I have long emphasised how conventional mathematical interpretation of symbols is so limited with its mere emphasis on the quantitative aspect (that directly concurs with the linear use of reason).
We refer to this as the analytic aspect of interpretation.
Equally, however every mathematical symbol possesses a distinctive qualitative aspect that arises from the direct intuitive recognition of symbols (that indirectly is expressed through a paradoxical i.e. circular use of reason).
We refer to this as the corresponding holistic aspect of interpretation.
The true dynamics of mathematical experience then arise through the combined interaction of both quantitative (analytic) and qualitative (holistic) meaning.
I then have sought through these blog entries to apply this dynamic approach to interpretation of the zeta zeros.
However this quickly led to the realisation that there are in fact two sets of such zeros - equally important - that are dynamically interdependent with each other.
I refer to the first (recognised) set as the Zeta 1 zeros. These concur directly with the Riemann zeros (i.e. non-trivial zeros) of the Riemann Zeta function.
Now once again I will attempt here to highlight the holistic significance of these zeros.
We are accustomed to think of the primes (in quantitative analytic terms) as the independent "building blocks" of the natural number system (≠ 1). So from this perspective, all natural numbers can be expressed as the unique combination of individual primes.
However the unrecognised complementary counterpart to this (in qualitative holistic terms) is the view of the primes, as the corresponding interdependent connections governing the collective relationship of all primes (≠ 1) with the natural number system.
So when we allow for both the distinctive quantitative (analytic) and qualitative (holistic) aspects of the primes, we realise that in dynamic interactive terms, they combine both extreme independence and interdependence in a relative fashion.
So the Zeta 1 (Riemann) zeros from this perspective, can be holistically viewed as representing the complete set of such interdependent connections, which the primes collectively maintain with the natural number system.
In fact I have illustrated in my blog entries how the frequency of Riemann zeros bear a remarkably close relationship with natural number factors.
In other words the accumulated total number of natural number factors (of the composite numbers) up to n (on the real scale), approximates very closely to the corresponding frequency of Riemann zeros up to t (on the imaginary scale) where n = t/2π.
Therefore once again we can see the complementary relationship involved. So we start with the primes (as analytic measures of independence) and then find them related to the complementary holistic notion of interdependence (i.e. as factors of natural numbers).
Thus for meaningful interpretation, in a dynamic interactive manner, the primes and Zeta 1 zeros must be viewed - relatively - in analytic and holistic terms with respect to each other.
So therefore when we view the primes in a quantitative manner, we must then view the Zeta 1 (Riemann) zeros in corresponding qualitative fashion i.e. as an expression of interdependence, that indirectly can then be expressed in an imaginary number manner!
However the true interdependence as between the primes and the zeros is demonstrated by the fact that we can equally validly, switch reference frames, so that now the individual zeros assume a direct quantitative (analytic) meaning and the corresponding primes (with which they are complementary) a qualitative (holistic) interpretation as the collective behaviour of the prime numbers.
So in dynamic interactive terms, a mutual independence and interdependence characterises the primes and zeros (both of which are seamlessly integrated from two directions with each other).
This is just another way of stating the ultimate synchronistic nature of the number system, where neither primes nor zeros precede each other, as it were, but rather both mutually arise in a seamlessly integrated fashion (enabling the subsequent consistent relationship of number in both quantitative and qualitative terms).
The unrecognised - certainly with regard to their significance - set of zeros, relate to what I refer to as the Zeta 2 zeros. Indeed ultimately the Zeta 1 zeros can have no strict dynamic meaning in the absence of the Zeta 2 zeros (and vice versa)!
In some ways these zeros are in fact much easier to understand than the recognised Riemann zeros.
However the insight as to what they represent comes from ordinal rather than cardinal understanding.
As we have seen we typically start by viewing the primes in an individual cardinal manner as quantitative "building blocks" to establish their quantitative relationship with the overall natural number system (again in cardinal terms).
The zeta (i.e. Riemann) zeros then emerge to show that there is something seriously missing with this approach, by providing what in effect is a shadow system of collective holistic relationships (that must meaningfully be interpreted in a complementary qualitative manner).
However, one can also start by attempting to see each individual prime as already necessarily composed of natural numbers (in an ordinal manner). So instead of each natural number being defined quantitatively as the product of cardinal primes, alternatively, in reverse fashion, we define each prime as already ordinally composed in a unique manner by its natural number members.
So for example, 3 is a prime which is uniquely defined (from this perspective) by its 1st, 2nd and 3rd members. Now indirectly we can represent this (in quantitative terms) by obtaining the corresponding 3 roots of 1!
Significantly, when we obtain the prime roots of any number, all of these roots (again ≠ 1) representing its ordinal members, will be unique for this prime .
So the Zeta 2 zeros simply express this unique representation for each prime of its ordinal number members. And this feature of behaviour represents the complementary ordinal counterpart to the established fact that every natural number (≠ 1) in cardinal terms is uniquely composed of prime factors.
So here we start with the qualitative notion of each prime, as representing a shared group (of ordinal number members).
Then the Zeta 2 zeros arise as the indirect quantitative expression (through the prime roots of 1) of this (inherent) qualitative nature.
So once again, we can see an important complementarity here with the Zeta 1 zeros, where the set of zeros - by contrast - carry a qualitative holistic significance.
However as before the frame of reference can be switched, so that the prime (representing dimension) carries a quantitative meaning, while the collection of its ordinal members (represented by roots) is qualitative. In fact this is simply illustrated by the fact that the sum of roots = 0, implying - literally - that their combined nature carries no quantitative significance!
So within each number and throughout the number system as a whole, we have the two-way interaction of both prime and natural number behavior (in quantitative and qualitative terms).
Thus numbers as individual members and composite groups, contain particle and wave aspects. The particle aspect refers to numbers is both cardinal and ordinal terms, while - relatively -the wave aspect relates to both Zeta 1 and Zeta 2 zeros, which dynamically keep interchanging with each other - ultimately - in a purely relative manner!
Then the inherent nature of number, from this informed dynamic perspective, approaches pure synchronicity (as between its analytic and holistic aspects) in a merely relative manner. This is mediated through the two-way relationship of both prime and natural number aspects.
The great poverty of Conventional Mathematics - in refusing to give any formal recognition to the qualitative (holistic) nature of number - is that it cannot possibly appreciate, within its greatly limited framework, this true dynamic nature of the number system.
So quite simply, nothing short of the most radical revolution possible with respect to Mathematics is now urgently required.
If you are a mathematician reading this, I urge you to wake from your slumbers and bring the "good news" to your colleagues - which unfortunately they may initially see as "bad news" - that our true mathematical journey has scarcely begun!