We are now in a position to outline in detail the remarkable complementarity of number types and psychological structures.

Numbers as we have seen have dual properties both as rational (mathematical) quantities and intuitive (psychological) qualities. These latter "qualities" relate to various paradigms of understanding combined with their associated structures.

The paradigms can in general terms be identified with the four levels of experience (linear, circular, point and radial).

The structures can in turn be identified with the various stages of development which emerge at each level.

The emergence of all these stages of development and their associated psychological structures can be exactly and uniquely related with corresponding number types.

This comprehensive number system therefore provides the most appropriate means of precisely ordering all the varied experience of development.

Finally, it provides a highly creative way of approaching key issues in both psychology and mathematics. In original terms, both conscious and unconscious are completely identical. Thus reason (associated with mathematics) and intuition (associated with transpersonal psychology) in like manner are identical.

Therefore the rediscovery of this fundamental unity of reason and intuition generates original insight which can unravel what otherwise seem the most intractable of problems.
 
 

THE LINEAR LEVEL 

Positive Real Rational Paradigm 

The linear level as we have seen is directly identified with the rational paradigm. The line - in mathematical terms - is one dimensional.

In like manner the rational paradigm is one dimensional where all experience is considered within just one dimension of experience. Understanding is both real and positive. In other words solely conscious criteria are used (real), with an objective direction (positive).

In mathematics a line can have a negative as well as a positive direction. However - as conventionally understood - in qualitative terms there is no difference as between these directions. Thus when both are combined - rather than generating a new synthesis - they simply exclude each other. Thus 1 - 1 (where both directions of the number are combined) = 0 (i.e. nothing). We can say accurately therefore that in mathematics there is a simple reduced interpretation of negative numbers - as with positive numbers - as one dimensional.

Likewise in psychology, understanding can have a negative (i.e. subjective) as well as a positive direction. However there is (within the rational paradigm) no qualitative difference as between these directions. Again when both are combined - they tend to exclude each other. Thus in the rational paradigm opposite viewpoints exclude each other. In this logic truth is absolute. The truth of one position - by definition - thereby excludes its opposite. The rational paradigm cannot combine and reconcile opposite positions for this - by its logic - nullifies its very basis (i.e. leads to nothing).

The rational paradigm therefore simply reduces the negative (subjective) direction of experience to the positive. This tendency manifests itself in extreme terms in behavioural psychology. Here we have the most determined effort to avoid subjective meaning by attempting to explain experience in solely objective terms.

This reductionism is of course the very basis of "rational" science, where correct procedure requires the elimination - insofar as is possible - of all subjective criteria.

Science strives to deal with a solely "real" world (conscious phenomena). Ultimately of course this position is untenable. At the early primitive level of instinctive behaviour, experience is more "imaginary" than "real". Likewise at the most fundamental sub-atomic level, reality is more "imaginary" than "real".

Science also strives to deal with a solely "positive" world (objective direction). Again this position is not ultimately tenable. The attempt to consciously screen out the "negative" world (subjective direction) leads at an unconscious level to strong subjective bias. Scientists are thus often unaware of the highly subjective nature of their key assumptions (in particular the belief that truth can be identified with what is solely objective).

The rational paradigm of course has great value. It represents an extreme in terms of psychological differentiation (the separation of opposites). Here the finite (material) is separated from the infinite (spiritual). The real (conscious) is separated from the imaginary (unconscious). Finally the positive (objective) is separated from the negative (subjective).

Whereas the strength of the rational paradigm is in terms of this (partial) differentiation, by the same token, its great weakness is in terms of the complementary need for (holistic) integration. The considerable fragmentation - so evident in Western society - owes much to the dominant role of the rational paradigm.
 

Rational Numbers: Quantitative and Qualitative

There are several different types of rational numbers (mathematical quantities).

There are equally several different types of complementary rational numbers (psychological qualities).
 

Binary Numbers 

In mathematical terms the two original number quantities are the binary digits 0 and 1.

These binary numbers are of immense importance. They are sufficient to construct a complete number system. On this system the miracles of our computer age are based.

The very essence of the binary system is a fundamental separation of opposites. 1 (what exists) is clearly separated from 0 (what does not exist). Likewise in computer electronics, this logic of separation is evident with 1 representing an on state and 0 representing an off state.

Implicit in these original numbers is the very logic of the rational paradigm where opposites are clearly separated.

In psychological terms the two original number qualities are also the binary digits 0 and 1. These binary numbers are also of immense significance, being sufficient to construct a complete psychological system. On it, the miracle of complete personality transformation is made possible.

The very essence of this alternative binary system is a fundamental unity of (apparent) opposites. 1 (the line) represents the rational approach. 0 (the circle) represents the intuitive approach. The rational approach literally posits the existence of 1's (i.e. independents units or phenomena of existence). The intuitive approach involves a dynamic complementarity of positive and negative directions of experience (where nothing in phenomenal terms exists). The rational approach is based (directly) on the conscious mind whereas the intuitive approach is based (directly) on the unconscious mind.

At the beginning of life, in psychological terms 1 and 0 are identical. Unity is inseparable in experience from nothingness. In other words the conscious is completely undifferentiated from the unconscious mind.

Thus, with the original numbers, we have a direct complementarity as between the rational binary number system (representing mathematical quantities) and the intuitive binary number system (representing psychological qualities).
 

Prime Numbers 

A prime number - in mathematical terms - is one with no factors (other than itself and 1). Thus whereas a number such as 6 can be represented as two dimensional - as the product of two numbers (i.e. 2 X 3), a prime number such as 7 is one dimensional (with no constituent factors).

Thus with prime numbers there is a complete separation as between numbers as quantities (within the 1st dimension) and numbers as dimensions (to which the number 1 is raised).

In mathematical terms, with prime numbers we cannot simultaneously combine quantities and dimensions.

A prime number in psychological terms, represents a situation where the infant is unable to separate conscious phenomena (within the dimensions of space and time) from the dimensions (in which these phenomena are experienced). This in fact is a perfect description of pure instinctive behaviour, where one is unable to give phenomena any kind of background perspective (a context in space and time). Thus experience here tends to be highly transient, with phenomena arising and disappearing in the spontaneous moment of creation.

Indeed this is commonly described as primitive behaviour bearing direct comparison with the mathematical notion of prime numbers.

Thus again we have a direct complementarity of mathematical and psychological numbers. In rational quantitative terms, prime numbers involve the complete separation of such numbers from their corresponding dimensions. In intuitive qualitative terms, prime numbers (primitive instincts) involve the complete (undifferentiated) complementarity of such numbers with their corresponding dimensions.
 

Natural Numbers 

Mathematically, natural numbers involve composite quantities. These composite numbers arise from combining prime numbers in more than one dimension. 6 - as we have seen - is such a composite number involving two factors or dimensions. These composite numbers, along with their constituent primes, generate the natural number system. The natural numbers arise from a given set of uni-dimensional quantities (i.e. the prime numbers) which can be combined to give higher dimensional quantities (i.e. composite numbers).

There is a remarkable correspondence here with physics. Superstring theory is based on the assumption that the most fundamental particles are in fact one-dimensional. These can be viewed therefore as the prime particles. Natural phenomena - which can be observed - arise out of a higher dimensional composite synthesis of these superstrings or prime particles. (Interestingly in superstring theory this higher dimensional reality - as in mathematics - is not bound by the conventional assumption of four dimensions).

Psychologically natural numbers involve a fully complementary pattern. Whereas in mathematics, natural numbers emerge from combining (prime) numbers and dimensions, in psychological development, natural structures emerge from the growing ability of the child to separate phenomena and dimensions. The child in other words discovers a background context - through a dimensional perspective of space and time - within which to experience phenomena. With this, the natural world literally emerges in experience.

However this composite process - which is inherently higher dimensional - becomes quickly reduced to one dimensional terms. The child learns to interpret experience from a rational (one dimensional) perspective where each object is given a separate (independent) existence. So the child begins to believe - through an increased reductionist tendency - that such objects enjoy a static absolute existence. The truth in fact - which is quickly lost - is that they are dynamic and highly arbitrary multidimensional readings of a mysterious pattern, the prime ingredients of which cannot be observed.

A similar tendency towards reductionism of course is evident in mathematics. Composite numbers (though involving several factors or dimensions) are ultimately expressed in one dimensional terms. Thus once again for example 6, though it is inherently a two dimensional number (i.e. has two prime factors), is expressed in one dimensional terms. Indeed this is true of all the natural numbers where both prime numbers (which alone are inherently one-dimensional) and composite numbers are treated alike.
 

Integral Numbers (The Integers) 

The integers - as mathematical quantities - involve an important advance over the natural numbers. They now include a negative - as well as a positive - direction. These can be represented by the line representing such numbers with 0 in the centre. The combination of a positive number with its negative counterpart is therefore zero (i.e. 0). Negative numbers, like positive numbers are treated in one dimensional terms. Reductionism is once more involved. Though in quantitative terms they are opposed, in qualitative terms they are treated the same.

Thus we have positive integers such as 2. Strictly this is a (horizontal) integer quantity within the fixed first dimension i.e. 2 ¹.

Then we have the corresponding negative integers such as -2 i.e. -2 ¹.

We can also have positive integers such as 2, this time representing (vertical) integer quantities to which the fixed number one is raised (i.e. 1² .)

We can also have negative integers such as 2, representing (vertical) integer quantities (i.e. 1^-2 ).

The integers - as psychological qualities - also involve an important advance. With the natural structures a child is not able to properly sustain the loss of objects. It is a case of "out of sight, out of mind" where the child believes that the object no longer exists when it is not directly present in experience.

However with the development of the negative direction, this changes. In other words the child develops a degree of subjective awareness, where the self to some degree is separate from the objects of experience. This subjective awareness enables the child to sustain the loss of objects. The child now believes that they continue to exist even when not directly present in experience.

Firstly the child develops the ability to posit whole quantities (within dimensions) through perceptions of objective phenomena.

Next the child develops the ability to negate these objects (without loss of permanence) through positing the self.

Then the child slowly learns to posit whole dimensions through growing conceptual ability to organise objects.

Finally - at this stage - the child learns to negate these dimensions (again without loss) through a growing concept of self.

Implicitly the child has by now developed an understanding of objects that is both positive and negative and even more surprising an understanding of dimensions as positive and negative. What this simply means is that perceptual and conceptual ability involve a dynamic relationship of the child with the world.

Conventionally the child is trained to interpret such experience in reduced static terms.

Whereas in mathematics, positive and negative directions converge and cancel each other out (i.e. a number added to its negative is zero), in complementary psychological terms, positive and negative directions diverge. In other words the external world and the internal self become increasingly separated from each other.
 

Rational Numbers: Concrete

Positive 

When we combine quantitative and qualitative integers arising from the previous stage we obtain fractions (rational numbers). A positive rational number - in mathematical terms - can be expressed as a fraction e.g. 1/4. More correctly this should be written as (1/4)¹ representing a quantity within a fixed unitary dimension.

Alternatively this can be written as 2^-2 . So mathematically all reciprocals - and by extension, all fractions involve the dimensions in negative form. So just as there is a negative - as well as positive direction - to quantities (within dimensions), there is also a negative direction to the dimensions themselves.

Expressing a number as a fraction involves fragmentation. If we break a unit e.g. a cake into four slices each slice - though a unit in its own respect - represents a fragment or fraction of the original whole.

When we combine the horizontal and vertical abilities of the integral stage (perceptions and concepts) we obtain a new form of experience or (qualitative) number. This is the concrete rational paradigm which - in psychological terms - involves the ability to analyse and fragment reality. In other words concrete rational understanding involves the ability to break objects into various parts, which enjoy a separate independent existence, while maintaining a relationship with the original whole.
Implicitly this requires the ability to formulate objects in negative dimensions.

The previous integral stage involved the ability to suffer the loss of object perceptions through giving such objects a negative as well as positive direction. It also involves the the further ability to sustain the loss of conceptual objects (i.e. dimensions). Thus the dimension or whole object remains intact in experience even when constituent parts are formulated. This enables one in turn to relate these constituent parts back to the original whole which is the very basis of rational analytical ability. Thus one can break a whole object - such as a cake - as in our example into four constituent slices which are understood as related to the original whole.
 

Negative 

Of course in mathematics we can also have rational numbers which are negative fractions e.g. -1/4. Again correctly this should be written (-1/4)¹ which is (-2)^-2 We now have a negative direction to both quantity and dimension.

Again we have the precise corollary on the psychological side.

This psychological number represents concrete rational development in a negative direction (i.e. with respect to the self).

One develops increasingly during this stage the ability to split one's ego into a number of parts. In other words one develops a number of separate personae arising from different roles and responsibilities which are related to one's general self.

Implicitly this involves the ability to formulate negative dimensions in experience (i.e. by being able to sustain the temporary loss of a self concept. In this manner one's personality becomes more flexible.
 
 
 

Rational Numbers: Formal 

Positive 

This is a more abstract generalised approach to number. We are now directly concerned with number as a qualitative dimension (rather than a quantity). A positive fraction such as a reciprocal can now be also be represented as 1/4, which this time is written correctly as 1^1/4 .

The corollary on the psychological side is formal rational understanding. Before this the child could analyse concrete objects. Now, with the emergence of adolescence, there is the additional ability to analyse formal objects. One is able to break concepts i.e. dimensions into parts which can be related back to the original concepts.

This ability - more than any other - is what defines the rational paradigm and is especially employed in theoretical science. Strictly speaking, it literally involves experience of fractional dimensions. However because this highly important qualitative definition of number is overlooked, a reduced quantitative - and linear - interpretation results.
 

Negative 

This is just the negative direction of the fractional dimension (e.g. -1/4 or more correctly 1^-1/4.

The psychological equivalent is the ability to reason in abstract terms regarding subjective rules of conduct e.g. abstract systems of philosophy.

One is thereby enabled to break holistic concepts of the subjective self into parts which can be related back to the original concepts. This ability for example would - in general terms - enable one to work out systems of behaviour consistent with being a "good" citizen. This time one is literally experiencing reality in negative - rather than positive - fractional dimensions
 
 

Integration of Rational Numbers

In mathematics the rational number system includes all the numbers that we have encountered so far.

Thus the rational number set includes the original or binary numbers (0 and 1), the prime numbers, the composite numbers leading to the natural numbers, the negative whole numbers, which together with the earlier numbers, gives us the set of all integers, the fractions (both positive and negative) both in a specific and generalised sense, which combined with the whole numbers provides the set of all rational numbers.

This treatment is highly reductionist in that all the whole numbers - at their varying levels - are then treated simply as special cases of rational numbers.

In psychology - as we have seen - we have an exactly matching complementary - system of numbers in the various stages of psychological development which unfold during the linear level.

Thus we have original development (where conscious and unconscious are undifferentiated), prime development (primitive instinctive behaviour), natural development (where the objects of the natural world are differentiated), integral development (which includes self consciousness), concrete rational development (where one can analyse situations in practical terms both in relation to the world and the self) and finally formal rational development (where one can analyse in theoretical terms both the world and self).

In Western psychology, the highest level of development - sometimes referred to as the centaur - involves the integration of rational mental development with the earlier more instinctive physical stages of development.

However just as in mathematics this integration is achieved in reductionist terms, with earlier "physical" numbers treated as a special case of the later rational "mental" numbers, the same reductionist tendency is evident in psychology.

When the rational paradigm becomes dominant as the appropriate model for interpreting reality, it is inevitable that earlier stages of development - which do not use the same model - will subsequently be reinterpreted from a rational perspective. In other words, in the light of the "superior" rational stages there will be an inevitable tendency to now repress earlier stages as inappropriate.

Indeed integration cannot be properly achieved within the linear level. In mathematics we know that there are many other number sets in addition to rational numbers. Likewise in (transpersonal) psychology there are many other number sets (stages of development) in addition to rational development.
 
 

TRANSITION FROM LINEAR TO CIRCULAR LEVEL

Negative Real Rational Paradigm 

We have already looked at the paradigm of the linear level, which is conscious, rational and based solely on the positive direction of experience.

As we have seen in mathematics, a line can also have a negative direction. However within the rational paradigm there is no qualitative distinction as between positive and negative directions.

Likewise in psychology, within the rational paradigm, in strict terms there is no qualitative difference as between positive and negative directions. There is thus an inevitable tendency to reduce subjective experience to the objective i.e. appropriate it through rules which are objectively understood.

However with the transition from linear to circular levels, this position changes. One begins to experience reality in terms of a rational paradigm which is now negative (rather than positive). What this means in effect is that the subjective level of experience (arising from a profound existential crisis) is now made explicit and not simply reduced to objective criteria. This negative paradigm in fact is dynamically opposed to the positive. In other words quite literally, the products of (positive) phenomenal consciousness are negated or eroded during this time. This is precisely why it is psychologically such a harrowing experience.

Just as in physics, anti-matter when brought in contact with matter will lead to mutual cancellation in pure energy, likewise in psychological terms when psychic anti-matter (the explicit negative or subjective direction of experience) is brought into contact with psychic matter (the positive direction) there will be a dynamic cancelling out process generating pure spiritual energy. This stage therefore represents the first truly transpersonal stage of development.

Expressed in formal terms one now becomes profoundly aware of the existence of "psychic anti-matter". In other words for every phenomenon that positively or consciously exists there exists a mirror image alternative which negatively or unconsciously exists.

What this simply means that all experience involves a dynamic relationship involving the positive direction of the world (in relation to self) and the negative direction of the self (in relation to the world). Whereas in reduced static terms these two directions are formally identical, in dynamic terms they are opposed.

Thus expressed in static terms, every perception, every concept of reality has a mirror image opposite (i.e. anti-perception and anti-concept) which is formally identical. By extension in science every theory has a mirror image opposite (anti-theory). Like absolute numbers in static terms this sign difference is irrelevant. However in dynamic terms these opposite signs become highly significant.

When we come back to numbers, this of course is directly relevant. Now we can see that every number has a negative interpretation in psychological as well as mathematical terms. This as we see leads to a relative interpretation of number.

In other words every number really is a dynamic creation in experience which simultaneously involves what is observed and who observes i.e. the number (in relation to self) and the self (in relation to the number).
 

Anti-Rational Numbers: Quantitative and Qualitative

Concrete

If for example 1/2 or more correctly (1/2)¹ represents a concrete rational number, strictly speaking this refers to one direction only i.e. the positive external direction of the number (in relation to mind).

The corresponding anti-rational number is also (in static terms) 1/2, this time referring to the complementary direction only i.e. the negative internal direction of the mind (in relation to the number).

It is the same in relation to the concrete rational number -1/2 or again strictly (-1/2)¹ . Again this as a rational number only refers to the positive direction of experience of the number. The anti-rational -1/2 refers to the negative direction of the number.

Of course because - in static terms - the positive and negative directions of the number coincide, in conventional mathematics this vital directional distinction is completely ignored. This in turn leads to a fixed absolute rather than a more correct dynamic relative interpretation of number. At this level, numbers are not positive and negative in an absolute sense. Rather each number has both positive and negative signs, which are activated alternately.

The corresponding equivalent here on the psychological front relates to the development of the concrete anti-rational structures.

Firstly we have the development of the anti-structures in relation to positive external experience. During this stage, objective sense phenomena are literally undone and eliminated in experience.

Next we have the development of the anti-structures in relation to negative internal experience. During this time subject sense phenomena (i.e. emotional feelings) are likewise eroded in experience. When we clearly recognise what happens during this time it is not surprising why these stages in ascetical mystical terms are referred to as the way of purgation. Of course - though not understood so well at the time - considerable generation of spiritual or psychic energy takes place within.
 

Formal

Again 1/2 (or correctly 1^1/2 ) can represent a rational number, now understood as a qualitative dimension. Correctly interpreted only the positive direction (general power or dimension in relation to mind), refers to this formal rational number. The negative direction - though identical in static terms - correctly speaking - relates to the corresponding anti-number.

- 1/2 now represents a negative (formal) rational number (correctly 1^-1/2). Again only the positive direction - in understanding - relates to the number, with the negative direction relating to the corresponding anti-number. Of course at this level of understanding psychological and mathematical understanding of number is inextricably linked.

The corresponding psychological numbers refer to the development of the (formal) rational anti-structures. In other words the deeper levels of conceptual understanding underlying the rational paradigm are greatly eroded during this time.

Firstly the conceptual base underlying objective understanding (e.g. conventional understanding of science and mathematics) is negated.

Finally the deeper conceptual base underlying internal subjective experience is negated in the generation of the negative rational anti-structures.
 
 

THE CIRCULAR LEVEL

Real Irrational Paradigm: Positive 

During the linear level the positive (conscious) direction of experience is explicitly differentiated in experience. Though the negative (unconscious) direction is implicitly involved, it is not formally recognised.

During the transition from linear to circular level, the negative direction is now explicitly differentiated. Thereby it is understood that there is a negative as well as positive direction to all experience.

Though in static terms these directions are identical leading inevitably to considerable reductionism, in dynamic terms they are directly opposed.

During the circular level there is therefore the emergence of a new dynamic paradigm which is based inherently on the complementarity of opposites.

The very logic on which this paradigm is based contradicts that of the rational paradigm. In the rational paradigm, opposites (e.g. truth and falsehood) are clearly separated. In this paradigm however, they are dynamically complementary and indeed ultimately identical. In terms of the rational approach, this paradigm seems deeply paradoxical and indeed irrational. It is to be found in the mystical expression of all religions, In Western philosophy its most complete expression is the Hegelian system. It is also highly relevant to many of the findings of modern physics e.g. Quantum Theory.

Whereas the rational paradigm is well suited to the detailed analysis of specific problems (within a fixed context) it is by the same token ill suited in terms of providing a holistic synthesis of problems (within a dynamic context).

By contrast the irrational (holographic) paradigm is ill suited to detailed analysis of specific problems. However it is ideal - being more intuitively based - for the holistic synthesis of dynamic problems.

Correctly understood therefore, the rational and irrational paradigms are themselves complementary. However because of the heavy dominance of the former, there remains a continual tendency to firstly reduce and then attempt to solve complex problems within its limited perspective.

The behaviour of algebraic irrational numbers in mathematics directly parallels the psychological situation. Particularly relevant is the square root of 2.

The irrational paradigm arises from the attempted expression of what is inherently two dimensional (and intuitive) in one dimensional (rational) terms. A vertical (circular) form of understanding is thereby reduced to horizontal (linear) terms where it appears highly paradoxical with two opposite poles.

It is exactly similar with irrational numbers in mathematics. If we express a number - such as the square root of 2 - which is inherently two dimensional - in one dimensional terms, i.e. by obtaining its square root, the result - as the Pythagoreans discovered - appears highly paradoxical. Once again we get two numbers with opposite poles (i.e. the root of 2 and - the root of 2). Also, these numbers are themselves irrational (i.e. cannot be understood in rational terms.

In fact they are really relative numbers. As is well known the value of irrational numbers can never be absolutely determined but rather approximated. Irrational numbers are really dynamic number entities subjected to an attempted reduced (rational) interpretation.

They are also inherently circular. The two roots of 2 lie on a circle. Higher roots will also lie on a circle. If for example, we obtain the eight roots of 2, we will obtain eight points that lie on a circle within the complex plane.

When we attempt to reduce higher dimensional (vertical) levels of understanding - such as two dimensional understanding - to one dimensional terms, we literally obtain an irrational or paradoxical circular form of understanding.

Similarly when we attempt to reduce a higher dimensional (vertical) numbers - such as the root of 2 - to one dimensional terms (through extracting square roots) we literally obtain irrational or paradoxical circular numbers (i.e. numbers that lie on a circle).

Conventionally mathematics is tied to a linear interpretation of number (through defining all numbers ultimately in terms of the 1st dimension).

However there is an equally important circular system of numbers which arises from converting higher dimensional vertical numbers - such as the root of 2 - to the 1st dimension (i.e. by extracting the successive roots of numbers). In contrast to the linear system which is rational, this latter system is irrational.

Correctly speaking all irrational numbers lie on curved rather than straight lines with straight forward limiting conversions (i.e. roots) lying on circles. Placing irrational numbers on a straight line is just an unfortunate form of reductionism.

Thus once again there is a perfect correspondence as between irrational numbers as mathematical quantities and irrational numbers as psychological qualities (i.e. as a paradigm of understanding).
 

Real Irrational Paradigm : Negative 

We can also distinguish an anti-irrational paradigm.

A degree of rigidity can be attached to the understanding of the paradigm of the complementarity of opposites. Even though the inherent understanding is directly intuitive, one to some extent will identify with the reduced rational forms of expressing this understanding. To say that all things are relative is itself an absolute statement. This then requires detachment from these secondary means of expression in the formation of new anti-structures. In mystical terms, St. John of the Cross refers to this as passive purgation. In my language it is the emergence of the negative or anti-irrational paradigm, where secondary phenomenal expressions of intuitive spiritual truth are eliminated from consciousness.

During the circular level, which is highly intuitive, there is a far greater dynamic as between the positive and negative directions of experience.

Thus the anti-irrational paradigm tends to be very closely associated with the irrational paradigm during this time. In other words phenomenal symbols behave in a very subtle fashion acting as temporary conveyors of an archetypal spiritual light, spontaneously arising and disappearing quickly from consciousness.
 
 

Concrete Irrational Numbers: Quantitative and Qualitative

The root of 2 or more correctly (the root of 2)¹ as we have seen is the best known (algebraic) irrational number in mathematics. It serves as an appropriate archetype for all such numbers. An irrational number has both a finite and infinite significance simultaneously (i.e. though its finite value can be approximated its true value transcends such approximations).

Formerly during the linear level, specialised development of (fractional) analytical understanding took place firstly in terms of concrete quantitative and later formal qualitative understanding.

However when one tries to combine concrete and formal rational understanding - combining non-unitary quantities and qualities - a highly interesting problem arises.

If one does not reduce experience - in contrast to the conventional approach - to solely quantitative terms, there necessarily is a transformation in such understanding whereby actual rational understanding is transcended altogether. This transformation has a precise mathematical complement.

The root of 2 can be written as 2^1/2. This is of course an irrational number. Indeed any rational number raised to a fractional rational power (excepting numbers which are already perfect powers) results in an irrational number. Thus in mathematics when we combine rational (horizontal) quantities and (vertical) qualities in this fashion we generate irrational numbers.

When we interpret this number in dynamic relative terms the root of 2 has also its own anti-number. This anti-number is not recognised in conventional mathematics.

The negative root of 2 is an example of a negative (concrete) irrational number (-2^1/2). Again this root has its own anti-number.

The corresponding number type on the psychological side relates to the (positive) concrete irrational structures (i.e. the supersensory structures). One initially tries to combine concrete and formal rational understanding. However when one does not simply reduce the qualitative to the quantitative, the rational paradigm (of the linear level) actually breaks down.

As we have seen a (horizontal) rational number raised to a (vertical) rational number (i.e. a non-trivial fraction) generates an irrational number in mathematical terms.

In precise complementary terms concrete rational understanding (i.e. horizontal perceptions) raised to - or experienced in the context of - formal rational understanding (i.e. vertical concepts) generates irrational structures psychologically.

When experience takes place without reductionism, backward integration of formal with concrete rational understanding thus proves impossible, and requires transcendence in a new irrational form of understanding.

Here there is the emergence of a highly intuitive view of reality where objective phenomena are interpreted in terms of the complementarity of opposites. As well as their natural finite nature, the dynamic fusion of these opposites in experience gives all symbols an archetypal infinite significance. Thus like irrational numbers in mathematics, these psychological numbers have simultaneous finite and infinite aspects.

Quite literally, as this experience is now of fractional dimensions, we can see how the conventional model of four dimensional space-time is so limited. Thus whereas the earlier concrete rational structures of the linear level, led to the ability to fragment the phenomena of experience, these irrational structures of the circular level, lead to the ability to actually fragment the dimensions themselves.

The corresponding anti-numbers or mirror structures lead to the erosion of these subtle phenomena in experience.

There is then the emergence of the (negative) concrete irrational numbers or subjectively based supersensory structures. This is a highly existentialist phase, where (finite) personal decisions are invested with universal (infinite) importance.

This phase in then followed by a corresponding anti-number stage where once again these subtle subjective phenomena of experience are eroded.
 
 

Formal Irrational Numbers: Quantitative and Qualitative

We are now dealing with the generalised abstract form of irrational number as dimension.

In mathematical terms this is symbolised by the root of 2 representing now an irrational dimension i.e. 1^{the root of 2}. We now have moved from an irrational quantity (within the qualitative 1st dimension) to the complementary notion of the quantitative unit raised to an irrational dimension. The irrational understanding refers to the (positive) objective experience of this relationship.

The anti-structure equivalent would refer to the (negative) subjective experience of the same relationship.

Then we can have a negative irrational dimension symbolised by -the root of 2 (or more correctly
1^{- the root of 2}.

The formal irrational understanding refers to the objective direction in experience of this number relationship, whereas the anti-understanding refers to the subjective direction of the same relationship.

In psychological terms experience of irrational dimensions refers to that highly illuminated understanding of the suprararational level. One literally now experiences reality in such irrational dimensions i.e. one interprets reality in terms of the complementarity of conceptual opposites.

The anti-irrational equivalent - leading to the profound dark night - involves the erosion of these subtle objective concepts in experience.

Next we have the emergence of negative irrational dimensions. In other words one interprets intimate subjective personal experience in terms of the dynamic complementarity of conceptual opposites.

Finally - at this level - we have the corresponding mirror structures i.e. the anti-irrational structures. Here all subtle phenomena - even in terms of refined inner experience are eroded in the passive purgation of the dark night.
 
 

TRANSITION FROM CIRCULAR TO POINT LEVEL

Imaginary Rational Paradigm: Positive 

Another profound transition now takes place. During the circular level though experience grows steadily more intuitive (and unconscious), one still tries to keep it under rational control through subtle and indirect one dimensional (i.e. conscious) translations. However ultimately with the erosion of all conscious understanding, this no longer proves tenable. Thus the crisis of the dark night - which is the culmination of the circular level - involves a pure and unreduced form of spiritual (unconscious) intuition.

Having been for so long kept under the control of reason, the unconscious is now at last free to express itself in its own terms. Now whereas the conscious (in relation to unconscious) has a positive sign, the unconscious (in relation to conscious) has a negative sign. However, giving it indirect conscious expression requires translation through one dimensional format. So once again we obtain the square root, this time in relation to a negative sign. This - in mathematics - involves obtaining an imaginary quantity. In precisely similar psychological fashion, we now obtain - in qualitative terms - an imaginary number.

What this means is that conscious experience is now seen as the projection of the unconscious mind. Though indirectly these projections involve conscious phenomena, their real significance lies not in these specific phenomena but rather in the dimensions from which they emanate. In other words they have a holistic archetypal significance.

Real experience involves in direct terms the experience of phenomena (against a given background of dimensions).

Imaginary experience by contrast involves in direct terms the experience of dimensions (against a given background of phenomena). With imaginary experience we literally create the dimensions in which we experience reality.

Conventional science is based on the "real" world where dimensions of space and time are given. However correctly speaking, these dimensions are not given at all, but rather directly created through experience. So we actually live in a "complex" world which is both "real" and "imaginary".

We now can see why the dark night there is such a congested suffocating experience. This is because - due to the high level of control of instinctive response, the unconscious is unable to express itself directly. Quite literally there is a prolonged - near total - starvation of dimensional experience. One is therefore not able to create a framework of either time or space. In the absence of such dimensions psychologically, matter becomes more and more compressed in experience leading to an enormous feeling of grief (i.e. gravity). The dynamics of the black hole are exactly complementary. The black hole represents a situation, where matter is unable to create a supporting framework of dimensions (i.e. space and time). It therefore gets squeezed out of space and time through an immense inward gravitational pull. The subsequent radiation of black holes can be seen as an attempt by matter to once more create the corresponding dimensions of space and time.

The spontaneous emission of projected phenomena, which occurs during this transition from circular to point level, represents a psychological attempt to restore dimensionality to experience. The meaning of phenomena is now no longer conscious but unconscious leading back to the holistic dimensions from which they have been projected. In other words phenomena now are imaginary speaking of a virtual reality.

During the previous transition we saw that all phenomena are in dynamic terms bi-directional with positive and negative aspects. Now one understands that all phenomena are bi-modal with real and imaginary aspects.

Thus for science, this means that every fact, every concept, every hypothesis has real and imaginary aspects. Perceptions (specific objects) and concepts (holistic dimensions) are in relative terms real and imaginary with respect to each other. If the perceptions are "real", then the corresponding concepts are "imaginary". Likewise, in reverse, if the concepts are "real", then the perceptions are - in relative terms -"imaginary". Conventionally science suffers from considerable reductionism where it is tacitly accepted that "real" facts have a correspondence with "real" theories. Actually - in dynamic terms - "real" facts have a correspondence with "imaginary" theories and "imaginary" facts have a correspondence with "real" theories. So just as there is a need for anti-theories in science, there is equally a need for "imaginary" theories. Interestingly, one important area, where this conventional view is coming unstuck is in the attempted reconciliation of Relativity and Quantum Mechanics. What is required here is a "complex" rational rather than a "real" rational framework.
 

Imaginary Rational Paradigm: Negative 

Now, we can distinguish a positive from a negative imaginary (rational) paradigm.

However just as in physics virtual particle and anti-particle pairings are very closely related, likewise in psychology, positive and negative imaginary paradigms are very closely related. Because experience is at such an unconscious level, there is very little rigidity in the indirect phenomenal translations. Thus the positive (projected) phenomena are quickly negated in a continual return to their source in the unconscious.

Thus just as every real theory has a corresponding anti-theory, every imaginary theory has a corresponding anti-theory.
 

Imaginary Rational Numbers: Quantitative and Qualitative

Concrete

The square root of -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 the square root of -1 can be expressed as +1 and -1 simultaneously.  - i is the corresponding negative quantity.

i can also represent the corresponding psychological unit quality. 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 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. 1ⁱ .

The corresponding negative number is 1^-i .

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, , 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 - 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. pi and -pi.

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.(pi) i is an example of a positive imaginary transcendental number; -(pi)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 pi¹ and pi ^-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 1^pi and 1^-pi. 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

pi(i) and -(pi)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 1^(pi)i and 1^-(pi)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 the square root of 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.