive awareness. However bi-directional paradoxical translation itself greatly facilitates this intuitive recognition.

Ken clearly does not provide an integral interpretation of the quadrants.
Also, insofar as he differentiates the quadrants he does so in a rigid absolute - rather than a balanced relative - manner. Not surprisingly this leads to a considerable amount of inconsistency.

For example perception is associated with the Right and interpretation with the Left. However this makes little sense from a dynamic perspective (where such distinctions have a merely relative significance). Indeed the "myth of the given" arises directly from the attempt to give perceptions meaning without the need for corresponding (conceptual) interpretation.

He then identifies his Right-Hand quadrants in "it" terms as the home of (empirical) science. However as in dynamic terms, scientific perceptions require corresponding conceptual interpretation, we could equally identify the quadrants in "it" terms as (theoretical) science. As Ken places mental concepts in his Left-Hand quadrants, then the theoretical aspect of science would be Left-Hand, and the empirical, Right-Hand respectively.

Likewise he identifies a value such as compassion with the Left-Hand quadrant. However again in dynamic terms, an (interior) value has no meaning in the absence of an (exterior) objective context. Thus the sight of a suffering child might well be associated with compassion. However in reduced linear terms this has two equally valid interpretations. We could say that the sight of the child (exterior) causes the compassion; equally we could say that compassion (interior) causes one to notice the child.
In other words, in dynamic terms the value cannot be exclusively identified with either quadrant.

There are other obvious inconsistencies. Ken tries to identify the Right-Hand quadrants with "it" and the Left-Hand with "I" and "We".
As he considers Mathematics to relate to the interior aspect this would be placed in his Left-Hand quadrants.
However Mathematics is considered a supreme expression of "it" understanding (though he identifies the Left-Hand as "I" and "We").

Likewise his attempts to identify morality with "We" makes little sense. Morality has certainly a (collective) "We" aspect. However it equally has an individual "I" aspect (as with existential morality). Morality also has an important "it" aspect. The programmatic approach of the institutionalized churches to moral behavior is based on a strong belief in "objective" morality.

He also identifies beauty with "I" which is very one-sided. There is a strong cultural "We" component to our notions of beauty. Indeed modern marketing and advertising have conditioned aesthetic perspectives to an unhealthy extent. Beauty clearly also has an "it" aspect where it is identified directly with (exterior) object symbols.
Once again, by definition a holon includes all four quadrants. So as science, mathematics, morality, and beauty are holons, it makes no sense to try and exclusively identify them with just one quadrant. However it requires a dynamic relative treatment to preserve this balance.

Ken then represents the disaster of modernity as the collapse of the Left to the Right. However if we associate the rapid growth of Mathematics with modernity (which he identifies with the Left), this position is not strictly tenable (even in his terms).
The real problem is that he fails to distinguish true interactive from (merely) absolute notions of the quadrants.
So properly speaking, the disaster of modernity represents the collapse of dynamic notions of Left and Right to (merely) reduced static interpretations (which can be identified with either Left or Right). Indeed in this respect, Ken's attempt to use an analytic approach, as a means of translating integration, is itself a reflection of the true problem of modernity.
Thus because of a lack of a dynamic approach, he continually comes down in favor of one side of a polarity (when the other is equally valid). From an integral perspective, his rather compartmentalized treatment of the quadrants is very confused, as he reduces dynamic interactions to rigid static interpretation.
 
 
 

Resolving the Dilemma
 

The inconsistencies that I have pointed out arise from trying to fix quadrant locations in somewhat static fashion.

This really indicates a deeper problem with Ken’s treatment. Basically the direction of movement is of the same sign in all four quadrants so that development unambiguously moves in a forward direction. If we imagine four drivers at a driver heading off in opposite directions (horizontal and vertical) each one when viewed separately (i.e. differentiated) will move in a positive forward direction. However integration requires that we consider these movements relative to each other (which Ken does not consider).

Quite simply in dynamic terms, it makes no sense to split up quadrants as Ken does trying to identify them with specific quadrants for - by definition – a holon belongs to all quadrants.
Thus for example as empirical science is a holon it necessarily belongs to all four quadrants.

However to appreciate the deeper significance of this we have to recognize a much more subtle way of viewing holarchies of development.
Ken defines the holarchies in all his quadrants in terms of transcendence and inclusion. Again this makes no sense from a dynamic perspective.

In dynamic terms we can only dynamically include one polar aspect of experience by dynamically excluding the other. This if we are to affirm the inclusion of the exterior aspect then we must exclude (i.e. dynamically negate) the interior aspect.
Equally if we are to include the interior (+) then we must exclude the exterior (-).

This is an extremely important point for inclusion and exclusion relate to the very means by which we differentiate and integrate in experience respectively.
Thus if we identify (phenomenal) inclusion in "higher" stages with differentiation, corresponding reverse (phenomenal) exclusion is then identified with integration.
So strictly speaking the differentiation and corresponding integration of experience entail different holarchies which - relatively - unfold in opposite directions from each other. Thus if we define differentiation in terms of a holarchy of inclusion, then - relatively - corresponding integration relates to a holarchy of exclusion.

This difference is well brought out by comparing Ken’s holarchical Ascent with that of St. John of the Cross.
Ken’s holarchy is defined in terms of transcendence and (phenomenal) inclusion leading to the (direct) differentiation of "higher" level holons.
However St. John’s Ascent is defined by contrast in terms of transcendence and (phenomenal) exclusion leading to the (direct) integration of "higher" level holons.

Though in - linear terms - we can define Ascent directly either in terms of the differentiation or integration of stages, in dynamic experiential terms, these take place - relatively - in opposite directions.
Thus within each stage of development, differentiation and integration take place in horizontal directions which are (dynamically) positive and negative with respect to each other.

This is also the case when we view development in vertical terms (as between stages).
Once again there is a problem with Ken’s approach as he defines development merely in terms of the same sign that is defined by transcendence in each of his quadrant holarchies.

However in dynamic terms, development equally involves immanence as well as transcendence.
Thus if transcendence is defined as the progressive movement towards more collective wholes, then immanence represents the corresponding opposite movement towards more unique parts.
Thus from one perspective, the "lower" part is transcended in the "higher" whole. However from the equally valid opposite perspective, the "higher" whole is made immanent in the "lower" part.
Thus wholes can reflect parts in collective fashion (transcendence); equally parts can reflect wholes in unique fashion (immanence).

Once again in dynamic relative terms, these two directions of development are positive and negative with respect to each other. Thus if we identify holarchical transcendence as positive (+), then holarchical immanence is thereby negative (-).
So in dynamic relative terms the four quadrants represent very different types of holarchies, which move in opposite direction from each other in both horizontal and vertical terms.

Thus if we define the UR quadrant as a holarchy of transcendence and inclusion then this will have a (+ +) designation (i.e. vertical and horizontal lines positive). The UL quadrant will then represent a holarchy of transcendence and exclusion with a ( + -) designation.
The LR quadrant will represent a holarchy of immanence and inclusion (- +). Finally the LL quadrant will represent a holarchy of immanence and exclusion with a (- -) designation.
Thus in dynamic terms any aspect of a holon can be associated with all four quadrants.

It must be remembered that this fixing of quadrant designations is itself arbitrary (and continually changes in experience). It is a bit like a juggling act involving four balls. If we stop the act at an a given moment any of the four balls can be found in the juggler’s hand.
 
 

Moving from Differentiation to Integration
 

I have identified two ways of differentiating the quadrants. Once again the first gives a (linear) analytic interpretation where meanings are fixed in unambiguous terms. However from a dynamic experiential perspective this leads to a considerable amount of inconsistency. It corresponds to - what I refer to as - an Analytic 2 translation and is not consistent with proper integral translation. As we have see Ken Wilber very much uses this approach in his interpretation of the quadrants. (With an Analytic 1 translation, the quadrants are not properly differentiated, even in unambiguous terms!)

The second way gives a more subtle dynamic relative interpretation. This is based on the important recognition that development in opposite quadrants (horizontal and vertical) moves – relatively - forward and backward with respect to each other (giving positive and negative directions). So every holon continually switches as between four distinct types of movement (which can be precisely defined in holistic mathematical terms).

This relative interpretation combines both linear and circular notions. From one perspective we can arbitrarily fix quadrant designations and obtain a more comprehensive differentiated interpretation.
Alternatively the paradoxical connections as between opposite quadrants provides the basis to move to a proper integral interpretation. This corresponds to an Analytic 3 translation.

The actual translation of the integrated aspect of quadrants requires concentration on the (paradoxical) circular connections as between opposite quadrants. So it is based on the (dynamic) complementarity of opposites (using a both/and logic).

I would distinguish three degrees of integral translation.

An Integral 1 translation requires establishing the complementarity of opposites in horizontal terms (within a given level). This leads to an entirely different type of scientific approach.
The purpose here is to establish two-way structural similarity as between all relationships with both physical (exterior) and psychological (interior) aspects. It can be applied in principle to any discipline. (I have chiefly concentrated on Mathematics, Physics, Psychology and Economics!)

An Integral 2 translation establishes complementarity of opposites in vertical terms (between specific levels). This for example can establish exciting structural connections as between "lower" and "higher" stages.

An Integral 3 translation requires establishing the complementarity of opposites in diagonal terms (both within and between levels). For example we could use this type of complementarity to establish important structural connections as between relationships in sub-atomic Physics and Transpersonal Psychology.

In (analytic) mathematical terms when we combine (i.e. add) both directions of the diameter lines we get 0 (zero). Likewise in dynamic holistic terms, when we simultaneously combine relationships from opposite perspectives it leads to an intuitive form of awareness (empty of phenomenal considerations). It is this simultaneous (bi-directional) understanding that is the basis for an integral - as opposed to a differentiated - translation.

Finally, I would distinguish two comprehensive translations, which I call Radial, where both (linear) analytic and (circular) holistic notions interpenetrate. This leads to an extremely diverse and creative form of scientific understanding.

In a Radial 1 translation some rigidity as between analytic and holistic connections remains.

In a Radial 2 these are largely eliminated leading to an increasingly transparent form of simple scientific awareness.
 
 

Conclusion
 

I have used holistic mathematical notions to provide a much more complex notion of quadrant interactions. My purpose is to demonstrate that - rather than one single interpretation - a range of methods is needed to properly translate the interactions involved. Not alone does Holistic Mathematics demonstrate the need for this wide range of interpretations, it can also provide the structural nature of the appropriate translations.
 
 
 
 

References

Dynamic Interpretation of Holons  This provides a more comprehensive outline of holon interactions (horizontal, vertical and diagonal).

The Marriage of Sense and Soul  (4-9) This offers a detailed criticism of Ken's four quadrant approach in the context of what he says in this book.

Integral Physics  This provides a detailed outline of an Integral 1 scientific approach as applied to fundamental relationships in Physics.