Where Science and Art Coincide (10)

In my last blog entry, I expressed how the Zeta 2 zeros represent the unconscious holistic basis of the natural number system (in ordinal terms).

Customary understanding of this system takes place in a reduced - merely analytic - manner viewed  solely from a quantitative perspective.

However properly understood, this ordinal aspect entails both (conscious) analytic and (unconscious) holistic aspects which interact in dynamic fashion.


These zeros in the first instance arise as solutions to the finite equation

ζ₂(s) =  1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime) = 0.

So we have a relationship here as between the value of s (as base quantitative value) and the natural number sequence 1, 2, 3, ...., t – 1 (representing corresponding dimensional values).

Now as I have frequently expressed, the relationship between base and dimensional values is always as quantitative to qualitative (and qualitative to quantitative) in dynamic interactive terms.

So in the expression a^b, a is the base and b the dimensional numbers respectively. Therefore if in this context a is interpreted as quantitative, then the exponent b is thereby - relatively - qualitative in nature.

 

In this sense the Zeta 2 zeros represent an equal and indispensable partner providing the qualitative appreciation underpinning customary quantitative understanding of the ordinal number system.

Though in many ways, much more difficult to intuitively grasp, the true nature of the Zeta 1 (i.e. Riemann) zeros can be expressed simply in a direct complementary fashion.


These zeros by contrast represent the inversion (with respect to the Zeta 2) of base and dimensional values.

So the Zeta 1 zeros arise as solutions to the infinite equation

ζ₁(s) = 1^–s  + 2^–s  + 3^–s  + 4^–s  +……..  = 0.


So the role of the Zeta 1 zeros can be succinctly expressed as providing the corresponding unconscious holistic basis of the natural number system in cardinal terms.

Thus once again though we are accustomed to especially appreciate the cardinal number system in a merely quantitative analytic manner, properly understood, it contains two interacting components that are quantitative and qualitative with respect to each other.

Therefore from this perspective the Zeta 1 zeros provide the (unrecognised) qualitative aspect that underpins our customary quantitative appreciation of the cardinal number system!


So the Zeta 1 and Zeta 2 zeros approach the fundamental identity of quantitative and qualitative aspects of the number system from two complementary perspectives.

In the case of the Zeta 2, the inherently qualitative identity of each ordinal number (within its specified grouping) is given an individual quantitative identity in an indirect manner. For example in the context of two members, 2nd  is indirectly given a quantitative identity as – 1.

Its qualitative nature is then expressed through the collective addition of the quantitative values with respect to all members of the group.

So again in the context of two members, the sum of quantitative values (representing the two roots of 1) = + 1 – 1 = 0.


In the case of the Zeta 1 zeros , it is the reverse!

Here the inherent quantitative  identity of each cardinal member is given an indirect qualitative identity. So for example the qualitative identity (representing a dimensional value) of the first pair of non-trivial zeros is given as + 1/2 + 14.134725...i and + 1/2 – 14.134725...i respectively.

Their quantitative identity is then expressed through the collective nature of all the zeros (which are finitely unlimited in nature).

In other words the non-trivial zeros can be collectively employed in a quantitative manner to fully correct the deviations that arise with respect to the general prediction of the frequency of prime numbers (up to any given natural number).

(Though not yet properly recognised, the Zeta 2 zeros can be used in a complementary fashion.

Thus the individual probability that a number is prime can be corrected through deviations associated with the Zeta 2 zeros. I have in other places shown how these deviations (from the average limiting value of the roots of 1, when the number of roots is very large) can be calculated. See for example "Alternative Prime Hypothesis!"

Thus in principle using this alternative approach in principle we should be able to approximate the exact probability of each number being prime so that the sum of these probabilities would then closely match the exact frequency of primes (up to a given number).


However, though it is initially valid to attempt both the Zeta 1 and  Zeta 2 zeros in a relatively independent manner, in the most comprehensive mathematical understanding (Type 3) we understand them increasingly in a relatively interdependent fashion.

Thus from this perspective, cardinal notions have no strict meaning independent of ordinal; likewise ordinal have no strict meaning independent  of cardinal!

So properly  understood both the Zeta 1 and Zeta 2 zeros are ultimately interactively determined in a manner approaching pure simultaneity.

In the most accurate sense, both thereby serve as the ultimate phenomenal partition bridgng both the finite (dualistic) and infinite (nondual) realms.


Once again in an attempt to provide additional perspective on their nature, the Zeta 2 zeros arise from the attempt to reduce higher dimensional qualitative notions ( ≠ 1) in a 1-dimensional manner.

And the is the manner in which customary ordinal notions are understood!

Therefore to express the qualitative notions of 1st, 2nd and 3rd (in the context of 3 as representing dimensions) in a reduced circular 1-dimensional quantitative manner, we obtain the three roots of 1.

And as the very notion of interdependence has no strict notion in 1-dimensional terms, the corresponding sum of these roots (expressing their qualitative interdependence) = 0 (in quantitative terms). However 0 now also has a qualitative meaning representing a psycho spiritual energy state (corresponding to pure intuitive understanding)

However the Zeta 1 zeros in reverse manner arise from the attempt to transform 1-dimensional quantitative notions in a higher dimensional qualitative manner.

And as the Zeta 2 zeros relate to the ordinal nature of the natural numbers (in the context of each prime) the Zeta 1 zeros relate to the ordinal nature of the composite natural numbers (in the context of the unique product of prime factors).

So as I have repeatedly stressed the process of multiplication inherently implies a qualitative aspect!

Therefore the Zeta 1 zeros simply represent the transformed expression of the qualitative nature of prime number multiplication.

So for example when we multiply 3 by 5 a qualitative - as well as quantitative - transformation takes place in the numbers involved. And the Zeta  1 zeros simply represent the attempt to express the qualitative nature of this transformation for all composite natural numbers (representing the unique product of primes).

Now in reverse fashion to the Zeta 2 zeros, the qualitative interdependent nature of the Zeta 1 zeros is represented with respect to each individual pair. Thus each zero can be seen thereby as approximating a singularity or pure energy state (without form).

Then the corresponding quantitative independent nature of the zeros is represented though their combined collection as a group (thereby enabling the correction of remaining errors with respect to the generalised prediction of prime number frequency among the natural numbers (up to a given level).