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

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

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

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

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

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

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

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

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

The linear and circular levels are therefore themselves complementary.

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

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

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

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

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

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

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

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

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

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

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