We have demonstrated how numbers can be viewed in both an external and an internal manner which in dynamic terms are complementary.
So once again from an external (analytic) perspective, the relationship between the primes and the natural numbers seems one-way, with all natural numbers (except 1) resulting from the unique combination of prime factors in cardinal terms.
Likewise from an internal (holistic) perspective, the relationship between the primes and natural numbers again appear one way - in a reverse manner – with every prime representing a unique combination of natural number members (again except 1) in ordinal terms.
So when one views the two-way relationship of the primes to the natural numbers (and the natural numbers to the primes) it becomes increasingly apparent that ultimately they are fully interdependent with each other.
So the ultimate nature of the primes – or more correctly this two-way relationship of both the primes and natural numbers - is shrouded in deep paradox, which is just an indirect rational manner of saying that it is utterly mysterious.
We therefore have only the appearance of some deterministic relationship as between both (i.e. the primes and the natural numbers) when we attempt to separate to a degree the external (Type 1) analytic from the corresponding internal (Type 2) holistic aspect of understanding.
So from the Type 1 perspective, we have the Riemann Zeta Function and the famed non-trivial zeros (arising as solutions for complex values of s) all seeming to line up on a vertical imaginary line through ½.
Now when we attempt to look at this in isolation it again gives the appearance of some definite unambiguous factor underlying the relationship of the primes to the natural numbers in quantitative terms. And hence the continuing obsession in the Mathematics profession to prove the Riemann Hypothesis!
Then when we look at this problem in relative isolation from the Type 2 perspective, we again have the appearance of an underlying factor explaining this relationship (in a complementary qualitative manner).
However when we realise that true understanding requires the incorporation of both the Type 1 and Type 2 approaches (in – what I refer to as – a Type 3 manner) then it becomes ever more apparent that the quantitative aspect of such explanation entails the qualitative and in like manner the qualitative aspect entails the quantitative!
In other words from the Type 3 comprehensive perspective, both the quantitative (analytic) and qualitative (holistic) aspects of this key relationship - connecting the primes with the natural numbers – are mutually contained in each other in a manner that is ultimately totally ineffable.
The implications of what I am saying here – when properly grasped – could not be more devastating for the conventional approach to Mathematics, for this applies not just to interpretation of the Riemann Hypothesis but intimately to every mathematical symbol and relationship!
So putting it simply, we have attempted to build Mathematics on the illusion that symbols and relationships can be (formally) given a mere quantitative meaning.
However in truth all such symbols and relationships possess an equally important qualitative (holistic) as well as quantitative (analytic) meaning.
So in the terms that I employ, Mathematics can be given a Type 2 – as well as Type 1 – interpretation for every symbol and relationship.
Now we can indeed attempt to obtain specialised knowledge with respect to either aspect (in relative isolation).
However a complete comprehensive mathematical appreciation ultimately requires the growing interaction of both types of understanding (Type 1 and Type 2) in a mutually interdependent fashion (Type 3).
Now contrast this vision with the present state of Mathematics!
Because there is no formal recognition of a Type 2 holistic aspect, equally there can be no recognition of a Type 3 (which entails the appreciation of both Type 1 and Type 2 as ultimately fully complementary).
What is even worse is that Conventional Mathematics, insofar as it deals with the quantitative aspect, is thereby rooted in an absolute - rather than relative – type appreciation that blocks access to the other aspects.
So we can have a closed quantitative system ìn Type 1 terms of an absolute nature, or an open system that is defined in a relative manner. And once again, as it stands, Conventional Mathematics represents the extreme specialisation of the former system.
This is why I have little faith that the necessary revolution that is now so crucially needed can emerge from within the Mathematics profession. However, in fairness there will always remain a small group, who can perhaps remain open to questioning basic assumptions.
When one redefines the Riemann Zeta Function in a Type 3 manner, the true nature of the Function and its associated Riemann Hypothesis become immediately apparent.
It also demonstrates emphatically why Conventional Mathematics is so unsuited to this task!
As I have stated, every mathematical symbol and relationship can be given both a Type 1 (analytic) and Type 2 (holistic) explanation.
So for example from the Type 1 perspective, the one value where the Riemann Zeta Function remains undefined in quantitative terms occurs where s (representing the power or dimension of the Function) = 1. The Function can thereby be only defined in quantitative terms for all other values (except 1).
Likewise from the Type 2 perspective, the one value where the Riemann Zeta Function remains undefined in qualitative terms occurs where s = 1.
In this context s = 1, refers to the linear i.e. 1-dimensional method of interpretation that formally characterises Conventional Mathematics.
So the Riemann Zeta Function remains uniquely undefined from a conventional mathematical perspective!
What this precisely entails becomes clearer when one looks at the nature of other values.
Now again in Type 1 terms we cannot give a linear interpretation to values of the Function for s < 0.
What this entails therefore is that such values represent an indirect quantitative representation of a (circular) holistic meaning (i.e. directly relating to Type 2 understanding).
So from a Type 3 perspective the true significance of Riemann’s Functional Equation is that it associates values with a direct Type 1 quantitative value (for s > 1), with a direct Type 2 qualitative meaning in complementary fashion for corresponding values of s < 0 (that are indirectly represented in a linear quantitative manner).
So the whole point about the Functional Equation from this perspective is that we can thereby match analytic (quantitative) with qualitative (holistic) values throughout the complex plane.
Thus a key importance thereby attaches in such a formulation to the points where quantitative and qualitative type interpretations exactly coincide.
Therefore the significance of the Riemann Hypothesis in this context is that it sets the condition for a coincidence of interpretations to take place.
The Functional Equation associates values for ζ(s) on the RHS with values for ζ(1 – s) on the LHS.
Where s > 1 on the RHS and s < 0 on the LHS, a clear distinction as between analytic and holistic meanings applies with values on the RHS directly interpreted in analytic and values on the LHS interpreted in a holistic manner respectively.
Then for values within the critical region 0 < s < 1, both analytic and holistic interpretation apply.
The condition then for the full coincidence of these values, where ζ(s)= ζ(1 – s) = 0, requires that the real part of s = .5.
So from this perspective, the famous Riemann Hypothesis is the condition required for the full coincidence of both quantitative (analytic) and qualitative (holistic) type understanding with respect to the non-trivial zeros. The non-trivial zeros represent the solutions for s to the equation which necessarily occur in pairs of the form .5 + it and .5 – it respectively!
However there is a catch here in that an unlimited number of non-trivial zeros potentially are involved, of which only a finite number can be known.
Thus strictly the truth to which the Riemann Hypothesis points (i.e. the ultimate identification of quantitative with qualitative type meaning) can only be approximated from a phenomenal perspective in a relative finite manner.
So the ultimate nature of this relationship of the primes to the natural numbers (and natural numbers to the primes) pertains in turn to the corresponding nature of quantitative to qualitative (and qualitative to quantitative) meaning, which is totally mysterious and can thereby be only approximated in a finite relative manner.
As the Riemann Hypothesis – again from this more comprehensive perspective – states the condition with respect to number, for the identification of analytic (quantitative) with holistic (qualitative) meaning, not alone can it not be solved from the conventional Type 1 perspective, but its true nature cannot be understood in this manner!
So once again the Riemann Zeta Function remains uniquely undefined for s = 1 (in both quantitative and qualitative terms).
This of course means that in thereby remains uniquely undefined in conventional mathematical terms (defined merely by its 1-dimensional manner of interpretation).
For all other values of s, e.g. in the simplest case where s = 2, a dynamic interaction arises as between two aspects of understanding which are quantitative and qualitative with respect to each other.
This type of understanding uniquely is missing from conventional appreciation based on the attempt to understand relationship in a merely absolute quantitative type manner!
So rightly understood, true interpretation of the Riemann Hypothesis should serve as the requirement for the most important revolution yet in the history of Mathematics where the qualitative aspect of interpretation is recognised as equally important to the quantitative and where both are then mutually incorporated in complementary fashion through a new dynamic interactive form of understanding intimately affecting every mathematical symbol and relationship.
So again in the context of the Riemann Hypothesis, we started with the Euler Zeta Function defined in a real quantitative manner.
This was then extended by Riemann to the complex plane (entailing both real and imaginary aspects) again in a quantitative manner.
What I am now clearly suggesting is that the necessary next evolution in understanding requires that the complex plane itself be defined in both a quantitative (analytic) and qualitative (holistic) fashion.
From a qualitative perspective the "real" relates direct to the analytic and the "imaginary" directly to the holistic aspect – indirectly expressed in a linear manner – respectively.
So, all complex notions therefore can be given both quantitative (analytic) and qualitative (holistic) interpretations. In fact, strictly speaking it is impossible to have quantitative without qualitative (or qualitative without quantitative) meaning!
When one clearly sees this, then the key mathematical issue pertains to the ultimate nature of this dynamic relationship between quantitative and qualitative aspects!
So once again the Riemann Hypothesis – which directly points to this issue – cannot be solved using conventional mathematical axioms as they are solely based on a (reduced) quantitative notion of such meaning.