We have explained how the (finite) Zeta 2 Zeros play a key unrecognised role with respect to the ordinal nature of the number system where they essentially act as a bridge as between the analytic and holistic appreciation of number, enabling both the relative independence and relative interdependence of such numbers be mutually reconciled in a dynamic interactive manner.
Putting it another way, they enable the actual (quantitative) recognition of ordinal numbers in specific circumstances with their potential (qualitative) recognition in general circumstances.
I demonstrated this at length yesterday, with respect to the simplest possible case relating to the prime number "2" where 1st and 2nd members can be identified. Note that in this context each prime in considered as a group (containing distinct individual members)!
Thus each member enjoys a certain relative independence with respect to each other member, while also (collectively) sharing a common relative interdependence with each other.
So when we recognise the relative independence of the 1st member, we equate it with the cardinal notion of 1 (i.e. + 1). Therefore when 1st is identified with the left turn (in relation to our crossroads example) it is consciously posited as + 1. Implicitly however, in dynamic interactive terms, this implies that we thereby negate consideration of the corresponding right turn (as 1st in this context).
So implicitly we must equally recognise that the right turn is thereby not the 1st turn (again in this interactive context).
Thus the very ability to consciously recognise 1st in this dynamic experiential context, is the corresponding implicit ability to unconsciously recognise that this thereby excludes 2nd.
And in the more intricate circumstance where several other options exist, again the ability to recognise 1st (in any given context) implicitly requires the corresponding ability to negate consideration of 2nd, 3rd, 4th and so on).
And this is why the roots of 1 are so important (especially where the prime roots are involved).
Thus the conscious recognition of any position (which in this continually changing context always = 1) implicitly requires holistic recognition of all the other - thereby - excluded positions which are given by the remaining roots of 1.
So again in our crossroads example the initial explicit analytic recognition of the left turn (as 1st turn) = + 1, implicitly requires the corresponding holistic recognition of the right turn (and not the 1st turn) = – 1.
Then when we change context through approaching the crossroads from the opposite direction the 1st is now identified with the right turn (which implicitly in this context excludes the left as "not 1st").
So in the whole context, where all options are considered as interdependent, quantitative units cancel and this represents the true holistic (qualitative) appreciation of the relationship.
Therefore whereas in a "part" (independent) context we can unambiguously identify either a left or right turn as 1st = + 1, in a "whole" (interdependent) context, we recognise the paradoxical nature of such identification with + 1 also – 1.
Thus we are enabled to combine the analytic understanding of relative independence of identification of left and right (and thereby 1st and 2nd) with the holistic understanding of relative interdependence where what is left and right (and thereby 1st and 2nd) continually change position depending on the frame of reference employed.
So putting it more simply, the "non-trivial" Zeta 2 zeros essentially - when appropriately interpreted - provide the "hidden" holistic basis of the ordinal number system. And the key importance of the primes in this context is that their set of "non-trivial" zeros are thereby unique in every circumstance.
I cannot stress enough the significance of this finding, which must always remain - literally - totally unseen from the conventional mathematical perspective (where the analytic aspect of number is solely recognised).
This is why my considerations here go far beyond any specific consideration of the Riemann Hypothesis, to state again quite emphatically that the current mathematical paradigm - especially in relation to a coherent understanding of the number system - is simply not fit for purpose.
And what I am really attempting to do here is to show how implicitly underlying all conscious understanding of mathematical relationships is an implicit unconscious basis (that is equally important).
So the holistic aspect of mathematical understanding essentially consists in bringing this (hidden) unconscious aspect fully into the light, where then remarkably, all mathematical symbols are seen to take on a profound new significance (i.e. in holistic terms).
However it does not end here, for the final task is to properly integrate both analytic and holistic appreciation in a coherent manner. Only then can we hope to enter a true golden age of Mathematics and we are at present many many light years from that goal!
Now one might initially question - even when accepting the rationale of my crossroads example - as to why this should have such profound significance for mathematical understanding.
However it is directly relevant for the very simple reason that all phenomenal understanding - including of course mathematical - is necessarily conditioned by polar opposites that continually interact in expereince.
Thus from one perspective, we have the interaction of external (objective) and inetrnal (subjective) poles. Thus the understanding of all mathematical "objects" necessarily entails mathematical "subjects" (i.e. in the mental categories we use to interpret such "objects")
So just like the N and S directions in our crossroads example, we have the continual switching in experience of external and internal aspects, creating ultimately holistic paradox in terms of all our carefully maintained mathematical propositions (at the analytic level).
Likewise - perhaps even more importantly in mathematical terms - we have likewise the continual interaction of wholes and parts that are quantitative and qualitative with respect to each other.
In fact, most of my consideration regarding the true relationship of the primes with the natural numbers (and the natural numbers with the primes) centres on this key polar interaction.
So quite simply, in conventional mathematical terms, wholes are reduced to parts (in a merely reduced quantitative manner). In fact this problem has now become so ingrained following centuries - even milennia - of use that it is not even recognised as an issue by the profession, where in fact it is by far the most important issue facing Mathematics (by a long long shot).
If you can grasp what I am saying, then you may already recognise that the fundamental relationship between the primes and the natural numbers is in principle exactly the same as between left and right turns at our crossroads i.e. purely relative.
Now once again, when we interpret our approach to the crossroads analytically (i.e. within a single isolated reference frame, heading N or S) we can unambiguously identify left and right directions.
However when we simultaneously combine both reference frames, left and right are seen as paradoxical i.e. purely relative terms (with no meaning independent of each other).
Now it is exactly the same in mathematical terms.
When we attempt to understand the relationship between the primes and natural numbers analytically within an isolated - merely quantitative - frame of reference, we can unambiguously identify the direction between them (with the primes in cardinal terms serving as the "building blocks" of the primes).
Then when we attempt to understand the relationship between the primes and natural numbers, now, within an isolated - merely qualitative - frame of reference, we can again unambiguously identify the direction between them (with the natural numbers in ordinal terms serving as the "building blocks" of each prime).
So again from this perspective, a prime is uniquely defined - apart from the last - by its 1st, 2nd, 3rd, 4th and 5th members (which can indirectly be analytically represented by the 5 roots of 1).
However when we then simultaneously try to combine both quantitative and qualitative frames of reference, clear paradox arises.
In the first case, the primes seem to be determining the natural numbers; in the second case the natural numbers seem to be determining each prime.
So seen holistically, the ultimate nature of the primes and natural numbers (and natural numbers and primes) is one that approaches a complete two-way synchronicity (utimately in an ineffable manner).
So properly seen in an enhanced dynamic interactive manner, the primes and natural numbers are fundamentally identical i.e. mutually reflect each other as perfect mirrors.