Let us briefly recap the present position.
In the Type 1 quantitative approach to number, we view the primes collectively as the building blocks of the natural number system, with each natural number (except 1) representing a unique combination of prime constituents in cardinal terms.
This Type 1 approach leads to a linear view of the number system interpreted in a merely quantitative manner.
In the Type 2 qualitative approach to number we view – in reverse fashion – the natural numbers individually as the building blocks of each prime number, so that for example 3 contains a 1st, 2nd and 3rd member, with now each prime - expressed though its roots - representing a unique combination of natural number constituents (except 1) in ordinal terms.
This Type 2 approach leads to a circular view of the number system interpreted in both a - relative - quantitative and qualitative manner (that are complementary).
So though the relationship between the primes and natural numbers (and natural numbers and primes) is unambiguous within each approach as separate, from an interdependent perspective, it is paradoxical.
Thus in the Type 3 approach, where both quantitative and qualitative aspects are identical with each other - or more correctly in dynamic phenomenal terms approach identity with each other - both the primes and natural numbers (and the natural numbers and the primes) likewise mutually reflect each other as identical.
And this mutual identity of primes and natural numbers in both qualitative and quantitative terms is expressed through the non-trivial solutions for the Riemann Zeta Function where ζ(s) = 0.
In other words the non-trivial zeros directly represent the Type 3 approach to number.
This Type 3 approach leads to a point view of the number system (combining both linear and circular notions as identical) in a manner that approaches a pure ineffable state.
The implications of all this could hardly be more fundamental.
In the deepest sense, the relationship between whole and part (which governs all phenomenal relationships) is rooted in the mathematical mystery of the connection of the primes to the natural numbers (and the natural numbers to the primes). When we attempt to view these as somehow separate, their relationship leads to two parallel worlds, which while consistent in each case, are fully paradoxical in terms of each other.
So in a very refined manner, the non-trivial zeros represent the solution to this great original paradox.
However it is a solution that still remains shrouded in great mystery, for there is no way to access the meaning of these zeros without entering deeply in experience into that very same mystery. So ultimately, it is only through the most advanced psycho spiritual development, where all phenomena can be viewed in an unattached manner as utterly transparent in pure contemplation of reality, that the meaning contained in these zeros can be fully appreciated.
Because they directly point to the full reconciliation of qualitative with quantitative type notions, this directly implies that such understanding requires the corresponding reconciliation to a very advanced level of intuitive with rational understanding (where both approach a perfect balance with each other).
And here is the other side of that great mystery which I wish to now stress!
As we have seen phenomenal reality (at all levels) is governed by two fundamental polarity sets.
Perhaps in the desire to maintain a more objective stance with respect to the nature of mathematical understanding, I have general emphasised - especially in the relationship of the primes to the natural numbers - the second polarity set relating to quantitative and qualitative (whole and part).
However the first relating to internal and external is equally important.
And again in the deepest sense, the answer to this fundamental mystery of the relationship between mind and matter is again revealed through the non-trivial zeros.
For what these zeros – when properly understood - directly imply, is that we cannot have objective knowledge of the ultimate nature of reality without this being directly united with corresponding subjective reality! In other words what we objectively can know about reality is ultimately inseparable from the psychological constructs we use to view this reality.
Though in one sense the non-trivial zeros were always deeply implicit at the heart of our number system, until Riemann’s pioneering work some just over 150 years ago, nothing was humanly understood regarding their incredible significance. Indeed their very existence remained unknown!
So is only through the evolution of knowledge to a sufficient degree, that we have at last begun to unearth number’s greatest secret.
Thus the non-trivial zeros constitute this great hidden treasure of the number system. Therefore we can now realise that it has long remained concealed under the veils of our conventional treatment of number. However we still have obtained but the most rudimentary glimpses into its great mystery.
So what at an objective level remains most deeply implicit in the number system (in its true original state) can only be made properly explicit through once again entering this state in its most advanced spiritual evolution. So ultimately both the origin and destiny of all evolution are realised in the same present moment (where both objective and subjective poles are fully united in experience). And it is this same present moment thet the mystery of the primes (in their relationship to the natural numbers) is finally resolved.
Now once again in the strictest sense, the non-trivial zeros do exist in phenomenal space and time. However they only do so in the most refined dynamic manner possible, as the gateless gate, as it were, that bridges the phenomenal world with ineffable reality.
So inherent in earliest phenomenal reality (once it becomes manifest) is the solution to the great paradox of number which thereby enables the universe to unfold with respect to both its quantitative and qualitative features.
And then in the actualisation of evolution in the pure contemplation of mystery, the explicit solution to this paradox of number again exists as the final divide between phenomenal and ineffable reality.
As we know a great deal of attention is now placed on the origins of the physical universe and the notion of a Big Bang starting it all.
However properly understood, the very notion of number is necessarily already implicit in any such phenomenal developments.
So the really important question – which seems to me largely ignored – is the explanation of the number Big Bang and how this capacity for order with respect to the number system (in both quantitative and qualitative terms) has emerged.
And right at the heart of this number Big Bang we find the non-trivial zeros.
As we saw in yesterday’s blog entry, The Riemann Zeta Function – when appropriately interpreted – provides the required framework through which the 3 aspects of the number system can be understood.
Therefore initially we can attempt to view all numerical results in a merely quantitative manner in Type 1 terms.
Secondly we establish two-way complementary results through the Functional Equation for analytic values of s > 1 on the RHS and corresponding holistic values for s < 0 on the LHS in Type 2 terms.
Finally we establish the mutual identity of both Type 1 and Type 2 systems in Type 3 terms through the midpoint of the critical region (0 < s < 1) where holistic and analytic interpretations directly coincide.
Then by setting ζ(s) = ζ(1 – s) = 0, we find the non-trivial zeros which define the Type 3 number system. This does indeed generate a set of complex numbers (with real part of a rational and imaginary part of a transcendental nature). However proper understanding of such numbers, relating to a common interdependence of quantitative and qualitative characteristics, is of the most dynamic nature possible. Here each number, in phenomenal terms, serves as but a fleeting mediator of an ultimate meaning that is ineffable. So once again we are at the other extreme here from the standard interpretation of number in a static abstract manner!
This Type 3 system then serves as the vital necessary condition for subsequent consistent interpretation with respect to both the Type 1 and Type 2 systems.
And without such consistency, phenomenal reality as we know it could not exist!