Levels of Development
 
 

Once more it is quite remarkable how the progressive roots of unity provide a strikingly accurate representation of the fundamental structure of the higher stages of development.
 
 

The Linear Level
 

The first root of unity yields one value (+1). No indirect translation from vertical to horizontal terms is required here. This yields the one way direction of the linear level, which recognises the validity solely of what is directly posited in consciousness.

This understanding evolves through many stages.

Firstly we have the original understanding of unity and nothingness based on the primary binary distinction of 0 and 1.

We then have prime understanding which represents a confused understanding of the phenomena and dimensions of experience.

Then we have natural understanding of the physical world representing an increased ability to combine phenomena within dimensions.

This leads on through the centre of personality - in the opposite direction - to corresponding awareness of self giving integral understanding (natural understanding in positive and negative directions).

With growing consciousness there is an increased analytical ability in creating parts out of the whole (in literally creating fractions).

There is firstly the concrete rational structures (in relation to the world)

Then there are the concrete rational structures (in relation to self)

Later there are the formal rational structures (in relation to the world).

Finally there are the formal rational structures (in relation to the self).

Though clearly understanding of the self and world takes place during this time, considerable confusion remains, with both directions being confused. In other words knowledge of the world and self is still largely conscious, and merely posited in an external fashion.

This is the psychic matter of rational understanding.
 
 
 

Transition from Linear to Circular Level
 

The two roots of unity yield two values +1 and -1. This second root yields the negative (i.e. linear mirror structures) which represents the transition from linear to circular levels. Quite literally, in what involves a profound existential crisis, during this time (linear) consciousness is considerably negated in experience.

There now develops the negative rational structures both at the concrete rational and formal rational levels of experience.

This is the psychic anti-matter of rational understanding.
 
 
 

The Circular Level
 

Combining these two directions of experience yields a dynamic relative form of understanding based on the inherent complementarity of the positive and negative poles of understanding. In terms of previous linear understanding, this appears highly paradoxical and indeed irrational.

We have the concrete irrational structures in a positive direction (the supersensory structures).

We then have the concrete irrational structures in a negative direction (the mirror supersensory structures).

Later we have the formal irrational structures in a positive direction (the suprarational structures).

Finally we have the formal irrational structures in a negative direction (the suprarational mirror structures).

Though the directions of experience are largely differentiated during this time, there remains considerable confusion of modes so that affective and cognitive experience are not yet properly separated.
 
 
 

Transition from Circular to Point Level
 

Obtaining further roots of unity yields imaginary as well as real solutions. The three roots of unity are 1, ( -1 + Ö-3)/2 and( -1 - Ö-3)/2. The first root corresponds to the transcendent form of spiritual union achieved at this time. The other two roots refer to the confused form of "complex" activity that also emerges. There is much projection from the unconscious in the form of virtual or "imaginary" projections which initially are initially understood in misleading "real" terms.

We now have the differentiation of "imaginary" as well as "real" understanding.
 
 
 

The Point Level
 

This involves the attempt to reconcile the two previous levels of the linear (based on the static separation of conscious opposites) and the circular (based on the dynamic complementarity of conscious opposites). We have here in psychological terms the relationship between line and circle. p in mathematics represents this relationship and represents the most famous example of what are referred to as transcendental numbers. All transcendental numbers in turn involve this inherent relationship of horizontal (linear) and vertical (circular) notions.

In like manner the transcendental structures represent this inherent relationship as between the rational (analytical) and holistic (intuitive) paradigms.

Just as the circular structures develop in two directions (positive and negative), the transcendental structures develop in two modes (real and imaginary).

The transcendental structures that develop in a real mode, I have referred to as (advanced) superstructures.

The transcendental structures that develop in an imaginary mode, I have referred to as substructures. There is considerable interpenetration of both types of structure in terms of direction (positive and negative) during this time.

We can also identify a more concrete stage of real transcendental development. This is the early point level, where understanding of reality is largely psycho-physical (i.e. where physical and psychological reality are understood as complementary).

There is later a more formal stage of real transcendental development, where now understanding of reality is largely psycho-mathematical (i.e. where mathematical and psychological reality are understood as complementary). This latter stage involves a very high level of both rational and intuitive specialisation.

The substructures relating to "virtual" reality operate in reverse fashion. Thus the earlier stage of imaginary transcendental development is more formal involving unconscious fantasies still subject to censorship by the "higher" self. Repression arising from undue rational control still exists at this time.

The later substructures by contrast are more concrete and immediate involving very little interference by the rational self.
 
 
 

Transition from Point to Radial Level
 

With the removal of consciousness, experience from a phenomenal perspective becomes progressively more transient and "imaginary".

The culmination of the point level involves the progressive harmonisation of both "real" horizontal opposites (i.e. internal and external experience) and "imaginary" vertical opposites (i.e. the "high-level" and "low-level" aspects of personality).

This is beautifully represented by the four roots of unity +1, -1, +Ö-1and -Ö-1 respectively. In other words with the culmination of the point level there is the full differentiation of both real (voluntary) consciousness and imaginary (involuntary) consciousness in both positive and negative directions.
 
 

There then follows a prolonged void - in terms of phenomenal consciousness - which represents the birth of a new kind of structure. In other words one no longer tries to reduce spiritual experience through indirect reductionist translations, however subtle. This represents the pure differentiation of process, where finite or conscious reality is no longer confused with unconscious or infinite reality.

Subsequently however it is possible to give this experience a mathematical interpretation.

The four poles of experience now have a transfinite as well as a finite interpretation. This structure is remarkably provided by obtaining the eight roots of unity. As well as the previous four roots +1, -1, +Ö-1and -Ö-1, we obtain four additional roots 1/Ö 2 (1 + i),

1/Ö2 (1 - i), 1/Ö2 (-1 + i) and 1/Ö2 (-1 - i) where i =Ö-1. The significance of these four roots is that the real and imaginary components are equal. In complementary psychological terms real and imaginary understanding is harmonised at this time.

These additional structures can be accurately represented by the two diagonal lines representing radii of the circle drawn through the centre equidistant from both real x and imaginary y axes. Since at each end of these two diagonals x and y coordinates are equal, using the Pythagorean Theorem, the length of diagonal in each of the four sectors of the circle is zero. 1² + i² = 0. These diagonal lines represent null lines with no finite value. They are in fact intimately related to rays of light and have a transfinite rather than a finite meaning.

We have seen before that an imaginary number defies the logic of conventional mathematics, where both directions are simultaneously equal. Strictly speaking therefore Ö-1 is both +1 and -1.

In similar fashion a transfinite number defies the logic of conventional mathematics, where both modes are now simultaneously equal. Strictly speaking a transfinite number

is one with both real and imaginary coordinates equal. In two dimensional terms such a number however has no finite meaning.

In exactly similar complementary terms, the transfinite "structures" - which represent pure psychic energy or intuition - represent the complete absence of structure when interpreted in correct unconscious (i.e. two dimensional) terms. In direct terms pure intuition is spiritual and nothing in conscious phenomenal terms.

This represents the pure direct activity of will. However it can be given a satisfactory reduced and indirect (one-dimensional) conscious interpretation as representing the complete harmonisation of both real (cognitive) and imaginary affective) modes.
 
 
 

The Radial Level
 

This represents the most complete level of development in the attempt to integrate both transfinite and finite processes.

Though transfinite experience is purely simple, when integrated with finite experience it displays four aspects. Thus we can have the (holistic) contemplation of transcendence with cognitive and affective aspects. Likewise we can have the (specific) contemplation of immanence with cognitive and affective aspects.

The related finite structures are - in mathematical terms - complex, with subjective conscious experience - in relation to objective - negative, and affective conscious experience - in relation to cognitive - imaginary.
 
 

During the radial level, experience has therefore related transfinite and finite aspects.

One attempts to integrate both aspects until they are indivisible. Though never totally possible, in peak moments of experience, this will be momentarily achieved.

The earlier radial level can best be represented by the diagonal line (sloping up from left to right). On one end of this diagonal both real and imaginary signs are positive. At the other end both real and imaginary signs are negative. This signifies that at a holistic spiritual level, experience is transcendent in quality. At the specific physical level experience is immanent in quality. Some level of split is still apparent so that one cannot yet properly integrate both aspects. Daily material activities do not yet fully reflect one’s deepest spiritual convictions.

The later stage can be best represented by the other (radial) diagonal sloping down from left to right. Here at both ends of the diagonal, real and imaginary signs are different and complement each other.

This in turn represents the fact that far greater interpenetration of transcendent and immanent experience is now possible. This means in effect that spiritual contemplation is progressively integrated with dynamic secular involvement in the highest expression of active involvement.