I keep pointing out the utterly complementary nature of fundamental mathematical relationships in dynamic terms.
When one properly appreciates such complementarity one begins to see the relationship as between the primes and the natural numbers in a completely new light.
Normal mathematical interpretation is strongly defined by the absolute nature of dualistic distinctions made.
Therefore though quantitative and qualitative constitute an extremely important complementary pairing in dynamic terms, Conventional Mathematics is based on the ultimately untenable dualistic assumption that we can seek to have quantitative knowledge as separate from the qualitative aspect.
Also though internal and external again comprise another fundamental complementary pairing in dynamic terms, Conventional Mathematics is again based on the ultimately untenable dualistic distinction that we can have external (objective) knowledge of mathematical relationships that can be abstracted from corresponding internal (subjective) interpretation.
So again Conventional Mathematics is strongly based on a linear (1-dimensional) approach where - in any relevant context - knowledge is based with reference to just one non-interacting polar reference frame.
This in fact - though rarely adverted to - is the most important thing we can say about such Mathematics.
Though it might seems as heresy to those accustomed to accepting the conventional wisdom, Conventional Mathematics - by its very nature - is crucially one-sided and therefore distorted with respect to the nature of truth thereby generated.
So for example we have been long conditioned to view the number system in an abstract manner as a set of absolute quantitative relationships frozen in space and time.
Such a mistaken interpretation stems directly therefore from the linear manner of interpretation involved, which thereby eliminates authentic dynamic notions.
In truth, the number system is inherently of a dynamic interactive nature. From one perspective, we cannot meaningfully form the external objective notion of number in the absence of corresponding mental constructs which - relatively - are of an internal nature.
Likewise we cannot meaningfully have independent quantitative notions of number in the absence of corresponding qualitative notions that relate to notions of shared interdependence
So in relation to the primes and natural numbers, we do not have just one static perspective that is of an absolute quantitative nature. Rather we have two aspects, quantitative and qualitative, that are in dynamic interaction with each other.
Of course just like the left and right turns at a crossroads, what is quantitative from one perspective is qualitative from the other; and what is qualitative from one perspective is quantitative from the other.
This interaction of quantitative and qualitative is expressed through the cardinal and ordinal nature of number in dynamic interaction with each other.
If we view the cardinal in quantitative terms, then the ordinal is thereby of a qualitative nature; however in reverse when we view the ordinal in quantitative terms, the cardinal is of a qualitative nature.
Thus depending on polar perspective, cardinal and ordinal have both quantitative and qualitative aspects.
However as the standard linear approach is geared merely to quantitative appreciation, we need a corresponding holistic manner of interpretation to convey the qualitative aspect of mathematical understanding.
So the two sets of zeta zeros (Zeta and Zeta 2) can be appropriately seen as the holistic counterparts to the quantitative analytic nature of both the cardinal and ordinal aspects of the number system.
In dynamic interactive terms therefore, the two sets of zeros perfectly complement in holistic manner the cardinal and ordinal aspects (understood in analytic terms).
However, again we can switch perspectives, so that the two sets of zeros now have an analytic interpretation with the cardinal and ordinal numbers directly complementary in a holistic manner.
Put more generally all mathematical relationships - when viewed appropriately in a dynamic interactive manner - possess both analytic (quantitative) and holistic (qualitative) aspects.
In fact coming back to yesterday's blog entry, I believe that I did not emphasise fully the complementary nature of the holistic (Type 2) formula for generating the frequency of primes.
With the standard cardinal approach the frequency of primes relates to just one possible set of prime numbers.
However with the corresponding ordinal approach, innumerable different sets can be chosen which equally approximate the number of primes.
So as in our example when t = 127, the formula estimates 24 primes up to n = 81 (where n = 2t/π).
However, strictly this does not relate to just one set of numbers. In fact if we keep choosing a random similar number of "converted" roots, their sum would approximate the same answer.
Thus though there is an obvious bias in the (linear) cardinal number system as between the primes and (composite) natural numbers, there is no such bias with respect to the (circular) ordinal number system, represented by the various roots of 1. This likewise extends to the "converted" roots of 1.
Thus if we for example were to repeatedly choose 40 "converted" roots at random from the 127, and then obtain the sum each of these selections of 40 roots, the answers would approximate close to each other (with the approximation improving as both population and sample size increase).
The reason for this once again is that the Zeta 2 zeros - in a similar manner to the Zeta 1 - properly represent in dynamic holistic manner, the paradoxical identity of opposite polarities.
So once again - now with respect to the ordinal natural number members of a prime group - notions of independence and interdependence, randomness and order, quantitative and qualitative etc. are mutually combined with each other. So from this perspective the very notions of prime and (composite) natural numbers as separate lose their meaning.
Thus when we estimate the number of primes using the holistic formula, (2t/π)/(log 2t/π – 1), any of the "converted" roots can be included.