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In my earlier "Transforming Voyage", I outlined the manner in which the various structures of personality, conscious and unconscious emerge at different stages of development.

I concluded there, that the most complete level of development involves the integration of the fully differentiated conscious (rational) and unconscious (intuitive) processes.

Though reason and intuition are seen at lower levels as separate - and largely unrelated - processes of understanding, at higher levels they become complementary and indeed ultimately identical.

Thus for example, at lower levels, psychology and mathematics are treated as very separate areas. Psychology, is understood as relating to subjective (internal), and mathematics to objective (external) reality. Whereas intuition - especially in terms of transpersonal psychology - may be seen as pertaining to the former, reason is seen as appropriate to the latter.

However, at the highest level, these boundaries break down altogether, and indeed, the closest relationships exist between those areas, which previously seemed most separate. Thus, at this level, psychology and mathematics, far from being unrelated, are understood as being strongly complementary.

I have referred before to this dynamic type of understanding, which explicitly relates both reason and intuition as transrational. Indeed the static (merely) rational approach, - as in conventional mathematics - where intuition is not formally recognised is therefore a considerably reduced form of understanding and in truth greatly distorted.

The purpose of this book is to adopt a transrational approach to the
various psychological structures of understanding. I am involved, therefore
in what might be termed psycho-mathematics, demonstrating how fundamental
psychological processes of development display remarkable (complementary)
similarities to basic mathematical relationships.

In particular, I am concentrating on the concept of "number".

I agree with Jung - who showed deep insight into its psychological significance - that number is the most important archetype we have for order.

Now, this is obvious objectively in terms of numerical quantities. More surprisingly, it is equally true of the psychological realm.

Here, number, again is the best archetype, however, this time in a qualitative sense.

In dynamic terms, therefore, numbers - like light - have dual complementary aspects. They can be understood as mathematical quantities. This is the objective rational approach, which is overwhelmingly the conventional interpretation.

Equally, they can be understood as psychological qualities. This is
the subjective and intuitive approach, which is rarely coherently articulated.

When in "Transforming Voyage" I was outlining the various stages and sub-stages of personality development (at each level of development), I found myself searching for a satisfactory way of classifying the many psychological structures involved. It slowly dawned on me that the number system provided the ideal vehicle for doing this. Initially, I was intrigued by the psychological undertones implicit in the very terms used for these numbers. Irrational, imaginary, transcendental, transfinite are all terms, strongly suggesting boundaries extending beyond the merely rational level. The more I examined the connections, the more amazed, I was at the underlying similarities as between various number types and psychological structures.

I will state my thesis briefly here. There are many number types, and
each number type can be clearly matched - in complementary dynamic terms
- with a corresponding basic psychological counterpart.

For example, we have positive and negative numbers. These can be identified with the two complementary directions of understanding i.e. external (objective) and internal (subjective) respectively.

Then we have real and imaginary numbers. These can be directly identified with the two complementary modes of understanding i.e. cognitive and affective, respectively.

We also have rational and irrational (algebraic) numbers. These again can be identified with the two levels of understanding i.e. linear and circular.

Within the rational set we have prime numbers, natural numbers, the real integers, fractions etc. Again, there are close matching parallels on the psychological side in terms of the stages of linear development.

Even in terms of more abstruse number types, such as transcendental
numbers and transfinite numbers, we can identify precise complementary
psychological structures. For each mathematical number type or structure,
interpreted in a rational horizontal manner, there is an exactly matching
psychological number type or structure, this time interpreted in an intuitive
vertical fashion.

Indeed, this whole exercise can be highly illuminating from both perspectives.
In some cases, starting with psychology, it is possible to establish fascinating
mathematical connections as between numbers and physical processes. In
other cases, starting from mathematics, it is equally possible to discover
psychological connections as between number types and see more clearly
into the fundamental nature of consciousness. So this transrational approach
is - at the very least - a very creative approach.

I have called this book "The Number Paradigms"". Each major psychological structure can be closely matched with a corresponding number type, which in turn leads to a fundamental world-view or paradigm.

Now, there are a large number of such number qualities - or paradigms - possible, all giving valid - though partial - information about reality.

When we reflect on it, most understanding, being identified with a particular paradigm, is inevitably highly limited in scope.

If we take the conventional paradigm for science and mathematics, we can identify it - in terms of our number types - as a real positive rational approach. It is real, in that it reduces the mode of experience to the cognitive. It is positive in that it reduces the direction of experience to the external. It is rational in that it reduces the level of experience to the merely linear.

Now, just as the real positive rational number system only represents one example of many possible number types, likewise the real positive rational paradigm only represents one example of the set of possible paradigms. It is highly ironic therefore in mathematics (and in science generally) , that though many different number types as quantities are recognised, that only one number type as quality (i.e. paradigm) is utilised. The development of science - considerable though it has been - has taken place almost entirely, horizontally, within the confines of a single philosophical approach (i.e. the positive real rational paradigm).

There is enormous scope for vertical expansion in mathematics - as in other fields - where understanding takes place within differing paradigms.

The use of an alternative (number) paradigm then would clearly alter the nature of understanding obtained.

For example, the positive real rational paradigm involves the cognitive objective approach of the linear level based on the separation of opposites. Numbers for example are understood in absolute terms. The positive real rational paradigm therefore treats numbers as static entities, consciously understood, independent of the mind.

By contrast a positive irrational approach to mathematics - though again
involving a cognitive objective approach - would this time operate at the
circular level of complementary opposites. Objective phenomena such as
numbers would now be understood in relative terms. Thus, the positive real
irrational paradigm would treat numbers as dynamic entities involving the
interdependence of both an objective (external) aspect and a subjective
(internal) aspect (i.e. the act of understanding) .

However, my main task in this book, is - starting from my outline of personality structures - in "Transforming Voyage" - to show how these various structures are complementary to basic number processes. This ordering in turn will provide a very valuable map for psychological development and a clear basis for further discussion on the precise nature of the different paradigms involved.