Wednesday, September 9, 2015

Zeta Zeros and the Changing Nature of Number (9)

I will now carefully attempt to simply illustrate the precise nature of the zeta zeros, Zeta 1 and Zeta 2 through drawing together the two extreme interpretations i.e. Type 1 (analytic) and Type 2 (holistic) in what represents Type 3 (radial) understanding of the number system.

It is perhaps easier to start with the Zeta 2 zeros of which the simplest example relates to the prime "2".

Now in Type 1 (analytic) terms, 2 has an absolute fixed identity as form that is merely quantitative in nature.
This corresponds directly with (conscious) rational interpretation of a linear (1-dimensional) nature.

As we have seen the very basis of such understanding is that,

2 = 1 + 1, so that the two units are considered as absolutely independent of each other (without a qualitative identity)

However in Type 2 (holistic) terms, 2 now has a completely relative identity as ineffable emptiness that is of a merely qualitative nature.
This corresponds directly with (unconscious) intuitive appreciation of a circular (2-dimensional) nature. Indirectly however it can be expressed in a paradoxical rational manner.

The very basis now of such understanding is that the two units of 2 are considered fully interdependent i.e. fully related, with each other, representing a pure energy state. So this is a now a formless qualitative identity, that is totally lacking in quantitative characteristics.

Now in actual experience it is impossible to fully separate these two extremes.

Implicitly Type 1 (analytic) understanding is dependent on an intuitive basis (enabling the relationship between numbers to take place).

Likewise implicitly, Type 2 (holistic) understanding is dependent on a rational basis as numbers must be first recognised as independent, before their qualitative interdependence can be appreciated.

Thus properly understood, the understanding of number necessarily entails the interaction of both Type 1 (analytic) and Type 2 (holistic) aspects in a dynamic interactive manner (i.e. Type 3 understanding).  

So conventional mathematical understanding is thereby of a greatly reduced nature, that merely recognises the Type 1 quantitative aspect in an explicit manner!

Again from the Type 3 perspective (representing the interaction of Type 1 and Type 2 aspects) number is necessarily relative in nature, with aspects that are relatively independent (as quantitative) and relatively interdependent (as qualitative) with respect to each other.

So we are now in a position to give a Type 3 interpretation to the number 2 (which combines both its Type 1 and Type 2 aspects).

So in Type 1 terms, 2 is quantitative in nature (with two independent units i.e. 1 + 1).

In Type 2 terms, 2 is qualitative in nature (where the "units" are now understood as interdependent and ultimately identical with each other).
Now this understanding corresponds directly to pure intuitive insight. However indirectly this can be expressed in a linear (1-dimensional) manner through circular paradox.

Now, such paradox conforms to the two roots of 1 i.e. + 1 and – 1 which precisely expresses the nature of interdependence in the 2-dimensional case (with just two poles of reference).

We have already explained the nature of such interdependence with respect to the turns at a crossroads. Thus left and right turns are rendered paradoxical when we consider the approach to the crossroads from both N and S directions!

So of the left turn is represented as + 1, the right turn is – 1 and if the right turn is + 1, then the left is – 1. So with two dimensions, the directions keep switching in relative terms as between + 1 and   – 1, so that ultimately with simultaneous recognition, these can no longer be identified as separate.

So in Type 3 terms, we have now identified two units in a relatively independent quantitative manner, and then through the two roots the same units in a relatively interdependent qualitative manner. 

Thus while a certain relative independence can be given to each root (in isolation) i.e. + 1 and   – 1, when combined together, their sum = 0 (representing their relative qualitative interdependence)

What we have in fact represented here is the nature of the 1st of the Zeta 2 zeros (i.e. – 1) which is then combined with the default root of 1 for meaningful interpretation.

So the very nature of the Zeta 2 zeros is to maintain consistency as between both the quantitative and qualitative aspects of each prime so that its relative independence of its units in quantitative terms can be fully balanced with the relative interdependence of the collective units in a qualitative manner.

So again we have this paradox!

For example 5 in Type 1 terms as a prime appears as an independent building block of the natural number system in quantitative terms.

However 5 in Type 2 terms, as a prime, already is uniquely defined by its ordinal natural number members in qualitative terms (which can be expressed in an indirect linear quantitative manner by the five roots of 1).

So the Zeta 2 zeros (in Type 3 terms) in effect provide the consistent (internal) reconciliation of this paradox of the primes (with respect to both its quantitative and qualitative aspects).

Put another way they provide the consistent means of switching between the Type 1 and Type 2 (and in reverse Type 2 and Type 1) interpretations of number.

The Riemann (Zeta 1) zeros can then be viewed in a similar (external) manner.

From the Type 1 perspective, each natural number is uniquely expressed as the product of primes in a quantitative manner.

However from the Type 2 perspective, each prime obtains a unique resonance, as it were, through its relationship with the natural numbers in a qualitative manner.

So the Riemann (Zeta 1) zeros in effect now provide the consistent (external reconciliation of the paradox of the primes (with respect to both quantitative and qualitative aspects).

Put another way, they again provide the consistent means of switching between the Type 1 and Type 2 (and in reverse Type 2 and Type 1) interpretations of number.

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