In my "Transforming Voyage", I outlined the development of the various stages of the point level. In the light of what has been said, I will now attempt to translate the various stages involved in psycho-mathematical terms.

The transition from circular to point level is characterised on the one hand by a transcendent form of spiritual union marking the death of directly conscious phenomenal understanding. On the other hand there is the start of a confusing process by which the unconscious starts to project itself outwards in (indirect) phenomenal form.

As I have already stated the fundamental structure of this period is given by the three roots of unity. The real root ("1’’) complements the attainment of psycho-spiritual union. The other two roots - which are complex - complement the confused form of phenomenal experience that now unfolds. Projections are in part interpreted in "real" terms. Indeed this is a very necessary phase enabling one to regain a foothold once more in conventional linear reality. However projections are also interpreted in their inherent "imaginary" form as fantasies or archetypal expressions of experience which is fundamentally unconscious. This leads in turn to a continual attempt to appropriate internally what initially has been projected externally. In other words "imaginary" phenomena are subject to a high level of negation. This again is complemented mathematically by the existence of two complex roots, one with a positive and the other with a negative imaginary sign. This also explains why phenomenal experience tends to be so transient and unstable during this time. Experience, is considerably under the influence of "imaginary" fantasy and in continual need of negation (i.e. reappropriation by the unconscious).

The mathematical roots of unity are particularly valuable in portraying the key structural features of the transition stages between each of the major levels.

The transition to the linear level or the 1st dimension, is characterised by the emergence of the ability to posit phenomena in conscious form (i.e. giving phenomena a positive unitary existence). This corresponds in psychological terms to what is the 1st root of unity (in mathematical terms).

The transition to the circular level or 2nd dimensions is characterised by the emergence of mirror (i.e. negative) structures complementing the already developed conventional (i.e. positive) structures. Each phenomenal unit of experience is understood now as dynamic and relative, having - expressed in reduced static terms - both a positive and negative direction. This corresponds to the two roots of unity (i.e. +1 or -1) mathematically.

The transition to the point level or 3rd dimension is characterised by the arrival at "real" spiritual union (i.e. through transcendence of "real" conscious phenomena) and - as we have just seen - the emergence of a confused form of "complex" experience, with the projected fantasy or "imaginary" element especially subject to negation, through mirror structure activity. This is complemented by the three roots of unity i.e. +1, (-1 + Ö -3)/2,

(-1 - Ö -3)/2.

We will briefly move forward to the transition to the radial level or 4th dimension. During the point level, there always remains some degree of confusion regarding "real" conscious and "imaginary" unconscious phenomena.

Thus the key task of the point level is to separate and fully differentiate both "real" and "imaginary" experience. With the transition to the radial level this task has largely been achieved.

"Real" conscious structures are fully balanced by their mirror structure counterparts. We have positive and negative "real" structures. "Imaginary" unconscious structures (projected in phenomenal form) are also fully balanced by their mirror structure counterparts. One can now fully appropriate such projections seeing them as expressions of one’s own unconscious. Thus we also have positive and negative "imaginary" structures.

So this complete differentiation of mode and direction in relation to structures during this transition is a prelude to their mature integration As this takes place the void slowly becomes a plenum-void. This situation psychologically, is replicated in mathematical terms by the highly important four roots of unity (+1, -1, Ö 1and Ö -1).

We are now returning to the point level.

During the transition to this level, there is this basic confusion, whereby one is not able to properly differentiate conscious and unconscious phenomena.

There now starts a process whereby the two modes of the personality separate. At the "high level" one attempts to translate reality in solely "real" terms (without interference from the "imaginary" unconscious).

At the "low level" one attempts to translate reality in solely "imaginary" terms (without interference from the "real" conscious). This temporary - though prolonged - separation of the two modes is a necessary step so as to enable their full differentiation.

At the "real" (superior) level of personality we now have a highly subtle new form of cognitive translation leading to the emergence of the transcendental structures (i.e. the advanced superstructures).

There is a dynamic process of evolution involved linking them with the past irrational structures of the circular level and the future transfinite structures of the radial level. There is a firstly a clear understanding of the structures of the previous circular level. This is followed by an emerging understanding of the structures of the present point level (i.e. transcendental). This is followed by full understanding of the transcendental structures with finally the erosion of these structures as one prepares for the next level.

The rational structures of the linear level are characterised by the discriminating consciousness and the clear separation of opposites.

The irrational structures of the circular level are characterised by holistic intuition and the inherent complementarity of opposites.

The (now emerging) transcendental structures of the point level are characterised by the understanding of reality as being inherently the relationship between the formal linear and (inherently) formless circular levels.

However to attempt this translation, the circular level itself must be reduced to rational terms. Thus though the circular level is inherently non dimensional, intuitive and formless, in reduced terms it is now two dimensional and formal. Thus the transcendental level - though highly advanced and subtle does involve a degree of confusion whereby intuition is interpreted in formal terms.

I will try and describe the nature of transcendental structures in slightly more concrete terms.

As an economist I have long been deeply aware of the unduly linear and fragmented approach of the economics' profession to economic reality. In the attempt to make the scientific model (of Newtonian science) fit reality, economists frequently resort to highly reductionist and inappropriate assumptions by which the inherently dynamic nature of economic activity is lost. Instead of economics being treated as a living process it is portrayed in static mechanical terms. Indeed in this interpretation, the cornerstone of the "free" market economy is "The Price Mechanism". This is a totally distorted and insidious notion of freedom largely divorced from any ethical implications..

Whereas the (rational) linear scientific approach is appropriate for examination and analysis of specific problems (in the context of an overall coherent framework), it cannot of itself provide such a framework.

The (irrational) circular holistic approach is far more suited for providing this framework and the overall understanding and evolution of the economic system. It provides a means of intuitively interpreting the dynamic consequences of change, of appreciating the relationship between economic activity and the environment, of understanding the interaction of subjective wants with material progress and interpreting key imbalances in the system such as the uneven distribution of wealth in society.

Though many people are concerned with such issues, by and large the conventional methods of economists exclude such considerations.

The (transcendental) approach of the point level involves a more subtle approach where one sees the relationship as between the analytical linear and synthetic circular approaches as being of paramount significance.

The intuitive circular approach is still understood as vital in terms of preserving an integrated economic perspective over time. However it is not geared to the study of specific problems at a particular point in time. Here, the (rational) linear approach is more appropriate. Now the balance between these is crucial. Without the circular intuitive approach, the finely tuned analytical methods of economists are so often used without any coherent context. An impersonal view is taken of the environment. Major economic issues are either ignored or treated in a reduced and distorted manner. Also economic models - because of lack of dynamic considerations - are frequently outdated.

Of course, the circular approach in itself is equally unbalanced. Too much concern with the overall situation can lead to a vague and passive approach and an inability to address concrete issues.

So it is not now a question of considering the circular level as either superior or inferior to the linear. Rather, both are now considered equal, with the balance between them the key consideration.

Just as we can have positive and negative transcendental numbers (e.g. p and -p ), we also can have positive and negative transcendental structures. In "Transforming Voyage" I referred to both as advanced superstructures.

The positive structures refer to that holistic philosophical view which interprets reality in terms of the integration of both the (linear) rational and (circular) holographic paradigms. The trouble in practice is - because it is based on extreme specialisation of the intuitively cognitive mode - one thereby greatly loses the ability to pick up concrete sense information from the environment. Thus the goal of reconciling the two paradigms is throughout this time, a potential ideal rather than an actual practice.

Mirror structure activity in relation to these structures enables the emergence of the corresponding (internalised) "negative" transcendental structures.

These relate to highly intimate and personalised decision making.

At the linear level one characteristically "objectifies" morality, seeing decisions in polarised terms therefore as (objectively) right or wrong.

At the circular level one sees morality in terms of the complementarity of opposites, where each decision involves a dynamic interaction involving the (personal) subject and (impersonal) object. This involves relativity and an authentic existential approach where the morality of an act derives from the genuine exercise of free choice in the face of uncertainty. Carried to extremes during the circular level - where all conventional supports are removed - morality becomes indistinguishable from the pure exercise of faith in all decisions.

At this stage of the point level, a morality is seen in more subtle terms as the relationship between the actual (linear) decision and the (circular) exercise of faith. Thus in this refined context a specific act can be right or wrong (in a relative sense). Far from this implying that anything goes, it demands a far more sensitive exercise of conscience, involving a unique act of freedom in taking a decision on each occasion.

Once again, however - because of erosion of conscious sense phenomena, this remains a (potential) ideal rather than an (actual) practice. In other words actual decision making is still subject considerably to unconscious projections.

SUBSTRUCTURES
 
 

These relate to the affective "low level" structures which emerge during the point level. Whereas the complementary "high level" development is concerned with the differentiation of "real" structures, "low level" development is concerned by contrast with the differentiation of "imaginary" structures.

At the linear level, just as a clear dichotomy exists as between the two directions of experience, a similar dichotomy exists also as between the two modes of experience.

Thus there is at the linear level a clear divide as between the "imaginary" (i.e. fantasy) and the "real" (reality). The very word "reality" that we use is strong testimony to this bias.

At the circular level, one attempts to end this divide through incorporating the "imaginary" within a higher level intuitive reality.

At the point level, both modes are restored so that they are seen as interpenetrating each other.

The "real" that excludes the "imaginary" - as at the linear level - is termed rational. The "real" that includes the "imaginary" - as at the circular level - is termed irrational.

In like terms, the "imaginary" that excludes the "real" - again as at the linear level - can be termed rational. In other words it is rational to interpret fantasy as unreal at this level (i.e. does not fit in with a scientific worldview). The "imaginary" that includes the "real" is therefore interpreted as irrational.

In other words the interpretation of the "real" world as in part an "imaginary" projection of one’s unconscious is irrational from the scientific perspective.

However, as I have already detailed this is precisely what happens with the transition from the two dimensional circular to the three dimensional point level. We have a mathematical verification of this in the complementary generation of irrational imaginary quantities when we obtain the cube root of unity. The complex roots are (-1 + Ö 3i)/2 and (-1 - Ö 3i)/2 (where i =Ö -1). Thus the imaginary part of these complex numbers is an irrational imaginary number. Also, we have both the positive and negative directions. Psychologically, we have both positive and negative irrational imaginary structures. The positive direction involves the projection of the unconscious in phenomenal form. The negative direction involves the internal appropriation of these projections.

As the point level unfolds new substructures develop. Just as the corresponding superstructures involve the emergence of transcendental structures which are "real", the substructures involve the complementary emergence of transcendental structures which are "imaginary".

There is now a more subtle form of fantasy involved, whereby the irrational projections of the unconscious are maintained in a very delicate balance with the rational mind. One attempts to interpret the relationship between the "imaginary" (fantasies) and "real" (rational mind) in imaginary form. This means that one projects phenomena freely from the unconscious (i.e. fantasies) while implicitly relying on rational control to keep the process in check. However due to over anxiety on the affective side, the process remains to some extent involuntary until the radial level. The overemphasis on the "real" at the superior level leads to a complementary overemphasis on the "imaginary" at the inferior level of personality.

Again the positive direction refers to the phenomenal projection of such fantasy. The negative direction refers to the internal appropriation of such phenomena.