- as null lines (with numerical value = 0) representing the creative void of pure potential;
- as lines with (equal) real and imaginary co-ordinates representing actual phenomena.

Though Ken Wilber's four quadrant approach in many ways is helpful and useful, I have criticised it on the grounds that it is too linear and static in its presentation.

As illustrated in my post last week, Ken essentially adopts a unary system.

A truly dynamic system however should be binary consisting of a circular aspect and a linear aspect (that is bi-directional). Ken's presentation represents a sophisticated version of the rational (linear) paradigm and does not lend itself to dynamic treatment.

Because the rational paradigm is based on a fixed frame of reference, one will tend to view reality from the perspective of a single quadrant (largely determined by personality characteristics). Even if one succeeded - in the manner advocated by Ken - in viewing reality from all four quadrants, a multi-differentiated (rather than an integrated understanding of reality) would result.

True integration understanding involves going considerably beyond L0 (the rational linear level).

Thus I see Ken's four quadrant approach as a preliminary step preparing the way for the more dynamic understanding that unfolds at the "higher" spiritual levels.

Ken restricts himself to four quadrants, dealing with horizontal (interior and exterior) and vertical (individual and collective) polarities. However there is an important third level of diagonal polarities not addressed by his quadrants. Thus to extend Ken's treatment we can say that in diagonal terms all holons have actual and potential (manifest and unmanifest) polarities.

I will now outline a summary of the dynamic treatment of holons.

Once again the (qualitative) binary system combines a bi-directional linear with a circular aspect.

Thus in terms of the horizontal direction of holons, we can say that all holons have interior and exterior aspects.

Which pole is interior (and which is exterior) depends on the frame of reference.

If we fix the first pole as interior then the second pole is exterior;

However if we fix this first pole as exterior then the second pole is interior.

In the conventional unary system, the directional sequence is taken in one direction only; in the binary system the directional sequence is taken in both directions.

Thus in linear terms - depending on the frame of reference - each pole is either interior or exterior.

A circular interpretation arises from viewing both poles in simultaneous (rather than sequential) fashion.

This leads to a paradoxical interpretation involving a both/and logic i.e. each pole is both interior and exterior (simultaneously).

This direct simultaneous understanding of the relativity of both poles cannot be grasped rationally but rather intuitively. Indeed the whole point of this latter circular understanding is the development of the ability - when appropriate - to surrender dualistic understanding.

Thus to sum up in horizontal terms:

All holons have interior and exterior aspects which can be viewed sequentially in bi-directional linear terms.

All holons likwise have a central (non-locational) aspect which can be viewed in circular (simultaneous fashion).

The dynamic activity of holons involves the interaction of linear and circular aspects.

This treatment is perfectly general and applies to all interpretation of horizontal poles e.g. subject-object, internal-external, inner-outer etc.

In holistic mathematical terms the poles are simply positive and negative with respect to each other.

Regarding the vertical direction, we can say that all holons have individual and collective aspects.

Again which pole is individual (and which is collective) depends on the frame of reference.

If we fix the first pole as individual, then the second pole is collective; however if we fix the first pole as collective, then the second pole is individual.

In a binary as opposed to a unary system, we have again a bi-directional linear interpretation.

We also have the circular definition where both poles are understand in simultaneous fashion (both individual and collective).

Thus to sum up in vertical terms:

All holons have individual and collective aspects that can be viewed sequentially in bi-directional linear terms.

All holons likewise have a central (non-locational) aspect which can be viewed in circular (simultaneous fashion).

The dynamic activity of holons involves the interaction - in vertical terms - of (bi-directional) linear and circular aspects.

This treatment is perfectly general and applies to all representations of vertical poles e.g. part-whole, quantitative-qualitative, higher-lower etc.

In holistic mathematical terms these vertical poles are once more simply positive and negative with respect to each other.

Regarding the diagonal direction, we can say that all holons have actual and potential aspects.

Again which pole is actual (and which is potential) depends on the frame of reference.

If we fix the first pole as actual, then the second pole is potential; however if we fix the first pole as potential, then the second pole is actual.

We have again a bi-directional linear interpretation.

We also have the circular definition where both poles are understand in simultaneous fashion (both actual and potential).

Thus to sum up in diagonal terms:

All holons have actual and potential aspects which can be viewed sequentially in bi-directional linear terms.

All holons likewise have a central (non-locational) aspect which can be viewed in circular (simultaneous) fashion.

The dynamic activity of holons involves the interaction - in diagonal terms - of (bi-directional) linear and circular aspects.

This treatment is perfectly general and applies to all representations of diagonal poles e.g. manifest-unmanifest, plenum-void, formal-fomless etc.

In holistic mathematical terms these diagonal poles are once more positive and negative with respect to each other.

Now even deeper mathematical connections can be made here.

In conventional mathematical terms, the horizontal (x) axis is represented as real, and the vertical (y) axis as imaginary.

In holistic mathematical terms it is exactly similar; the horizontal poles of holons can be represented as real and the vertical poles as imaginary respectively. However it is important to understand that these now have a dynamic - as opposed to static - interpretation.

Of course if we decide to fix the horizontal poles as imaginary, then the vertical will be - relatively speaking - real.

The diagonal poles can be represented in dual fashion

I will now apply this holistic mathematical translation to the understanding of "an atom" at the different levels of understanding. The treatment equally applies to any holon.

At L0 - the rational linear level, understanding is one-directional. This applies in horizontal, vertical and diagonal terms.

In horizontal terms, the atom has an objective (exterior) meaning (independent of mind).

It exists in a dimensional framework that is defined in positive terms only. (Space and time move in a solely forward direction).

In vertical terms, the atom has a quantitative (individual) meaning (independent of a holistic qualitative prospective). In other words it exists as a distinct part.

Its dimensional framework is defined solely in real terms only. Space and time have a mathematically "real" as opposed to "imaginary" existence.

Finally in diagonal terms, the atom has a finite meaning (independent of any infinite considerations). In other words the atom is given a solely actual (as opposed to a potential existence).

Its dimensional framework is defined in similar terms. Space and time have a strictly limited finite interpretation. There is no appreciation here of a continual underlying (infinite) present moment.

At HL1, the circular level (subtle realm), understanding changes considerably especially in relation to the horizontal aspect.

Understanding of the atom is now bi-directional. This entails a dynamic interaction of the (observed) atom with the (observing) mind so that the two poles (exterior and interior) keep switching in experience. Just as the observer has (exterior) physical and (interior) psychological characteristics, likewise with the atom. We can no longer view it statically as an object but rather in dynamic terms as an interacting subject-object. When we attempt to interpret this dynamic interaction in (static) linear terms we come up with a bi-directional interpretation.

We can take the relationship as involving (interior) observer and (exterior) observed (which is the conventional approach); equally we can take the relationship as involving (exterior) observer and (interior) observed.

This likewise brings about a highly important change in the understanding of space and time.

The dimensions now have dual interpretations (in exterior and interior terms). In other words movements are entirely relative in dynamic terms, so that what goes forwards, must equally go backwards. In other words movement in space and time now takes place in both positive and negative directions.

This is not apparent in linear terms. Again if we view the movement of a car up a highway (against a fixed frame of reference), it will be positive (in space and time); likewise if we take the movement of another car down the same highway it will be positive (in space and time).

However if we now view these movements together in relative terms, then the movement will be both positive and negative. In other words as one moves forward the other moves backwards (in relative terms).

Thus in the bi-directional linear interpretation of the atom movement in space and time will be positive (though taking place in opposite directions).

However when we simultaneously view this in circular (complementary) terms, movement will be paradoxically both positive and negative in space and time. This cannot be grasped rationally but rather intuitively and leads to the spiritual realisation of an underlying central present moment.

Thus at HL1, the atom has a dynamic relative existence (involving the interaction of positive and negative poles). Likewise in dimensional terms, space and time now have both positive and negative directions.

When we view this dynamic interaction in (static) linear terms, we must give a bi-directional interpretation.

At HL2 the point level (causal realm), we have additional important changes in the understanding of the vertical polarities.

Just as we viewed the atom in horizontal terms as an interaction of exterior and interior aspects, we now view the atom as an interaction involving individual and collective aspects.

This could be expressed alternatively as saying that the atom (and indeed every holon) has both a quantitative and qualitative interpretation.

Now conventional science concentrates exclusively on this fragmented quantitative aspect; it greatly ignores the equally important qualitative (holistic) aspect. Quite simply the qualitative is reduced to the quantitative in conventional science.

Now the implications of correctly interpreting the qualitative aspect is of the first magnitude and involves a total revolution in the way we view space and time.

In conventional terms we define dimensions exclusively in "real" terms. This simply reflects the quantitative bias of conventional science.

Now when we distinguish the qualitative (as opposed) to the quantitative aspects of objects, then we must introduce "imaginary" as well as "real" dimensions.

Thus when we correctly interpret reality from the perspective of HL2, we realise clearly that we are living in a mathematically "complex" ("real" and "imaginary") world.

Thus the atom from this vertical perspective has now both quantitative (part) and qualitative (whole) characteristics that dynamically interact. It is neither a "part" nor a "whole" as such but rather the dynamic interaction of part and whole (and whole and part). In mathematical terms this represents a real-imaginary interaction pattern.

Our dimensional framework must be defined in similar fashion. Holons now exist in both "real" and "imaginary" space and time.

This is not so far-fetched as it might seem. In life we are aware of the role and importance of the imagination. However in scientific terms we divorce this unconscious world (fantasy) from (conscious) concrete reality. Indeed it is the imagination which provides the essential qualitative context within which quantitative "real" experience takes place. Remarkably fantasies (and projections) have an exact holistic mathematical translation in "imaginary" terms.

The point level (causal realm) involves the specialised understanding of "imaginary" fantasy. The correct understanding of the relationship of qualitative to quantitative leads to the important realisation of how the "imaginary" can be incorporated with the "real" in "complex" mathematical terms.

Thus in vertical terms an atom has a quantitative (part) and qualitative (whole) existence which dynamically interact. Space and time now likewise interact in real and imaginary terms.

This leads in linear terms to a bi-directional interpretation.

Again we can take the relationship as in conventional terms involving the (qualitative) observer and (quantitative) object. Equally however we could take it as involving the (quantitative) observer and (qualitative) object.

Once more in isolation these will lead to "real" interpretations.

However in dynamic interaction we have relatively - "real" and "imaginary" interpretations. In other words the atom keeps switching between a "real" (quantitative) and "imaginary" (qualitative) state.

This circular interpretation of objects as both "real" and "imaginary" is of course paradoxical (in rational terms). It can only be grasped intuitively and leads to the underlying realisation of a continual present moment (immediate and eternal).

The most comprehensive stage of life which I refer to as the radial level involves the mature differentiation and corresponding integration of all previous levels. It therefore provides the correct standpoint for a full scientific appreciation of holons.

The atom now has - in addition to horizontal and vertical a diagonal interpretation.

This can be expressed by saying that it has an infinite (or transfinite) potential as well as an actual finite existence. Conventional science reduces this infinite (potential) to finite (actual) terms.

Thus in diagonal terms the atom can be viewed as a dynamic interaction involving a finite (actual) and infinite (potential) state. This could be equally expressed as the interaction of the manifest and unmanifest (or form and formlessness).

The relationship between the finite and infinite is remarkably expressed by "The Theory of Everything" (in the simple diagram representing the eight roots of unity). Here we literally give expression to reality - at all levels - as a reduced expression of unity.

It will have a horizontal "real" axis going through the centre of the circle (with positive and negative directions). It will also have a vertical "imaginary" axis intersecting the horizontal at the centre (again with positive and negative directions).

Finally it will have two diagonal axes drawn an equal distance from horizontal and vertical - also going through the centre with positive and negative directions.

Using the Pythagorean Theorem we can show that these diagonal lines (in all four quadrants) have a numerical value = 0. Alternatively they can be expressed in terms of equal "real" and " imaginary" co-ordinates.

This provides the basis for a perfect holistic mathematical explanation of the diagonal radial level.

From one perspective the diagonal lines have a potential (infinite) meaning representing a creative void (nondual reality).

Equally the diagonal lines have an actual "complex" (finite) meaning representing (equal) real and imaginary aspects. In other words (actual) phenomena - at this level - express themselves in dynamic relative terms as both "real" (quantitative) and "imaginary" (qualitative).

What I find truly remarkable is the manner in which these diagonal lines clarify the nature of the main phases of the radial level. (Despite its great importance the precise nature of this understanding is largely overlooked in the mystical literature).

The first diagonal sloping up from left to right combines real and imaginary polarities. In one quadrant they are both positive (and in the other both negative).

This defines the earlier more passive phase of the radial level. One has - at a global (integrated) level - a refined spiritual worldview (transcendent). However - at a local ( differentiated) level one increasingly sees the value of worldly action (immanent) as an expression of this spiritual commitment. At both global and local levels holistic (imaginary) and analytical (real) understanding are combined (which are ++ and - - with respect to each other). However the global (transcendent) and the local (immanent) viewpoints are not yet fully harmonised (Thus there is still some separation in diagonal terms).

The second diagonal sloping up from right to left combines real and imaginary polarities in both quadrants (with opposite signs). This signifies mature cosmic understanding where universal spiritual conviction and immediate worldly action mutually interpenetrate.

This latter phase represents the most complete expression of authentic living and is emphasised in Western (rather than Eastern mysticism).

Thus the great heroes (and heroines) of the Christian tradition e.g. St. Paul, Francis of Assisi, Ignatius of Loyola, Teresa of Avila, Catherine of Sienna and more recently Mother Teresa of Calcutta, were undoubtedly great contemplatives. Equally however, they were also great actives, filled with superhuman energy and a heroic commitment to serve in society.

At the radial level science has a spiritual (infinite) as well as physical (actual) interpretation. The physical (actual) interpretation contains two parameters which relate to real (analytical) and imaginary (holistic) interpretations respectively (both of which are equally emphasised). In other words science in its actual manifestation has now a "complex" (rather than a "real" meaning).

Conventional science by definition is analytical (real) in its approach with little or now appreciation of the complementary holistic (imaginary) aspect.

Science at this radial level is correctly understood in binary terms as the dynamic interaction between circular and linear aspects.

The circular intuitive aspect in direct terms (horizontally, vertically and diagonally) provides integration.

The linear rational aspect (which is bi-directional in horizontal, vertical and diagonal terms) provides corresponding differentiation.

Thus at the radial level of comprehensive understanding an atom (and indeed any object) is understood in terms of a dynamic interaction pattern with horizontal, vertical and diagonalaspects. This results from combining a circular (intuitive) with a bi-directional linear interpretation.

In horizontal terms, the atom represents an interaction as between positive (exterior) and negative (interior) polarities. Movement is both forwards and backwards in space and time.

In vertical terms the atom represents an interaction as between real (quantitative) and imaginary (qualitative) polarities. Movement in space and time is now both real and imaginary.

In diagonal terms the atom represents an interaction as between finite (actual) and infinite (potential) polarities. Movement in space and time is now simultaneously real and imaginary.

Opposite poles (horizontally, vertically and diagonally) are differentiated in bi-directional (rational) linear terms.

These poles are then integrated in intuitive circular terms.

Once more the conventional scientific approach is linear and one-directional (in horizontal, vertical and diagonal terms). There is little recognition of the circular intuitive aspect. It falls far short of providing an adequate translation of reality.

My purpose in providing this necessarily general model of holons is to demonstrate that creation at all levels has a fundamental - and beautifully simple - coherent binary structure that can be precisely described in (holistic) mathematical terms.

Once again, just as all (static) information states can be encoded in (quantitative) binary digital terms, quite simply all (dynamic) transformations states can be equally encoded in (qualitative) binary digital format.

I hope to return in future weeks to illustrate how this approach can connect physics and transpersonal psychology in a remarkable yet unexpected manner.