Yesterday, we looked at the - as yet unrecognised - significance of the Type 2 non-trivial zeros.
We saw that they provide a ready means of assigning a unique identity to each individual member of a number group in ordinal terms.
So for example we can look on 4 as a collection of individual ordinal members i.e. 1st 2nd, 3rd and 4th respectively.
The 4 roots of 1, obtained through solving the equation 1 – s^4 = 0, then provide a relative quantitative identity (in circular number terms) to each of these individual members i.e. i, – 1 – i and 1 respectively.
Then the combined addition of these four numbers = 0, expresses their qualitative interdependence.
As 1 is always a root with respect to any dimensional value (t), we divide 1 – s^4 = 0 by 1 – s to obtain 1 + s + s^2 + s^3 = 0 which provides the non-trivial solutions.
Then to convert to a complementary form to the Zeta 1 equation we multiply by s
So s^1 + s^2 + s^3 + s^4 = 0.
More generally,
s^1 + s^2 + s^3 + s^4 + …+ s^t = 0.
The key significance of solutions to this equation (where t is a prime number) is that it ensures a unique identity to each ordinal member.
So the non-trivial zeros have a special significance in this context, as the means of providing a unique identity to the (natural number) ordinal members of a prime group.
Not alone has each member a unique individual identity in quantitative terms, but also a perfect holistic collective identity in qualitative terms (illustrated through the sum of roots =0).
Therefore the key (unrecognised) function of these Zeta 2 zeros is that they enable the seamless integration of each individual natural number (in ordinal terms) with their overall collective prime number identity.
For example the 1st, 2nd, 3rd, 4th and 5th members of 5 (as represented by the individual 5 roots of 1) are given a unique identity in quantitative terms; then equally, they have a holistic qualitative identity in their combined relationship with each other (represented as the sum of the 5 roots).
So in this way, the (ordinal) individual natural number members (of 5) in quantitative terms, share an overall qualitative holistic relationship with the cardinal prime number 5 (considered as a whole number set).
As the prime number grouping increases the dynamic interactivity required to reconcile the individual (quantitative) members with their overall collective (qualitative) shared interdependence so greatly increases that both aspects (quantitative and qualitative) can no longer be explicitly distinguished from each other. So in the seamless integration of both, qualitative approaches identity with quantitative meaning.
In this sense, therefore the natural numbers and the primes are likewise identical with each other. So, we can then no longer distinguish the individual members (as ordinal natural numbers) from the overall collective grouping (as a cardinal prime).
In fact any static identification of what is quantitative or qualitative loses meaning as switching between both aspects now occurs so rapidly as to be instantaneous!
However this can only be conceived in a relative rather than absolute sense.
One might be tempted to propose that by allowing the prime number to be infinite that we can thereby embrace all its natural number members (likewise in an infinite manner).
However this is a strictly meaningless proposition, as we would have no means of obtaining the infinite roots of an equation. So therefore there is no way of establishing a unique identity for each individual member or likewise of establishing an overall collective identity (in infinite terms).
But by making the dimensional number t larger and larger in finite terms, we can approach ever more closely to this identity of natural numbers with each prime (in a relative approximate manner).
So again from a very important perspective, this more refined treatment of number (allowing for both quantitative and qualitative aspects) exposes clearly the reductionist nature of the standard approach to infinite notions.
Put simply, ultimately at the interface of finite and infinite notions, we always face inevitable uncertainty. Indeed the uncertainty arising from the interaction of quantitative and qualitative (and qualitative and quantitative) is but a direct expression of this prior relationship as between finite and infinite (and infinite and finite).
And as mathematical activity (implicitly or explicitly) entails in any context the relationship between finite and infinite notions, it too is rooted inevitably in uncertainty.
So mathematical truth, as so graphically demonstrated in the very nature of the non-trivial zeros, is necessarily of a relative approximate nature.
Now, through reductionist procedures, we may certainly create the illusion of an absolute Mathematics; however ultimately this illusion is built on shifting sand without any solid foundation.
I have gone on at some length about the Zeta 2 approach to the non-trivial zeros for two major reasons.
Firstly its true significance (with respect to the relationship as between the primes and the natural numbers) remains totally unrecognised.
Secondly the equally important significance of the Zeta 1 non-trivial zeros can only be properly appreciated with respect to the complementary nature of the Zeta 2.
As we have seen, when one properly allows for both quantitative and qualitative aspects, there are two ways of viewing the relationship between the primes and the natural numbers.
From the standard Type 1 perspective, we can view the primes (in their collective nature) as the building blocks of the natural number system (in a cardinal manner).
However from the unrecognised (shadow) Type 2 perspective, we can equally view each prime (in its individual whole nature) as composed of natural number building blocks (in an ordinal manner).
In my own development - precisely because I have specialised for so many years now in this unrecognised holistic aspect of Mathematics - I had already become aware some time ago of the enormous significance of the Type 2 perspective (which requires very little in the way of abstract mathematical techniques).
However, it is very much the opposite with respect to the standard Type 1 approach where highly specialised complex techniques have been developed to deal with all aspects of the Riemann Zeta Function.
These would be largely inaccessible to all but a small number of professional practitioners. However, in my opinion, intuitive insight into what it is really all about still remains remarkably thin on the ground.
For example it has been patently obvious to me for some time that the Riemann Hypothesis is not capable of proof (within standard mathematical procedures). Put another way, its very nature greatly transcends conventional mathematical interpretation.
No amount of further improvements with respect to sophisticated mathematical procedures are going to change this situation. In fact they will lead even further away from any fundamental intuitive insight into what the problem truly entails.
The repeated failures with respect to attempts to “prove” the Riemann Hypothesis are in fact clearly indicating profound limitations in our very understanding of what Mathematics is about.
So the Riemann Hypothesis is really pointing to the ultimate identical nature of the quantitative and qualitative aspects of Mathematics. But this can never be appreciated while formally approaching interpretation in a mere quantitative manner.
We hear often for example practitioners state that some big new idea is required before real progress towards a proof of the Riemann Hypothesis can be made!
Well that big idea is that the qualitative aspect of interpretation must now be included in Mathematics not only to make sense of the Riemann Hypothesis but ultimately to make sense of all Mathematics!
We have been trading for far too long on the quantitative illusion i.e. that Mathematics can be formally interpreted in a merely quantitative manner.
However that illusion has run straight into the rocks protecting the Riemann Hypothesis and the truths underlying the very nature of the number system.
When one begins to accept the qualitative aspect with respect to all mathematical procedures, a marvellous new sense of mystery accompanies exploration into the deepest recesses of the mathematical system.
What we have then in the Type 1 and Type 2 approaches, two complementary visions of the origins of number. However growing appreciation of such complementarity, eventually leads one to the realisation that in the end we must surrender all phenomenal attempts at understanding its nature.
So reason can cooperate with intuition in drawing one into its sublime secrets, but in the end the final realisation of what it is (where everthing is now understood as interdependent), simply involves a surrender to that very mystery.
And in the ultimate questions regarding the number system - indeed regarding all phenomenal reality - is likewise found the ultimate answers in the pure experience of mystery. And here, the primes and the natural numbers finally melt together in an ineffable embrace.
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