In Western society, scientific understanding employs predominantly the rational (analytical) paradigm which recognises solely the conscious process. This involves the separation of object from subject in an impersonal detached view of truth. Mathematics represents the extreme version of this specialised use of reason.

Understanding in Eastern society - especially spiritual - has traditionally placed more emphasis on the unconscious (holistic) paradigm. Here object and subject are directly united in an experiential view of truth. Transpersonal psychology (i.e. mystical or esoteric understanding) represents the extreme version of this specialised use of intuition.

For many the great attraction of mathematics is the absolute nature of truth involved, free - in formal terms - of all "vague" intuitive interference.

Others seek truth experientially, through direct unity with reality, finding the formulations of "specialised" science to be purely relative, and ultimately illusory.

Thus historically, these two approaches - offering two extreme views of truth - have developed largely apart, with radically different perspectives and language of communication. Consequently, little success has been achieved in terms of their integration.

However - remarkable though it may seem - mathematics and transpersonal psychology are in truth fully complementary. The amazing marriage of both awaits, which has the power to dramatically alter our whole vision of reality.

The holistic approach which seeks to make explicit this inherent unity of reason and intuition, I refer to as transrational.

This can be applied at different levels. For example, the relationships of physics and psychology are - when appropriately understood - fully complementary. Transrational physics thus entails a dynamic way of structuring reality so that for every relationship in physics, a corresponding (mirror) complementary relationship in psychology can be readily identified. Transrational psychology would then represent the other side of the coin starting with psychology to identify complementary relationships in physics.

However at an even deeper level, the fundamental structure of reality - in both physical and psychological terms - is mathematical. Thus a potentially more fruitful synthesis exists as between mathematics and (transpersonal) psychology.

Incredible as it may appear the great richness and subtlety of transpersonal psychology can be precisely structured (indirectly) in terms of mathematical relationships.

What this really entails is that there are indeed two complementary mathematical systems.

The first is the conventional system of (quantitative) mathematical relationships based (directly) on the specialised use of reason.

The second - and largely unknown system - is a complementary set of (qualitative) mathematical relationships - which though indirectly capable of rational expression - is based (directly) on the specialised use of intuition.

Thus, when appropriately understood (i.e. from a transrational perspective), all (quantitative) mathematical relationships can be translated in (qualitative) psychological terms. This ultimately provides a coherent means of raising mathematical understanding to the mystical level of pure contemplation (as with the Pythagorean ideal).

In complementary fashion, all (qualitative) psychological relationships can be translated in reduced (quantitative) mathematical terms. This in turn, provides a means, of precisely structuring all the various stages of (transpersonal) psychological development in mathematical format.

These ideas may initially seem somewhat unusual. This however is only because we are not accustomed to looking at reality in this manner. The following chapters will attempt however to make explicit some of the implications of this (potentially) revolutionary approach.
 
 
 

THE HIDDEN DIMENSIONS 

Introduction 

I will start with a short statement that has profound significance.

"Mathematical understanding is always a psychological experience"

Put simply all understanding involves - in dynamic terms - complementary dual
components. There is an objective or external aspect i.e. what is understood (in relation to the understanding mind) and a subjective or internal aspect i.e. the understanding mind (in relation to what is understood).

In conventional understanding, however mathematics is interpreted in reduced static terms where propositions are given an absolute objective validity. Indeed - as I have already stated, the great appeal of mathematics for so many people is this belief in its "absolute" nature.

Because of this one-sided and inherently distorted approach, the complementary psychological aspect of mathematical understanding is thereby lost.

Let me illustrate this with an analogy. A scissors comprises two complementary blades which overlap. The cutting action of the scissors comes from the dynamic operation of these blades, working in unison. Clearly therefore, to attempt to explain the scissors in terms of one blade only, is highly misleading.

Yet in regard to the scissors of mathematical understanding, only one blade i.e. the external aspect is formally recognised. The other blade representing the internal psychological aspect is thus largely ignored. Most importantly the dynamic action of these two aspects - which truly constitutes mathematical understanding - is overlooked.

My task therefore is to firstly recover this "hidden" psychological aspect, and then to incorporate both aspects in a new dynamic transrational interpretation of mathematics.
 

Linear Number Systems

Horizontal Numbers

Implicit in the conventional definition of a number is a hidden dimensional characteristic. If we take for example 2, this really is 2¹. 2 is a number quantity within the (fixed) 1st dimension. Conventionally in mathematics numbers are defined with respect to the 1st dimension. If initially a number is defined in another dimension its reduced quantitative value will be given in relation to the 1st. Thus 2² = 4 (i.e. 4¹).

Now the straight line is the appropriate geometrical representation of the 1st dimension. Therefore the approach that interprets numbers within the invariant 1st dimension is one dimensional or linear.

We will refer to this conventional linear approach as horizontal.
 

Vertical Numbers 

However there is also a complementary definition of number. Whereas the former represents a number quantity (raised to a unitary number quality or dimension), the latter represents the number quality or dimension (to which a given number "1" is raised).

Thus 1¹, 2¹, 3¹ and 4¹ represent the first four natural numbers (in the conventional number system). Here I, 2, 3 and 4 represent quantities (raised to a unitary qualitative dimension).

1¹, 1², 1³ and 1⁴ represent the corresponding first four natural numbers (in the alternative number system). Here 1, 2, 3 and 4 represent qualities (to which the unit quantity is raised).

Thus relative to the former system which is directly quantitative (and indirectly qualitative), this latter system is directly qualitative (and indirectly quantitative). In contrast to the horizontal quantitative system, we can refer to this latter qualitative system as vertical linear.

In terms of our scissors analogy, the (conventional) quantitative system represents the (direct) mathematical aspect, whereas the complementary qualitative system represents the (direct) psychological aspect of number.
 

Circular Number Systems 

As we have seen, neither blade in isolation, but rather the dynamic co-ordination of both blades constitutes the action of the scissors.

Likewise neither number system in isolation, but rather the integration of both constitutes dynamic mathematical understanding.

To achieve this integration we need a means of translating

(a) the (psychological) qualitative, in complementary (mathematical) quantitative terms and

(b) the (mathematical) quantitative, in complementary (psychological) qualitative terms.

These indirect translations lead in both cases to the emergence of fascinating circular number systems.

Thus the linear (psychological) qualitative system, has an indirect reduced translation as a circular (mathematical) quantitative system.

Likewise the linear (mathematical) quantitative system, has an indirect raised translation as a circular (psychological) qualitative system.

It is through the integration of these linear and circular number systems, that the inherent complementarity of mathematics and psychology can best be appreciated.
 

Quantitative Circular Numbers 

As we have seen the first four natural numbers in the (psychological) qualitative system are 1¹, 1², 1³ and 1⁴ respectively.

To translate these numbers in indirect (mathematical) quantitative terms, we must express each number in terms of an invariant 1st dimension (i.e. as solutions to the equation xⁿ = 1. where n is a natural number). This simply implies taking the successive roots of each number.

The one root of 1, which could be written  is of course 1 or +1. This number lies on the corresponding circle of unit radius.

The two roots (i.e. square root) of 1 or 1^1/2 are +1 and -1. Thus expressed in one dimensional linear (quantitative) terms, the 2nd (qualitative) dimension has complementary aspects (i.e. two numbers of opposite sign). These two numbers again lie - as both ends of a diameter - on the corresponding circle of unit radius.

The three roots (i.e. cube root) of 1 or 1^1/3 are 1, (-1 +(-3)^1/2)/2 and (-1 -(-3)^1/2)/2. We now have three roots which again lie on the corresponding circle of unit radius (where horizontal axis represents real, and vertical axis represents imaginary number systems respectively).

The four roots of 1 or 1^1/4 are 1, -1, i and -i respectively (i = the square root of -1). These four roots again lie on corresponding circle of unit radius (with four points representing the extremities of real and imaginary axes).

Higher order roots will also lie on the same circle. If we obtained for example the 100 roots of unity
(i.e. 1^1/100), these would lie as 100 equidistant points on the same circle).

Therefore there is a coherent (mathematical) quantitative system which is circular corresponding to the (psychological) qualitative system which is linear.
 

Qualitative Circular Numbers  

Let us take as an example the simple equation x = 1. In conventional practice when we square both sides we get x² = 1. However when we reverse this operation - by extracting the square root - we get x = +1 and x = -1. Thus, whereas x formerly had one solution, now it has two!

This lack of reverse symmetry in operations highlights an important problem.

When we start with x = 1 and square both sides, strictly speaking the result is x² = 1² . However in conventional terms, 1² is interpreted (solely) in reduced quantitative terms as 1(i.e. 1¹). In other words the qualitative transformation involved in moving from the first to second dimension is thereby lost.

If we are to interpret it correctly, when we move from 1 (i.e. 1¹ ) to 1², there is a psychological transformation involved. 1² represents a "higher" order unity than 1¹. This second dimension in fact corresponds to (circular) understanding based on the principle of the complementarity of opposites. This exactly matches the corresponding mathematical translation (i.e. through obtaining the square root of unity). In like fashion this and all higher dimensional numbers can be referred to - in psychological terms - as circular.
 
 

Summary 

Understanding - in dynamic terms - is like a scissors involving the interaction of complementary aspects.

The conventional (quantitative) mathematical number system is horizontal and linear (where numbers are ultimately interpreted within the fixed 1st dimension)

The complementary (qualitative) psychological number system is - in relative terms - vertical and linear (where numbers represent dimensions or powers to which the given number "1" is raised).

Both quantitative and qualitative systems can be translated indirectly in terms of each other. These translations give rise to circular number systems which are complementary.

The (qualitative) psychological system, when indirectly translated in mathematical terms gives rise to a quantitative circular number system.

Also the (quantitative) mathematical system, when indirectly translated in psychological terms gives rise to a qualitative circular number system.
 
 

ONE DIMENSION - THE LINEAR LEVEL
 

Experience - in dynamic terms - involves the continual interaction of two complementary processes of understanding.

Rational understanding corresponds to the line and is directly suited to quantitative interpretation of reality. Intuitive understanding corresponds to the circle and is directly suited to qualitative interpretation.

Psychologically, therefore, reality - at all levels - can be described in terms of the interaction of line and circle.

Once more we have a direct parallel in mathematical terms. The two binary digits 1 and 0 are based (with only minor modifications) on the symbols of line and circle respectively. These two symbols are sufficient to construct the entire number system. As is well known the marvels of the computer age are intimately related to this mathematical binary system.

However dynamically speaking - as we have illustrated - this binary system has two complementary aspects i.e. both qualitative psychological and quantitative mathematical, which continually interact. Whereas all information processes can be translated through the quantitative binary system, remarkably all transformation processes (physical and psychological) can be translated through the qualitative binary system.

Since they are so fundamental, let us look more closely at these two symbols of line and circle.

A straight line has a beginning and an end extending - as it were - without returning to its source. In conventional understanding this extension takes place solely in a positive forward direction. This one-way linear notion is firmly embedded for example in the conventional understanding of time. Time moves forward and never returns to itself. The idea that time could have also a negative direction - and move backwards - would seem incredible to most people and merely the stuff of science fiction. Time that has passed, in this view, therefore is gone forever consigned to an unrecoverable past.

The line lends itself to finite discrete terms. It is possible therefore to break it into a number of separate parts. Again we see this in relation to time itself. Different activities are organised in terms of discrete time allocations. Efficient organisation is then interpreted in terms of further fragmentation of the time unit.

Indeed this is probably the key feature of the rational linear level. It is ideally suited to partial analysis based on multiple fragmentation of the original unit of reality. Literally through this process, separate line fragments i.e. "1's" (independent units) are created.

The circle, by contrast lends itself to infinite continuous understanding. A circle has no (distinct) beginning or end. Any point on a circle can serve as the beginning or the end.

In itself intuition is totally immediate and obvious. When translated however in rational format - as it attempts to reconcile what are polar opposites from the linear viewpoint - it appears circular and paradoxical. Thus from this circular intuitive perspective time has no beginning or end and continually returns to itself. In other words, all time is contained in an ever repeating present moment.

This is in sharp contrast to the conventional linear perspective. However even in practical terms it is possible to demonstrate the important role of the circular viewpoint.

If I set out on a journey around the "whole" world - tracing out a circular path - I will ultimately arrive back at the stating point. Thus the end of this "whole" journey is indistinguishable from the beginning. Furthermore there is no distinction holistically between positive or negative directions. Irrespective of which direction I take the start and end points (of the whole journey) will be the same.

Now if we confine ourselves to just part of the journey, the circular perspective will break down. On any leg of the journey, starting point and destination point are separate and can best be represented in linear terms. Also the direction of the journey now becomes relevant. Clearly, to travel on any leg in an opposite direction will lead to arrival at a different destination point.

Thus we have two complementary approaches to understanding.

The linear (quantitative) approach based on reason is partial and analytic (directly suited to the breaking of a whole into parts). When reductionism is avoided these parts are quantitatively different from the whole.

The circular (qualitative) approach based on intuition is holistic and synthetic (directly suited to the creation of a whole from the parts). Again when reductionism is avoided, this whole is qualitatively different from the parts.

Putting it another way the linear approach is suited directly to differentiation, and the circular approach directly to integration of experience. However there is no direct correspondence as between the two approaches. The local character of the rational linear approach is broken by moving to intuition; likewise the global character of the intuitive circular approach is broken by moving to reason.

Of course a balanced dynamic understanding requires both approaches operating in mutual harmony.

In practice, experience is highly reduced with most problems interpreted in merely rational linear terms. Not surprisingly therefore, understanding frequently lacks any coherent overall structure. In other words it lacks integration.
 

The Linear Level 

I will now outline the manner in which experience is interpreted at the linear level. We will do this in terms of the understanding of number using the fully differentiated rational paradigm.

As we have seen, understanding at this level is one dimensional. This entails that there is no distinction as between the quantitative and qualitative interpretation of numbers. In other words the psychological (qualitative) is simply reduced to the mathematical (quantitative) aspect.

As we have seen we can convert the psychological (qualitative) to mathematical (quantitative) format by extracting the appropriate root. As the linear level is one dimensional this entails taking the 1st root of unity.

Now 1¹ = 1 (i.e. +1). We have here a precise mathematical way of demonstrating in terms of the linear level the identity of quantitative and qualitative number systems. The qualitative system therefore - at the linear level - can be conveniently reduced to the quantitative and eliminated from further consideration.

Let us now clarify the precise manner in which the dynamics of understanding are interpreted at the linear level.

Understanding of all phenomena involves a dynamic mind-matter interaction. We have external phenomena (in relation to perceiving mind which is internal). Equally we have the perceiving mind (in relation to the external phenomena). In other words both positive and negative directions are involved.

However - as we have seen, the linear level only recognises the positive direction of experience. This entails the assertion that external phenomena have an objective (static) existence independent of mind. Though the mind is necessary to register such phenomena, in terms of this level, their absolute identity is not altered.

Thus typically mathematicians will maintain that numbers - which in experience arise through mental perceptions - have an absolute objective identity unaffected by our interaction with them. Indeed - as I have stated before - the great attraction of mathematics for so many people is this (mistaken) belief in the absolute nature of the objects utilised.

Thus in terms of mathematical quantities (i.e. the mental perceptions we identify as distinct numbers), only the positive direction of experience is recognised. In other words numbers are simply treated as (static) absolute objects.

The qualitative dimensional characteristic of phenomena comes directly from conceptual understanding.

With linear understanding all phenomena are interpreted within a fixed unitary dimension. Let us now - in the context of number - outline precisely what this means.

Particular number perceptions relate directly to (number) quantities. For example 1, 2, 3 and 4 are number quantities. However these numbers have something in common which is the general quality or dimension of number. This qualitative understanding of number (i.e. where the number represents in direct terms a dimension) is given by the concept of number. Again in terms of the dynamics of experience both quantitative and qualitative aspects are inextricably linked. We cannot identify particular numbers in the absence of the general number concept. Likewise we cannot (ultimately) identify the general concept in the absence of particular numbers.

Now in conventional mathematics the number concept is strictly one dimensional. In other words there is literally - in terms of this interpretation - only one number concept. Whereas individual number perceptions as quantities can vary, the number concept is understood as remaining fixed. Various number perceptions such as 1, 2, 3 and 4 are interpreted in terms of the same number concept. Indeed all numbers (ultimately) are interpreted in terms of this one fixed number concept (i.e. numbers have a dimension or power of 1).

The concept or dimension of number of course is qualitatively distinct from particular number quantities. As a number dimension it does not refer to any number as such but rather the general property of number. Strictly speaking, it has - in direct terms - a potential rather than an actual existence.

However in reduced terms - in terms of the linear level - this qualitative notion of number appears identical to the quantitative. From the conventional perspective the number concept will be defined as the set of all (particular) numbers. So what is qualitative - and relatively distinct - becomes reduced to a global set of quantities. In other words the complementary qualitative aspect of number is lost with only the quantitative aspect remaining.

The linear level - which represents the conventional approach to mathematics - is therefore a highly reduced interpretation of reality. Only the positive (i.e. objective) direction of experience is recognised. Thus, the qualitative aspect of experience is translated in terms of the quantitative.

So, with reference to mathematics, numbers are interpreted as (static) absolute objective entities.

In addition, the qualitative dimensional aspect of number (i.e. the number concept) is interpreted in reduced terms as the set of all number quantities.

(Though I have used the important case of number for illustration purposes, the dynamic approach outlined above applies to all perceptions and concepts).
 
 

Transition from One to Two Dimensions  

Negative Direction 

The first step in terms of the recovery of the complementary psychological aspect of mathematics is an explicit understanding of the negative direction of phenomena.

In conventional terms, mathematical quantities can have a negative as well as a positive sign.

Likewise in dynamic psychological terms all phenomena can have a negative as well as a positive sign (i.e. direction).

In psychological terms, the field of potential reality i.e. the unconscious, involves undifferentiated polar opposites. The differentiation of phenomena (i.e. bringing phenomena to consciousness) thus requires the positing of these phenomena in experience. This in turn involves the separation of inherently complementary poles. Objective is thereby separated from subjective experience and given an absolute independent existence. The more "successful" this differentiation, the more rigid phenomena become in experience with the subjective pole being negated i.e. repressed in the unconscious.

Mathematics itself is a fine example of this extreme differentiation. The belief in the objective validity of mathematical phenomena is so entrenched that the complementary subjective aspect is almost entirely repressed. This makes it extremely difficult for example to engage "believers" in open discussion on the inherently relative nature of all mathematical understanding.

The negation of phenomena involves a growing recognition of the true role of the unconscious with its twin complementary poles. Therefore moving away from conscious experience to discover the full potential of the unconscious requires the corresponding psychological negation of phenomena. In other words, phenomena which are posited (and thereby made conscious) must be likewise negated and undone (removed from consciousness) to once more enter the unconscious. In Christian ascetical literature this is referred to as the way of purgation which inevitably precedes illumination (i.e. the outpouring of the refined unconscious).

I have referred to this transition as the development of linear mirror (or negative) structures. Indeed it involves a form of psychic anti-matter which dynamically fuses with psychic matter creating spiritual energy..

In mathematics, when equal numbers of opposite sign are added we get 0 (i.e. nothing). Likewise in psychological terms, when phenomena of opposite sign (psychic matter and psychic anti-matter) are combined we get 0 (i.e. nothing in phenomenal terms). However whereas the conventional mathematical notion of nothingness is a static one, the psychological notion is dynamic, referring to pure (spiritual) intuition.

When we apply this to mathematics we get a significant re-interpretation.

Each particular number perception has both a positive and negative direction. Like matter particles every number has a corresponding anti-number. Thus both poles of experience are now made explicit. If the objective perception (in relation to mind) represents a (particular) number then subjective mind (in relation to objective perception) represents the corresponding anti-number. In terms of the dynamics of experience it is this very fusion of number and anti-number that generates intuitive insight enabling the mind to switch to the corresponding number concept.

In the first instance this concept of number is an inherently spiritual or intuitive notion representing the very potential for number existence. It is only when expressed in reduced quantitative manner that the number concept collapses - as it were - to rational conscious terms.

This explains why the qualitative intuitive notion of the number concept is lost in conventional understanding. Though the understanding is still inherently dynamic - with the negative pole undeveloped - little fusion of number and anti-number takes place. Therefore the limited amount of intuitive insight generated in the process is used solely to switch between perceptions and concepts which are now both interpreted in reduced rational fashion.

Just as all perceptions have both positive and negative directions, so do all concepts. Thus the concept of number has a corresponding anti-concept. Alternatively the 1st dimension of number has a corresponding anti-dimension. (i.e. the negative 1st dimension). Again if the objective concept (in relation to mind) is the positive 1st dimension, then the mind (in relation to objective concept) is the corresponding anti-dimension (or negative 1st dimension).

The development of these anti-concepts in psychological terms represents the more deep-rooted undoing of the positive direction of experience (i.e. the undoing of objective rigid concepts).

In terms of the dynamics of experience, the fusion of concepts and anti-concepts is the means by which (reverse) intuitive insight is generated enabling a switch to potential number perception. Once again in conventional linear interpretation, the potential intuitive aspect collapses to leave a merely actual rational perception.

Thus if we take the number "2" to illustrate. The positive direction gives the rational objective number "2" The negative direction gives the corresponding anti-number "2". The fusion of number and anti-number generates a transformation in intuitive insight which is of a qualitatively different order. This enables a shift to take place from the horizontal number quantity to the vertical number quality or dimension (i.e. concept). Conventionally however, this vertical number dimension is interpreted in reduced horizontal terms (as the set of number quantities).

Again if we take the corresponding number concept (to which the number "2" is related). The positive direction gives the rational objective concept. The negative direction gives the corresponding anti-concept. Again the fusion of concept and anti-concept causes a transformation in qualitative intuitive terms enabling a switch back to the horizontal number perception "2". However again this qualitative aspect is lost in a reduced rational interpretation.

However, though there is an enhanced appreciation at the circular level of the inherently dynamic and bi-directional nature of all phenomena, confusion as to the precise relationship as between concepts and perceptions still exists. Though they are qualitatively different, one still attempts to view both in "real" terms. This problem is not solved till the point level.

Again, intuition is generated through the dynamic fusion of psychic matter and psychic anti-matter (i.e. the two directions of experience). This holistic potential aspect of experience enables the switch from perception to concept, and also from concept back to perception to take place. However when little intuition is generated it quickly collapses to actual rational format with a merely quantitative interpretation remaining. Indeed when understanding is especially rigid, so little intuition is generated in the dynamics of experience, that even switching ability is greatly limited.
 
 

SUMMARY 

Reason is symbolised by the line and intuition by the circle. Both aspects are dynamically linked in understanding, with reason directly geared to (partial) analysis and intuition to (holistic) synthesis of experience.

The linear level represents the extreme specialisation of reason. In mathematics, numbers are consistently interpreted in one dimensional terms. Here, the qualitative (intuitive) aspect of understanding is translated in terms of the quantitative (rational) aspect, leading to a reduced and - ultimately distorted - understanding.

During the linear level only the positive direction of experience is developed. With the transition to the two-dimensional circular level, the negative direction is explicitly developed. Both number perceptions and concepts now have corresponding anti-number perceptions and concepts respectively.
 
 
 

TWO DIMENSIONS - THE CIRCULAR LEVEL
 

The circular level involves a significant advance where understanding now becomes two dimensional. This involves the two complementary directions of understanding (i.e. positive and negative), now mutually combined.

Reality - at all levels - is now explicitly understood in terms of the complementarity of polar opposites.

At the linear level - as we have seen - (from a philosophical perspective) numbers are interpreted strictly in one dimensional terms. They are understood solely as mathematical quantities. The directly qualitative dimension of numbers - given by the power or exponent - is then interpreted in reduced quantitative terms.

2² for example in this approach has a solely (reduced) quantitative interpretation as 4 (i.e. 4¹). Through this translation, the dimension of the number is literally reduced to 1.

At the circular level, numbers now have both a mathematical (quantitative) and psychological (qualitative) interpretation which are themselves complementary.

Here the (horizontal) number system - representing mathematical quantities - can be converted indirectly in terms of the (vertical) number system, representing psychological qualities.

In turn the (vertical) number system - representing psychological qualities (i.e. dimensions) can be converted indirectly in terms of the (horizontal) number system, representing mathematical quantities.

In both cases these translations give rise to circular number systems.

We have firstly therefore the conversion of the horizontal to the vertical number system. Here there is a transformation from a merely quantitative to an indirect qualitative understanding of number. Again whereas the original quantitative understanding is linear, the indirect qualitative translation is circular.

If for example we take the (horizontal) number quantity "2". This is we have will be understood initially in linear terms (i.e. with respect to the 1st dimension).

However this understanding recognises solely the positive direction of experience which is the number (in relation to the perceiving mind). However, in dynamic terms there is also the corresponding anti-number giving the negative direction of the number which is the perceiving mind (in relation to the number).

When these two directions are explicitly recognised we now get a qualitative understanding of each number as representing the dynamic interaction of complementary poles (i.e. positive and negative directions respectively).

In contrast to the linear quantitative understanding - where the positive direction is clearly differentiated from negative - this qualitative understanding is circular (where both directions are now understood as ultimately identical).

In direct terms this qualitative understanding is purely intuitive - representing the immediate fusion of opposites - and thereby non-dimensional. However, when expressed in reduced rational terms this understanding is two dimensional (i.e. comprising both positive and negative poles).

This leads to a new interpretation of number which is strictly relative. Each number is now seen in dynamic terms with interacting positive and negative poles (i.e. involving objective and subjective understanding respectively).

Numbers are now subject to a new uncertainty principle. The attempt to focus on the merely quantitative aspect of number, of necessity, entails the corresponding attempt to screen out the related qualitative aspect. This - as we have seen - accurately represents the conventional mathematical approach to number, which is misleadingly interpreted in static absolute terms. We now see that knowledge of the quantitative aspect - which is subject to a dynamic uncertainty principle - can only be approximate.

We have secondly, the conversion of the vertical to the horizontal number system. Here there is the corresponding transformation from a merely qualitative to an indirect quantitative interpretation of number.

In our example the (horizontal) linear quantitative interpretation is given by the number perception "2" (2¹).

The (vertical) linear qualitative interpretation is - in relative terms - represents - in direct terms - the concept through which this number is interpreted (1²).

Again whereas the original (vertical) psychological interpretation is linear, the corresponding (horizontal) mathematical interpretation is circular.

This latter translation is simply obtained by obtaining the square root of unity, which yields two complementary solutions, once again representing the two directions of experience. Again, from the perspective of linear understanding - where number signs are clearly differentiated - this mathematical result appears deeply paradoxical and circular.

Thus we see here a simple - yet remarkable - correspondence as between psychological and mathematical experience. The psychological principle of the complementarity of opposites, in moving from the 1st to the 2nd dimension, is exactly replicated by a corresponding mathematical principle of the complementarity of opposites, in the reverse movement from the 2nd to the 1st dimension (i.e. through obtaining the square root of a number).

Thus we have in understanding at the circular level both a continual raising (i.e. transforming) and reducing of experience. Linear understanding is transformed or raised through intuition before being reduced once more in indirect rational terms.

Initially this "circular" translation is interpreted as representing a higher form of reason.

Linear (one dimensional) reason is based on the (static) principle of the separation of opposites (where the existence of the positive pole thereby excludes the corresponding negative pole).

Circular (two dimensional) reason is based on the (dynamic) principle of the complementarity of opposites (where the existence of the positive pole thereby includes the corresponding negative pole).

In this sense the circular level simply represents a higher form of reason than the linear level.

Indeed in Hegelian philosophy, "reason" is identified with this two dimensional circular use. "Understanding" - which is considered inferior - is reserved for linear one dimensional purposes.

Likewise, in reverse fashion, when we obtain the square root of unity, though two complementary answers result, both indeed are rational (i.e. +1 and -1).

The situation is different however when we decide to mix both horizontal and vertical number systems (i.e. where neither the quantity nor dimension are one).

Take the simplest case i.e. 2² as an example. This is actually a mix of both quantitative and qualitative systems. Written fully, it is 2¹ (number quantity) ^. raised to the power of 1² (number quality). Therefore - in unreduced terms - it contains as an expression, aspects which are both (directly) mathematical and psychological.

From the standpoint of either linear or circular reason, such an expression is from - a psychological perspective - therefore irrational. In other words it cannot be solely understood directly in linear, or indirectly in circular rational terms.

In reverse complementary fashion, when we extract the square root of 2, the resulting mathematical expression i.e.2^1/2 is also irrational.

Some 2,500 years ago, the Pythagoreans were rightly disturbed when they discovered the existence of irrational numbers such as 2^1/2. They had at the time - within the context of a psychological system that was rational, no means of explaining why irrational number quantities could arise. Subsequently in Western mathematics this problem has been avoided by adopting a merely reduced quantitative - and thereby distorted - interpretation of mathematical relationships.

By recognising that numbers as (vertical) powers or dimensions inherently represent a mathematical system that is qualitatively different from numbers representing (horizontal) quantities, we now can actually explain why irrational numbers arise.

Again when appropriately understood (i.e. in transrational terms), there is an underlying structure to - potentially all - mathematical relationships which is exactly complemented by a corresponding structure in psychological terms.

The relevance of the above affects - not just the understanding of numbers - but all perceptions and concepts.

Indeed this explains very well the dynamics of the evolution of the circular level.

Firstly there is a period (supersensory) through which one attempts to transform all linear perceptions in intuitive terms. Understanding - in psychological terms - thereby becomes higher dimensional and vertical, where symbols obtain a circular meaning as archetypes of a holistic reality where subjective and objective comprise a dynamic union. When expressed in reduced (one dimensional) terms such union appears deeply paradoxical.

Later there is another period (suprarational) through which - from the perspective of transformed intuitive understanding - one seeks a more profound union by transforming all concepts likewise in purely intuitive terms. Again the reduced interpretation in terms of the principle of the complementarity of opposites appears paradoxical in terms of linear reason.

These intuitive circular periods - in themselves - are exhilarating and exciting. The problem arises when one attempts to combine linear - required by the demands of everyday life - with this new circular understanding. Quite simply- in terms of each other- linear and circular understanding are irrational. Most of the confusion and conflict of the circular level actually results from this clash of inherently different systems of understanding, which ultimately cannot be resolved at this level.
 
 

SUMMARY 

The circular level is two dimensional and based on the complementarity of opposites. It is directly based on intuition representing the pure fusion of opposite directions which is non-dimensional. However it has an indirect rational interpretation - where complementary poles are made explicit - which is two dimensional.

Every horizontal (linear) mathematical system can be translated indirectly as a (circular) psychological system. This leads to a new relative interpretation of number subject to the uncertainty principle.

Equally every vertical (linear) psychological system can be indirectly translated as a (circular) mathematical system. This leads to a new realisation that the underlying structure of (transpersonal) psychological understanding is actually mathematical.
 
 
 

FOUR DIMENSIONS - THE POINT LEVEL 

Introduction 

We are moving here directly from two to four dimensions.

There is a good reason for this. As we have seen the conscious process is basically one dimensional and the unconscious two dimensional (when expressed in rational terms).

Psychological development initially proceeds by the conscious positing of phenomena. This is the basis of the linear level and one dimensional. These phenomena are then negated and returned to the unconscious. This represents the transition from linear to circular level.

The circular level then consists in making the unconscious explicit in indirect rational terms). This involves the principle of the complementarity of opposites and is two dimensional. These (indirect) translations are in turn returned to the unconscious before being made once more explicit in a new "higher" level.

Making the unconscious explicit involves increasing dimensions - as we have seen - by a factor of two. Since the implicit contents are by now two dimensional, this entails that when indirectly expressed in conscious terms they will now require a four dimensional format.

So the underlying structure of the point level - expressed in (reduced) conscious rational terms - is four dimensional. This is a highly important structure. As we will see these four dimensions in fact represent the complex number system (with real and imaginary axes in both positive and negative directions).

It also implies that the true structure of our four dimensional space-time, in physical and psychological terms, is quite different from what is conventionally assumed. This structure is highly symmetrical. It can most accurately be represented - at this level - in complex terms with two "real" dimensions and two "imaginary" directions of space (i.e. positive and negative directions) .

It is beautifully simple. This underlying supersymmetrical structure of reality is just a reduced expression of unity (mathematically expressed - at this level - by the four roots of one).
 

Transition from Two to Four Dimensions  

There is an inherent conflict at the circular level which remains unresolved. This is a clash as between two systems of rationality which are qualitatively different. These can be represented by the horizontal and vertical number systems respectively. The horizontal and vertical systems - as we have seen - can be translated into each through circular number systems.

However whereas, linear systems represent rationality based on the separation of opposites, the circular systems represent rationality based on the complementarity of opposites. Linear and circular systems cannot be reconciled in terms of each other. This dilemma is well represented by the two main branches of physics. Relativity Theory, represents the culmination of classical ideas ultimately conforming to the linear system. Quantum mechanical behaviour, by contrast, corresponds very closely to the circular system. Neither however can be reconciled in terms of each other.

A higher principle is required.
 

Singularities 

The horizontal and vertical number systems are qualitatively different. Alternatively linear and circular number systems are qualitatively different.

There is however one point i.e. the intersection of horizontal and vertical axes, which is common to both systems. Alternatively the point at the centre of the circle is also at the centre of the line diameter (of this circle). If we continually reduce the diameter of the circle, both line and circle will contract, ultimately coinciding at a single point i.e. (circle of zero diameter). This central point could be looked on as a singularity.

This pattern is remarkably replicated in both psychological and physical terms.

The culmination of the circular level is represented by the spiritual "dark night". In psychological terms one becomes highly sensitive to the conflicts as between (direct) linear and (indirect) circular systems of understanding. One attempts to find that elusive centre or mid-point of the personality, free of conflict. This involves a high level of purgation where all conscious phenomena (direct or indirect) are removed. The personality greatly implodes and contracts resulting in psychic death. This culminates in the arrival at a singularity representing the pure centre of being. This is referred to in mystical literature as the "void" or "nada" (i.e. nothing). Here all conventional laws of behaviour break down as one learns to live entirely by faith. In other words one moves essentially to an infinite principle of being.

This pattern is replicated in physical terms by the "black hole". Extraordinary as it may be seen, this can be looked on as an attempt by physical nature to resolve a complementary problem (the conflict between relativistic and quantum mechanical systems of behaviour). Matter in trying to find its own ground centre greatly implodes resulting in physical death culminating at the arrival at a singularity. This represents its own pure centre. Here the laws of physics - geared to finite reality- break down.

In other words the ultimate ground of matter is in fact infinite. The task for physics as for psychology (esp. transpersonal) is to incorporate both finite and infinite aspects of behaviour in a new coherent system of understanding. This can only be done in terms of a new manner of understanding reality.
 

Birth of the Imaginary  

With the arrival at the singularity a fascinating transformation takes place which reveals the limitations of the previous level.

At the circular level one attempts to view the world in "real" terms. At this level there is continual conflict as between two competing systems of rationality i.e. the "real" linear and circular systems respectively. Now with the arrival at the singularity, the world subtly changes and is reborn in a new and initially confusing manner.

Consciousness involves the separation of opposites. The positive pole - in being cut off and thereby made separate - is made conscious by the simultaneous process of returning the negative pole to the unconscious. Much repression can result from this one-sided attempt to understand reality in "real" conscious" terms.

The "dark night" involves a high level of compression of repressed material in the unconscious. This is why arrival at the singularity is so painful.

When one finally let's go - living by pure faith - one gives up any remaining attempt to control and thereby translate reality in solely "real" terms. This relaxation enables projection of - hitherto repressed - material from the unconscious. Now this material carries a negative sign. Also we have seen that the unconscious - which is inherently two dimensional - can only be translated indirectly in reduced one dimensional terms. This involves the attempt to obtain the square root of what is negative. Mathematically, this is imaginary.

Thus the projection of unconscious psychological material in indirect phenomenal form (i.e. fantasy) is imaginary in the exact mathematical meaning of the term.

Thereby the "dark night" radiates - thereby becoming less compressed - through a long period of intense phenomenal projection.

Again we have a complementary pattern in physical terms. Largely due to the work of Stephen Hawking, it is now accepted that black holes can radiate. This is due to the activity of virtual particles in the vicinity of the event horizon. Such particles are extremely short lived - with closely associated pairings - that spontaneously arise and disappear in the moment of creation. However it is possible for positive and negative poles of such particles to separate resulting in the black hole radiation.

This resembles closely fantasy projection. With pure fantasy, positive and negative poles are united lending a holistic archetypal quality to experience which is purely "imaginary". However when positive and negative poles diverge, fantasy is projected outwards in phenomenal form reducing the compression of the "dark night".

So just as in physical terms, the activity of virtual particles is a direct expression of "imaginary" dimensions (of space and time), fantasy - in psychological terms - is also a direct expression of "imaginary" dimensions.

Real and Imaginary Numbers  

Despite the subtlety of the circular level, a conflict remains which cannot be resolved at this level. We have a horizontal linear number system directly representing quantities and a vertical linear number system representing qualities. The quantitative mathematical system (i.e. horizontal) can be indirectly expressed in qualitative terms through a circular psychological system.

Likewise the qualitative psychological system (i.e. vertical) can be indirectly expressed in quantitative terms through a circular mathematical system.

Unfortunately neither of these indirect translations can be reconciled in terms of each other at the circular level.

At the point or singularity lying at the centre, however both are harmonised.

Thus in physical terms, the black hole singularity represents the point where (quantitative) matter and (qualitative) dimensions are identical.

Likewise in psychological terms, the "dark night" singularity represents the point again where matter and dimensions are identical.

Normally phenomena of experience (matter) are understood in the context of a background of space and time (dimensions). However if the phenomena are indistinguishable from the dimensions, then clearly nothing (in phenomenal terms) can be experienced. Thus we can represent a situation, where matter and dimensions are identical - both physically and psychologically - by a singularity or point.

The problem with the circular level is that one tries to reconcile reality in solely "real" terms. Arrival at the point represents a unification of the quantitative and qualitative aspects of experience. This enables a new translation of reality to take place in both "real" and "imaginary" fashion. What was formally a circular system in "real" terms now can be translated as a linear system in "imaginary" terms.

In terms of this new perspective, we now can reformulate our number system.

This can be done both in (mathematical) quantitative and (psychological) qualitative fashion.

Formerly we had a horizontal (quantitative) and a vertical (qualitative) linear number system. Both of these were understood in "real" terms.

Now we can translate the vertical (qualitative) system into quantitative terms through switching it to "imaginary" format.

Thus 1¹ in the former "real" qualitative system is now i (where i = the square root of 1) in the new "imaginary" quantitative system. 1² in the former is 2i in the latter; 1³ is 3i, 1⁴ is 4i etc.

Thus we obtain the well known complex number system with horizontal "real" (both positive and negative) and vertical "imaginary" (both positive and negative) axes. This can be represented by the Argand diagram and represents the four roots of unity. Thus whereas the mathematical structure of reality at the circular level is represented by two roots, at the point level it is represented by four roots.

Thus underlying the true nature of physical reality is a set of perfectly symmetrical mathematical relationships.

From one perspective we could say that reality can be conceived solely in terms of four spatial dimensions with two "real" (one positive and one negative) and two "imaginary" (one positive and one negative).

However "real" time = "imaginary" space. Therefore we could alternatively characterise this ultimate structure as comprising two "real space (one positive and one negative) and two "real" time (one positive, one negative) dimensions.

Alternatively since "real" space = "imaginary" time, we could represent this structure as comprising two "imaginary" space (one positive, one negative) and two "imaginary" time (one positive one negative) dimensions.

Also the negative direction of a dimension simply represents the positive direction of matter. Thus in dynamic terms, neither dimensions nor matter pre-exist. Rather they are mutually created through dynamic interaction. Thus for example matter (in relation to space) is posited through the negation of space (in relation to matter). In turn space (in relation to matter) is posited through the negation of matter (in relation to space).

Thus at this level of symmetry, matter and dimensions can be translated into each other (through positive and negative directions). Likewise space and time can be translated into each other (through real and imaginary modes).

At the level of physical observation of reality this fundamental underlying symmetry is broken. However a truly holistic understanding - not at the level of broken but rather underlying unbroken symmetry of reality - is perfectly given by successive roots of unity (that are squares of 2).

We also can derive a complementary psychological system of qualitative relationships.

Here we maintain our former "real" qualitative vertical system 1¹, 1², 1³ etc. (in both positive and negative directions), and translate the "real" quantitative horizontal system into "imaginary" qualitative format (again in both positive and negative directions).

Thus 1¹ becomes 1ⁱ, 2¹ becomes 1²ⁱ, 3¹ becomes 1³ⁱ etc.

So we again have a complex number system, where in this case numbers directly represent qualities (i.e. dimensions).

Again this psychological complex number system has a fascinating interpretation.

Whereas, the "real" system relates directly to the cognitive mode of reason, the "imaginary" system relates by contrast to the affective mode of sense.

The activity of affective sense is vitally important in science and mathematics in the very recognition of phenomenal symbols. However this activity is invariably interpreted in reduced cognitive fashion.

In a direct sense the affective mode is directly based on the unconscious. In the initial moment of sense recognition, subject and object are fused in an intimate moment of personal meaning. Affective sense, is thereby based directly on personal rather than impersonal meaning. However in (conventional) science, the information provided through the senses is quickly reduced to the cognitive mode, where it is interpreted in an impersonal and detached manner. This thereby preserves the illusion that science deals with the "real" world.

However, when we properly recognise the nature of the affective mode, we recognise that it - relative to the cognitive - "imaginary". The appreciation of its "imaginary" nature requires initial separation from the cognitive, in the intense experience of projected fantasy.

Thus, at the four dimensional point level, one recognises clearly that - in psychological terms - one lives in a world that is qualitatively "complex" (with two "real" and two "imaginary" dimensions).

Indeed all (conscious) experience takes place in terms of four basic psychological structures (which very closely resembles Jung's four functions). The "real" positive direction is given by the cognitive mode reason (i.e. externalised thinking). The "real" negative direction is given by the cognitive mode of judgement (i.e. internalised thinking).

The "imaginary" positive direction is given by the affective mode of perception or sensation (i.e. externalised emotion). The "imaginary" negative direction is given by the affective mode of feeling (i.e. internalised emotion).

Thus we have here a qualitative psychological number system which exactly complements the corresponding quantitative system. The underlying nature of both physical and psychological reality can now be structured in terms of the four roots of unity (representing the complex number system).
 
 

The Point Number System 

As we have seen, the point represents the centre or singularity where real and imaginary systems intersect. This is true in physical terms (i.e. black hole singularity); it is also true in psychological terms (i.e. dark night singularity).

Finally it is true in mathematical terms (intersection of "real" and "imaginary" number axes). It is this mathematical point - underlying both physical and psychological singularities - which gives rise to a unique number system.

In the context of the understanding of number, let us now probe more closely this point system.

Understanding starts with the affective mode of sense enabling recognition of number symbols to take place. This sense recognition is necessary, even when the most abstract of symbols are employed. Firstly this takes place in a positive direction (symbol in relation to mind). It then takes place in complementary negative direction (mind in relation to symbol).

This fusion of positive and negative directions enables the switch to the concept of number to take place. Now this concept will be of opposite cognitive mode. Thus a switch takes place from what is an "imaginary" to - what is in relative terms - a "real" mode.

Central to this switch as between "real" and "imaginary" is the fusion between positive and negative directions generating a flash of pure intuition (i.e. mental energy). Now this intuition is nothing (in phenomenal terms) and yet is central to both modes. At this point of pure insight, "real" and "imaginary" modes are equal.

This point is therefore 0 and in dynamic terms is central to the creation of all numbers "real" and "imaginary". It really is an infinite rather than a finite notion. 0 as used in conventional terms in mathematics is reduced and static. What is inherently infinite is thereby nothing in finite terms. Thus from a merely finite perspective, 0 merely represents the absence of any (specific) number. From the correct dynamic perspective however, it represents the potential for the existence of all numbers.

Thus the point number system underlies the existence of all numbers and is the essential backdrop for all actual numbers ("real" and "imaginary").

In physical terms the point or singularity thereby - in like manner - represents the potential for existence of all actual creation (matter and dimensions). Properly understood actual reality is "complex" with interaction between "real" (matter) and "imaginary" (dimensions) taking place through a central point or singularity (i.e. quantum vacuum).

In psychological terms the point or singularity, represents the potential for existence of all phenomena (quantitative and qualitative) in experience. Experience of reality is "complex", with switching between "real" (cognitive) and "imaginary" (affective) aspects taking place through a central point (i.e. pure insight).

Thus conversion from "real" to "imaginary" (and "imaginary" to "real") in mathematical, physical and psychological terms comes through this point or singularity.
 

Transcendental Numbers 

There remains a significant problem which cannot be properly resolved at the point level.

Reality can now be structured in terms of two "complex" number systems. Thus we have a quantitative "complex" system which provides the underlying structure for physical reality. We also have a qualitative "complex" system providing the underlying structure for psychological reality.

The difficulty arises when we try and express each system in terms of each other.

Now - as we have already seen - in terms of indirect translation - quantitative and qualitative systems are linear and circular with respect to each other.

Thus finding the relationship as between quantitative and qualitative systems amounts to expressing the relationship as between line and circle.

Mathematically, pi represents this relationship and is the most famous of all transcendental numbers. Likewise the structures that emerge from the attempted reconciliation of line and circle (i.e. quantitative and qualitative systems) are - in mathematical terms - transcendental.

Now transcendental numbers are irrational. However they differ from (algebraic) irrational numbers in that they cannot be the roots of polynomial equations (with rational coefficients). (Algebraic) irrational numbers arise from the attempted reduction of vertical numbers - initially expressed in higher dimensions (i.e. powers > 1) in one dimensional terms. The square root of 2 is the simplest - and most important - example of such a reduction.

Now, transcendental numbers cannot be reduced in such fashion. The reason is very revealing. A transcendental number is neither horizontal (quantitative) or vertical (qualitative) Rather it expresses a hybrid relationship that is both horizontal and vertical (i.e. both quantitative and qualitative). This is borne out by the very nature of such a number. It has both finite (quantitative) and infinite (qualitative) aspects. Thus pi for example has a finite value (3.14 approx). However it also has an infinite aspect in the continual sequence of its decimal expansion. Therefore a transcendental number is essentially indeterminate with only an approximate value.

Now once again, reality - at the appropriate level of understanding - will have its fundamental structure revealed in (mathematically) transcendental terms.

This has a remarkable implication for physical reality. Normally we try to clearly separate quantitative from qualitative aspects of understanding. Thus conventionally we believe that material objects (quantities) have a place within dimensions of space and time (qualities). However, at this transcendental level of understanding, we cannot separate objects and dimensions. Thus every object - appropriately understood - is approximate and of hybrid composition representing a dynamic interaction pattern which is both material and dimensional (i.e. quantitative and qualitative). As modern physics shows, matter inherently is largely empty space; in turn "empty" space is teeming with particle life. The (attempted) separation of matter and dimensions is therefore no longer tenable.

This also has profound implications in psychological terms.

In conventional terms we try to clearly separate perceptions (quantities) and concepts (qualities). Thus for example a particular number perception such as "2", is placed within the appropriate number concept . The nature of both perception and concept thereby remain unaffected. Again however, at this transcendental level of understanding, we cannot (clearly) separate perceptions and concepts. Thus every psychic object is again an approximate hybrid composition, resulting from the dynamic interaction of perception and concept. We cannot therefore understand a particular number perception such as "2" in the absence of its appropriate dimension (i.e. the number concept). At an advanced level of understanding, psychic matter becomes highly transparent and elusive (i.e. illusory) and seen ultimately as empty space. Likewise the concepts or dimensions are now seen as representing in turn the potential for actual existence.

Most importantly at this level, the attempt to separate physical and psychological reality becomes increasingly untenable. Reality is now seen in psycho-physical terms. Every physical relationship has an exact complement which is psychological.

Likewise every psychological relationship has an exact complement which is physical. I have been attempting here to make these complementary relationships explicit.

In other words the lens through which physical understanding is filtered is psychological; the lens, in turn through which psychological understanding is filtered, is physical.
 
 

SUMMARY 

An inherent clash remains at the circular level as between linear and circular understanding which cannot yet be properly resolved.

Ultimately - in physical and psychological terms - both are reconciled through the arrival at a pure point or singularity which is nothing in phenomenal terms.

From this fundamental ground phenomena are now projected in (mathematically) "imaginary" terms.

The point level involves the gradual unification of "real" and "imaginary" aspects of experience in both positive and negative directions.

Before this unification is achieved there is the emergence of - what in mathematical terms are highly subtle transcendental structures. At the "higher" level of personality one attempts to understand the relationship as between line and circle (i.e. rational and intuitive understanding) in "real" terms. At the "lower" level of personality one attempts to understand this relationship in "imaginary" terms.

Underlying all of this is a central point system, representing the underlying potential for phenomenal existence.
 
 
 

EIGHT DIMENSIONS - THE RADIAL LEVEL
 

As we have seen each of our main levels can be precisely structured in basic mathematical terms. Each level reflects a reduced expression of unity and is given by the simple mathematical equation xⁿ = 1. The value of n = 2^k, where k = 0, 1, 2 and 3 respectively.
 

Linear Level 

When k = 0, we get x¹ = 1, which accurately represents the basic structure of the linear level. This power or dimension - which is the qualitative number - refers fundamentally to the interpretation of time. The most striking characteristic of the linear level is the view that time is strictly one dimensional moving in a positive direction only. Indeed, the belief that the arrow of time moves irreversibly forward is perhaps the most characteristic feature of linear thinking and indeed conventional science.

The other three space "dimensions" actually relate to the quantitative characteristics of physical phenomena. In this sense, any object that exists is by definition three dimensional (in spatial terms).

Therefore, physically at the linear level we have only one (positive) dimension (i.e. time moving in a forward direction). This (qualitative) dimension of time - understood in an absolute sense - is clearly separated from the three (quantitative) "dimensions" of space.

Likewise in complementary psychological terms, at the linear level, we have in qualitative terms only one dimension. Here, all phenomena belong to the one general concept or class (i.e. of phenomena). Time in experience, arises directly therefore, from this mental structure used to organise reality. Again the one (qualitative) dimension of time (i.e. conceptual structure) is clearly separated from the three spatial "dimensions" representing observed phenomena in space (i.e. perceptions). Time again is understood in absolute terms. In other words conceptual understanding is still static.
 

Circular Level 

When k = 1, we get x² = 1, which accurately represents the structure of the circular level. When expressed In reduced one dimensional terms (i.e. linear) we have now two values for x i.e. +1 and -1. In other words the circular level - expressed in one dimensional linear format - is bi-directional with complementary positive and negative poles.

In physical terms this means - at this level - that time has now two directions one positive and one negative. In other words, time is now purely relative.

In dynamic terms, if time is moving forward for object A (with respect to object B), then - relatively speaking - time is moving backwards for B (with respect to A).

For example it takes about 8 minutes for light to travel from the Sun to the Earth.

Thus the Earth is eight minutes forward in time (relative to the Sun). However when this light does reach Earth, eight minutes has already passed on the Sun. Thus the original source of light on the Sun is now eight minutes backwards in time (relative to the Earth).

Notions of linear time are only possible when we fix our frame of reference. Thus when we take the time that light takes to travel from Earth to Sun, or Sun to Earth (separately) the measurement in both cases will be positive in time.

However - taken relatively - measurement for both is of necessity forward and backwards in time.

Therefore from a dynamic perspective, for creation as a whole, time is purely circular with all objects moving simultaneously in complementary fashion, both in a positive and negative direction in time. There is a synchronicity or global coincidence of opposites therefore in the universe, whereby all objects fundamentally are connected through the present moment.

Speculation therefore as to the overall age of the universe represents a basic confusion, whereby partial linear notions are applied to the whole system. In dynamic terms the universe did not come into existence in linear time. Rather it continually exists in a central moment or point instant (through which relative notions of time are rooted).

We have at this level of relativity two dimensions of time (i.e. positive and negative). We now also have two complementary dimensions of space (again with positive and negative directions).

This is reflected through the manner in which conceptual understanding changes.

At the linear level, concepts are strictly one dimensional. All numbers for example are understood to belong to the one concept of number.

However at the circular level, concepts acquire two directions. The positive direction is the objective concept (in relation to mind) . The complementary negative direction or anti-concept is the mind (in relation to the concept). In psychological terms concept and anti-concept - in direct terms - involve experience of positive and negative time (relative to each other).

It is the fusion of concept and anti-concept that creates that essential light or insight enabling the switch to particular perceptions in experience to take place. This intuitive light - through which concepts are related - represents an immediate present moment continually regenerated in experience.

The implications of this understanding are far reaching. For example all mathematical understanding at this level is strictly relative. Every theory has a corresponding anti-theory, which has an opposite truth value.

Thus the Pythagorean Theorem has both a positive and a negative truth value which in static absolute terms - where direction is ignored - are formally identical. However in relationship to each other theory and anti-theory are dynamically opposed and strictly relative. Thus, any mathematical theory - however valuable - represents - in dynamic terms - no more than an existing social consensus.
 

Point Level 

When k = 2, we get x⁴ = 1. This accurately expresses the structure of the point level.

When expressed in (reduced) linear terms, we now get four solutions for x. Two of these are real (+1 and -1) and two imaginary (+i and --i). Whereas the circular level represents the complementarity of polar opposites in "real" terms, the point level represents the complementarity of polar opposites in both "real" and "imaginary" terms.

This structure is especially important in physical terms providing the "true" mathematical symmetry of space-time.

We now have two "real" time dimensions (positive and negative) and two "imaginary" time dimensions (again positive and negative).

At this level "real" time = "imaginary" space (and "real" space = "imaginary" time).

Thus we can readily convert time into space and space into time. So we could equally characterise reality as comprising four dimensions of space (again with two "real" and two "imaginary").

At the circular level, we showed that movement of objects in time - relative to each other. However there is still at this level, a certain attempt to fix the dimensional background. Both objects and dimensions are characterised in "real" terms.

However, now the dimensions themselves are understood as "imaginary" in relation to "real" objects.

Thus if the movement of objects (with respect to dimensions) is both positive and negative in "real" time, then the corresponding movement of dimensions (with respect to objects) is positive and negative in "imaginary" time. Likewise, given this frame of reference, the movement of objects (with respect to dimensions) is both positive and negative in "imaginary" space, with the corresponding movement of dimensions (with respect to objects) positive and negative in "real" space.

This is the fundamental problem underlying the reconciliation of relativity theory and quantum mechanics. The former is directly associated with gravity and global dimensional characteristics of reality. The latter is directly associated with energy and minute material (i.e. particle) aspects of reality. However the synthesis of both cannot be conceived in solely "real" terms for the world is now mathematically "complex" (with dynamically interacting "real" and "imaginary" aspects).

This structure is equally important in psychological terms.

We now also have - in raised (transformed) terms four dimensions, again with two "real" (positive and negative) and two "imaginary" (again positive and negative).

These relate to the four basic structures (or functions in Jungian terms), through which conscious understanding of the world takes place.

The two "real" dimensions are reason and judgement. The former represents the positive direction and the latter - relatively - the negative direction of cognitive thought.

The two "imaginary" dimensions are perception and feeling. Again the former represents the positive direction, and the latter the negative direction of affective emotion.

Thus in dynamic terms the two modes of understanding (i.e. cognitive and affective) are at this level "real" and "imaginary" with respect to each other. Rational concepts of reality (i.e. the "real" cognitive mode) are not in direct correspondence with "imaginary" sense observations of reality (i.e. the "imaginary" affective mode).

We can now perhaps appreciate how closely associated are the physical and psychological aspects of understanding at this level.

Relativity theory has emerged from - in direct terms - the (cognitive) conceptual approach to the world (later tested by observation).

Quantum mechanics by contrast has largely emerged from the (affective) sense approach in the (attempted) direct observation of sub-atomic particles (later formulated in theoretical terms).

However in psychological terms, these two modes are "real" and "imaginary" with respect to each other.

Thus at the point level, the "high" level differentiation of psychological structures ("real" and "imaginary") directly complements the "low" level differentiation of physical structures (again "imaginary" and "real"). It is this "raised" psychological understanding which is necessary to properly interpret "reduced" physical understanding at this level.

Physical information (about reality) can therefore no longer be separated from psychological transformation (of reality).
 
 

RADIAL LEVEL  

When k = 3, we get x⁸ = 1. This precisely represents the structure of this final (and most complete) level.

When expressed in reduced linear terms, we now have eight solutions for x. These are two real solutions (+1 and -1), two imaginary solutions (+i and --i) and four complex solutions 1/(2^1/2)(+1 + i),
1/(2^1/2)(+1 - i), 1/(2^1/2)(-1 + i) and 1/(2^1/2)(-1 - i).

It is to the mathematical significance of these latter solutions that we now turn.
 

Diagonal Number System 

We have already looked at both the horizontal and vertical linear systems corresponding to "real" and "imaginary" numbers respectively.

We now have in addition a diagonal number system which - in reduced terms - can be characterised as "complex" (i.e. both "real" and "imaginary").

This number system simply represents the diagonal lines that intersect x and y axes in both positive and negative directions. Any point on these lines is thereby an equal distance from both x and y axes.

If we take the diagonal line radius for example in the quadrant (where x and y axes are positive, we can form a right angled triangle with these axes with x =1/(2^1/2) and y = (1/(2^1/2i)) (where i = the square root of  -1).

Now using the Pythagorean theorem, the square on the hypotenuse of this triangle (i.e. the diagonal line) is (1/(2^1/2))² +  (1/(2^1/2))i)² = 0. Thus the length of this diagonal line (or any part of it) = 0. This applies also the diagonal lines in the other quadrants.

We have therefore a diagonal number system which has no finite value. However it does have a transfinite value.

Thus the horizontal axis represents "real" numbers, the vertical axis "imaginary" numbers and the diagonal axes "transfinite" numbers.

This diagonal number system is an extension of the point system. In the point system it is recognised that the number concept is - dynamically speaking - nothing, representing the potential for the existence of particular numbers. Also physical dimensions (of reality) in themselves are nothing. In the absence of matter, they cease to exist (the black hole). Ultimately dimensions represent the potential for material existence. Also psychological dimensions (i.e. concepts) in the absence of material perceptions cease to exist (the dark night).

Now we have a double point system, where particular numbers are also dynamically nothing. This simply is the recognition that a particular number quantity has no meaning in the absence of its number quality or dimension (i.e. number concept).

Just as number concepts (i.e. dimensions) represent the potential for the existence of number perceptions (i.e. quantities), number perceptions in turn represent the potential for the existence of number concepts. Thus, in their actual existence, there is a finite aspect to the existence of all number quantities and dimensions. However, there is equally a transfinite aspect to number quantities and dimensions in their potential existence. Finite existence of numbers relates directly to conscious understanding, and transfinite understanding directly to unconscious (i.e. intuitive) understanding. In the dynamics of understanding both are employed. However, it is customary - in conventional understanding - to reduce understanding to the realm of the conscious and finite. Here, infinite ( i.e. transfinite) notions are misleadingly treated as a linear extension of the finite. Correctly understood in dynamic terms, the transfinite notion of number is inherent in all finite numbers as the ever present potential ground of their actual existence.

The implications of this transfinite system are enormous. Physical reality can no longer be conceived in solely finite terms. Material phenomena actually exist in finite space and time. However this existence is always against an invisible backdrop representing the ever present potential for (finite) existence which is transfinite in origin. This notion has gradually forced its way into physics through recognition of "singularities", "quantum vacuums" etc. However there is still a marked reluctance to accept this inherently transfinite (i.e. infinite) nature as an indispensable aspect of reality.

We have in physical - as in mathematical terms - two complementary "voids". One is a holistic void which can be expressed as the state of physical reality without dimensions (i.e. where space and time are nothing). Within this void the potential for all dimensional existence is (directly) contained. Coming from the relativistic perspective, this is represented by the famous black hole "singularity".

The other is a specific void which can be expressed as the state of physical reality without matter (i.e. where particles no longer exist in phenomenal terms). Likewise in this void the potential for all material existence is (directly) contained. Coming from the quantum mechanical perspective, this is represented by the "quantum vacuum".

Ultimately these two voids themselves are fully complementary. This is the combined absence of matter and dimensions. Paradoxically however, this state of pure potential, is a state of pure actuality in the outpouring of energy (and the corresponding creation of finite matter in space-time).

This state of pure physical voidness should not be envisaged in linear terms - as a beginning state of the universe that can be traced backwards in time. Rather it is an ever present state continually underlying all dynamic - and thereby relative - (physical) processes in creation.

Again we have an exactly matching psychological interpretation.

As mystics of all ages testify the ultimate spiritual nature of reality lies concealed beneath the veils of finite phenomena and is infinite.

This realisation of this pure spiritual state also represents a void which has two complementary aspects.

From the detached holistic (relativistic) perspective, this void represents the absence of all dimensions in the emptying of all conceptual knowledge (which literally gives dimensional form to experience). This is the famous dark night "singularity" (i.e. nada or nothing). More strictly it represents a transcendent void (arising from a spiritual approach that goes beyond all rational conceptual form).

From the specific (quantum mechanical) perspective - which is more intimate - the void represents the absence of all specific particles (in the emptying of all attachment to virtual phenomena i.e. projected fantasy). This is the immanent void arising from a complementary spiritual approach (that goes beneath all phenomena).

These psychological voids are fully complementary. This is well recognised in mystical literature, where the spiritual void (representing absence of phenomena), becomes a plenum-void (full of light and fulfilment). What represented mere potential is now actualised in the outpouring of spiritual energy. For those living at the radial level, this infinite spiritual outpouring is experienced as the ever present source and end of all finite relative expressions of the world.

It is now that that most exciting synthesis of all is realised, where physical and psychological understanding of reality are unified. The ultimate origin of the physical universe - continually present in experience - directly coincides with the ultimate goal of this universe - also continually present - in pure spiritual fulfilment. This represents the culmination of creation.

Therefore In the most complete experience of mystical ecstasy, evolution realises its own destiny. This could be aptly described as the experience of everything.
 

Diagonal Numbers 2 

Though the infinite in its purest form has no form or content, yet paradoxically it can be given phenomenal expression in an indirect reduced fashion. The implications of this translation are profoundly revealing.

Now, we have seen that in two dimensional terms the diagonal (i.e. 45^o null lines) can be represented as having a finite value of zero.

However when we express the points on this line in (reduced) one dimensional terms, we obtain a fascinating new number system.

Any point on the diagonal line is giving by the corresponding co-ordinates on the x and y axis. Thus if - for example - the horizontal "real" value on the x axis is .5, and the vertical "imaginary" value on the y axis .5i, then the corresponding diagonal value is a "complex" number representing the sum of these numbers

(i.e. .5 + .5i). Now the absolute numerical value of "real" and "imaginary" numbers will - by definition - be always the same. However signs can differ so we have a different line in each quadrant of the circle. So with the same numerical values in addition to 5 + .5i, we can also have 5 - .5i, - 5 + .5i and -5 - .5i.

In this reduced sense the transfinite numbers can be given a fascinating interpretation as the combination of equal "real" and "imaginary" components. (Incidentally this explains Cantor's apparent paradox of an "infinite" number of transfinite numbers!)

Thus what is a null line (from a two dimensional perspective) and = 0, has from a reduced (one dimensional perspective) a whole series of values.

Thus whereas earlier we saw that imaginary numbers (in reduced one dimensional terms) correspond to a logic of complementary opposites (both positive and negative directions), transfinite numbers - again in reduced one dimensional terms - correspond to a new logic of complementary opposites (both "real" and "imaginary" aspects). Thus in the movement of the "real" understanding of a number perception (in relation to the number concept) to the corresponding "imaginary" understanding of the number concept (in relation to the number perception), there is an intuitive flash of light or implicit recognition enabling the switch to be made. This intuition is itself transfinite (nothing in phenomenal terms). The visible reduced phenomenal manifestation of this intuition is the combination - expressed in reduced terms - of the "real" perception and "imaginary" concept in a dynamic synthesis of understanding.
 

Physical Implications 

This new number system - represented by the four complex roots of unity has a fascinating physical application.

We have dealt with (horizontal) quantities representing matter and (vertical) qualities representing dimensions. The thorny question as to how dynamic interaction as between matter and dimensions takes place then arises. The secret to this lies in the very nature of our diagonal number line which literally intersects horizontal and vertical.

This interaction takes place through physical forces four of which have been identified (electromagnetic, gravitational, strong and weak). Many mysteries still surround their precise nature. Also the attempt to unify all four forces has proven remarkably elusive.

Just as the four roots of unity provide the symmetry that enables conversion as between matter and dimensions and space and time, the eight roots of unity provide a higher level symmetry which also enables the conversion as between matter, dimensions and forces.

Perhaps, the best known of the physical forces is the electromagnetic of which natural light is an important manifestation.

Now the true nature of light cannot be directly investigated, but rather its indirect interaction with physical phenomena. However, in terms of these indirect interactions, light has a dual nature (i.e. as particles and waves). Now from a classical perspective, light should be either composed of particles or waves. However from a quantum mechanical perspective, light is composed of both particles and waves. Indeed this is often highlighted as the central paradox of quantum physics!

However our diagonal line number system - representing the intersection of horizontal and vertical gives a simple explanation.

Light in physical terms can be represented by a similar diagonal (i.e. null) world line which in two dimensional terms is zero. This represents the path of intersection of dimensions and matter (i.e. a means by which they interact). Now these points of intersection can be looked on as the fusion of "real" matter and "imaginary" dimensions, which - in correct two dimensional terms - is infinite. However in reduced one dimensional terms this interaction must be expressed as the (separate) sum of both a "real" quantitative and "imaginary" qualitative component. Thus - at the level of (reduced) scientific investigation, light - will of necessity display two complementary aspects. The "real" quantitative (material) aspect is represented by light particles and the "imaginary" qualitative (dimensional) aspect - in relative terms - by light waves.

Our complex roots would strongly suggest that the other forces can be represented in similar fashion. Thus each of the four forces represents the basic manner in which material and dimensional interactions can take place which in each case involves an equal "real" and "imaginary" component.

Thus if (1/2^1/2)(I - i) mathematically represents the structure of the electromagnetic force, then the gravity force could be represented by if (1/2^1/2)(- I + i). Thus to convert the gravitational to the electromagnetic force, we just interchange real and imaginary components.

The weak and the strong forces would then be given by i.e. (1/2^1/2)( I + i), and

(1/2^1/2)(-I - i) respectively. To convert the weak to the strong force, we just interchange positive and negative components.

Now as we have seen the square root of 2 is the very symbol of the irrational. Thus the inclusion of the square root of 2 in our solutions, signifies how - in the context of linear (rational) understanding - the complementary dual nature of force phenomena is indeed irrational.
 

Psychological Implications 

This new number system - represented by the four complex roots of unity equally has a fascinating psychological application.

The question now arises as to how interaction takes place as between psychic matter (i.e. perceptions) and psychic dimensions (i.e. concepts).

Again this comes through our diagonal number system which - in reduced terms - has four aspects. We could look on these likewise as four forces (this time interpreted in psychological terms).

Mystics speak of the contemplation of immanence and the contemplation of transcendence. In spiritual terms, the contemplation of immanence - which is localised - directly complements electromagnetic radiation (i.e. physical light). It represents the ultimate refinement of that light or illumination through which the mind can immediately see. By contrast the contemplation of transcendence.

- which is globalised - directly complements the gravitational force through which the mind universally sees. The weak and strong forces then relate more to these forces (applied to the inner self). The strong force - in psychology as in physics - really represents an intense form of inner gravity (i.e. authentic faith). The weak force can represents the outward expression of this inner light.

Of course in direct terms this pure spiritual light is transfinite. However in reduced linear terms it has a fascinating interpretation that is mathematically complex (i.e. real and imaginary components).

Spiritual light has both particle and wave forms. The contemplation of immanence in direct terms represents the particle form (as in literally illuminates the distinct grains of experience). Its wave form thus remains hidden. It can therefore - in complementary manner to electromagnetic radiation - be represented as (1/2^1/2)(I - i).

The contemplation of transcendence can then be represented as (1/2^1/2)(- I + i) where now the wave aspect is directly revealed.

Again the two are related by a real to imaginary (imaginary to real) transformation.

Spiritual gravity (contemplation of transcendence) can thus be seen as simply the imaginary aspect of spiritual light (contemplation of immanence)

Finally the strong and weak are related through a positive to negative (negative to positive) transformation.
 
 

SUMMARY 

The radial level can be characterised as a double point system, which is the pure marriage of potentiality with actuality. Thus simultaneously it involves the fundamental subfinite ground (from which all physical reality emerges) and the ultimate superfinite experience (where reality realises its own destiny).

We now have the emergence of fascinating diagonal number systems.

From one perspective these numbers are nothing (i.e. have no finite value). They do have however an infinite or transfinite value which characterises the true nature of light (both in physical and spiritual terms).

From a reduced linear perspective diagonal number systems are complex (with equal "real" and "imaginary" terms). These in turn explain for example why physical light has both a "particle" (real) and "wave" (imaginary) form, and why spiritual light also has a "particle" (contemplation of immanence) and "wave" (contemplation of transcendence) form.

The mathematical eight fold translation of the radial level is especially useful in terms of providing models of supersymmetry in both physical and psychological terms.