The Original Number System

In my last entry I outlined the basis for a new vision of the number system which works on three different levels.

On the customary analytic level of rational (conscious) understanding we have the cardinal and ordinal aspects of the number system (interpreted in a merely quantitative type manner).

Then at a "higher" holistic level of intuitive (unconscious) type appreciation we have the understanding of both the Zeta 1 and Zeta 2 zeros as providing a complementary appreciation of the number system in terms of both physical and psychological energy states.

Finally at the most comprehensive level of understanding we have the growing marriage of both analytic and holistic type appreciation, where the zeta zeros are seen as vital in terms of enabling consistency to be maintained as between the quantitative (Type 1) and qualitative (Type 2) aspects of the number system.

Now this dynamic system is based on the two-way interaction of the primes and natural number numbers.

However strictly it does not itself explain the prior existence of - what I call - the two original numbers i.e. 1 and 0.

So the dynamic relationship as between the primes and natural numbers (and natural numbers and the primes) which in turn is consistently mediated through the corresponding two-way interaction of the Zeta 1 and Zeta 2 zeros is itself based on the prior relationship of 1 and 0 (again in quantitative and qualitative terms).

Again from a comprehensive perspective 1 and 0 have both analytic and holistic interpretations which dynamically interact with respect to all phenomenal relationships.

1 as a cardinal number (as analytically defined) is directly associated with the linear notion (i.e. 1-dimensional) in holistic terms. 0 as analytically defined is directly associated with the circular notion (i.e. 0-dimensional) from a holistic perspective.

In both physical and psychological terms differentiation is directly associated with the (conscious) linear, whereas integration is associated with the corresponding (unconscious) circular notion respectively.

Now the zeta zeros relate directly to these extreme notions in both analytic and holistic terms.

From the analytic perspective, the Zeta 1 zeros occur as conjugate pairs of the form a –  it and a –  it respectively. The Riemann Hypothesis postulates that they are of the form 1/2 –  it and 1/2 –  it respectively.

This would imply that the quantitative sum of each pair = 1.

Then from the analytic perspective the sum of the non-trivial roots for any group n (of 1) =  – 1.

Thus when we combine both we get 0.


From the holistic perspective (in both physical and psychological terms) the initial starting point of evolution - which equally represents the starting point of the number system - entails a totally confused state of integration (where no linear differentiation has taken place).  This equally entails thereby a total state of circular confusion.

Therefore in psychological (and physical terms) the requirement that all Zeta 1 (non-trivial) zeros lie on a straight line points to the corresponding need for complete (linear) differentiation of what earlier commenced in total confusion.

Therefore correctly understood, complete appreciation of the truly dynamic interactive nature of the  Zeta 1 zeros (approaching pure energy states) requires full differentiation with respect to unconscious experience i.e. where all hidden primitive (prime) elements are brought fully to conscious light.
Put another way this requires the full recognition of the hidden (unconscious) shadow with respect to the conscious experience of number.  

Put yet another way full holistic appreciation of their nature requires implies that the Zeta 1 zeros  lie on a straight line.

Now the reason why it is an imaginary - rather than real - straight line is due to the fact that expression of what is inherently of an unconscious nature (indirectly in rational conscious manner) requires imaginary rather than real translation.


With respect to the Zeta 2, the full holistic integration of what has been differentiated at a conscious linear level entails a purely circular appreciation of relationships. Here understanding again becomes so dynamic that rigid attachment to symbols is completely eroded thereby enabling pure holistic appreciation of their interdependent nature.   

And of course ultimately both of these are interdependent. For pure circular integration at the "higher" transcendent level of experience equally requires pure linear differentiation at the corresponding "lower" immanent level.

Thus full integration in psychological terms at the "higher" conscious level, ultimately is inseparable from full differentiation (i.e. of all unrecognised shadow material) at the "lower" unconscious level. 

Thus the two-way relationship of the primes and natural numbers is itself intimately based on a prior two-way relationship of the original numbers 1 and 0 (in both quantitative and qualitative terms).


There are also other fascinating connections with 1 and 0.

The only point for which the Zeta 1 Function remains undefined is with respect to s = 1.

However, Riemann's Functional Equation establishes an intimate connection as between ζ(s) and ζ(1 – s) .

This means therefore that  such an intimate connection binds ζ(1) and ζ(0).

In fact when deriving the value for ζ(0), we saw that it represents the pure complementary extreme of linear logic i.e. pure circular logic based on the complementarity of opposites.

This relates to the value of the Zeta 2 Function when s (for the Zeta 2) = – 1.

This gives a value for the Zeta 2 Function of 1/2.

The corresponding value for the Zeta 1 i.e.ζ₁(0) is then obtained as a linear expression of the inherently circular value associated with ζ₂(– 1).

So all the non-intuitive values  for ζ₁(1 – s) where s ≤ 1, in fact initially represent corresponding circular values associated with  ζ₂(– s)  that are then converted in a linear Zeta 1 manner. 

It can also be easily demonstrated that the only values for which the Zeta 1 and Zeta 2 Functions coincide is where s = 0 (with respect o the Zeta 1) and s = 1 (with respect to the Zeta 2).

In this  case 

ζ₁(0) = 1 + 1 + 1 + 1 +....

and  

ζ₂(1) = = 1 + 1 + 1 + 1 +....

Now strangely though both may seem the same ζ₁(0) is defined  whereas ζ₂(1)  is undefined, just as in complementary fashion ζ₁(1) is undefined and ζ₂(0) is defined!

The common denominator is that when s = 1 (in both cases representing in the case of the Zeta 1 a dimensional and in the case of the Zeta 2 a base value respectively).