I wish to show here as to how there is an intimate connection as between the Zeta 2 Function and the true mathematical meaning of the imaginary number i.
As I have stated on so many occasions all mathematical symbols are used in a reduced (i.e. merely quantitative) manner in conventional terms.
However the true context for such symbols is inherently dynamic and interactive, involving the two-way interplay of both quantitative (independent) and qualitative (interdependent) aspects.
In psychological terms quantitative recognition is of a (rational) conscious nature whereby objects are directly posited in experience.
Qualitative recognition - by contrast - is directly of an (intuitive) unconscious nature, whereby conscious recognition of objects is dynamically negated. So this negation of specific quantitative features of objects is the very means by which corresponding qualitative recognition takes place in a holistic manner.
The very positing of an object as a distinctive unit in (conscious) experience implies the number 1.
Now in an important sense the entire natural number system can be seen as an extension of 1.
So for example 2 (in this quantitative fashion) = 1 + 1.
Then the next number 3 = 2 + 1 and so on.
Therefore by repeatedly adding 1 to each new member , we can continually extend the natural numbers to ultimately include any member of our choosing.
Thus from this quantitative perspective, + 1 can be seen as the essential building block of all (determinate) natural numbers.
However in the dynamics of experience the conscious continually interacts with the unconscious aspect. So the specific (independent) features of object recognition must be temporarily negated in an unconscious manner to enable holistic recognition of a qualitative nature to take place.
So once again, properly understood, number itself is of a dynamic interactive nature entailing aspects that are relatively independent and interdependent with respect to each other.
This relativity with respect to number is of such an intimate nature that we have missed its significance completely.
For millennia now we have attempted to preserve the fiction that number can be successfully understood with respect to its - mere - quantitative characteristics.
However the very notion of relating numbers with each other implies interdependence (which is a qualitative notion).
So the cardinal notion of number (as an independent entity) has no strict meaning in the absence of its corresponding ordinal notion (which implies in any context the relationship between a group of numbers).
Thus cardinal and ordinal are quantitative and qualitative with respect to each other with a relative - rather than absolute - identity.
This is why we need two distinctive aspects of number that dynamically interact. When these are initially dealt with in relative isolation, I refer to them as Type 1 and Type 2 respectively. Then when they are combined as complementary I refer to this as Type 3 understanding of number.
This leads in turn to both Zeta 1 and Zeta 2 Functions (which again can initially be interpreted in relative isolation from each other). However comprehensive understanding of the Zeta Function (Zeta 3) requires that both aspects be fully combined in complementary manner whereby they are ultimately seen as identical with each other in an ineffable manner.
We have seen how from the Type 1 aspect how + 1 can serve (through addition) as the building block of all other natural numbers (2, 3, 4,.....). Once again this Type 1 (cardinal) appreciation relates to the standard quantitative interpretation of number (now however understood in a relative - rather than absolute - fashion).
What is fascinating is that from the perspective of the Type 2 (ordinal) aspect, all the numbers (2,3,4,...) can be combined in a reverse manner, so that their sum = – 1.
What this entails is that in the context of the n roots of 1, the sum of the 2nd, 3rd 4th ... up to the nth root = – 1.
However though these roots do indeed have an indirect quantitative meaning in circular terms (i.e. with respect to the circle of unit radius in the complex plane), this relationship has a qualitative (holistic) rather than quantitative (analytic) significance.
Indeed this pertains to the central nature of the relationship as between whole and part.
From the standard (quantitative) perspective, the relationship between whole and part is reduced to one-directional unambiguous interpretation (of a merely quantitative nature).
Thus from this perspective the parts are contained (quantitatively) in the whole.
However the true relationship is of much more refined subtle nature. So from the quantitative perspective, the parts are quantitatively contained in the whole; however from the complementary qualitative perspective, the whole is contained in each part.
So, when we recognise the Type 2 aspect of number we can quite literally how the whole (in the sum of the non-trivial ordinal roots of unity) = – 1.
So the qualitative aspect of number - literally - implies the temporary negation of the corresponding Type 1 aspect.
Thus where + 1 serves as the quantitative building block of the Type 1 (cardinal) aspect of number, thereby generating the natural numbers, from the complementary Type 2 (ordinal) aspect these natural numbers in turn serve as the building blocks for generating – 1.
In other words, when properly understood number keeps dynamically switching as between its Type 1 and Type 2 aspects in a paradoxical fashion implying both quantitative and qualitative interaction.
Now the key significance of the Zeta 2 Function, is that its non-trivial zeros directly correspond to the various roots of 1 (other than + 1). In other words the Zeta 2 Function is directly concerned with the holistic (qualitative) aspect of number behaviour, which when contrasted with the Type 1, is of a circular paradoxical nature (based on the complementarity of opposites).
So once we explicitly negate the Type 1 aspect (based on absolute either/or logic), we start generating a new type of circular understanding (based on a relative both/and logic).
Thus again using our crossroads analogy, when we employ a single frame of reference (in terms of the direction in which it is approached) a turn is unambiguously either left or right. So this represents linear (1-dimensional) interpretation based on a single frame of reference.
However when we allow the crossroads from two (opposite) directions, a turn is both left and right (depending on context). So we are now viewing left and right in a holistic manner (as interdependent) in a 2-dimensional manner. This two-way interdependence now leads to (circular) paradox, whereas previously each turn was viewed independently in an unambiguous (linear) manner.
As it stands the supreme limitation of Conventional Mathematics is that it is formally based solely on linear (1-dimensional) logic. So when paradoxes arise (as for example with Cantor's theory of sets) it attempts to interpret these through the very same logical framework that generates the paradoxes in the first place.
When one appreciates its nature, one steadily grows in the realisation, that as it stands, Conventional Mathematics has no means for properly dealing (so as to avoid gross reductionism) with the fundamental notion of interdependence (which is qualitative in nature).
And the key relationship of the primes with the natural numbers (and the natural numbers with the primes) cannot be understood in the absence of the true notion of interdependence.
So the mystery of the number system comes down to the manner in which its quantitative and qualitative aspects are mediated - in relative two-way fashion - through the primes and natural numbers!
As we have seen interdependence at a minimum requires incorporating 2-dimensional - rather than 1-dimensional understanding.
In other words it requires both positive (+ 1) and negative (– 1) polarities in relation to each other through the complementarity of both quantitative and qualitative aspects of understanding.
Now the holistic aspect does find its way however into Conventional Mathematics in a fascinating reduced manner.
In other words the very use of the imaginary notion i, represents the attempt to express the (unconscious) holistic notion of negated (unitary) form in a reduced linear manner. So just as in quantitative terms, when a number is expressed with respect to the power of 2, we express it in 1-dimensional terms by obtaining its square root, likewise with – 1.
So from a qualitative perspective, the imaginary number i expresses the indirect attempt to express meaning which is properly of a holistic nature (in accordance with the 2nd dimension) in a reduced linear manner.
Now clearly this is extraordinarily useful in quantitative terms. However its equally important qualitative significance is completely missed in conventional mathematical terms.
For example the fact that Riemann's Zeta Function is defined with respect to the complex plane clearly implies both real (analytic) and imaginary (holistic) aspects of understanding.
From a deeper perspective. it can then be seen that the Function is designed precisely to demonstrate the intimate links as between the complementary analytic (quantitative) and holistic (qualitative) aspects of number i.e. its cardinal and ordinal aspects.
And this key insight which then quickly unlocks the very nature of the Riemann Hypothesis (as the condition for the ultimate identity of both aspects) is not accessible from a conventional mathematical perspective.
So once again not alone can the Riemann Hypothesis neither be proved (nor disproved) within the present mathematical paradigm; more importantly, it cannot be properly understood from this limited perspective.