From the onset let us abandon any notion that the significance of the non-trivial zeros with respect to Riemann's Zeta Function can be understood in a linear rational manner.
And as their true nature greatly transcends such understanding this directly poses a dilemma for conventional mathematical interpretation (which is formally based on linear reason).
Therefore, as Hilbert rightly suggested, these zeros, when appropriately appreciated, relate to what is truly the greatest secret underlying the nature of the phenomenal - not alone the mathematical - universe, their very nature serves as an entry into the direct experience of ineffable mystery.
Though in fairness, there is now a growing appreciation of the possible physical significance of these zeros, there is little or no realisation yet of their enormous potential psychological implications.
Indeed I have little doubt that in future times the non-trivial zeros will serve an extremely important holistic scientific role in preparing spiritual aspirants to attain a pure state of meditation consistent with maintaining the highest degree of involvement in phenomenal concerns. In this way the zeros will be seen as an invaluable aid in the quest for the fullest expression of life!
Putting it bluntly the present (standard) interpretation of what constitutes Mathematics is of an extremely limited nature.
So in effect we have taken an important special case of linear (1-dimensional) understanding, where qualitative is reduced to quantitative interpretation and misleadingly elevated this as solely synonymous with valid Mathematics.
Nothing however could be further from the truth. In fact a potentially unlimited set of possible alternative interpretations of mathematical relationships exists (each one of which possesses a partial relative validity). And in all of these cases both quantitative and qualitative type aspects are related in a dynamically interactive manner.
So properly understand all mathematical activity is of a dynamic relative nature, where quantitative and qualitative aspects of understanding (which are complementary) interact.
Quite simply therefore appropriate interpretation of such activity must also formally recognise that like the blades of a scissors, Mathematics possesses two equally important aspects that are quantitative and qualitative (and qualitative and quantitative) with respect to each other.
Now, when seen from this wider dynamic perspective, there is indeed an especially important limiting case where the focus is placed primarily on quantitative meaning.
But rather like the role of Newtonian Physics, this special case should be seen as but a useful approximation with respect to mathematical reality, which inherently is of a dynamic nature and necessarily subject to uncertainty.
Indeed because of the extreme quantitative bias that defines the formal presentation of mathematical ideas, we have increasingly lost any clear notion in our society of what the qualitative aspect of Mathematics might even entail!
Though from one perspective, I can greatly admire the incredible abstract ability and indeed sheer brilliance of so many professional mathematicians, I would also see that in the main they remain blind to obvious deficiencies in the procedures that they adopt without question.
So the first requirement in approaching the nature of the non-trivial zeros is acceptance of the key fact that the number system itself must be interpreted in a dynamic interactive manner (with quantitative and qualitative aspects that are complementary).
Therefore we will first start by considering both of these aspects as relatively independent of each other.
So we have - what I term - the Type 1 aspect that is geared to the quantitative interpretation of mathematical symbols (in the standard analytical type manner using linear reason).
Then we equally have the unrecognised Type 2 aspect where the same mathematical symbols are now interpreted in a new holistic manner (using a more paradoxical circular type reasoning).
All mathematical understanding entails both reason and intuition. In brief the quantitative aspect relates to the (linear) rational and the qualitative directly to the holistic intuitive aspect respectively.
However this intuitive aspect can then be translated indirectly in a circular rational manner (using a paradoxical form of logic).
So the Type 1 aspect entails the relative specialisation in (linear) reason though intuition must also necessarily be involved.
The Type 2 aspect entails the relative specialisation of intuition, indirectly expressed in a paradoxical rational manner, though (linear) reason must also be employed.
Thus both aspects are complementary, with the fullest Type 3 expression entailing the most refined interaction of both intuitive and rational understanding.
Now the key relationship of how the primes are related to the natural numbers (and natural numbers to the primes) can be coherently constructed in isolation through both the Type 1 and Type 2 aspects. However as we shall see, they lead to interpretations - while consistent within their own frames of reference - that are in fact paradoxical in terms of each other!
So it is through the remarkable resolution of this paradox (i.e. of the two way-relationships of the primes and natural numbers) that the non-trivial zeros attain their key importance.
They in fact lead to a new distinctive Type 3 interpretation of number that is inherently of the most dynamically interactive possible, where both quantitative and qualitative aspects are reconciled.
From this new perspective, the very significance of the non-trivial zeros relates to the fundamental requirement of reconciling two modes of interpretation (that are uniquely distinct in isolated terms). Clearly when seen in this light, the true nature of the non-trivial zeros cannot be approached in a conventional mathematical manner. Not alone is this defined solely in Type 1 terms (with mere focus on quantitative interpretation) but in an absolute - rather that relative - independent fashion.
I have stated the direct consequence of this numerous times before without a hint of hyperbole. Not alone can the Riemann Hypothesis neither be proved (nor disproved) in conventional mathematical terms; its true significance cannot even be approached from this perspective.
In fact the non-trivial zeros serve as the fundamental requirement for the most advanced from of mathematical understanding (which I refer to as Type 3).
Both the Type 1 and Type 2 aspects can then be clearly seen as merely relatively independent expressions of what is interactively combined in Type 3.
Sometime in the future, Mathematics will be directly understood in Type 3 terms.
We are a long way off from that day. However the very fact that I am discussing such a development in this blog, indicates that the huge transformation process thereby required with respect to Mathematics has already started.