Real Irrational Structures - Introduction
 

Some 2,500 years ago the Pythagoreans were shocked by the discovery of irrational number quantities such as 2. In interpreting reality they were employing what was inherently an integrated mathematical approach in the belief that the (quantitative) mathematical nature of reality complemented a (qualitative) paradigm used to interpret this reality. Initially all seemed well. The rational paradigm was matched by a mathematical world where only rational quantities were known to exist.

The discovery of 2 and irrational quantities however destroyed this belief for there was now no matching (qualitative) paradigm with which to interpret this strange new world of quantities.

This ended dramatically the quest for an integrated mathematical approach.

Subsequent development in Western mathematics has taken place in a reduced fashion with the quantitative becoming increasingly divorced from the qualitative. Irrational, imaginary and even transfinite quantities are interpreted within the confines of an ever more specialised and limited rational paradigm.

The qualitative and "vague" intuitive dimension - if recognised at all - is seen as the province of mystical religion and transpersonal psychology and nothing to do with real mathematics. However all the esoteric paradigms of transpersonal psychology can be precisely translated in dynamic holistic mathematical terms offering an entirely new vision of reality.

Let us recap development so far. The linear level (gross realm) culminates in the specialised development of positive rational structures.

The transition from linear to circular level then involves the development of complementary negative rational structures.

The key feature of the circular level (subtle realm) is that understanding is now seen in relative terms as the dynamic interaction of both positive and negative directions of experience. When these directions are fully fused we have an expression of the developed unconscious (i.e. pure intuition). So from this perspective the circular level involves the development of (unconscious) intuitive rather than (conscious) rational structures.

However pure intuition cannot be sustained for long so that one attempts to translate this experience in reduced rational form.

The very basis of conventional reason is that opposite poles are clearly separated. If a proposition is true for example, then by this logic it cannot be false.

However when we express spiritual intuition in rational terms it appears deeply paradoxical. Here opposite poles are complementary. Thus if a proposition is true, then by this logic it must also be false. Putting it less dramatically it now enjoys a relative - rather than absolute - truth value.

Indeed by the logic of the conventional reason (of the linear level), the paradoxical reason of the circular level is irrational. This is because we are expressing an order of truth which is inherently two dimensional (involving complementary opposite poles) in reduced one dimensional terms (where these poles are clearly separated).

Now this problem is exactly replicated in mathematics with irrational numbers such as 2^1/2. If we express this number in the second dimension (by squaring it) it now is rational i.e. if x = 2^1/2 then x² = 2.

However if we try to reduce this number subsequently to the first dimension (by obtaining its square root) it becomes irrational. Furthermore the result has two complementary poles + 2^1/2 and - 2^1/2.

An irrational number has both finite and infinite aspects. It is finite in that its value can be approximated (i.e. reduced) to a discrete value. Thus correct to four decimal places 2^1/2 = 1.4142. However it is infinite in it's full decimal sequence which continues indefinitely. An irrational number is thus inherently dynamic and relative with discrete (finite) and continuous (infinite) aspects.

In psychological terms irrational experience is similar where phenomena have both finite and infinite aspects. Thus partial phenomena arise in experience which are consciously understood. This is the finite rational aspect. However they are now invested with a transcendent light whereby they are seen as archetypes of an invisible whole. This is the infinite intuitive aspect. The combination of both aspects gives all phenomena a dynamic relative existence.

There is yet another remarkable connection as between these (quantitative) mathematical and (qualitative) psychological irrational notions.

In mathematics if we raise any rational number to a fractional power we get an irrational number. (I am ignoring unreduced trivial cases such as 4^1/2 = 2, which is a rational number for 4 itself can be expressed as 2² so that the right hand reduces to 2¹).

Now we have seen that in the later formal rational stages of the linear level, that conceptual ability - correctly interpreted - literally involves experiencing reality in fractional dimensions.

Thus when one tries to subsequently relate concrete (horizontal) perceptions to formal (vertical) concepts (i.e. interpret rational objects in the context of fractional dimensions) a decisive transformation takes place by which the experience becomes irrational). Thus in dynamic psychological terms we have here a surprising complementary result to mathematical behaviour. Indeed it is this very psychological dynamic that leads to genuine transpersonal development. Because of the more intuitive nature of experience, the transpersonal candidate does not successfully reduce concepts to merely rational format. This leads to an inevitable conflict as between the horizontal and vertical levels of experience.

Whereas the linear level - especially in the later rational stages - is geared to the development of merely positive structures (i.e. conscious), the circular level relates to the progressive development of structures in both their positive and negative directions (involving the explicit development of both conscious and unconscious).

In other words we have here the development of both reason and intuition in what might be termed superstructures.

The circular level is constantly changing. As the quality of intuition increases, the use of reason becomes ever more subtle and refined.

We can in - as at the rational level - identify both concrete and formal stages of irrational development. However there is a marked cyclical quality to these stages.

The positive stages are characterised by illumination. This involves the outpouring of spiritual intuition with subsequent translation through subtle rational structures. The negative stages in stark contrast are characterised by purgation with complementary inpouring of intuition (in darkness) and substantial erosion of the subtle structures formerly developed.

Spiritual development during the circular level is transcendent in quality.

This is due to the fact that the unconscious - though considerably activated - remains under the subtle control of the cognitive function (i.e. reason). Thus unwittingly considerable emotional repression can take place. In dynamic mathematical terms we are still dealing here with "real" rather than "imaginary" structures.