The Number Spectrum of Consciousness

 

GENERAL INTRODUCTION

 
 

The purpose here is to provide a radical new approach to the mapping of the full spectrum of consciousness. It is based on the application of holistic mathematics, where every quantitative (mathematical) relationship has a complementary dynamic qualitative interpretation which in essence is psychological. 

Indeed - in this qualitative sense - the fundamental structure of all psychological experience is mathematical. Holistic mathematics therefore provides a remarkably precise means for both interpreting and classifying the full spectrum of consciousness. 

The spectrum of consciousness is typically classified in terms of various levels. There is general agreement in mystical traditions on four major levels e.g. gross, subtle, causal and ultimate. In my own mathematical approach these are referred to as the linear, circular, point and radial levels. As I will demonstrate the fundamental structure of these levels is intimately connected to a simple mathematical procedure involving successive roots of unity. 

Within each level we have various stages which can all be classified in terms of the qualitative number systems. 

There are many number types in mathematics 

Thus we have the rational numbers (as quantities) made up of the binary, prime and natural nos. and the fractions. 

The corresponding qualitative numbers have a direct dynamic interpretation which corresponds fully to the various stages of the linear level. 

Then we have the negative and irrational numbers whose qualitative counterparts relate to the stages of the circular level. 

The imaginary and transcendental nos. - qualitatively understood - define the stages of the point level. 

Finally the transfinite and complex nos. - again qualitatively interpreted - define precisely the stages of the radial level. 

However the best way of appreciating this approach is to see it in action.

So read on and enjoy! 

 

All understanding can be interpreted in terms of process, mode and direction.

The two fundamental processes of understanding are the unconscious and conscious respectively.

Holistic (synthetic) knowledge is based (directly) on the unconscious, whereas partial (analytical) knowledge is based (directly) on the conscious.

The two fundamental modes of understanding are the cognitive and affective which are initially closely related with conscious and unconscious respectively.

Rational (impersonal) knowledge is based directly on the cognitive mode and emotional (personal) knowledge based (directly) on the affective mode.

The two fundamental directions of understanding are the external and internal respectively.

Objective knowledge relates (directly) to the external direction; subjective knowledge relates (directly) to the internal direction.

 

Mathematical Terminology

 

In dynamic mathematical terms, conscious and unconscious processes are "finite" and "transfinite" with respect to each other.

In like manner cognitive and affective modes are "real" and "imaginary" with respect to each other.

Finally, external and internal directions are "positive" and "negative" with respect to each other.

Initially, all these directions are considerably confused in experience. The linear level is concerned with the explicit differentiation of direction (only) whereby the external is separated from internal and literally posited in experience.

The linear level is thereby one directional (dimensional) in nature.

 

 

Pre-Rational stages: Binary

 

In qualitative terms the conscious and unconscious can be represented by 1 and 0 respectively. Through consciousness unitary phenomena are actually posited in finite experience. Through the unconscious these are dynamically negated again leaving nothing in finite terms. However this represents a transfinite reality (which is pure potential for existence). 

In the beginning both the conscious and unconscious are fully undifferentiated.The very basis of the qualitative binary state is that 1 = 0. However we have here a totally confused binary state. Indeed the essence of psychological transformation can be expressed as the subsequent movement towards a fully fused (i.e. fully differentiated) binary state. 
 

First Transition 

The first psychological stage involves a crucial transition with the emerging differentiation of conscious and unconscious. Finite (actual) reality begins to separate from transfinite (potential) reality. This process begins in the womb and is accentuated in the first months of infancy. Fleeting undefined events take place in experience arising and disappearing in the moment of creation. 

However there is as yet little differentiation of cognitive and affective modes or objective or subjective directions in experience. 

 

Pre- Rational stages: Prime

A prime number e.g. 7 has no factors (other than itself and 1) and is inherently one dimensional. 6 by contrast is two dimensional as it has two factors (2X3).Prime numbers involve therefore the complete separation of quantitative and qualitative (i.e. dimensional) characteristics. 

The first and root prime number is 2. This really comes from a reduced interpretation of the unconscious. Now in dynamic interaction the two poles (positive and negative) are fused and 0 (i.e. +1 - 1). However when separated both poles - from a conscious perspective - carry a positive sign. So in moving from 0 to 2 there is a fundamental confusion of conscious and unconscious. Put another way the unconscious (0) has a reduced (conscious) qualitative interpretation as 2 where both directions (external and internal) are given conscious expression. In this sense in contrast to the conscious approach which is one dimensional, the unconscious - in reduced linear terms - is two dimensional. 

All (other) prime numbers which are odd are intimately linked to this root prime number 2. Mersenne primes represent one famous example of this approach.  Thus 3 (which is prime) = 2² - 1.

Now psychologically, the prime structures relate directly to primitive (i.e. instinctive) understanding. The essence of such understanding is that the qualitative (dimensional) aspect of experience is directly confused with the quantitative. Alternatively the affective (personal) aspect is directly confused with the cognitive (impersonal). Indeed this always remains the basis of (blind) instinctive behaviour where personal desire is directly embodied in impersonal objects. 

Thus psychological prime behaviour (where quantitative objects are confused with qualitative dimensions) directly complements mathematical prime numbers (where both are fully separated).

Second Transition 

The next psychological stage involves a further transition whereby affective (personal) reality becomes gradually separated from (impersonal) matter. In other words there is the start of the recognition of a separate self and some extension of experience in space and time. However the directions of experience are still greatly confused. A clear sense of self is not yet possible. 

 

Pre-Rational Stages: Natural

All natural numbers can be derived from a unique combination of prime factors. 

Properly understood therefore natural numbers involve both a (reduced) quantitative aspect (its one dimensional value) and a qualitative aspect (the number of factors involved). Thus 2X2X2X3 has a reduced quantitative aspect (i.e. 24). It also has a qualitative aspect (i.e. 4) representing the number of prime factors. 

When the infant learns to relate the qualitative and quantitative (i.e. cognitive and affective) - in what is literally a psychological multiplication experience - composite objects begin to appear. Though much confusion exists the infant begins to place objects within a dimensional framework. 

However such natural phenomena are initially highly transient relating only to what is immediately present in experience. The infant is still unable to properly distinguish whole from part so that objects are invested with a magical universal quality. 

We also have here experience of natural space and time relating to a free flowing present moment activated through immediate sensory awareness. 

Third Transition 

The next psychological stage is the highly important one where the negative direction of experience is implicitly differentiated. This leads to a knowledge of self distinct from knowledge of objects. 

An object is posited (in relation to self). This causes imbalance in terms of the neutral unconscious introducing the counterbalancing negative charge. This causes a switch to self (in relation to object) which is now posited in experience. However by now the separation of conscious (from unconscious) through positing of phenomena is sufficiently strong that they continue to remain in memory when not immediately present in experience. 

All object phenomena now enjoy both a positive identity (existence when immediately present) and a negative identity (an existence that continues when removed from present awareness). The existence of objects now begins to extend in space and time. 

A deeper level of awareness arises when the self assumes both a positive identity (in relation to present objects) and a negative identity (that can suffer the loss of present objects). In other words the self too begins to extend in space and time achieving a more stable independent existence not immediately tied to the transient present. 

This is represented mathematically by the set of integers. Starting from the centre which is 0, the natural numbers have both a positive and negative direction. Likewise in experience starting from the 

 

Summary of Pre-Rational Stages

Psychological Interactions have finite and transfinite aspects (conscious and unconscious processes),

real and imaginary aspects (cognitive and affective modes) and positive and negative aspects (objective and subjective directions). 

The first stage of human evolution involves differentiating all of these aspects separately (i.e. bringing them to consciousness). 

Originally all of these aspects are totally confused and undifferentiated. In this original binary state existence coincides with non-existence (1=0). 

The first crucial transition involves the process by which finite and transfinite aspects begin to differentiate with actual reality separating from potential reality. 

We now reach the prime state where quantitative objects (horizontal) are still confused with qualitative dimensions (vertical) exactly complementing the notion of prime numbers in mathematics. 

The second transition involves the emerging differentiation of these real and imaginary aspects. Psychologically this entails the separation of cognitive (impersonal) from affective (personal) experience. 

We next reach the natural state where positive and negative directions of experience are still largely confused. 

The third transition involves the differentiation of these aspects. Psychologically this entails the development of an independent subjective self separate from a world of stable objects. This state is complemented by the set of integers which now consist of numbers with positive and negative directions.

 

Rational Stages - Introduction

As we have seen the initially the finite and transfinite aspects, the real and imaginary aspects and finally the positive and negative aspects of reality are completely undifferentiated. 

The pre-rational stages can be defined in terms of the preliminary attempts to separate and differentiate these aspects, which is only partially successful. 

With the next major grouping of stages, this differentiation - which is heavily centred on the conscious process is carried to an extreme level of specialisation. However this inevitably entails a high level of reductionism due to consequent repression of the unconscious. 

Thus rationality becomes identified with what is finite and actual (as opposed to what is transfinite and potential). Spiritual experience - pertaining to the infinite - tends to be translated in reduced finite terms. 

Likewise rationality becomes identified with what is "real" and quantitative (as opposed to what is "imaginary" and qualitative). The very use of the word "reality" bears testimony to this tendency. Affective experience - pertaining to the imaginary - tends to be translated in reduced cognitive terms. 

Finally rationality becomes identified with what is "positive" and external (as opposed to what is "negative" and internal). Subjective experience - pertaining to the internal - likewise tends to be translated in reduced positive (i.e. objective) times.

This rational approach represents a distinctive and highly important paradigm best exemplified by the conventional scientific and mathematical approach. 

It is generally referred to - in number terms - as the rational paradigm. However this is too imprecise. In fact - expressed fully in number terms - it is the finite positive real rational paradigm. What is entirely missed however is that just as the set of finite positive real rational numbers only comprises one small subset of the set of all possible numbers as quantities, likewise the finite positive real rational paradigm comprises just one small subset of the set of all possible numbers as qualities (i.e. paradigms). 

We are now at the stage of examining this important - though limited - rational paradigm.

 

Concrete Rational: External

Mathematically the very structure of a rational number such as a fraction is highly revealing. The simplest example involves a reciprocal. Thus for example the reciprocal of 4 is 1/4 which can be written as 4^-1 . Thus whereas 4 is expressed with reference to the positive 1st dimension (i.e. 4¹) its reciprocal is expressed with reference to the negative 1st dimension. 

This is directly relevant to the psychological dynamics which underline this concrete rational stage. To illustrate let us probe the dynamic of understanding a cake that is divided into four slices. 

Understanding of phenomena always involves interacting quantitative and qualitative aspects. 

Here, the quantitative aspect is given by the perception of the cake (in relation to the concept of "cake"). The qualitative aspect is given - in relative terms - by the concept of "cake" (to which the perception of the cake is related). 

The positive direction of the object phenomenon of the cake is given by the perception of cake (in relation to self) 

The negative direction of the (quantitative) object phenomenon of the cake is given by the self (in relation to the perception of the cake). 

The positive direction of the (qualitative) object dimension of "cake" is given by the concept of "cake" (in relation to self). 

The negative direction of the (qualitative) object dimension of "cake" is given by the self (in relation to the concept of "cake"). 

Thus object quantities (perceptions) and object qualities or dimensions (concepts) have in dynamic terms - both positive and negative directions. 

To understand the cutting of our cake into four slices a remarkable psychological transformation is required whereby every object is able to enjoy a dual identity both as whole and part. Thus each slice can be recognised as a whole unit (in its own right) and also as a part of the original whole unit (i.e. the cake). Thus each slice is 1 (in terms of itself and 1/4 (in terms of the original unit of the cake). 

Now we saw at the previous stage, that the ability to sustain objects in existence (when not immediately present in experience) involves sufficent development of the negative direction of experience (i.e. the self). 

The ability now to sustain the qualitative object (i.e. the concept of "cake") in existence (when not present in experience) comes from sufficient development of the self (in relation to the concept of "cake"). 

Quite literally just as the external direction of the concept relates to the positive 1st dimension, the subjective internal direction relates to the negative 1st dimension. 

Thus when the attention moves from the cake to the four slices resulting from the cutting (now understand as whole units), the concept of cake remains in memory. One therefore is able to relate these slices back to the concept of "cake" now seeing them as parts (of the original whole). 

Unfortunately though the dynamics outlined above are vital for understanding, they are merely implicit and are interpreted (explicitly) in reduced conscious fashion. Thus though the negative internal direction implicitly is vital to stabilise object concepts, explicitly the experience is interpreted in solely positive terms. 

Likewise though the qualitative dimensional aspect of experience implicitly is vital to relate whole to parts, explicitly it is interpreted in merely reduced quantitative terms. 

Indeed the failure to explicitly recognise this qualitative dimension ultimately leads increasingly to parts (without wholes) and the literal fragmentation of all experience. 

This concrete rational stage really has both affective and cognitive aspects. On the one hand the child becomes increasingly able to respond affectively to external sense stimuli which through fragmentation become more self contained and localised. 

On the other hand s/he becomes better able to control the environment cognitively, with an increasing ability to practically analyse problems. In both cases only the positive real aspect of experience.

 

Concrete Rational: Internal

Again we can take -1/4 as a simple example of a negative rational number (fraction). 

This can be written -4^-1. 

Here the direction of both the number quantity and number quality (i.e. dimension) is negative. 

In psychological terms this relates to both perceptions and concepts which are inner directed, enhancing self development. 

Just as the previous stage involves the ability to break (whole) objects coherently into parts, this complementary stage involves the ability to break the (whole) ego into parts. In other words one learns to play different roles which become parts of an overall self identity. Again affective and cognitive aspects are involved. The affective aspect involves the ability to deal with a wider range of emotional feelings. The cognitive aspect represents the ability to exercise judgement in terms of practical decision-making. 

Again the ability to integrate (part) roles with the overall (whole) personality involves a similar set of psychological dynamics as outlined in the previous section. 

 

Formal Rational: External

 

There is a fundamental distinction as between perceptions and concepts which is rarely appreciated. 

Experience necessarily entails a dynamic interaction of (specific) perceptions with (holistic) concepts. This is true of all scientific and mathematical data. For example we cannot meaningfully understand a specific number perception (e.g. 2) without relating it to the corresponding concept of number. 

In conventional rational understanding concepts are reduced to perceptions with their fundamental qualitative difference thereby overlooked. 

Let us illustrate this point carefully with respect to the understanding of a number. 

The perception of a number represents a specific number datum or content (e.g. 2). 

The concept of number by contrast represents the general class or form of number. 

The meaningful understanding of a number therefore necessarily entails this dynamic interaction of content and form (i.e. of quantity and quality). 

However a merely reduced quantitative appreciation of number exists in conventional mathematics. 

To recognise a specific finite number phenomenon (e.g. 2) it must be rationally posited in consciousness (and thereby differentiated from other finite number phenomena). 

To move to the general concept of number the specific number phenomenon must be dynamically negated in consciousness (enabling integration with other numbers). It is this fusion of positive and negative poles which causes a crucial switch from the actual finite understanding of number as a specific quantity to the potential infinite understanding of number as a general quality. Thus correctly understood the number concept - though related potentially to all number phenomena - initially does not relate actually to any specific phenomenon. 

However in conventional rational understanding this potential qualitative notion of concept (which is inherently spiritual and intuitive) quickly collapses as it were to a reduced actual quantitative notion. In other words the number concept now is simply interpreted as relating to all actual number phenomena. 

In this way what is a dynamic interactive process involving both actual conscious and potential unconscious understanding is interpreted in reduced static terms as being merely conscious. Alternatively what inherently involves the subtle interaction of reason and intuition is explained as merely rational. 

Thus there is both horizontal and vertical aspects to all understanding. Whenever one switches from perception to concept (from content to form), understanding necessarily switches as between the extensive rational horizontal and the intensive intuitive vertical aspects of understanding. However once again this crucial vertical aspect of understanding is entirely missing in conventional rational understanding.

When one interprets reality from the flatland of the reduced (solely) horizontal perspective, understanding becomes literally linear (one dimensional). 

This is well illustrated for example in mathematics by the quantitative number system. If we use numbers such as 2 or 2,000 it is implied that these number are one dimensional (i.e. raised to the power of 1). Thus both numbers - strictly speaking - should be written 2¹ and 2000¹. Whenever a number is initially expressed with respect to another dimension e.g. 2², its value ultimately will be given in (reduced) one dimensional terms (i.e. 4¹ or more simply 4). 

However, this equally applies to all concepts used in (conventional) science. 

If for example in physics we can have many different particle phenomena. However it is assumed that all these relate to the one concept of particle. 

Thus - in this view - though we can have many partial phenomena (as quantities), we have only one holistic phenomenon (as quality) which is the concept of particle. 

In other words the concept of particle - and by extension all concepts in physics - are understood in strictly one dimensional terms. 

Just as number quantities in mathematics are defined with respect to the 1^st dimension, likewise all phenomenal quantities in science are defined with respect to an invariant 1^st dimension (i.e. the general class to which these phenomena belong). 

This is fundamental. The objects of experience are (directly) given by perceptions. The dimensions in experience are (directly) given by concepts. 

When we interpret concepts in one dimensional terms physical reality not surprisingly appears as one dimensional. The three space dimensions are essentially measurements we apply to objects (i.e. quantities). Time is the remaining qualitative aspect of experience and universally interpreted as one dimensional. 

Our physical view of reality therefore mirrors the limited and reduced manner in which we rationally interpret reality. 

Revolutionary New World-view 

When one gives equal consideration to the qualitative - as well as quantitative - aspects of experience, the possibilities for a remarkable new world-view in terms of both physical and psychological reality becomes possible. 

This is where the qualitative vertical number system is of immense value. While going greatly beyond mathematics - as conventionally understood - paradoxically it greatly enhances the applicability of mathematics to reality. When correctly appreciated, one realises that whatever is possible mathematically is actually replicated - at a certain level of experience - in the real world (physical and psychological). In this "new" world both physical and psychological aspects are dynamically complementary with an underlying structure which is mathematical.

We have seen that corresponding to the quantitative number system 

1¹, 2¹, 3¹, 4¹…… is a complementary qualitative number system 

1¹, 1², 1³, 1⁴…… 

I now wish to give this second qualitative system its dynamic interpretation in the context of experience. 

Conventional experience freezes interaction by keeping the concept fixed. All perceptions of a certain class are interpreted absolutely within that (same) conceptual class. Thus psychologically - as we have seen - all particles for example are interpreted within the same concept of "particle". Likewise physically all objects are interpreted within the same dimension of time. 

However in dynamic terms perception and concept are equal partners in dynamic interaction. 

Thus the perception of the number 1, has a dynamic interaction with the number concept (to which 1 is related). 

The relationship works both ways. 

We have the (quantitative) perception of the number 1 (in relation to the concept of number). This is 1¹. 

We have also the (qualitative) concept of number (to which the number "1" is related. This is also 1¹. 

Next we have the (quantitative) perception of the number 2 (in relation to the concept of number). This is 2¹. 

In reverse fashion we have the (qualitative) concept of number (to which the number "2" is related). This is 1². 

In like manner we can generate complementary numbers for 3, 4 etc. 

What this really entails is that each number perception - in dynamic terms - has a unique relationship with the corresponding number concept (and the concept in turn with the perception). For each number perception (quantitative) there is a corresponding unique number concept (qualitative). 

By extension all quantitative phenomena in mathematics and physics have unique relationships with corresponding qualitative conceptual classes. Thus each particle phenomenon - in dynamic terms - is related to a unique particle class. 

Amazingly, when correctly appreciated we live in a multidimensional reality where there are as many (qualitative) dimensions as (quantitative) phenomena. 

This viewpoint - though not usually articulated in this fashion - is the illumined mystical vision where heightened intuitive awareness is maintained in harmonious balance with rational understanding. 

Dynamics of Development 

Whereas the concrete rational stages are correctly represented in terms of the quantitative number system, the formal rational stages - relating to conceptual understanding are more correctly represented by the corresponding qualitative system. 

We studied in detail - at the concrete stage - the breaking of a cake into 4 (quantitative) slices is represented by ¼, i.e. (¼)¹. 

The corresponding breaking of the concept of cake into four (qualitative) sub concepts is represented by 1^1/4 .

What is remarkable about this is that even this simple conceptual analysis involves operating in fractional dimensions. 

However because of reductionism this is not appreciated. Conventionally, all understanding - both physical and psychological - is interpreted within one qualitative dimension of time. 

However as we have already seen, objective and subjective experience involve - what in dynamic relative terms - are positive and negative dimensions. 

Now we see that conceptual analysis involves operating in what literally are fractional dimensions. As we will see all the other options permitted by the number system in terms of irrational, imaginary, complex, transfinite numbers are actually replicated at appropriate levels of experience. 

As before this stage of development has both affective and cognitive aspects. 

Normally with conscious development, affective sense development precedes cognitive rational development. 

The affective phase here would relate to a more holistic capacity in terms of general sense involvement with reality. One is now more able to screen out immediate attachment to objects to obtain an overall impression of the environment. However as the thrust of development becomes increasingly rational, conscious affective experience is subject to much repression becoming less spontaneous. 

This often leads to the release of the primitive unconscious through fantasy in an attempt to compensate. 

The cognitive phase relates of course to a more holistic mental capacity, whereby one can increasingly reason using abstract mental objects. Pure logic and mathematics best exemplify this analytical ability. 
 

Formal Rational: Internal

We are now switching to internal development which - relative to external - is dynamically negative. With reference to our cutting the cake into four slices this could be written 1^-(1/4). 

 

 

Spiritual (Integration of Rational Stages)
 

The highest stage of conventional (psychological) development represents the integration of mind and body (i.e. rational with previous primitive stages of development). 

This inherently is a spiritual stage. 

However there is a fundamental weakness in the conventional psychological treatment. Basically specialisation of conscious development - which accelerates during the rational stages is suited to differentiating and fragmenting reality. 

By the same token it is unsuited to the complementary task of integrating and unifying reality which is based on the unconscious. Thus only a limited form of integration - geared to existing social conventions - is possible at the linear level. More meaningful integration must await "higher" levels. 

This spiritual stage can be characterised in mathematical terms. 

We have already referred to the horizontal number system as (directly) rational and the vertical number system as (directly) intuitive. Now spiritual understanding arises from the dynamic interaction of conscious and unconscious and is maximised where intuition and reason are equally developed. This can be represented by a diagonal line which intersects the plane comprising horizontal and vertical number axes. 

We will give this diagonal number system - representing the spiritual - a fascinating mathematical interpretation at a later stage. 

Of course in conventional mathematics, reductionism once more abounds. Each earlier number set is simply seen as a subset of a later set. Thus prime numbers are a subset of the natural number system, They in turn are a subset of the integers which in turn are part of the rational numbers. 

This reductionist tendency is also very pronounced in conventional psychology, where earlier "lower" stages of development are evaluated from the perspective of the "higher" rational stages.

 

Summary of Rational Stages

The rational stages of development represent specialisation of conscious understanding. This is extremely useful for fragmenting and analysing reality and is the unquestioned basis of conventional science. 

However the (real positive) rational paradigm is very seductive in that it creates the illusion that reality has an independent existence (which mental constructs represent in separate fashion). 

However just as the set of real positive rational numbers represents just one limited subset of the total set of number quantities, the real positive rational paradigm represents just one limited example of the total set of number qualities (i.e. paradigms). 

Many other more subtle (number) paradigms exist. Here physical and psychological reality are inextricably linked in relative complementary fashion. 

We will examine these presently. 

The concrete rational stages are more content based where one learns to analyse the objects of experience in a logical manner. 

The external phase has both affective and cognitive aspects. The affective aspect involves increased ability to analyse sense experience whereas the cognitive aspect involves a more detached ability to analyse mental perceptions. 

The internal phase also has both affective and cognitive aspects. In the former case one learns to analyse emotional feelings and in the latter case one uses (moral) judgement in terms of practical decisions.

The formal rational stages are more holistic (i.e. concerned with qualitative form) where one now learns to analyse pure mental objects logically. 

The external phase again has both affective and cognitive aspects with the former related to a more refined form of sense awareness and the latter to a purer mental ability dealing with abstract form. 

The internal phase involves the affective aspect of more refined emotional feeling and the cognitive aspect of abstract mental judgement.

There is a crucial difference as between the concrete and formal stages which is completely confused in conventional rational understanding. The concrete (i.e. part) is qualitatively different from form (i.e. whole). In mathematical terms they are horizontal (rational) and vertical (intuitive) with respect to each other. 

However in conventional understanding intuition is simply reduced to rational explanation. The whole is thereby simply reduced to the parts and interpreted quantitatively as the sum of the parts. 

The starting point for holistic mathematics is the recognition that the whole is qualitatively different from the part bearing a relative complementary relationship to each other. Thus we have two dynamic number systems representing quantitative (parts) and qualitative (wholes) respectively. 

Rational (conscious) development is readily suited to the differentiation and analysis of the "parts". It is not suited to the complementary task of integration and synthesis of the "whole" of experience. This is directly based on intuitive (unconscious) development which due to reductionism is largely screened out of conventional development. 

This poses a major problem in terms of meaningful integration of mental and body structures which relates directly to spiritual development. 

However we have a mathematical way of characterising this central spiritual development. As we have seen the rational is horizontal (where understanding takes place extensively within a fixed linear dimension or paradigm); the intuitive is vertical (where understanding takes place intensively within a variety of paradigms); the spiritual which involves the balanced use of reason and intuition is diagonal (taking place dynamically extensively and intensively within each and every paradigm). 

 

The Negative Rational Stages - Introduction

The basics of experience can be explained quite simply in dynamic mathematical terms. 

The unconscious - in its pure form - is dynamically neutral with complementary poles, carrying equal positive (+) and negative (-) signs. 

Consciousness involves the separation of these poles. One pole - which is revealed in experience - is dynamically posited in phenomenal form. The other pole which remains concealed is simultaneously dynamically negated and repressed in the unconscious. 

This causes an imbalance in terms of the neutral unconscious causing a reverse inward move in consciousness towards the self which is then also posited in consciousness. Thus experience keeps switching in this manner as between positing of the world and positing of the self. 

Thus though the unconscious remains vital - even in conventional experience - as a dynamic switching device (between world and self) it remains merely implicit and not formally recognised. In other words the unconscious is not given an opportunity in the rational paradigm to properly develop. 

Some individuals are especially sensitive to the imbalance that this causes. While searching for spiritual integration - usually on the threshold of adulthood - they find something fundamental missing and a true existential crisis develops. They become increasingly disillusioned and unable to fit in with conventional expectations. After perhaps a considerable period of blind searching the crisis is resolved - sometimes suddenly - in a mysterious and unexpected fashion. 

This phase is well documented in the Christian ascetical tradition and commonly referred to as purgation. Terms such as "learned ignorance" and "cloud of unknowing" have also been used. What it involves is literally the dynamic undoing or negation of conscious phenomena in experience. 

The conscious mind is developed by moving out from the unconscious in the dynamic positing of phenomena. The unconscious mind in turn is developed through a reverse backward movement from the conscious in the dynamic negation of these phenomena. Here consciousness of phenomena is literally undone and holistic intuitive capacity increased. 

I have referred to these holistic structures as mirror structures, reverse structures and anti-structures. Most simply they - in terms of direction - are negative structures which complement the positive structures of rational consciousness. 

So one now understands that all phenomena in experience have both external (positive) and internal (negative) directions. The external direction represents the dynamic appearance of phenomena in experience. The internal direction represents the consequent (dynamic) disappearance of the same phenomena from experience. Needless to say phenomena now enjoy a relative rather than an absolute existence. 

The implications of this world-view are startling. Experience is now two dimensional. Every fact, every concept, every hypothesis is complemented by a corresponding anti-fact, anti-concept, anti-hypothesis. 

To illustrate, the Pythagorean Theorem is complemented by an anti-theorem. In absolute static terms both theorem and anti-theorem are identical. However in dynamic relative terms they are opposite. Thus at this level the truth of all mathematical (as well as scientific) knowledge is strictly relative.

The basic principle at all these "higher" levels is that physical and psychological reality are complementary. "Higher" level psychological understanding reads "lower" level physical reality. When one level changes (as psychological understanding does here), the other level must necessarily change in complementary fashion. 

All physical objects therefore now have (in dynamic terms) two directions (external and internal). Matter is complemented by anti-matter. Close to its holistic fundamental ground of energy, particles become increasingly dynamic switching direction continually as between matter and anti-matter states. Clearly at this level of material organisation the conventional paradigm is no longer adequate and unable to provide the appropriate intuition to interpret the phenomena observed. 

 

Negative Rational Stages - Evolution

In existential terms, this is a very distressing time. Though considerable intuitive development takes place in darkness it is only revealed at a later stage. 

Rather one is mainly aware of what is lost. Because consciousness is literally undone at this time one moves backwards in time losing touch with many former relationships and skills. 

Purgation pertains mainly to the more superficial levels of concrete rational development. St. John of the Cross refers to this as the active night of sense. In religious terms one now dies considerably to the sensible world. 

Thus, corresponding to both the affective and cognitive aspects of external concrete rational development, we now have - in the erosion of associated phenomena - the explicit development of the corresponding negative direction. 

Purgation would also take place in relation to the internal self (practical feelings and judgements). However realistically at this time it would not be too intense. Psychologically it would be very difficult to support too much erosion of consciousness at this early stage.

Some negation of rational formal understanding would also take place mainly at the external level. In St. John's terminology this is the active night of spirit. However deeper conceptual understanding is likely to remain substantially intact. Again for psychological reasons, the dynamic erosion of consciousness, which involves much suffering and grief normally takes place gradually in several stages.

 

Summary of Negative Stages

 
 
 

The crucial requirement for radical transpersonal development is the explicit dynamic development of the negative direction of experience. Partial conscious phenomena are thereby considerably eroded enabling intense unconscious development of holistic intuitive ability. However far from being a pleasant experience, it usually involves a lengthy existential crisis with considerable psychological distress. 

Though some erosion of all conscious structures is likely to take place, the external senses are principally affected. 

As a result of this, the static nature of the rational paradigm breaks down to be replaced by a dynamic relative approach whereby physical and psychological reality become increasingly complementary. 

 

 

Real Irrational Structures - Introduction

 

Some 2,500 years ago the Pythagoreans were shocked by the discovery of irrational number quantities such as 2. In interpreting reality they were employing what was inherently an integrated mathematical approach in the belief that the (quantitative) mathematical nature of reality complemented a (qualitative) paradigm used to interpret this reality. Initially all seemed well. The rational paradigm was matched by a mathematical world where only rational quantities were known to exist. 

The discovery of 2 and irrational quantities however destroyed this belief for there was now no matching (qualitative) paradigm with which to interpret this strange new world of quantities. 

This ended dramatically the quest for an integrated mathematical approach. 

Subsequent development in Western mathematics has taken place in a reduced fashion with the quantitative becoming increasingly divorced from the qualitative. Irrational, imaginary and even transfinite quantities are interpreted within the confines of an ever more specialised and limited rational paradigm. 

The qualitative and "vague" intuitive dimension - if recognised at all - is seen as the province of mystical religion and transpersonal psychology and nothing to do with real mathematics. However all the esoteric paradigms of transpersonal psychology can be precisely translated in dynamic holistic mathematical terms offering an entirely new vision of reality. 

Let us recap development so far. The linear level (gross realm) culminates in the specialised development of positive rational structures. 

The transition from linear to circular level then involves the development of complementary negative rational structures. 

The key feature of the circular level (subtle realm) is that understanding is now seen in relative terms as the dynamic interaction of both positive and negative directions of experience. When these directions are fully fused we have an expression of the developed unconscious (i.e. pure intuition). So from this perspective the circular level involves the development of (unconscious) intuitive rather than (conscious) rational structures. 

However pure intuition cannot be sustained for long so that one attempts to translate this experience in reduced rational form. 

The very basis of conventional reason is that opposite poles are clearly separated. If a proposition is true for example, then by this logic it cannot be false. 

However when we express spiritual intuition in rational terms it appears deeply paradoxical. Here opposite poles are complementary. Thus if a proposition is true, then by this logic it must also be false. Putting it less dramatically it now enjoys a relative - rather than absolute - truth value. 

Indeed by the logic of the conventional reason (of the linear level), the paradoxical reason of the circular level is irrational. This is because we are expressing an order of truth which is inherently two dimensional (involving complementary opposite poles) in reduced one dimensional terms (where these poles are clearly separated).

Now this problem is exactly replicated in mathematics with irrational numbers such as 2^1/2. If we express this number in the second dimension (by squaring it) it now is rational i.e. if x = 2^1/2 then x² = 2. 

However if we try to reduce this number subsequently to the first dimension (by obtaining its square root) it becomes irrational. Furthermore the result has two complementary poles + 2^1/2 and - 2^1/2. 

An irrational number has both finite and infinite aspects. It is finite in that its value can be approximated (i.e. reduced) to a discrete value. Thus correct to four decimal places 2^1/2 = 1.4142. However it is infinite in it's full decimal sequence which continues indefinitely. An irrational number is thus inherently dynamic and relative with discrete (finite) and continuous (infinite) aspects.

In psychological terms irrational experience is similar where phenomena have both finite and infinite aspects. Thus partial phenomena arise in experience which are consciously understood. This is the finite rational aspect. However they are now invested with a transcendent light whereby they are seen as archetypes of an invisible whole. This is the infinite intuitive aspect. The combination of both aspects gives all phenomena a dynamic relative existence. 

There is yet another remarkable connection as between these (quantitative) mathematical and (qualitative) psychological irrational notions. 

In mathematics if we raise any rational number to a fractional power we get an irrational number. (I am ignoring unreduced trivial cases such as 4^1/2 = 2, which is a rational number for 4 itself can be expressed as 2² so that the right hand reduces to 2¹). 

Now we have seen that in the later formal rational stages of the linear level, that conceptual ability - correctly interpreted - literally involves experiencing reality in fractional dimensions. 

Thus when one tries to subsequently relate concrete (horizontal) perceptions to formal (vertical) concepts (i.e. interpret rational objects in the context of fractional dimensions) a decisive transformation takes place by which the experience becomes irrational). Thus in dynamic psychological terms we have here a surprising complementary result to mathematical behaviour. Indeed it is this very psychological dynamic that leads to genuine transpersonal development. Because of the more intuitive nature of experience, the transpersonal candidate does not successfully reduce concepts to merely rational format. This leads to an inevitable conflict as between the horizontal and vertical levels of experience. 

Whereas the linear level - especially in the later rational stages - is geared to the development of merely positive structures (i.e. conscious), the circular level relates to the progressive development of structures in both their positive and negative directions (involving the explicit development of both conscious and unconscious). 

In other words we have here the development of both reason and intuition in what might be termed superstructures. 

The circular level is constantly changing. As the quality of intuition increases, the use of reason becomes ever more subtle and refined. 

We can in - as at the rational level - identify both concrete and formal stages of irrational development. However there is a marked cyclical quality to these stages. 

The positive stages are characterised by illumination. This involves the outpouring of spiritual intuition with subsequent translation through subtle rational structures. The negative stages in stark contrast are characterised by purgation with complementary inpouring of intuition (in darkness) and substantial erosion of the subtle structures formerly developed. 

Spiritual development during the circular level is transcendent in quality. 

This is due to the fact that the unconscious - though considerably activated - remains under the subtle control of the cognitive function (i.e. reason). Thus unwittingly considerable emotional repression can take place. In dynamic mathematical terms we are still dealing here with "real" rather than "imaginary" structures. 

 

Concrete Irrational: Positive

 

This can be referred to as the supersensory stage, corresponding in a subtly intuitive manner to the concrete rational stage of the linear level. 

As stated there is a marked cyclical character to each of the stages of the circular level. The stage typically commences with a peak experience of pure spiritual illumination. This is often identified in mystical literature with a dramatic conversion to spiritual life (following a long painful search for meaning). An intense experience of platonic love could - in certain circumstances - equally mark the beginning of this state. 

There then follows an attempt to translate this experience in phenomenal terms.This has both affective and cognitive aspects. The basic principle of this translation is that of the complementarity of opposites, where meaning is now strictly paradoxical and relative. In other terms by the logic of the rational stages, understanding is now irrational. Affective sense symbols, while enjoying a partial (visible) existence equally serve (in the light of spiritual intuition) as archetypes of a holistic (invisible) reality. 

At the cognitive level one understands macro physical reality in relative terms. One would for example have little difficulty in intuitively accepting Einstein's Special Theory of Relativity. 

However one seeks an even deeper meaning of relativity. At this stage space and time are seen in symmetrical terms as both having positive and negative directions. Thus both space and time are strictly relative. When time for example moves forward for the world (with respect to the self), it equally moves backwards for the self (with respect to the World). In dynamic terms time moves in opposite directions for World and self. The conventional scientific view completely ignores this dynamic dialectical relationship between object and subject assuming that time moves forward simultaneously for both world and self. 

Correctly understand in terms of this stage time is merely relative flowing out from an ever present centre. A highly spiritual person experiences time in this manner. It equally means that time has only meaning in a partial relative context and is not applicable to the universe as a whole. The universe is always created out of a continual present. Pure intuition is thus an experience of this absolute truth. 

During this stage one experiences phenomena as irrational. This is because - as we have seen - one attempts to translate what is vertical and two dimensional (intuition) in reduced one dimensional (rational) terms. 

Mathematically this is expressed in horizontal terms by (2)^1/2. 

Because of the cyclical nature of this stage illumination ultimately fades with dark clouds appearing on the horizon. 

What typically happens is that psychic energy tends to move to the new "higher" regions of personality so that earlier rational skills fall into disuse. One experiences therefore increasing difficulty in terms of one's normal duties. Indeed this is a renewal of negative structures (i.e. purgation) in relation to the earlier rational structures reaching deeper internal levels of personality. An acute dilemma develops. One has progressed considerably up this steep slope. However in the process one has removed any means of getting back to dry land. 

A new deeper existential crisis inevitably develops.
 

 

Concrete Irrational: Negative

This stage equally starts with a dramatic peak experience of spiritual intuition. 

However far from being a pleasant experience this is extremely dark and harrowing throwing into clear relief the misery of the ego based self. 

Intuition itself has two binary states. Illumination is revealed in brilliant light and experienced as union (1). However the complementary state - developed through intense purgation - is concealed in dense darkness and experienced as nothingness (0). In more advanced spiritual states, these two aspects coincide simultaneously in a pure state of transformation. 

So just as all information processes - in computer terms - can be represented through one (static) binary system (where 1 and 0 are separated i.e. 1 0), all transformation processes - physical and psychological - can be represented through a complementary (dynamic) binary system (where 1 and 0 are fully complementary i.e. 1=0). 

Reality in the fullest sense - at all levels - can be represented as the continual interaction of both information and transformation processes which can basically be represented by a (dynamic) double binary system.

We now have the development of negative structures whereby the renewed undoing of remaining internal sense knowledge takes place (active night of sense in St. John's terminology). However, the radical undoing of the irrational supersensory knowledge of the previous stage also takes place. The key problem in the spiritual life is that of possessive attachment. At the linear level (positive direction) one becomes directly attached to phenomena. The subsequent negative direction is necessary to undo this attachment. 

At the circular level (positive direction) one becomes indirectly attached to phenomena. Though symbols serve as archetypes of a spiritual holistic order, one tends to identify meaning too rigidly with the objects employed. The subsequent negative direction of understanding is thus necessary to undo attachment to these (subtle) concrete irrational structures. 

This is what St. John refers to as the passive night of the senses. It is a time fraught with much difficulty. Psychosomatic conditions typically develop and one is prey to neurotic symptoms perhaps culminating in a temporary breakdown. 

There then follows a deeply existential phase whereby one learns to translate internal reality in a new relative fashion. These are the (internal) concrete irrational structures (in a positive direction). 

All emotional decisions are now subject to uncertainty. At the rational level one tends to take refuge in second hand conventional notions of objective truth, which lessen the need for true responsibility of choice. However one now faces a terrible freedom, where every decision is unique and taken in accordance with a deeper innate notion of meaning. 

Here again at a micro personal level we have the complementarity of opposite poles. Every decision involves a unique dialectical relationship of a (subjective) self and an (objective) situation. It is necessarily relative, subject to uncertainty and must be exercised in faith. This time one attempts to translate poles - which are separate at the one dimensional phenomenal level - into a higher two dimensional reality (where both are united). In other words one attempts to transcend through faith - which is intuition exercised in darkness - one's limited personal circumstances. 

This phase once more has two associated aspects. There is the affective phase where existential realisation in relation to emotional feeling is intense. There is then a later more detached cognitive phase whereby one is enabled to clarify intellectually the existential position. 

Passive undoing of these internally based structures is not likely to take place at this time. 

Again we have complementary "low level" (micro) physical understanding at this stage related to interpretation of quantum particle reality. 

The existential decision - as we have seen - is born out of deep awareness of a fundamental psychological uncertainty principle. Subject and object interact in dynamic terms. If one seeks to make the objective situation absolute, then subjective responsibility is reduced with behaviour becoming mechanical and dutiful. If however one seeks to maximise moral freedom (i.e. fix the momentum of a decision), then - in the absence of conventional rules - decisions can be become too unrestricted and "vague". 

The uncertainty principle in complementary terms is the very cornerstone of quantum physics. One can attempt to measure precisely the position or the momentum of a particle but not simultaneously. 

Also the irrational - in the form of complementary aspects of a relationship - is inherent to quantum behaviour. 

Thus the observer (subject) and what is observed (object) are inextricably linked in quantum experiments. This is really the recognition that matter at this level has two directions (i.e. positive and negative). 

Also we have the "same" phenomenon such as light manifesting itself in complementary particle and wave form. 

Thus the irrational experience of this stage is invaluable in terms of providing the appropriate intuition with which to interpret key findings of quantum physics. 

In order to offset too much introversion the later phases of this stage are likely to signal a return to more active involvement. This involves a renewal of the more linear rational purgation in the exercise of worldly responsibilities (which one now finds burdensome and difficult). 

In mathematical terms this irrational (supersensory) stage - which relative to the preceding stage has a negative direction - can be denoted by (-2)¹.

 

Formal Irrational: Positive

 
In terms of these "super" stages of the circular level, we now have reached the conceptually based formal structures (suprarational).

Once more this stage commences with a new peak experience of brilliant illumination. Here the boundaries between self and the world momentarily dissolve and the meaning of creation is revealed as love. The quality of this spiritual intuition depends in large measure on the degree of purgation preceding. Typically it is deeper, more intense and long lasting than that of the corresponding concrete stage.

There then follows later an attempt to translate this pure intuition in indirect rational terms. Once more this involves the complementarity of opposites and the language of paradox in what are irrational structures. 

This time the translation - intimately affecting the very dimensions of reality - goes to the heart of all formal systems of thought. Concepts - wherever used - are understood as strictly relative. There is now a more profound realisation of the futility of trying to model any scientific system in absolute terms. The very concepts we must necessarily use to formulate this system are - in themselves - inherently relative. 

It is highly instructive to apply this new understanding to mathematics, where truth is now only approximate. 

There is a fundamental confusion in conventional mathematics regarding finite and infinite notions whichlies at the centre of all notions of absolute proof. Finite and infinite - in direct terms - are rational and intuitive notions respectively. Because of the reduction of intuition to reason in conventional mathematics, the infinite is misleadingly treated - in effect - as a linear extension of the finite. Now a general proof in mathematics is supposed to apply to "all" cases which potentially is infinite. It does not in fact apply to actual cases which are finite. The assumption that it does is due to a reductionist tendency which is so deep rooted as to be rarely questioned. 

We have already seen that the (horizontal) concrete and (vertical) form are qualitatively different. The (formal) proof of a proposition does not establish it's application to a given (concrete) case. Thus the Pythagorean Theorem - to illustrate - has a strictly relative validity. What this entails in practical terms is that mathematical understanding - though highly useful - represents no more than an existing social consensus. The very fact that I am able to sincerely question the nature of this consensus illustrates that it is necessarily approximate.

At this stage all rational systems must be defined in dynamic relative terms. 

As an example I will redefine the number system in such fashion. 

Each number perception has both a positive and negative direction. In other words a number can be posited to be (finitely) determined, and thereby made actual. Alternatively it can be negated to become (finitely) indeterminate, and merely potential. The number concept essentially relates to this latter aspect of number which is infinite. 

The number system consists of - what are relatively - both determinate and indeterminate numbers. One cannot posit "all" numbers. Thus the number system - from a dynamic perspective - is finite but unbounded. All other systems - in dynamic terms - are likewise finite and unbounded. 

Because of considerable development of pure intuition - in the radical erosion of conscious phenomena - one at this stage develops holistic vision on a grand cosmic scale with the desire to intellectually integrate diverse areas of experience. This spirit is well embodied in Hegelian philosophy. However there is a danger here - exemplified by Hegel - of a subtle form of reductionism whereby spiritually intuitive truth is identified too rigidly with indirect rational forms of translation.

As a result of the psychological changes of this stage, physical reality once more changes in complementary manner. 

There are interesting parallels here with Einstein's General Theory of Relativity. 

Einstein showed here how light - assumed to travel in a straight line could - due to gravity - be bent in the presence of matter. Ultimately with sufficient gravity - as in a black hole - light will be completely bent in a circular fashion (and unable to escape the black hole). 

Psychological reality can be understood in reverse complementary manner. At the rational stages matter (i.e. phenomenal reality) is understood in a straight line fashion (i.e. the linear level). Then at the circular level, due to the presence of spiritual light (i.e. intuition), this matter begins to bend with phenomena understood in increasingly relative terms. When this spiritual light is sufficiently intense - as in the "dark night" stage, reality becomes completely circular with all phenomena - in understanding - becoming immersed in the unconscious.

Though the main focus is on the cognitive aspect, there is an associated affective phase. Due to considerable detachment from the senses, images become increasingly holistic and transparent conveying a pure form of spiritual joy. 

As the stage progresses, one can experience considerable difficulties in terms of adopting to conventional responsibilities. This leads to a renewal of active purgation (negative rational structures) at a deeper internal level. One is likely to suffer communication difficulties, feel ignored and misunderstood, and develop a growing sense of grievance.

In mathematical terms we are dealing here with the vertical qualitative aspect of number, where dimensions are irrational. In correct dynamic terms therefore experience at this suprarational stage actually takes place in irrational dimensions! 

Space and time are two directional (with positive and negative poles). 

Thus expressing these dimensions in (reduced) one directional terms is equivalent to obtaining the square root of 2 and irrational. 

 

Formal Irrational: Negative
  

The spiritual dilemma grows ever deeper. While scaling vertical intuitive states of development, one steadily loses contact with horizontal rational states. In order to cope with the psychological problems this entails, one is driven to a purer level of commitment through an extremely intensive level of spiritual purgation. 

This is the really critical stage on the transpersonal journey. It is the famous "dark night of the soul" or simply "dark night" though this is somewhat vague. St. John refers to it more accurately as the "dark night of spirit". 

What it entails is a radical erosion of all deeply rooted conceptual structures. 

Initially purgation was limited to the more concrete phenomena. Now in addition all formal phenomena must be uprooted so that nothing remains in consciousness. This involves the negative direction of formal rational understanding (esp. in relation to inner judgement (the active night of spirit). More importantly it involves the negative direction of the recently acquired formal irrational understanding (i.e. suprarational structures in the passive night of spirit). Not surprisingly it is most arduous psychologically involving a type of suffering that can only be appreciated by those who have directly experienced it. 

The stage typically gets underway through being plunged into such deep darkness that one feels completely lost and desolate. One experiences in a deeper way the misery of the human condition where one's own suffering reflects the suffering of humanity which now seems to grow without limit. 

After a while though the darkness does not lift one adjusts better to this new underworld and bravely tries to carry on without consolation. For a while one may still conscious of a very faint light enabling a degree of spiritual hope. 

A very refined form of internal translation now takes place. This is the development (in relation to judgement and feeling) of the internally based irrational structures. Decisions now seem extremely finely balanced and relative. One daily agonises with one's conscience waiting for a dim intuitive signal confirming that one is doing right. However as the light grows dimmer, the agony becomes more acute and one develops severe scruples. 

The affective aspect of this phase involves a generalised form of compassion whereby one identifies with suffering in all forms. There is however a feeling of complete helplessness. One can see no way of actually relieving this misery. 

However the stage inevitably gets even more difficult. The negative direction of these internally based structures now is developed, so that all remaining light is taken away. One is completely alone in a terrifying darkness. Though living increasingly by pure faith there is a growing conviction that one has finally lost one's way. Severe communication problems arise and psychotic symptoms in the form of severe depression are likely to surface. These only seem to confirm that one's spiritual quest has finally ended in tragic failure. 

One now heads towards a terrible crisis. 

 

Imaginary Rational Numbers: Quantitative and Qualitative

 

Concrete

Ö -1 = i, is the fundamental imaginary number quantity. It has no real valued solution. This is because conscious understanding is based on an either/or logic. However unconscious understanding, to which imaginary numbers are more closely related, is based on a both/and logic. Thus the value of Ö -1 can be expressed as +1 and -1 simultaneously.

- Ö -1 is the corresponding negative quantity.

 

Ö -1 can also represent the corresponding psychological unit quantity. This represents the attempt to express the unconscious through indirect conscious terms. Psychologically this represents (concrete) projected unconscious phenomena. The real meaning of such phenomena involves a both/and logic. The projection is only correctly understood through internal assimilation. Thus the imaginary phenomenon is correctly understood only when there is a fusion of both positive and negative directions leading to the disappearance of the phenomenon in a return to its home in the unconscious.

One of the most common confusions psychologically is the interpretation of projections in solely positive terms. Thus what is really of unconscious significance is reduced to solely conscious terms.

We have the development first of the concrete imaginary (rational) structures. This resembles in many ways the concrete rational stage of the linear level. One develops an interest in analytical practical pursuits. However these prove very transient. One is not able to hold on to phenomena possessively. They are quickly eroded and made unconscious due to the very high level of mirror structure activity characteristic of this stage. False hopes are continually vanquished during this time.

The positive structures relate to external pursuits, job ambitions, outdoor pursuits etc.

The negative structures - which are more difficult to assimilate - relate to the projection of subjective values and emotions.

 

 

Formal

 

In mathematical terms these relate to imaginary numbers as dimensions e.g. 1Ö ^-1 .

The corresponding negative number is 1^- Ö -1 .

Mathematically we can have imaginary dimensions that can be positive and negative.

These dimensions are replicated in the physical universe, when one interprets reality in terms of the appropriate imaginary paradigm.

 

Psychologically, the corresponding qualitative numbers refer to the formal imaginary (rational) structures where one now projects more deep rooted ideas and concepts from the unconscious.

The positive structures relate to objectively based concepts; the negative structures relate to more subjectively based ideas. Again the high level of mirror structure activity attaching to these projections quickly renders them imaginary. One now actually lives in a world where dimensions are experienced as imaginary. The world no longer seems to exist "out there". Rather it progressively appears to be - which it now dynamically is - a projection of one’s inner psyche. What one formerly experienced as a dream (i.e. fantasy) has now become reality. It is such a strange inversion of conventional reality, that it can only be understood through experience.

  

 

THE POINT LEVEL

Real Transcendental Paradigm

 

We have already encountered the rational (analytical) paradigm of the linear level, where opposites in experience are separated and the irrational (holistic intuitive) paradigm of the circular level where opposites in experience are complementary.

However these two paradigms are themselves complementary. So a fresh task is to try and integrate these earlier two levels.

This leads to an even more subtle paradigm which understanding is expressed as the relationship between the linear (rational) paradigm and the circular (irrational) paradigm.

Qualitatively this represents experience of what are referred to as transcendental quantities in mathematics.

Just as this paradigm represents the psychological relationship between line and circle, p , the most famous transcendental number expresses the complementary mathematical relationship between line and circle.

Also we have seen how the irrational paradigm arose from the unreduced application of rational understanding. To attempt to place rational objects within rational dimensions, leads to a transformation whereby experience becomes irrational. In other words one moves from the linear level (which is absolute) to the circular level (which is relative).

The transcendental paradigm - in like manner - also arises from the unreduced application of irrational understanding. To attempt to place irrational (or rational) objects within irrational dimensions, leads to a further transformation whereby experience becomes transcendental. In other words one moves from the circular level (which is relative) to the point level (which is the more subtle relationship between what is both absolute and relative). This profoundly important psychological relationship has - as we have seen - an exact mathematical counterpart in Gelfond’s Theorem.

The paradigms of both the linear and circular levels in their pure form are ultimately untenable. The linear level - based on conscious specialisation of reason - is ideal for differentiating reality through the application of partial analytical ability. However - by the same token - it is unsuited to the task of integrating reality through the application of holistic synthetic ability, which is directly based on intuitive understanding.

In turn the circular level - based on unconscious specialisation of intuition - is ideal for holistic understanding. However - when carried to extremes - one then loses the ability to differentiate reality in the first place. One is then left with literally nothing to integrate.

The linear and circular levels are therefore themselves complementary.

The task therefore of the point level is to grow in the understanding of this central dynamic interpenetration. In other words, neither the linear or circular levels - in isolation - are emphasised in experience, but rather the relationship between both.

Once again the most famous transcendental number p - which in mathematical terms involves this relationship of line and circle - serves as the perfect symbol of this paradigm.

Transcendental numbers can of course have positive and negative directions e.g. p and -p .

In like manner the transcendental paradigm has positive and negative directions. However at this highly subtle level of experience - where both structures and mirror structures are both highly developed, these two directions are very closely associated. In other words one moves so flexibly as between transcendental experience of the (external) world and transcendental experience of the (internal) self, that these two directions become ultimately inseparable.

 

 

Imaginary Transcendental Paradigm

 

The main distinction at the point level is - not so much between different directions - but rather as between different modes.

As we have seen, at this level, there is a very clear distinction in the personality as between "higher level" superstructures where reason is subtly employed and "lower level" substructures where primitive emotion is projected from the unconscious.

Whereas the superstructures are initially identified as "real", the substructures are identified as "imaginary". It is the development of these substructures that constitutes the imaginary mode of the paradigm.

These substructures involve both an immediate (conscious) physical instinctive aspect and an (unconscious) spiritual or holistic archetypal significance. Once again we have here the relationship between line and circle. The key to successful appropriation of such experience is to learn to value neither the instinctive or archetypal significance of fantasies (esp. erotic) in isolation but rather the relationship between both aspects.

Again, positive and negative directions of experience at this substructure level of experience are highly differentiated enabling flexible switching as between self and the world.

When this stage is successfully negotiated, the immediate instinctive response to objects directly coincides with an appreciation of their eternal universal significance.

 

Needless to say in mathematics we also can have real and imaginary transcendental quantities. p i is an example of a positive imaginary transcendental number; - p I is the corresponding example of a negative imaginary transcendental number. Once more we have complementarity as between mathematical and psychological number structures.

 

 

Real Transcendental Numbers: Quantitative and Qualitative

 

Concrete

 

The mathematical example here is given by p ¹ and -p ¹ (positive and negative directions).

The psychological equivalent relates to the advanced superstructures and understanding of the relationship between the (analytical) linear and (holistic) circular levels. Though these structures are highly subtle and intuitive they do require the use of

phenomenal symbols (for the purpose of expression) which inevitably involves a certain (rigid) linear aspect.

As we have seen these transcendental structures lead to the formulation of a dynamic psycho-physical worldview (where psychological and physical reality are readily seen as mirrors of each other). This in turn is due to the high level of development of mirror structure activity. Thus as soon as phenomena are posited in experience, they are quickly negated (through mirror structures). This leads to a highly dynamic interaction between objective and subjective experience which greatly facilitates this new world view.

There are at least eight identifiable phases to this stage with dynamic interaction increasing through each phase.

 

Because during this concrete stage affective and cognitive experience still overlap to some extent, these transcendental structures are developed in relation to both modes.

Thus we have transcendental structures which are both intellectual and emotional respectively.

 

Formal

In mathematical terms we can list the transcendental numbers as 1p and 1^-p . Here the transcendental numbers relate to the dimensions (to which the number 1 is raised).

In psychological terms the transcendental numbers relate to the later advanced superstructures of the point level and the dimensions in which experience takes place.

There now emerges a highly abstract deeply conceptual understanding of the relationship between linear and circular levels.

This understanding I have referred to as psycho-mathematical or transrational. Here the linear paradigm (mathematics) and the circular paradigm (transpersonal psychology) are harmonised in an amazing marriage of reason and intuition. One now clearly realises that every psychological relationship has a mirror image counterpart which is mathematical; in turn every mathematical relationship has a mirror image which is psychological. This offers the potential for highly creative insight into the fundamental structure of reality, now seen in complementary mathematical and psychological terms. (Indeed this whole book is an attempt to outline the fruits of such a marriage).

 

 

Imaginary Transcendental Numbers: Quantitative and Qualitative

Concrete

p i and -p i are examples of imaginary transcendental numbers in both positive and negative directions.

The corresponding psychological number structures relate to the substructures of the early point level. The task at this stage is to try and relate both the line and the circle in terms of projected fantasy. In other words there is a phenomenal element (representing the linear level) and an instinctive element (representing the circular level). Confusion arises - especially in the earlier phases of this stage - from separating the phenomena involved from their unconscious origins.

 

Just, as at the complementary superstructure level (of "real" structures) there is still some overlap of affective and cognitive modes. Thus we are still able to identify projections which attach to intellectual ideas and those which attach more directly to sense and emotion.

 

 

Formal

 

In mathematical terms, these transcendental numbers relate to dimensions which are imaginary. These can be represented by 1p ⁱ and 1^-p i respectively.

 

The corresponding psychological equivalent relates to the substructure activity of the later point level, which approximates to pure fantasy.

At the "higher" superstructure level of personality, emotion is almost entirely removed, enabling lucid insight of fundamental reality (simultaneously both intuitive and rational).

One is thereby enabled to provide a dimensional background appropriate to an understanding which harmonises both reason and intuition.

At the "lower" substructure level of personality, in complementary terms reason is almost entirely removed. In other words the blocking mechanism of super-ego control is removed from the experience of projections enabling a free unrestricted type of fantasy.

This literally represents experience of imaginary dimensions appropriate to the phenomena now experienced. As we have seen this involves a complementary marriage in the personality where masculine and feminine principles are fully harmonised.

Correctly understood therefore, at the advanced stages of the point level, experience of reality literally takes place in - what mathematically - are transcendental dimensions (both real and imaginary respectively).

 

 

TRANSITION FROM POINT TO RADIAL LEVEL

Transfinite Paradigm: Real

The culmination of the point level represents a void or experience which is completely formless. This absence of finite manifest phenomena represents a pure experience of what is spiritually transfinite. Indeed it is simply pure intuition without conscious reductionism.

In mystical traditions, this is referred to as the contemplation of transcendence. One has now reached a state of development where one intuitively experiences the whole of reality (without distraction of partial phenomena). This can fruitfully be referred to as the wave aspect of spiritual light.

 

The transfinite is also recognised in mathematics but in a very confused fashion. This is hardly surprising. The infinite or transfinite notion relates directly to what is intuitive (and only indirectly to what is rational). At best - as with the methods of Cantor - transfinite quantities can only be approached therefore in an indirect fashion. Without corresponding intuitive insight, understanding of the results obtained is doomed to misinterpretation.

We have seen earlier - at the transition from linear to circular level - that every number has a negative as well as a positive aspect (both quantitatively and qualitatively).

We saw later - at the transition from circular to point level - that every number has an

imaginary as well as a real aspect (again both quantitatively and qualitatively)..

We now see - at this transition from point to radial level - that every number has a transfinite as well as a finite aspect.

What this simply means is that the understanding of numbers always involves implicit as well as explicit recognition. Explicit recognition relates to rational conscious understanding. Implicit recognition relates to intuitive unconscious understanding. Just as natural light is necessary to see an object in physical terms, intuition or spiritual light is equally necessary to see an object in psychological terms.

Thus 2, for example, as a phenomenal perception represents the finite aspect of that number.

The transfinite aspect of the same number relates to the intuition or mental light (i.e. insight) necessary to see - in understanding - this finite phenomenon in the first place. In the dynamics of experience, finite and transfinte aspects are inextricably linked. Thus when the two are abstracted, paradox inevitably follows.

 

Transfinite Paradigm: Imaginary

The transition from point to radial level initially involves experience of the void and the transcendent pole of spirituality. This represents the culmination of the real transcendental structures of the point level (i.e. superstructures) and leads to the real aspect of the transfinite paradigm.

However the void gradually gives way to a plenum-void. There is now also the culmination of the imaginary transcendental structures of the point level (i.e. substructures) and leads to a complementary imaginary aspect of the transfinite paradigm.

This is referred to in the mystical literature as the contemplation of immanence. This can be looked on as the particle aspect of spiritual light.

Just as contemplation of transcendence enables one to look at the universal whole (to which each part is related), the contemplation of immanence enables one to look at the unique parts (to which the whole is related).

Transcendence provides - in direct terms - the spiritual underpinning of dimensional experience (to which phenomena are related); immanence provides in complementary terms - the spiritual underpinning of phenomena (within such dimensions).

Of course in full mystical experience these two poles directly coincide. However some separation is inevitable even into the radial level.

 

In mathematical terms, the real transfinite aspect of number relates directly to the dimensional aspect of number. In other words the general number concept (to which a particular number relates) has an (intuitive) transfinite as well as (rational) finite significance.

The imaginary transfinite aspect of number relates directly to the quantitative aspect of number (within a given dimension). Each particular number again has an (intuitive) transfinite as well as (rational) finite significance.

 

 

THE RADIAL LEVEL

 

Finite Complex Paradigm

 

The key task of the point level is the differentiation of the two modes of experience i.e. cognitive and affective which are - in mathematical terms - real and imaginary with respect to each other.

With both modes fully differentiated they now emerge harmoniously together, in experience, in complex terms.

Thus at the radial level, the world - in indirect linear terms - is "complex" rather than "real". Linear experience of the "real" world is therefore a reduced and distorted interpretation.

In conventional terms if I buy for example a can of oil, it will be considered a "real" can. What actually happens in such experience is that sense perceptions are deemed to directly correspond with rational concepts. They are thereby reduced to, and interpreted within the "real" rational paradigm.

In truth sense perceptions directly belong to the affective mode, which is qualitatively different from the cognitive rational mode. Relative to the cognitive mode - which is real the affective mode of sense impressions is imaginary. Thus properly understood the experience of a can of oil - or indeed any phenomenon - at the radial level, is understood as dynamically complex.

In mathematics, the real number system is a limited subset of the more general complex number system. In like manner, we can say in psychological terms that the real experience of the linear level, is a limited subset of a more general and integrated manner of understanding which emerges during the radial level i.e. complex understanding. Thus the experience of a "real" can of oil is simply a limited and reduced approximation of what in dynamic terms is more correctly understood as a "complex" can. In the first case the affective mode is subsumed under the cognitive mode and thereby misinterpreted. In the second case the affective mode - while interacting with the cognitive - maintains, in relative terms, its own identity.

Indeed the experience of this can of oil is even more subtle. Both real and imaginary aspects have positive and negative directions.

Thus the rational concept of "can" (in relation to the sense perception of a can) - which is the real aspect - has both positive and negative directions. The positive direction represents "can" (in relation to self). The negative direction represents the self (in relation to "can").

Likewise the sense impression of the perception of the can (in relation to the rational concept of "can") - which - in relative terms - is the imaginary aspect, has also positive and negative directions. The positive direction represents the can (in relation to self). The negative direction represents the self (in relation to the can).

 

We have here the perfect psychological counterpart to the complex number system in mathematics. Both the real (horizontal) and imaginary (vertical) axes - representing numbers have positive and negative directions. Likewise in psychological terms, the real (horizontal) axis representing the cognitive mode of reason, and the imaginary (vertical) axis representing the affective mode of intuitive sense, representing qualitative numbers, have both positive and negative directions. If the positive direction is objective, then in relative terms, the negative direction is subjective.

 

The other remarkable feature about complex experience is that dimensions seem to disappear. What in fact happens is that they become imaginary objects. In the complex experience of the radial level real objects have an alternative aspect as imaginary dimensions. Real dimensions in turn have an alternative aspect as imaginary objects.

Objects and dimensions now ceaselessly interpenetrate - in equal parallel - both in real and imaginary format.

The notion of the existence of one universe comes from the attempt to freeze dimensions by taking them as given. Thus in conventional science and even in relativity physics there is this limiting assumption of four dimensions of space and time. All objects therefore are deemed to exist in this one universe of space and time.

However in proper dynamic terms, there are as many dimensions as objects. Thus there exist an infinite number of parallel intersecting universes. In other words each phenomenon - at whatever level of existence - has a unique identity, yet interacts with all other objects. Each object therefore constitutes a single universe which interpenetrates with all other universes. The basic structure of this networking system of universes is given by the complex number system.

 

 

Transfinite Complex Paradigm

 

We have described conscious experience of the radial level in complex terms.

Unconscious experience - which in developed form is infinite or transfinite, can be described in like manner.

The real aspect of transfinite experience is the wave aspect of spiritual light (contemplation of transcendence).

The imaginary aspect of transfinite experience is the particle aspect of spiritual light (contemplation of immanence). Of course real and imaginary are now entirely relative terms.

The real aspect provides a universal spiritual dimension of wholeness to experience. The imaginary aspect provides a special uniqueness to everything in creation regardless of how minute.

There are positive and negative directions likewise involved. The positive direction of the real level relates to experience of the world, and the negative direction experience of self. At the radial level these directions are very closely harmonised. At one moment, the external world is spiritually illumined, at another moment the inner self. This becomes a such a dynamic pulsating display of light and darkness, that ultimately the light is indistinguishable from the darkness in what mystically is a luminous or dazzling darkness.

 

Simple Paradigm (Interpenetration of Finite and Transfinite)

 

The final stage involves the increasing interpenetration of both finite and transfinite experience. Both now serve as mirrors of each other in a dynamic fulfilling transforming manner. In mystical terms this is referred to as transforming union, where the tremendous spiritual energy - inherent in all phenomena - is greatly released. No further divisions remain in understanding. Directional split (i.e. external and internal polarities), has ceased. Likewise modal split (i.e. cognitive and affective polarities) has been eliminated. Now finally and most crucially process split (i.e. finite and transfinite polarities) nears completion. Experience becomes a dynamic seamless web of interconnecting events, both fully differentiated from, yet fully integrated with each other. Language now utterly fails, for this experience simultaneously involves the linear logic (of the separation of opposites) and the circular logic (of the complementarity of opposites. This is life, pure and simple.

 

Finite Complex Numbers: Quantitative and Qualitative

 

In mathematics, a complex number has both a real and imaginary part (a + bi).

The simplest example would be 1 + i. By varying the sign of these parts we get four such numbers 1 + i, - 1 + i, 1 - i, - 1 - i. When the eight roots of unity are obtained these four numbers are in fact generated (divided by Ö 2), in addition to the two real (1 and - 1) and two imaginary roots (i and - i).

The psychological counterpart relates to the remarkable finite structures that emerge at the radial level. Both directional and modal split in the personality have now been healed. If external (objective) experience of the world is positive, then - in dynamic relative terms - internal subjective experience is negative.

Likewise, if cognitive experience of the world is real, then again - in relative terms - affective experience is imaginary.

Thus when we take the four primary structures (thinking, judgement, perception and feeling) we can express them coherently, in qualitative psychological terms, as follows.

Thinking (objective direction, cognitive mode) can be represented by 1.

Judgement (subjective direction, cognitive mode) can be represented by -1.

Perception (objective direction, affective mood) can be represented by i.

Feeling (subjective direction, affective mode) can be represented by - i.

Now we also have four hybrid structures, which can be represented in complex form.

The combination of thinking and perception gives empirical or practical reason which can be represented by 1 + i.

The combination of thinking and feeling which gives a more personalised form of thinking, represented by 1 - i.

The combination of judgement and perception gives a practical form of judgement, represented by - 1 + i.

Finally, the combination of judgement and feeling gives a detached form of feeling, represented by - 1 - i.

 

Of course, these signs are entirely relative, depending on what direction is initially defined as positive and what mode defined as cognitive.

Conventional experience of the linear level involves considerable reductionism with only the positive direction of the real mode considered.

In other words, because of lack of mirror structure activity, both objective and subjective experience are posited in experience with a tendency to reduce the subjective to the objective direction. For example the overwhelming view is that time moves forward for both the world and the self. In truth - in dynamic relative terms - if time moves forward for the world, then it moves backwards for the self.

Likewise because of lack of virtual structure activity, both cognitive and affective modes are understood in experience as real. Again the conventional view is that both thought concepts (cognitive) and sense impressions (affective) are understood as relating to the real world. Again in truth - in dynamic relative terms - if thought constructs are real, then sense impressions are imaginary.

Thus with respect to finite experience, at the radial level, one clearly understands that we live in a (mathematically) complex rather than a real world.

 

Transfinite Complex Numbers: Quantitative and Qualitative

 

These greatly complement the finite structures.

Again there is a remarkable mathematical explanation involved here.

The finite complex solutions (of the eight roots of unity), give rise - as we have seen to four complex solutions where real and imaginary coordinates are equal. The diagonal lines connecting the centre of the circle to these four points, in each case gives rise to a right angled triangle where they can actually be represented as zero. Using the Pythagorean theorem, the square on the hypotenuse (the diagonal line), is equal to the squares on the two other sides (the segments of the real and imaginary axes respectively). Because these line segments are equal in length, the diagonal line (in two dimensional terms) in each case is zero.

In mathematical terms, these diagonal or null lines indeed represent rays of physical light. Phenomenal experience is conventionally posited in conscious one dimensional terms. These lines - being inherently two dimensional - relate to matters own holistic base, i.e. the fundamental ground or potential from which phenomena emerge.

In complementary psychological terms, these diagonal or null lines represent rays of spiritual light. These lines here relate to the holistic basis of phenomenal experience which is inherently intuitive.

Indeed these diagonal rays relate - both in mathematical and psychological terms to what is properly transfinite.

So we now have the emergence of transfinite complex numbers in quantitative and qualitative terms. (The complex aspect comes from the relationship of this light with finite reality now understand in complex terms).

Just as earlier we could define imaginary numbers - in dynamic terms - as representing what is simultaneously both positive and negative, we now can give a fascinating dynamic interpretation of transfinite numbers as simultaneously both real and imaginary.

In psychological terms the positive real pole represents transcendent experience of the world (in relation to self). This is the transfinite intuitive basis of thinking.

The negative real pole represents transcendent experience of the self (in relation to the world). This is the transfinite intuitive basis of judgement.

The positive imaginary pole represents immanent experience of the world (in relation to self). This is the transfinite intuitive basis of perception.

The negative imaginary pole represents immanent experience of the self (in relation to the world). This is the transfinite intuitive basis of feeling.

 

 

Simple Numbers

 

This represents the culmination of both mathematical and psychological approaches to number. By this advanced stage rational and intuitive experience so interpenetrate each other as to be indistinguishable. We now can no longer separate numbers (representing mathematical quantities) from numbers (representing psychological qualities).

Mathematical and psychological understanding are now dynamically identical.