The importance of this rational system is obvious as it forms the very basis of (conventional) scientific measurement. 

As we have seen in this system all numbers are ultimately interpreted in terms of an invariant first dimension. Psychologically, this implies that all numbers are understood in terms of one fixed number concept (i.e. all numbers belong to the same number concept). This abstract and impersonal interpretation of number - though highly useful - can easily lead to fragmentation and a loss of individual uniqueness. 
 

Linear Qualitative System 

The importance of this alternative intuitive system is rarely appreciated and indeed effectively ignored. In complementary fashion, it forms the basis of psychological measurement. 

In dynamic terms, understanding involves the mutual interplay of both (objective) perceptions and (subjective) concepts. Thus the understanding of number itself is inherently dynamic involving this two-way interaction. 

The (reduced) objective interpretation of number which is horizontal and static results from the consequent freezing of the subjective aspect (i.e. by holding the number concept fixed). 

This - as we have seen - yields the rational interpretation of number. 

However it is equally valid to freeze the objective aspect (i.e. by holding the number perception fixed). Here the one number perception is raised through varying number concepts giving a strongly intuitive and vertical interpretation of number. 

The former rational approach facilitates the understanding that all (separate) number quantities share the common quality of number (i.e. belong to the one number concept). 

The latter intuitive approach facilitates the understanding that all (separate) number qualities (or dimensions) belong to just one number quantity. In other words each number is multifaceted with a unique interpretation in each context. 

Remarkable as it may seem actual understanding involves as many dimensions as objects. This is not just true in hypothetical mathematical terms but underlines all physical and psychological understanding. It is the innate appreciation of this complementarity - as with numbers - which gives objects their personal uniqueness. 

To sum up, the rational objective understanding of number focuses merely on the quantitative aspect yielding a common impersonal interpretation. 

By contrast the intuitive subjective understanding focuses on the qualitative aspect yielding a unique personal interpretation of number. 

A true dynamic understanding of number necessarily involves both blades of this scissors in dynamic interaction. 
 

Circular Quantitative System 

This circular number system simply results from obtaining successive roots of unity. 

For example the four roots of unity are +1, -1, +i and -i and lie as equidistant points on the circle of unit radius. Again, though the existence of this number system is of course known to mathematicians it's true significance has been overlooked. 

Just as the linear quantitative system is ideal for analysis of the specific parts of the overall scientific system, the circular quantitative system is ideal for the complementary task of obtaining a holistic synthesis of this same system. 

As we will see later the fundamental structure of physical reality can be represented simply as reduced interpretations of unity (i.e. roots of unity). In particular the symmetrical relationships provided by both the four roots and eight roots of unity are especially useful in terms of this holistic synthesis. 
 

Circular Qualitative System 

This circular number system is the psychological counterpart of the quantitative system. 

Whereas the former system represents a reduction of unity (through obtaining roots), this represents a raising or transforming of unity through moving to higher powers or dimensions of understanding. 

Thus - for example - the qualitative number corresponding to 1^1/2 is 1². 

The former represents a rational understanding where positive and negative poles (i.e. +1 and -1) are separated. 

The latter represents an intuitive understanding where both poles are directly fused together in a transformed "higher" understanding. 

Thus phenomena which seem in conflict at the linear rational level, become reconciled in terms of (higher) level intuitive understanding so that finally through progressive psychological transformation only unity remains. 

The circular qualitative system is the ideal basis for obtaing a holistic synthesis of psychological understanding. As we will shall see later the various key levels of mystical transformation (i.e. "high level" pscyhological understanding) can be perfectly modelled in terms of this circular number system. 

Finally, these circular number systems entail that the fundamental holistic structures of both physical and psychological reality are themselves complementary. Physics and pscyhology now truly merge in a wonderful synthesis.