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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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