The transcendent (top-down) aspect of integration generally unfolds through the refined use of reason in disciplining - what may be perceived as - the "lower" senses (in the form of unconscious primitive impulses).

As we have seen the Zeta 2 zeros are directly associated with the holistic nature of such transcendent development.

However the immanent (bottom-up) aspect of integration unfolds in a reverse complementary manner in the refined use of emotional feeling through unconscious projection, that gradually empties the unconscious of all such involuntary primitive impulses.

And as we have seen the Zeta 1 zeros are likewise directly associated with such immanent development.

This means in effect in this partial relative context, that the Zeta 2 and Zeta 1 zeros are directly associated with both cognitive (rational) and affective (emotional) aspects of understanding respectively.

So we started in Type 1 (conventional mathematical) terms, by considering number interpretation in a solely rational manner.

Then in Type 2 (holistic mathematical) terms, we extended number interpretation to likewise include an intuitive - as well as rational - aspect.

However, now at the Type 3 (radial mathematical) level, we must include the additional fact that numbers reflect both cognitive and affective aspects in relationship to each other.

Thus if we look at the Zeta 2 zeros as representing the masculine (cognitive), then the Zeta 1 zeros thereby represent the complementary feminine affective principle.

Of course ultimately in Type 3 terms - where the interdependence of both Zeta 1 and Zeta 2 zeros is fully recognised - both sets will now be seen as equally reflecting cognitive and affective aspects.

As we have seen the Zeta 2 zeros provide the means (though the non-trivial roots of 1) of differentiating natural number ordinal positions (with respect to a given prime).

So for example for the prime number 3, (where 3 represents a group of individual members) we can thereby uniquely differentiate 1st, 2nd and 3rd in this context. This is achieved in an indirect quantitative manner through the 3 roots of 1 (of which all except 1 are non-trivial) However this equally implies that we can successfully integrate these 3 members in qualitative manner (expressed indirectly through the sum of the 3 roots = 0).

The Zeta 1 zeros then provide the corresponding means of differentiating primitive impulses (with respect to the confused composite elements of the unconscious).

Primitive impulses reflect the confused activity that results due to qualitative being directly identified with quantitative appreciation.

So the very process of resolving such confusion requires the successful (analytic) differentiation of the quantitative aspect equally combined with successful (holistic) integration at a qualitative level.

In an exactly similar manner, resolving the true nature of the prime numbers requires the successful (analytic) differentiation of each prime (in cardinal terms) equally combined with their successful collective (holistic) integration in a qualitative manner.

And once again the Zeta 1 zeros directly correspond with this latter holistic aspect of the primes (representing number dimensions).

However we can now quickly demonstrate the requirement for integration of both sets of zeros.

From the Zeta 2 perspective, we must assume the existence of each prime before differentiating its group members (in a natural number ordinal manner).

From the Zeta 2 perspective, we must assume the existence of each (composite) natural number (before differentiating its individual members as primes).

Thus the Zeta 1 and Zeta 2 zeros are mutually implied by each other.

This even helps to clarify a notable feature regarding my own experience of these developmental stages.

With respect to the Zeta 1 zeros, for many years, I was more concerned with the even numbered dimensions (especially 2, 4 and 8) than odd (except for 3).

This ultimately reflected an undue emphasis on the transcendent aspect of contemplative development. It is only much later I realised that this corresponding lack of immanent type development (with which the Zeta 1 zeros - and the successful differentiation and integration of the primes - are associated) was impeding my ability to appreciate the nature of the prime numbered dimensions (associated with the Zeta 1 zeros) .

In dynamic interactive terms, the (analytic) differentiation of the individual primes in cardinal terms is ultimately inseparable from their corresponding (holistic) integration in a collective manner.

The Zeta 1 zeros can then be seen as relating directly to this latter collective integral aspect (that perfectly shadows the individual nature of the primes).

Likewise from a dynamic interactive perspective, the (analytic) differentiation of the individual natural numbers in ordinal terms (i.e. the members of each prime grouping) is ultimately inseparable from their corresponding (holistic) integration in a collective manner.

The Zeta 2 zeros can then be seen as relating to this latter integral aspect aspect (that perfectly shadows the individual nature of each natural number).

So ultimately the primes and natural numbers are thereby seen as perfect mirrors of each other that are ultimately identical (i.e. fully interdependent) in an ineffable manner.

However the fact that the zeta zeros now mutually reflect both cognitive and affective aspects of interpretation raises the issue as to whether Mathematics reflects scientific or artistic appreciation.

We will return to this important issue in the next blog entry.

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