For a fuller understanding of the precise holistic mathematical nature of polarities (horizontal, vertical and diagonal), we now need to introduce the circular number system.
 

The circular number system
 

The analytic approach

We start again by looking at numbers in terms of their conventional analytic interpretation.

For example the well known natural numbers are 1, 2, 3, 4,…

What is generally overlooked is that these number quantities implicitly contain an important qualitative dimension (power or exponent) which is 1. So these numbers are all defined in one-dimensional terms. 1

For completeness therefore the natural number system could be thus represented as

1¹, 2¹, 3¹, 4¹, ….   (i.e. where 1 represents the fixed qualitative dimension).

If we introduce x (strictly x¹) representing a natural number it can take on any of these values

i.e. x¹ = 1¹, 2¹, 3¹, 4¹, ….

So taking the first four numbers to illustrate

x¹ = 1¹, x¹ = 2¹, x¹ = 3¹ and x¹ = 4¹ respectively.
 

So in the conventional linear (i.e. one-dimensional) manner, by which such the natural number system is defined, the number (as quantity) keeps varying, whereas the number as quality (i.e. dimension) remains constant as 1.

However we can define an alternative complementary number system where the rules for numbers representing quantities and (qualitative) dimensions are reversed. So now the number as dimension keeps varying, whereas the number as quantity remains constant as 1.

Thus x^1,2,3,4… = 1¹, 1², 1³, 1⁴, ….
 

So again taking the first four numbers (as dimensions) to illustrate

x¹ = 1¹, x² = 1², x³ = 1³ and x⁴ = 1⁴ respectively.
 

To express the (reduced) linear value of x (i.e. defined with respect to the fixed dimension of 1) we obtain successive roots of 1.
 
 

One Root of Unity
 
 

The 1^st root of 1 = 1 (i.e. 1¹).
 
 
 

Two Roots of Unity

 
 

The second (the square root) is either + 1 or - 1.

Remarkably when we plot these roots geometrically (second or higher), they all lie as equidistant points on the circle of unit radius. 2

So when we take two roots, they will lie as the end points of the diameter of the circle.
 
 
 

Four Roots of Unity
 

The four roots of 1 are + 1, - 1, + i and - i (where i is an imaginary number that represents the square root of - 1).

When we take four roots the (four points) will lie as the end points of the horizontal and vertical line diameters (intersecting at the centre of the circle).
 

In conventional terms the real roots + 1 and - 1 are represented on the horizontal line (the x axis).

The imaginary roots + i and - i are represented on the vertical line (the y axis).

In geometrical terms these four roots are precisely represented by the four quadrants of a circle.
 
 
 

Eight Roots of Unity
 

When we take the eight roots the (eight) points will lie - at an equal distance - as the end points of the horizontal, vertical and diagonal line diameters (intersecting at the centre of the circle). Again in conventional terms the real roots (+ 1 and - 1) are represented on the horizontal line (the x axis). The imaginary roots (+ i and - i) are represented on the vertical line (the y axis). Four additional complex roots are represented as the end points of the two diagonal lines (that lie at an equal distance from both the horizontal and vertical axes). The remarkable property of these diagonals is that they are null lines with a length = 0.  3

In geometrical terms these eight roots are precisely represented by the eight (equal) sectors of a circle.
 

The holistic approach

Once again the mathematical notions we dealt with above (in analytic terms) have corresponding dynamic holistic equivalent interpretations that are immensely important for a precise scientific understanding of reality that is truly integral.

Analytic science intimately depends on the quantitative approach provided through the linear interpretation of number.

However what is scarcely recognised - certainly in any explicit fashion - is that integral science equally depends in an intimate manner on the qualitative approach provided through the corresponding dynamic holistic interpretation of number. 4

Indeed - to put it simply - appropriate understanding of this system provides a dynamic Theory of Everything that can (potentially) encode all development processes.

The appreciation of this holistic circular number approach provides a deeper context for understanding the mathematical notions we have dealt with in the last Chapter while also providing extremely powerful additional integral notions.
 
 

Linear Understanding - One Directional

Just as the line - representing the natural number system - extends positively in one-direction, likewise linear understanding extends positively in one direction (leading to unambiguous asymmetric connections regarding development).

So linear understanding is thereby one-dimensional (or one-directional) in a precise holistic mathematical sense. 5

Also as we saw the presence of the qualitative dimension of 1 leaves the number quantity unchanged and is thereby effectively ignored.

Likewise in terms of linear understanding, its one-dimensional qualitative emphasis is effectively ignored and thereby reduced to the quantitative. (Correspondingly - as with scientific theory -  its one-dimensional quantitative emphasis is reduced to the qualitative).

So for example the typical scientist will generally treat the truth of empirical quantitative observations as invariant with respect to the qualitative perspective of the observer. Indeed conventional science is based on an assumed direct correspondence of quantitative and qualitative aspects. 6
In other words perceptions and concepts (qualitative) are assumed to directly correspond with object phenomena (quantitative). 7
 
 

Circular Understanding 1 - Bi-Directional (Horizontal)

We are now looking at the dynamic holistic interpretation of the circular notion of number. This provides the first of three fundamental structures for a proper integral appreciation of reality.  8

As we saw in analytic terms, when we attempt to express this relationship in (reduced) one-dimensional terms it splits into opposite polarities.

Thus x, i.e. the square root of 1² is either + 1 or - 1.
 

Equally from a dynamic holistic perspective, when we express this relationship in (reduced) linear fashion, again it splits into two opposite polarities which in complementary fashion are now both + 1 and - 1.

So what we are doing here is expressing a higher order unity where (opposite polarities coincide) in reduced one-dimensional terms.

Therefore in linear fashion we dualistically express the nature of this reality as the relationship of (equal) poles. So what in nondual fashion - as emptiness - represents the coincidence of opposite poles i.e. spiritual union (which is ineffable), in reduced linear fashion - as form - is expressed as the complementarity of both positive and negative poles (which are dualistically identified).

So in experiential terms there is always a dynamic relationship as between the bi-directional expression of form where opposite poles are identified, and the nondual experience of emptiness where - through direct coincidence - they lose any separate identity in an ineffable spiritual experience.

Perhaps Taoism provides the clearest traditional expression of this relationship where the primal undifferentiated unity of reality (Tao) splits in phenomenal reality into opposite poles that are positive and negative with respect to each other (i.e. yin and yang).

So I am demonstrating here a fundamental holistic mathematical relationship that represents the dynamic expression of circular duality. As always with such relationships there is a direct correspondence in analytic terms (i.e. as the two roots of unity).

Putting it more precisely we can say, in dynamic holistic mathematical terms, that reality - which in its primary spiritual state is nondual (0) always splits - at a phenomenal level - into opposite (horizontal) "real" poles that are positive (+ 1) and negative (- 1) with respect to each other.

As we have seen these horizontal poles can be expressed as exterior/interior (interior/exterior).

So all relationships - at whatever level of reality - have exterior and interior aspects, which dynamically interact.
 

Again the relationship between poles can be treated in two distinct ways.

For differentiation we dualistically separate exterior and interior aspects and thereby obtain a (linear) asymmetrical interpretation of phenomenal relationships (i.e. based on isolated reference frames). 9

For integration we attempt to unite exterior and interior aspects and thereby obtain a (circular) symmetrical interpretation (i.e. based on simultaneous reference frames). From a dualistic perspective the interpretation is paradoxical. However this serves as a necessary catalyst for the reconciliation of such paradox through nondual spiritual intuition. So once again in dynamic terms paradoxical form and nondual emptiness ceaselessly interact.

This understanding for example leads to a distinct notion of the physical dimensions of reality that is directly related to the mathematical interpretation. In other words from an integral perspective, the structure of physical dimensions is mathematical in a precise dynamic holistic sense.

Space has both positive and negative aspects; likewise time has positive and
negative aspects. 10
So movements in space and time always take place simultaneously both forward and backward with respect to each other.

Indeed it is the reconciliation of such paradoxical notions that leads to the nondual spiritual awareness of the present moment continually renewed.

So positive and negative directions of the line diameter emanate from a central point. Likewise positive and negative directions of both space and time equally emanate from a central point in relative fashion and continually return to this central nondual point in the absolute present moment. 11

Likewise there is direct complementary as between physical and psychological notions. So space and time in psychological terms equally have positive and negative directions. (What is positive from one perspective is negative from the other and vice versa).

Though real bi-directional understanding i.e. that relates to the interdependence of the horizontal poles (exterior/interior) can be associated to a degree with earlier levels, in its mature state it unfolds with the first of the higher mystical stages (corresponding to the psychic/subtle realm).

Associated with the understanding of this level is a distinctive type of integral scientific understanding, which I refer to as Integral 1 science. 12
 
 

Circular Understanding 2 - Bi-Directional (Horizontal and Vertical)
 

I will firstly refer again to the manner through which conscious and unconscious dynamically interact in experience.

For a phenomenon to become conscious it must be posited in experience. This leads to the separation of polarities so that the positive is (solely) made explicit with the negative remaining unrecognised.

For a phenomenon to be "emptied" and thereby made unconscious it must be subsequently negated. This leads to the fusion of both positive and negative poles.

So whereas the appearance of form requires the separation of opposites (with the positive solely recognised) the return to emptiness in the unconscious results from the process of dynamic negation leading to their mutual identity.

Thus the conscious direction of experience (i.e. the process through which phenomena are made conscious) is directly associated with positive form (+ 1).

The unconscious direction of experience (i.e. the process through which phenomena are emptied and made unconscious) is directly associated with negative form (- 1).

The problem then arises as to how this unconscious aspect of experience (which in mature terms directly embodies the spiritual nature of phenomena) can be indirectly expressed in reduced (linear) conscious fashion.

This involves expressing the negative direction - which in dynamic terms leads to the identity of both poles - in reduced linear fashion (where poles are separated).

This exactly entails the holistic equivalent of - what in conventional analytic terms is referred to as - an imaginary number (i.e. the square root of – 1 = i).

So in precise holistic mathematical fashion, the indirect conscious expression of what is – in a direct sense – of unconscious origin is imaginary. 13

Once again this has both positive and negative expressions (just as the square root of
– 1 = + i or – i).

Now in conventional mathematical terms the four roots of unity are + 1, - 1, + i and – i.

The real roots (+ 1 and – 1) are plotted on the horizontal x- axis. The two imaginary roots (+ i and – i) are plotted on the vertical y – axis.

Likewise in holistic mathematical terms the two imaginary roots are plotted on the vertical axis where they represent the extremely important vertical polarities of whole and part (and part and whole).

In psychological terms, imaginary phenomena represent projections that are primarily of an unconscious nature though indirectly expressed in conscious form. 14

Here we can distinguish two types (which in dynamic terms are directly related).
 

Mature projections

These become especially important at the higher mystical stages of development where extremely refined phenomena of form are used are used to convey spiritual meaning (that is ultimately ineffable).

When the dynamic interaction of (phenomenal) form and (spiritual) emptiness is free and unrestricted, very pure imaginary understanding can take place where indirect phenomenal expressions are directly suggested by the Spirit.
 

Immature projections.

Paradoxically these likewise become extremely important with advanced mystical development.

Increased purity in relation to (indirect) refined phenomenal expression of the Spirit heightens sensitivity to any remaining instinctive rigidities and confusion dating from the earliest prepersonal stages. So - to the extent to which they remain unrecognised - these become involuntarily projected into consciousness attempting to attach rigidly to phenomenal forms of expression.

Indeed higher level purification – especially at the causal level – is largely devoted to cleansing of remaining rigidity in relation to these "lower" imaginary projections. With sufficient cleansing, the involuntary nature of these projections is removed so that both (spiritual) intuition and (physical) instincts can be harmoniously combined.
 

Contrasting real and imaginary

Just as in analytical mathematical terms we have real and imaginary numbers, likewise we now can define these fundamental notions appropriately in corresponding dynamic holistic fashion.

So the "real" in holistic mathematical terms relates directly to form (as the direct expression of consciousness). Phenomena of this nature tend to be unambiguously defined and – initially at any rate because of this very fact – somewhat rigid in experience.

The "imaginary" by contrast relates directly to emptiness and indirectly to form (i.e. as the indirect phenomenal expression of the unconscious).

Phenomena of this kind always tend to be somewhat ambivalent and paradoxical (from a dualistic perspective).

Because they directly point to what is ineffable (i.e. spiritually empty) any rigid identification with their indirect phenomenal expressions tends to greatly reduce their meaning. Therefore the purer their expression, the more short-lived and elusive they become in experience. So "imaginary" symbols become virtual archetypes, as fleeting footprints of the divine Spirit, which tend to dissolve as soon as they are formed. 15
 

Relevance for everyday experience

Though imaginary expressions are subject to considerable confusion in terms of everyday experience, they remain extremely important.

Indeed in a direct holistic mathematical sense, all phenomena are complex.

In other words because conscious and unconscious are always involved, phenomena have a double meaning. At one level they can be given a "real" unambiguous interpretation; alternatively they can be given an "imaginary" paradoxical interpretation (where they serve as indirect expressions of a meaning that is ultimately purely spiritual and ineffable).

Now it is the very nature of experience that both meanings always interact. So all phenomena are complex i.e. have both real and imaginary aspects.

Putting it another way all phenomena have both dual and nondual interpretations.

The dual interpretation is provided by the real aspect.
The nondual interpretation (which in a direct sense is ineffable) indirectly is embodied in phenomenal form as the imaginary aspect. 16

When the imaginary aspect is well-developed, symbols become spiritually transparent to a significant degree where they serve as pure archetypes of the divine.
When the imaginary aspect is undeveloped and confused, instinctive desires attach rigidly to phenomena with which they - misleadingly - become directly identified.
 
 

Dynamic Relationship of Whole and Part

It is now time to demonstrate the intricate dynamics of how wholes and parts interact in experience.

Firstly we need to recognise that "real" phenomena always are manifested as holons (i.e. whole/parts or alternatively part/wholes) where a degree of reductionism is necessarily involved. In other words in any given context, either the parts are reduced to the whole or alternatively the whole is reduced to the parts. 17

The question then arises as to how recognition of a "real" holon switches from a whole/part (where the whole aspect predominates) to a part/whole (where the corresponding part aspect is dominant).

This amounts equally to explaining how experience keeps switching from quantitative to qualitative recognition (and in reverse fashion how it switches from qualitative to quantitative recognition).

The appreciation of this dynamic in turn is essential to explaining satisfactorily the interaction which underlies the relationship between perceptions and (corresponding) concepts on the one hand and concepts and (corresponding) perceptions on the other, which necessarily underlie all experience of phenomena.

Thus from one important perspective, to switch from quantitative recognition of individual phenomena to a collective universal qualitative recognition, imaginary understanding must be present, where to a degree the individual phenomena serve as (transcendent) archetypes of this common universal pattern. 18

Likewise here - in reverse fashion - to switch from this qualitative conceptual to individual quantitative recognition, "imaginary" understanding must again be present where the collective concepts as archetypes are made immanent in the individual data. 19

Indeed in the truest sense this is the very means by which intuitive energy is brought to phenomenal experience investing both perceptions and concepts with spiritual significance.

Now clearly if the "imaginary" aspect of understanding is undeveloped, little scope for its indirect expression in phenomenal terms will exist. Understanding will thereby be very rigid and greatly lacking in creativity.

Thus from one perspective difficulties will arise in "seeing" conceptual qualitative patterns to individual quantitative data.

From the other perspective one will find it very difficult to relate conceptual structures in a meaningful fashion to quantitative phenomena.
 
 

The Four Quadrants Revisited
 

We will now look again at the Four Quadrants in the light of four-dimensional understanding i.e. bi-directional in horizontal and vertical terms.

Again in conventional terms, the horizontal x-axis is defined as real and the vertical y-axis as imaginary.
Now this in fact is quite arbitrary. We could equally plot the horizontal as imaginary and the vertical as real.

In dynamic holistic terms, we have continual interaction as between the horizontal and vertical polarities i.e. between real (conscious) and imaginary (unconscious) understanding.
 

It would perhaps be instructive to explain this at greater length.

Again the very nature of "real" understanding is that reductionism is necessarily involved.

A holon is by definition a whole/part (or alternatively part/whole). However where "real" understanding is entailed, the whole is either reduced to the part aspect (or alternatively the part to the whole aspect).

Scientific empirical understanding represents a good example of the former. Here whole phenomena through repeated quantitative sub-division become fragmented into many parts. Thus when we look at the whole from this (quantitative) perspective it is typically understood as the sum of its parts. In other words the distinction as between whole and parts (qualitative and quantitative) is thereby lost.

Scientific theory frequently represents an equally good example of the latter where the emphasis is on an abstract conceptual explanation of a generalised series of
phenomena (ultimately - as with a physical TOE - all of reality). The real danger here is that (quantitative) data i.e. parts then can become reduced to the (qualitative) conceptual theory (whole). So once again - in reverse terms - the distinction as between parts and whole (quantitative and qualitative) is lost.

Conventional science in fact is based on a double correspondence (which equally reflects a double reductionism).

From the deductive perspective, theories (qualitative wholes) are assumed to correspond to empirical facts (quantitative parts). So here we use the whole to interpret the parts.

From the corresponding inductive perspective, empirical facts are assumed to correspond to conceptual theories. So here the parts are used to generate the whole.

However such science – by its very nature – is unable to preserve the crucial distinction as between wholes and parts (and parts and wholes) and can only proceed by fixing its frame of reference in terms of one isolated pole.

So from an empirical perspective, the reference frame is in terms of the (quantitative) parts, which thereby reduces the (qualitative) whole. From the conceptual perspective, the reference frame is the (qualitative) whole which thereby reduces the (quantitative) parts.

Therefore when related to each other we have – as I have stated – a double reductionism where both wholes and parts thereby become robbed of their true spiritual meaning and rapidly become devoid of mystery.
 

Now let us briefly look at the other extreme where phenomena act as pure mediators of a spiritual light. Here the crucial distinction as between part and whole (and whole and part) is preserved without reductionism.

From one perspective, the parts are transcended in the whole to reveal the pure spiritual light (that is empty of empirical quantitative phenomena).

Equally from the other perspective, the whole is made fully immanent in each part again revealing the spiritual light (that is now empty of theoretical qualitative phenomena).

So the spiritual fullness here experienced – which is equally transcendent and immanent – is based on emptiness of any rigid attachment to phenomena (of either a quantitative or qualitative kind).

Thus the dynamic integral task is to show how one moves from the rigid experience of real holons (where reductionism is necessarily involved) to the flexible spiritually transparent awareness (where rigidity is significantly eroded).
 

This is where imaginary understanding - again defined in a precise holistic mathematical fashion - becomes especially relevant.

Though the formal interpretation of phenomena in science is greatly reduced and devoid of mystery, in actual experience the Spirit is present (sometimes to a significant degree). So there are important aspects to all phenomenal experience that science fails to explain.

Through spiritual interaction – even to a limited extent – symbols have an imaginary (as well as real aspect). In other words they express - in necessarily indirect fashion - the desire of the unconscious for ultimate meaning.
 

What is fascinating is that the very notion of a holon (and holarchic development generally) is inapplicable to the imaginary aspect of experience.

Because in conventional "real" terms the phenomenal experience of a holon necessarily involves reductionism, the unconscious tries to compensate by momentarily separating both aspects which threatens the very identity of the phenomenon. To this extent the phenomenon becomes a symbol or archetype of a meaning that is ultimately spiritual (without adequate phenomenal expression).

Again this separation can take place in two ways. Firstly the whole is momentarily separated from the part and briefly acts to mediate in a typically transcendent fashion the Spirit (i.e. a whole appreciation that is not confused with empirical parts).

Secondly - in relative terms - the part is momentarily separated from the whole, again briefly acting to mediate in an immanent manner the Spirit (i.e. a unique part appreciation that is not confused with conceptual whole notions).

Such "imaginary" appreciation in its purest expression only properly unfolds with the more advanced stages of mystical development (especially at the causal realm).
Here little attachment to real conscious phenomena remains. Therefore the unconscious is greatly freed to express the ultimate longings of the Spirit (indirectly in conscious form).

So phenomena now become extremely fleeting and spiritually transparent as ever purer virtual archetypes of the Spirit in both transcendent and immanent terms.

Now the growth in this "imaginary" understanding lessens any remaining rigidity in "real" conscious terms.

Thus the interaction switches very fluently as between holons (as whole/parts) and holons as (part/wholes) i.e. the collective and individual aspects of "real" holons. This again is due to the continual intervention of imaginary understanding where symbols act as temporary mediators of Spirit. And this relates to both the exterior and interior aspects of experience.

So in this sense all of the quadrants - representing the interaction of horizontal and vertical polarities - have both "real" and "imaginary" aspects which keep interacting.

In other words the dynamic interaction of conscious and unconscious with respect to both sets of poles greatly increases with advancing spiritual development.

All of this is however deeply relevant to normal everyday experience (and not just advanced mystical development).

The problem is that the "imaginary" aspect – through not being properly recognised – is frequently greatly confused with "real " conscious understanding intervening in a somewhat blind instinctive fashion. The "imaginary" instinct  (expressive of unrecognised unconscious desire) then latches on directly to "real" conscious phenomena and becomes confused with them.

For example, the philosophy of materialism, which is so prevalent in affluent countries, reflects largely the unconscious "imaginary" desire for spiritual meaning that is misleadingly projected on to consumer goods and services. However the possession of "real" material phenomena cannot satisfy "imaginary" spiritual desire.

Rather spiritual desire can only be fulfilled through freedom from attachment to what is material.
 

To sum up  – at this stage – we are now able to give horizontal and vertical polarities mentioned in Chapter 1, a fuller holistic mathematical expression as "real" and "imaginary" with respect to each other. However because in dynamic terms the horizontal and vertical aspects keep switching positions, this means that horizontal and vertical polarities have both "real" and "imaginary" interpretations (representing the continual interaction of conscious and unconscious in experience).

Of course - as always - a complementary interpretation applies in physical terms. So the dynamic interaction of "real" and "imaginary" is here expressed as the interaction of "real" with "virtual" particles which is especially apparent at the sub-atomic level (where particle activity is especially dynamic).

Indeed now these polarities have both linear and circular interpretations (in both "real" and "imaginary" terms).
 
 

Importance for Mathematics and Science

Though imaginary numbers are used in mathematics and in science, there is a great lack of any proper philosophical appreciation of their nature.

The reason for this is very interesting and indeed very easy to state.

Mathematics - as conventionally understood - is based strongly on unambiguous analytic notions that use the linear either/or logical system.

However - as I have been demonstrating - the alternative dynamic holistic understanding of mathematical symbols, directly relevant for integral scientific understanding, is based predominantly on the alternative circular (both/and) logical system.

As the very nature of the "imaginary" is intimately related to the alternative both/and logical system, not surprisingly its true nature cannot be grasped within a linear either/or system.

However the appreciation of the dynamic holistic approach provides the appropriate means to indirectly appreciate the nature of the "imaginary" (even from an analytic perspective) and then apply it successfully in scientific terms.

We have already identified how the mathematical operations of addition and subtraction have dynamic holistic equivalents in terms of the positing and negating of all phenomena.
So the very interaction of (horizontal) exterior and interior polarities necessarily entails the continual positing and negating of phenomena.

Now it can be shown that in similar manner that the operations of multiplication and division have corresponding holistic equivalents in terms of positing and negating that operate on a vertical level. Also this dynamic holistic process of multiplication and division is an essential aspect of the interaction of (vertical) whole and part polarities. 20
 
 

Circular Understanding 3 - Bi-Directional (Horizontal, Vertical and Diagonal)

This represents the most refined form of bi-directional understanding, associated in turn with the most refined experience of the interaction of subtle (fleeting) forms and spiritual emptiness.

The diagonal polarities in fact represent the simultaneous operation of both horizontal and vertical polarities.

Perhaps this requires a little more explanation.

Though the “real” polarities of form (exterior and interior) are first to be reconciled in dynamic complementary manner, this necessarily takes place in an approximate incomplete fashion.

The reason for this is due to the fact that conscious and unconscious are interdependent. Therefore it is not possible to fully reconcile (horizontal) polarities in conscious (real) terms without corresponding (vertical) reconciliation at an unconscious (imaginary) level.

Later the attempt to achieve (vertical) reconciliation at an unconscious (level) entails some unresolved (horizontal) conscious "real" rigidities, which in turn sets limits to such unconscious development.

Therefore the final stage of development relates to the simultaneous attempt to reconcile both horizontal and vertical polarities (which are interdependent in dynamic terms).

This represents the most purely volitional stage of development. In other words, the proper bi-reconciliation of polarities - in both horizontal and vertical terms - requires the pure exercise of will, with considerable detachment from all rigid identification of either an affective or cognitive kind.
 

Now the diagonal polarities of form and emptiness need some clarification.

Form can be either "real" (conscious) or "imaginary" (i.e. indirectly conscious).
So the most dynamic - and refined - experience of phenomena, entails the continual interaction of both the "real" (conscious) and "imaginary" (unconscious) aspects which are - relatively - horizontal and vertical with respect to each other. So the diagonal polarities of form and emptiness refer to this highly dynamic experience, that entails the proper balancing of both the horizontal (real) and vertical (imaginary) aspects of all phenomena.

The key implication of such dynamic balancing is that very little rigid attachment to either aspect is possible, thus enabling a fluid flexible interaction to be continually maintained. Therefore - because of this lack of rigid attachment - (fleeting) phenomena now can become transparent to an extraordinary degree serving to continually mediate the pure light of the Spirit. We could say therefore that in this extremely refined dynamic interaction of form and emptiness, phenomena now become superconductors of the Spirit.
 

Diagonal Polarities (Double Interpretation)

What is truly remarkable is how - when appropriately interpreted - holistic mathematical notions can provide an exceptionally precise and subtle (indirect) interpretation of the ultimate nature of this key relationship as between form and emptiness.

When, in analytical terms, we obtain the additional roots of i and - i, we obtain four complex formulations where both the real and imaginary aspects are equal. Now in analytic terms these are represented as the co-ordinates of the two diagonal lines in our circle of unit radius (drawn at an equal distance from both horizontal and imaginary lines).

So we have the horizontal axis corresponding to the real values (+ 1 or - 1).
Then we have the vertical axis corresponding to the imaginary values (+ i and - i).
Finally we have the two diagonal axes corresponding in each case to two equal real and imaginary values (with opposite signs of both).

However it can be easily shown through a simple application of the Pythagorean Theorem that these diagonal lines in each case = 0.

Therefore - even in analytic terms - these diagonal lines can be given a double interpretation:

(a) as complex lines (with equal real and imaginary co-ordinates) and

(b) as null lines with a value = 0.

It should be said that the analytic interpretation of these null lines has found a very important place in modern physics e.g. Theory of Relativity and Twistor Theory, explaining in profound terms the very nature of light.

However - as always - we can equally give these mathematical notions an alternative dynamic holistic interpretation.

What this entails is that the diagonal lines (now representing our fundamental diagonal polarities of form and emptiness) can equally be given two interpretations.

(a) from the perspective of form as involving equal "real" and "imaginary"aspects. What this implies is that pure diagonal understanding of polarities is so refined that a perfect balance is maintained (in the understanding of phenomena) as between "real" conscious and "imaginary" unconscious understanding. In other words the dynamic interaction as between conscious and unconscious now reaches its purest most unobstructed expression.

(b) from the perspective of emptiness as entailing null lines i.e. empty of any phenomenal characteristics. Thus with the most complete expression of the dynamic interaction of the diagonal polarities, where rigid attachment of either a direct conscious or indirect unconscious nature ceases, experience itself attains to the purest state of spiritual emptiness.

Now what is extremely important to appreciate is that, from a dynamic perspective - which is the very nature of reality - both aspects are inextricably linked.

Therefore - even with the most extreme experience of contemplation - form and emptiness necessarily interact so that the balanced expression of both aspects should be maintained.
Indeed in this regard I would have to be critical of what I see here as an undue tendency in eastern esoteric traditions to emphasise emptiness over form (i.e. states over structures) at the higher mystical stages.

From a dynamic perspective therefore the very term nondual reality - when used to express the most complete spiritual contemplative experience - is a misnomer (for in dynamic terms such an experience is necessarily both dual and nondual in a highly refined manner).
Furthermore where this nondual bias predominates, there is less likelihood that sufficient emphasis will be given to the need for (subsequent) mature committed involvement in the world (where both the capacities for advanced levels of activity and contemplation are equally developed).

We can perhaps appreciate better this dynamic relationship between form and emptiness (in its most complete expression) by recognising that the purest state of contemplation equally requires the ability to remain detached from all phenomena (of either direct conscious or indirect unconscious origin).

This capacity resides in the pure exercise of volition i.e. the will in continually desiring the primary spiritual good (rather than attachment to lesser rigid phenomenal expressions).
However in the advanced state of contemplation the capacity of will is so finely developed that no (secondary) rest in phenomena takes place in experience (entailing rigid attachment).
However properly speaking this state of spiritual emptiness does not represent the absence of form but rather the absence of rigid attachment to form (which inevitably arises when the dynamic relation as between form and emptiness is unbalanced). So the state of emptiness equally entails the capacity for continued free identification with (phenomenal) form.

So paradoxically the attempt to achieve a pure state of emptiness, without recognition of the equal need for corresponding dynamic interaction with the structures of form, inevitably leads to an unconscious (unrecognised) attachment to form. This thereby damages the quality of spiritual experience and the subsequent ability to become properly immersed in the world of form.
 

Dynamic Relationship of Form and Emptiness

We have already looked at the dynamic relationship as between the horizontal and vertical polarities (which explains how holons switch between their whole/part and part whole aspects respectively).
It is now time to briefly explain how the interaction of the diagonal with both horizontal and vertical polarities takes place.

As we have seen (phenomenal) form can be manifested in two ways

(i) directly in "real" terms as conscious or

(ii) in "imaginary" terms as the indirect conscious expression of what directly relates to meaning that is unconscious.

Now the very recognition of phenomena ("real" or "imaginary") requires that we momentarily separate "real" from "imaginary" (and "imaginary" from "real").
However the movement to pure emptiness requires that we immediately cancel out such momentary attachment (which using dualistic language simultaneously equally combines both "real" and "imaginary" aspects).

In practical terms this implicitly entails a continual correcting process whereby one instantly recognises that what is experienced as "real" conscious has equally an "imaginary" unconscious aspect. Likewise what is "imaginary" (unconscious) has equally a "real" (conscious) aspect. So when the dynamic interaction as between both is highly developed, little rest (and thereby little attachment) to either aspect (as separate) takes place, and one continually dwells - as it were - in the intersection between both (which is pure emptiness).

This dynamic explanation of "high-level" psycho-spiritual activity has a corresponding "low-level" physical correspondent in terms of the most primitive of sub-atomic particles.
What happens here is that the fundamental empty ground of existence (diagonal) so rapidly interacts with particles - that have an extremely short-lived "real" identity as independent particles and an "imaginary" virtual identity indirectly expressive of an interdependent common ground - that any separate identity ("real" or "imaginary") becomes extremely elusive and ultimately evades any possible measurement. So eventually we approximate a pure physical void (which is equally the potential for all phenomenal existence).
 

Return to Eight Sectors
 
 

We will now look again at the Eight Sectors in the light of eight-dimensional understanding i.e. bi-directional in horizontal, vertical and diagonal terms.

Again in conventional manner, the horizontal x-axis is defined as real, the vertical y-axis as imaginary and the two diagonal axes as (equally) both real and imaginary. However the diagonal axes equally have a corresponding interpretation as null lines = 0. (See 3 again for explanation)
Now the actual plotting of these axes is arbitrary (once we remain consistent with our original reference point).

In dynamic holistic terms, we have continual interaction as between the horizontal and vertical polarities i.e. between real (conscious) and imaginary (unconscious) understanding.
Likewise we have continual interaction as between the diagonal (both conscious and unconscious) and horizontal and vertical polarities, where experience is momentarily either conscious or unconscious (in terms of separate phenomenal recognition).
In other words both conscious and unconscious are directly related to Spirit (as the central intersection of both). So in experience, conscious and unconscious are dynamically harmonised through the central role of Spirit. Alternatively we could say that both linear and circular modes of understanding are dynamically harmonised through the Spirit.

We have explained earlier how in horizontal dynamic terms, relating to exterior and interior (and interior and exterior) aspects, inclusion always entails exclusion (and exclusion entails inclusion). This means for example in holarchical terms that experience continually switches between what are - in fact - complementary opposite directions.

Likewise we have already explained how in vertical dynamic terms, relating to whole and part (and part and whole) aspects, that holarchy (where the lower is included in the higher holon) always implies partarchy (where the higher is included in the lower holon).
Equally of course partarchy implies holarchy so that again experience continually switches as between opposite directions in whole and part (and part and whole) terms.

Now in dynamic diagonal terms, relating to form and emptiness (and emptiness and form) aspects again we have bi-directional interpretations that in relative terms are complementary with each other.

Therefore from one perspective, the Spirit is involved in transcendent terms, whereas from the other, in corresponding immanent terms. This means in effect that holarchies and partarchies (both of inclusion and exclusion) have transcendent and immanent expressions.

In a comprehensive eight-sectoral approach we need to combine eight distinct (linear) asymmetrical holarchical interpretations which are related to each other (horizontally, vertically and diagonally) in a circular bi-directional fashion.

For a comprehensive differentiated understanding of development we take each of the (linear) asymmetrical interpretations in a relatively independent manner.

However for corresponding integral appreciation, opposite sectors must be bi-directionally combined (that are dynamically complementary with each other).
 

Importance for Science and Mathematics

The full eight-sectoral approach entailing bi-directional understanding in horizontal, vertical and diagonal terms enables the most detailed (holistic) mathematical structuring of the true dynamic nature of space and time.

We have already dealt with the four dimensional interpretation of space-time i.e. where space and time in dynamic terms have "real" and "imaginary" aspects with positive and negative directions in each case.

However the four complex roots (with an alternative explanation as null lines) lead to fascinating additional insights.

Indeed we can use this new understanding of complex space-time (equally from another perspective null lines) to reveal the very nature of physical light.
 

Now - as is well known - light manifests itself in two ways i.e. as both waves and particles.

In terms of our (holistic) mathematical approach, this can be easily explained in complex terms by saying that light has both real and imaginary aspects (which in dynamic terms keep switching).
Thus when light particles are observed, the particles are thereby "real" (and the corresponding waves "imaginary"). However when light waves are observed the waves are now "real" (with the corresponding particles "imaginary").
It is the very nature of observation that these two aspects must separate (so that only one aspect is thereby revealed). Where of course this occurs, light necessarily interacts with other (non-light) phenomena.
So in dynamic holistic mathematical terms, light is a complex phenomenon with real and imaginary aspects which keep switching (depending on the frame of reference). This is why - in "real" terms - light can be observed as either particles or waves (with in each case the corresponding "imaginary" aspect remaining hidden).

We can also say that the speed of light is constant (and travels at approximately 186,000 miles per second). However once again this measurement requires comparison with a phenomenal reference frame (that moves slower than the speed of light).

So in terms of phenomenal observation we never can detect light (in itself) but rather its interaction with other non-light phenomena.

If we ask what light is (in itself) its nature remains a mystery. However we can indirectly point to the nature of this mystery by saying that we have here the coincidence of both its real and imaginary aspects (i.e. both particle and wave characteristics). Of course we cannot directly observe this coincidence for by its very nature such observation requires their (relative) separation.

Now what we have here exactly corresponds with our definition of the diagonal polarities (in holistic terms) which are equally both "real" and "imaginary".

Also - as we know - these diagonal polarities have an alternative explanation as null lines with a magnitude = 0.

Now this interpretation as null lines intimately applies to the very nature of light (when defined in terms of its own frame of reference).

So if we somehow could move at the speed of light, we would find that now - from its own frame of reference - that time stands still. Put another way we could travel an infinite distance in zero time.
And this precisely is the meaning of null lines (in dynamic holistic terms).

So the very nature of light corresponds exactly - both in terms of our diagonal polarities - with both the complex formulation (i.e. complementary particle and wave aspects) and the null interpretation (i.e. as travelling in zero time from its own frame of reference). So the actual nature of physical light - like its spiritual counterpart - is utterly mysterious and can only be manifested indirectly through interaction with other phenomena.

However the implications here are even much more remarkable. Natural light is simply a manifestation of one of the four basic forces (i.e. the electromagnetic force).
Now each of the other forces equally has a complex formulation (with an alternative null formulation). In other words the other forces can manifest themselves in similar ways as both waves and particles.

So what I am proposing is that the four diagonal polarities (each with a complex and null interpretation) provide a precise holistic mathematical formulation of the nature of the four forces as their ultimate inherent nature where they are fully symmetrical and unified.

Of course this remarkable integral formulation - by definition - will not be obvious in the phenomenal world of form (where forces can only manifest themselves in an asymmetrical fashion).

This diagonal formulation also points directly to the very nature of forces as representing the intersection of object phenomena and dimensions. Thus again the complex nature of the physical forces indicates that, by their very nature, the forces have both (quantitative) phenomenal and (qualitative) dimensional characteristics.

So with the full eight-dimensional model, we can precisely interpret in holistic mathematical fashion, the relationship between object phenomena in terms of their positive and negative (exterior and interior) aspects. We can also interpret the dynamic nature of dimensions of space and time in positive and negative terms. Furthermore we can establish the important relationship as between object phenomena and dimensions (as - relatively - "real" and "imaginary" with respect to each other).

Finally we can model the fundamental nature of physical forces (in - alternatively - complex or null terms) and furthermore establish the dynamic relationship as between these forces and both dimensions and object phenomena.

Again - because of diagonal complementarity as between "lower" physical and "higher" spiritual reality - we can apply these very same notions to precisely formulate the approach to pure contemplative experience (as spiritual light).

Indeed spiritual light is remarkably complementary in structural terms to physical light.

It cannot be known directly in experience but rather indirectly through interaction with phenomena.
As with its physical counterpart, spiritual light manifests itself in two ways as transcendent and immanent. So in this sense it has a complex nature. Now if we want to know what spiritual light is in itself, there can be no phenomenal intermediary, so here, what are known as (separate) transcendent and immanent aspects must directly coincide as Spirit.
Therefore light now has a null nature i.e. is ever present and thereby "travels" an infinite distance in zero time.

Finally the existence of the other forces in physical terms points to a similar situation in spiritual terms. So light can manifest itself outwardly or inwardly. Paradoxically - as with the mystical dark night it can also remain hidden where we have the psycho-spiritual counterpart to the gravitational force.

So this very way of looking at reality - which is deeply in keeping with its ultimate integral nature - gives access to powerful new insights that would otherwise remain hidden.
 

Diagonal Operations: Powers and Roots

We have already dealt with the horizontal operations of addition and subtraction and the vertical operations of multiplication and division (giving them in each case a dynamic holistic mathematical interpretation).

Likewise the diagonal operations of powers and roots can be given a holistic mathematical interpretation, which also have very important applications in terms of the dynamics of understanding. 21
 
 

Brief Introduction to Radial Reality
 

We have distinguished as between linear understanding (where opposite poles are separated leading to dualistic interpretation) and circular understanding (where poles are understood as complementary ultimately leading to nondual awareness).
Whereas linear understanding is directly suited for the differentiation of reality, circular understanding is required for the dynamic task of its corresponding integration.
The earlier stages of development are largely suited to the differentiation of experience, which culminates at the middle of the Spectrum. Though integration is necessarily - also to a degree - involved at these stages it takes place largely within a predominantly differentiated worldview.

The later stages are by contrast largely devoted to corresponding integration of reality - which if successfully undergone - culminates in a high degree of nondual spiritual awareness.

The final stages - as I define them - then entail the mature interpenetration of both linear and circular understanding.

So for convenience we can look on development in 3 broad stages.

1) the gradual specialisation of linear understanding (differentiation leading to activity).

2) the gradual specialisation of circular understanding culminating in nondual spiritual awareness (integration leading to passive contemplation).

3) the gradual interpenetration - in mature terms - of both linear and circular understanding (the marriage of activity with contemplation).
 
 

Outline of Theory of Everything 2
 

In the first Chapter we introduced the basic outlines of our holistic mathematical approach to an integral Theory of Everything i.e. polarities (that are dynamically positive and negative with respect to each other) defined in horizontal, vertical and diagonal terms with both linear (1) and circular (0) interpretations.

Once again all dynamic processes can be precisely encoded in terms of these 3 fundamental polarities (given both linear and circular interpretations).

In this Chapter we have however gone considerably further to demonstrate the precise holistic mathematical nature of the polarities (horizontal, vertical and diagonal).

Quite remarkably, their dynamic structure is provided by the holistic interpretation of a simple mathematical relationship, which - quite literally - represents a reduced expression of oneness.

Because of its great importance perhaps it would be worthwhile to clarify further its nature.

In analytic terms we first demonstrated the basic fact that all numbers have both quantitative and qualitative (i.e. dimensional) characteristics.
By clarifying their precise nature, we showed that all numbers can be defined with respect to both a linear and circular number system.

So for example in the linear system, 2 is implicitly defined with respect to a (qualitative) dimension of 1 (which remains fixed).
So strictly speaking - from a linear perspective - 2 is 2¹.

However in the circular system, the rules are reversed so that now the (qualitative) dimension varies with respect to the number quantity of 1 (which remains fixed).

So in the circular system, 2 is strictly written as 1².

So 2 (in linear terms) is - strictly -  2¹ whereas 2 (in circular terms) is 1².

Likewise 4 (in linear terms) is 4¹ whereas 4 (in circular) terms is 1⁴.

Now we can represent numbers (in linear terms) literally as intervals marked off on a straight line.

However to express the circular nature of our alternative number system, we need to express each number in (reduced) linear terms (i.e. with respect to a dimension of 1).

This means - for example - with 2 that we need to obtain the square root of 1 (i.e. 1²) which yields two roots + 1 and - 1.

These then can be given coherent geometrical expression as two equidistant points on the circle of unit radius i.e. as the end points of the line diameter through the circle.

In similar fashion the circular nature of 4 is shown by obtaining the four roots of unity (i.e. 1⁴), which geometrically will be represented by four equidistant points on the circle (defined in the complex plane) which when joined with straight lines gives us the four quadrants of the circle.

So therefore we have in each case the linear definition of a number (as unit intervals marked out on a straight line) and the circular definition (as equidistant points on the circle of unit radius).
 

Now whereas the linear interpretation of number is of primary importance from an analytic scientific perspective, the circular approach assumes an amazingly powerful significance when given its appropriate dynamic holistic interpretation.

So we will now briefly explain circular duality (i.e. the holistic interpretation of 2).

As we have seen in dualistic terms all reality is conditioned by opposite polarities (i.e. positive and negative) which are - to a degree - necessarily separated.

However spiritual union requires the harmonisation - and ultimately mutual identity - so that these opposites no longer separately exist.

Using dualistic language, 1² - in holistic terms - literally relates to a higher dimensional experience of union where opposite polarities directly coincide. 22

So 2 here represents the union of opposites (positive and negative) which at the linear
one dimensional level are necessarily separated.

Therefore to indirectly express the nature of this higher dimensional unity in reduced linear language, we mathematically obtain the square root of 1.
This means that the higher dimensional spiritual union (where opposites coincide) splits dualistically at a phenomenal level into two opposite polarities (which are positive and negative with respect to each other).

Therefore all reality is conditioned by opposite "real" polarities i.e. that are expressed in a "real" conscious phenomenal manner.

Now it is the very nature of dynamic experience that the "higher" dimensional and reduced phenomenal experience necessarily interacts. So there is ceaseless interaction therefore as between the spiritual state of emptiness where opposite polarities directly coincide in nondual manner, and the reduced structures of (phenomenal) form where polarities separate in dualistic fashion.
Put another way there is ceaseless interaction as between circular and linear notions of unity.

We have identified our first set of "real" polarities in  horizontal (exterior/interior) terms that operate within a given level.
In proper dynamic interpretation they point both to the "higher" empty state (where these opposites  coincide) and the "lower" (reduced) formal structure where the opposites separate.
So all dynamic processes - in varying degrees - represent the interaction of these opposites with respect to their transformed empty (spiritual) state and reduced formal (phenomenal) structures.

The second set of polarities requires a much subtler interpretation.
Once again the first set relate to "real" interpretation (i.e. phenomena with a direct conscious interpretation).

However all phenomena have likewise an unconscious interpretation (i.e. as the indirect phenomenal expression of unconscious desire).

We have shown that the negative direction of experience is dynamically related to the unconscious (in direct fashion) leading to nondual merging of opposites.

Therefore to express this "higher" dimensional reality in reduced linear fashion we now must take the square root of - 1. In holistic terms this gives the precise correspondent of what is - in mathematical terms - imaginary.

All of this is deeply relevant to explaining the dynamic interaction as between the vertical polarities of whole and part  (and part and whole) operating between levels.

As we have explained their interaction cannot be explained successfully in solely "real" terms, as the dynamic interaction of conscious and unconscious in involved.
So to properly explain the relationship between wholes and parts (in a manner where qualitative and quantitative aspects are not reduced to each other) requires defining reality in complex terms i.e. with both "real" and "imaginary" aspects which ceaselessly interact.

So the higher dimensional reality i.e. empty spiritual state here is of order 4 (i.e. where both horizontal and vertical polarities - which are dynamically "real" and "imaginary" with respect to each other - are reconciled).
The "lower" phenomenal structures - which exemplify this state - entail opposite polarities (in "real" and "imaginary" terms) that are to a degree separated.

Finally the 3rd set of polarities involve the most refined (diagonal) relation as between form and emptiness (and emptiness and form). These point to the pure intersection as between "real" (conscious) and "imaginary" (unconscious) understanding which is directly spiritual.
Once again the four additional roots of unity provide a remarkable explanation of the nature of this relationship pointing directly to the pure empty (spiritual) state (as null lines) and the associated phenomenal structures (where perfect balance as between "real" and "imaginary" aspects is maintained). 23

We are also able to show the precise dynamic relationship - though it is extremely subtle and requires considerable clarity of intuition - as between the relationship of these diagonal polarities and both the horizontal and vertical types earlier described.

Indeed the holistic mathematical interpretation of these diagonal lines gives rise directly to a (closed) TOE that points to an integral interpretation of both the (physical) ground and (psychological) goal of reality. 24

So from one perspective it is incredibly simple.

All dynamic processes in their purest phenomenal expressions are mathematical - indeed binary - in a precise holistic manner.
The structures (and indeed states) which clarify most accurately - in phenomenal terms -  these relationships, result from the holistic interpretation of an incredibly simple (circular) number system which literally represents a reduced expression of oneness.

Thus the structures of reality at all levels can be encoded as a reduced expression of oneness which in turn ceaselessly interact in dynamic terms with the transformed empty (spiritual) state of creative nothingness. 25

Once again these dynamic processes can be precisely encoded using the holistic binary system i.e. linear (1) and circular (0) understanding operating with respect to 3 sets of polarities, that in turn have precise mathematical interpretations (in dynamic holistic terms).

So we have in dynamic relative mathematical terms, real (horizontal), imaginary (vertical) and complex (diagonal) polarities that ceaselessly interact with respect to their binary i.e. both linear (1) and circular (0) interpretations. 26

Remarkably in dynamic interpretation the relationship as between both states and structures with respect to all polarities can be maintained with - at the most advanced stage -  both interpretations coinciding i.e. diagonal polarities.

(Indeed an even more comprehensive formulation - incorporating the holistic interpretation of all the major number types can be given. These in turn provide the most scientific manner possible of precisely ordering all the various stages and sub-stages of development. I will describe this at a later stage in "Integral Theory of Everything 3"). 27

This integral Theory of Everything can be given a more limited closed interpretation (in terms of the ultimate symmetries expressing the ground and goal of existence).

Equally it can be given a more open interpretation in describing the dynamic interaction as between both the symmetrical and asymmetrical aspects of processes at all levels of reality.
It is to this latter use that we now turn in applying the Theory of Everything to the Spectrum of Development.
 
 

                                                 Notes
 

1. Numbers and Dimensions

2. Mathematical Dimensions and Space-time

3. Demonstration that Diagonal Lines are of Zero Magnitude

4. Carl Jung and Holistic Mathematical Approach

5. Time is space; space is time

6. Mystical Interpretations of a Rock

7. Double Correspondence in Science

8. Horizontal Polarities - Holistic Mathematical Interpretation

9. Asymmetrical and Symmetrical Relationships

10. Bi-directional Nature of Space and Time: Two Applications

11. Involution and Evolution

12. Integral Science

13. Imaginary experience of a House

14. Conventional and Mathematical Notions of Imaginary

15. Illustration of Real and Imaginary Dimensions

16. Levels, Planes, Structures, States and Body-Minds

17. On Holons

18. Relationship Between Real and Imaginary

19. On Holarchy and Non-Holarchy

20. Holistical Mathematical Interpretation of Multiplication and Division

21. Powers and Roots

22. Dimensional and Non-Dimensional Expressions

23. Holistic Mathematical Mapping of Basic Levels

24. A Closed Integral TOE

25. Emptiness and Form

26. Relationship Between the Holistic and Binary Systems

27. Closed and Open TOE's (Analytic, Integral and Radial)