This is based - in terms of developed consciousness - on the fundamental complementarity of the conscious (rational) and unconscious (intuitive) processes.

When understood in such dynamic transrational terms, (rational) mathematics and (intuitive) psychology are themselves complementary.

We have therefore what can be called a psycho-mathematical understanding of both areas. When suitably formulated, seen from one direction, every (horizontal) mathematical relationship has a (vertical) psychological counterpart which is complementary. Seen, alternatively from the other direction - which I have been chiefly dealing with - every (vertical) psychological relationship has a (horizontal) mathematical counterpart.

Thus as well as helping to integrate two areas of experience - which are often understood as diametrically opposed - the transrational approach can be extremely valuable in terms of generating creative insight in relation to both mathematical and psychological issues.
 
 

I am chiefly concerned in this book with showing how psychological development, with all its different structures can be remarkably modelled through close identification with the mathematical number system. Also. this very process of identifying various structures with corresponding number types, helps greatly to clarify the fundamental nature of these structures.

In reverse terms, the precise psychological clarification of structures greatly facilitates understanding of the nature of the corresponding number types (e.g. primitive structures and prime numbers), which can even be directly helpful in tackling important mathematical problems relating to these numbers. As the complementary relationships between both areas become more closely established, far more significant connections, than I have been able to demonstrate, will inevitably suggest themselves.
 
 

Thus to summarise so far, the two processes of understanding conscious and unconscious, which underline all understanding, can be directly associated with the fundamental numbers (i.e. 1 and 0 respectively).
 
 

The first level of understanding (i.e. the linear) culminating in rational understanding is concerned with the specialisation of the (real) conscious structures.

In complementary fashion, the structures of this level can be identified with the (real) rational numbers.
 
 

The first two stages of psychological development (i.e. the primitive structures) can be related directly to the prime numbers. This identification is particularly useful in terms of clarifying the precise nature in turn both of prime numbers and primitive structures.

The next two psychological stages (i.e. the composite) have very close parallels with the integers. We have firstly the positive integer set (i.e. natural noose). and then the full set (including 0 and the negative integers).
 
 

The two concrete stages relate to the concrete rational numbers (positive and then negative).

Finally the two formal stages relate to the formal rational numbers (positive and negative).
 
 

The combined stages would then relate to the integration of all number types so far, within the real rational system.
 
 

Now, though both psychological and mathematical understanding is inherently dynamic, at the linear level a gross level of reductionism is involved, whereby both are interpreted in distorted static terms.

Thus in terms of the processes of understanding, which relate to finite and infinite realms, the unconscious (infinite) is reduced to the conscious (finite) process.

In terms of the modes of understanding, which relate to quantitative and qualitative realms, the affective (qualitative) is reduced to the cognitive (quantitative) mode.

In terms of the directions of understanding, which relate to positive and negative realm, the internal (negative) is reduced to the external (positive) direction.
 
 

Thus the prevailing mathematical paradigm could be aptly described as a positive real rational approach. It is positive, seeing its direction as objective (external) rather than subjective (internal). It is real in that it sees the mode utilised as cognitive rather than affective. It sees itself as rational in that it deals with the conscious linear rather than other more unconscious intuitive levels.

Thus, though the understanding of all mathematical relationships implicitly employs complementary poles in terms of process, mode and direction of understanding, in explicit conventional terms, they are ignored. Therefore, relationships which are inherently dynamic and relative, are presented from a highly reduced (and thereby distorted) static interpretation.

Now, in psychological terms, there are other "higher" levels of understanding (i.e. the circular, point and radial), that go beyond the limitations of the linear level. Also we have in mathematical terms, other number types that go beyond the limitations of the rational (i.e. irrational, imaginary and transfinite).

However, Western psychology rarely recognises these other levels. Also - even though these other number types are now accepted as valid members of the system - they are invariably understood in reduced linear terms..

What I will attempt to show in later chapters is that these "higher" number types - as at the linear level - are fully complementary with corresponding "higher" psychological structures. Again this will help greatly to clarify both the true nature of these numbers and their corresponding psychological structures.