Introduction
 

In all that follows, it is important to remember that numbers have dual aspects which are complementary.

Numbers have - in direct terms - a horizontal or rational interpretation as mathematical quantities. This is the manner in which numbers are conventionally understood.

Equally, numbers have a vertical or intuitive interpretation - in direct terms - as psychological qualities. In other words, each number type represents a distinctive manner of viewing reality i.e. a paradigm. Though this latter understanding was very important in earlier times e.g. among the Pythagoreans, it was subsequently ignored, with the result that mathematics and esoteric psychology have developed as almost entirely separate disciplines.

Mathematics, as conventionally understood, represents the specialised use of reason and in formal terms is totally divorced from intuition.

Esoteric or transpersonal psychology, in turn represents the specialised use of intuition, often separated from rational formal means of expression.

Not surprisingly these two ways of describing reality often appear as opposite poles with little in common.

Remarkably however, when properly understood they are totally complementary.

Whereas each number has a direct rational (horizontal) interpretation as a mathematical quantity, it has an direct intuitive (vertical) interpretation as a psychological quality or dimension. This provides a ready means of linking mathematics with (transpersonal) psychology.

Equally, whereas each number type has a direct intuitive (vertical) interpretation as a psychological quality, it has an indirect rational (horizontal) interpretation as a mathematical quantity. This in turn, provides a ready means of linking (transpersonal) psychology with mathematics.

Numbers can be understood as mathematical quantities (within a given dimension of "1"). Ultimately all numbers are expressed conventionally within the 1st dimension. Thus 2 for example implicitly means 2¹ .

Numbers alternatively can be understood as the dimensions (to which the given number "1" is raised). Thus 2 in this case would implicitly mean 1² .

Now the significance of this second definition is - in direct terms - qualitative rather than quantitative. However conventionally in mathematics, a reduced indirect (rational) interpretation alone remains. Thus the vital significance of this vertical system has been lost. The task of this book is to attempt to restore a truth that is simple, yet quite astonishing in terms of its potential implications.
 
 
 

Binary Understanding
 
 

I have stated before that the two fundamental numbers one and zero, are linked directly to the conscious and unconscious processes respectively.

Even the very symbols used to represent these two numbers are remarkably apt.

The conscious process is connected with linear understanding, whereas the unconscious process is connected with circular understanding.

Now - with only minor modifications - the straight line is used to represent one (i.e. 1), whereas the circle is used to represent zero (i.e. 0).

It is interesting to note that the Greeks - who are so much responsible for laying the foundations of our rational (linear) culture - never made explicit use of 0 in their number system. Apparently, its mathematical use has Babylonian origins, which again undoubtedly is related to their more intuitive (circular) culture.

Now in static terms, 1 and 0 are clearly separated, just as existence and non-existence are seen as being mutually exclusive. Unity is what defines existence (i.e. something that exists is a definable unit), whereas zero defines non-existence (i.e. something that does not exist is zero or nothing).

However in dynamic terms, 1 and 0 are clearly related as complementary opposites. Here, existence and non-existence are, as it were, both sides of the same coin.

This can perhaps be appreciated by seeing nothing (0) as involving both the positing and negating of something (1). Even in simple mathematical terms, (+)1 - 1 = 0. Thus the existence of nothing (dynamically), presupposes the existence of something already existing (statically).

Likewise, in reverse fashion, for something to exist statically, we require the existence of nothing dynamically. 0 = (+)1 - 1. The two poles of phenomenal existence - as is so well expressed in Taoism - emerge form what - in dynamic (phenomenal) terms - is nothing.

Non-existence is thus from a dynamic perspective the potential for existence and the indispensable ground for all actual existence. Indeed, this idea has now entered modern physics through the study of the behaviour of elementary sub-atomic particles. There is even speculation that reality is ultimately "a free lunch" with all matter emerging from a quantum vacuum, or underlying ground of nothingness.
 

Conventional mathematics operates with symbols involving a static and thereby reduced point of view. What now needs to be provided is a means by which a more dynamic approach can be adopted which opens up amazing possibilities in terms of creative insight both inside and outside mathematics.

I am proposing - like the treatment of light in physics - a dual understanding of number, with both aspects fully complementary.

Theoretically and practically, numbers in mathematics are solely treated as quantities. The key point is that all numbers equally have a complementary interpretation as qualities. Let me now elaborate further on this crucial point.
 

When a number is given, there is always a concealed qualitative aspect. This is the dimensional characteristic or power of the number. This mirrors normal perception of objects which can only be understood in the context of dimensions (i.e. space and time). The understanding of objects is therefore not absolute but rather relative depending on experience of space and time.

Conventionally, this important problem of relativity is avoided in mathematics by ultimately reducing all numbers to the same dimension (i.e. the 1st dimension).

If, for example I speak of the number 2, it is tacitly assumed, that this number is defined with relation to the 1st dimension (i.e. 2¹). Thus when quantitative operations are carried out on numbers, they will therefore be reduced to the 1st dimension.

Even when numbers are initially presented in other dimensions, they will be ultimately reduced to this 1st dimension. Thus if I start with a number in the 2nd dimension e.g. 2², its reduced value (in the 1st dimension) is 4.

Now - when correctly interpreted - the number as dimension actually represents the qualitative aspect of number. Conventional mathematics, gets around this qualitative aspect, by in effect treating all numbers - where operations are concerned - as one dimensional quantities.

Putting it another way, mathematics adopts a merely horizontal interpretation of numbers. This is where numbers are considered solely as various quantities within a given dimension.

Here different numbers are raised to the same power of 1.

However, it is equally valid to adopt a merely vertical interpretation of number. This is where a given number is considered as an invariant unit quantity within differing dimensions. Here the same number (i.e. 1) is raised to different powers, this time representing qualities..

Whereas the horizontal approach represents the conventional definition of number, this vertical approach involves a psychological interpretation of number, where the very way in which number itself is understood is transformed. Just as each number (within a given dimension) represents a different number quantity, each number representing a power or dimension (to which a given number is raised), in direct terms, represents a different number quality and literally a different dimension or paradigm for interpreting numbers.

In other words, there is just not one way to understand numbers, but rather many depending on the particular stage of psychological development one is using as a basis for interpretation.

This vertical dimension of number arises from the process of psychological transformation whereby at each stage of development, a different type of understanding emerges.

It is this vertical qualitative dimension of number that I am mainly exploring in this book.

Whereas the horizontal quantitative approach towards number is mathematical and directly rational (and indirectly intuitive), the vertical qualitative approach is psychological and directly intuitive (and indirectly rational).The diagonal approach, which I term transrational, combines both.

And as soon as we adopt this dynamic two way interpretation of number both as horizontal quantities and vertical qualities, (conventional) mathematics and (transpersonal) psychology are understood themselves as fully complementary.

Seen in this dynamic light, mathematical concepts have tremendous value in terms of both clarifying and ordering psychological experience; equally psychological concepts can greatly illuminate the underlying nature of mathematical relationships.

At this stage we are dealing with the fundamental numbers 1 and 0. Now, there is a complementary relationship between these numbers. When we express 0 as dimension with respect to 1 as quantity, i.e. 1⁰, we get 1; likewise when we reverse this time taking 1 as dimension and 0 as quantity i.e. 0¹, we obtain 0.

In dynamic complementary terms, 1 (as a horizontal quantity), is 0 (as a vertical quality). Also 0 (as a horizontal quantity), is 1 (as a vertical quality).
 
 

Conscious and Unconscious Processes
 
 

This ties up directly with the very nature of conscious and unconscious processes. (Pure) reason is the highest expression of the conscious process, where unity is the basic defining characteristic. In the West, where the rational paradigm predominates, all the major religions define God as One and the experience of ultimate reality as of oneness.

(Pure) intuition, on the other hand is the highest expression of the unconscious process where nothingness - in phenomenal terms - is the defining characteristic. In the Eastern spiritual tradition, where the intuitive paradigm holds greater sway, the idea of God as being separate from the creature means little, and ultimate experience is more often understood in terms of a void (in terms of material experience).
 
 

So we can readily see that ultimate reality can be experienced in complementary terms, as unity or a void, depending on whether a rational or intuitive approach is used. Of course when we combine both intuition and reason, this highest experience will be seen in terms of both unity and nothingness. Thus terms such as the dazzling-darkness (West) and plenum-void (East) have been used to represent this paradoxical and fully creative mystical experience. Indeed the most famous of all Buddhist sutras gives precise expression to this experience.

"Form is not other than Void, Void is not other than Form."

Ultimately 1 and 0 are dynamic complementary symbols representing the two fundamental processes of reason and intuition, which underline all experience.

Thus, psychological experience can be seen as the interaction of these two processes.

Likewise in mathematical terms 1 and 0 are fully adequate to build a comprehensive number system (i.e. binary). The importance of this can be seen in the miracles of the present computer age which is so heavily based on the binary system of logic.
 
 

However, this binary system is used in a (static) manner where 1 and 0 are separate and mutually exclusive symbols understood in a reduced quantitative sense. In other words they are interpreted in a merely rational manner.

When both intuition and reason are explicitly used, the binary system becomes dynamic, where 1 and 0 are complementary and interchangeable terms used in both a quantitative and qualitative manner. 1 as (rational) quantity is 0 as (intuitive) quality. Equally 0 as (rational) quality is 1 as (intuitive) quantity. Thus the two complementary aspects of different numbers (within a given dimension), and (a given number) within differing dimensions, have both a quantitative and qualitative interpretation when reason and intuition are explicitly combined.

Putting it another way, in conventional computer terms, the binary system is a static one-way mutually exclusive system. If 1 represents an on state, then 0 represents an off state.

In dynamic terms, the binary system is a dynamic two way non-exclusive system. If 1 represents an "on" state horizontally, then it also represents an "off" state vertically where it is 0. By the same token if 0 represents an "off" state horizontally then it also represents an "on" state vertically where it is 1. It is this latter system which properly models the psychological system. 1 horizontally, representing reason, is 0 vertically, representing intuition. In complementary terms, 0 horizontally, representing intuition, is 1 vertically representing reason.

Understanding, therefore in dynamic terms can be interpreted as the continual interaction of reason and intuition (conscious and unconscious), switching as between "on" and "off" states both horizontally and vertically.