Riemann showed how to extend the existing Euler zeta function ζ(s), defined for real values of s > 1 to every value (except for s = 1) in the complex plane.
This was indeed a truly important achievement, which greatly enhanced existing knowledge of the primes.
However a crucially significant further development is now required, enabling understanding of the Riemann zeta function to take place in a dynamic interactive holistic manner entailing twin complementary aspects that are analytic (quantitative) and holistic (qualitative) with respect to each other.
Now it takes some considerable training in a very distinctive type of mathematical understanding before one can begin achieve familiarity with holistic notions.
For example - as we have seen - all the primes and natural numbers - can be given a holistic as well as he accepted analytic type interpretation.
For example, "1" is especially important in this regard where it now relates to unambiguous type understanding that is conducted with respect to - literally - 1 pole of reference.
As we have seen external (objective) and internal (subjective) constitute the first of the two sets of fundamental poles, within which all phenomenal understanding - including of course mathematical - takes place in a necessarily dynamic interactive manner. So strictly we can have no objective reality in experience abstracted from corresponding subjective mental interpretation (which are dynamically relative to each other).
However, conventional mathematical interpretation is based on absolute identification with just one pole. So mathematical symbols and relationships are given an absolute type validity (as if separate from interpretation). Alternative interpretation in the form of mathematical constructs is likewise given an absolute type validity. So the underlying assumption is that both interpretation and objective reality can be in absolute correspondence with each other, which strictly speaking is an untenable position (which requires the reduction of one pole in terms of the other).
Perhaps even more tellingly, mathematical symbols and relationships are equally given an absolute type in a merely independent quantitative manner (with the corresponding qualitative pole invariably reduced in merely quantitative terms). So the whole notion of "interdependence" in this conventional approach is crucially distorted. Though directly of a qualitative nature, it can only be given - again - a reduced meaning in a quantitative manner.
And of course both the objective and quantitative aspects are considered in conventional terms as - once more - in absolute correspondence with each other (which requires reducing one pole in terms of the other)
So this reveals the extremely important fact, that in holistic mathematical terms Conventional Mathematics is formally defined by its 1-dimensional approach.
This thereby gives the (misleading) appearance of an absolute mathematical world of fixed symbols and relationships, whereas in truth all such symbols and relationships are conditioned by fundamental complementary poles, that dynamically interact in two-way fashion with each other.
Even this on its own has a huge implication for true appreciation of the Riemann zeta function and its associated Riemann Hypothesis).
From the accepted analytic (quantitative) perspective, the one point at which the Riemann zeta function remains undefined is for s (where s represents a dimensional number or exponent) = 1.
Then from the unrecognised holistic (qualitative) perspective, the one point for which the Riemann zeta function remains undefined is alsofor s = 1. But 1 in this holistic context refers to the linear rational (i.e. 1-dimensional) approach that formally characterises all accepted Mathematics.
And quite simply, neither the Riemann zeta function, nor its associated Riemann Hypothesis can be properly defined in this rigid absolute manner, for in truth the number system is inherently of a dynamic interactive nature (with complementary quantitative and qualitative aspects).
Now of the "higher" natural number dimensions, 2 and 4 are especially important (which indicate how the two sets of fundamental polarities are related to each other in a holistic mathematical manner.
All other dimensions can then be seen as representing unique dynamic configurations with respect to the the two fundamental sets (i.e. external/internal and whole/part.
As we have seen "2" in this holistic mathematical context represents the interdependence of two units, which can be viewed as + 1 and – 1 (in dynamic relation to each other). This implies that + and – signs keep switching, as polar reference frames likewise switch in experience.
In analytic terms, + carries the connotation of positive , whereas – carries the connotation of negative (when attached to a number) and both of these exclude each other in an absolute manner.
In holistic terms, + carries the connotation of "to posit" (which simply means to make known in a conscious manner);
Therefore from a holistic mathematical perspective, all conventional mathematical interpretation is based solely on the + sign (i.e. explicitly in a conscious manner).
By contrast, again in holistic terms, – carries the connotation to "negate" (which entails the reverse direction of the unconscious) i.e. through dynamically negating in a psychological manner, what has already been posited in conscious terms.
Therefore in dynamic terms, both conscious and unconscious operate in a complementary dynamic manner.
One differentiates in experience through positing (+) in a conscious manner; one then integrates in experience, through negating (–) in an unconscious manner.
So in analytic (conscious) terms, both + and – are understood as distinctly separate from each other as is the case with number in Mathematics.
However in holistic (unconscious) terms, both + and – are understood as fully united with each other (as complementary opposites).
So once more as in our crossroads example, when approaches it from just one direction (either N or S) left and right turns and thereby positive e.g left (+ ) and negative i.e. not-left, or right (–) are clearly separate from each other.
However, when one simultaneously views the approach to the crossroads from two directions (both N and S), left and right turns are rendered paradoxical. So both left and right turns (and thereby + and –) are now understood as fully complementary with each other.
So the former understanding relates to analytic (conscious) appreciation in a (linear) rational manner; the latter understanding relates to holistic (unconscious) appreciation in a (circular) intuitive fashion.
Interestingly, the 2 roots of 1 yield + 1 and – 1. So again in standard analytic terms these are understood as separate; however in true holistic terms these are now understood as forming a unified pair (as fully interdependent with each other).
The imaginary notion of the square root of – 1 i.e. i also has a very important holistic explanation.
As we have seen, in dynamic interactive terms the process of (unconscious) negation, requires that conscious phenomena be already posited in experience. In fact this is very similar to the situation at a sub-atomic level ,where an anti-matter particle is brought into contact with a corresponding matter particle.
So we have an immediate fusion here in the form of physical energy. In like manner the unconscious operates through the recognition in experience of a corresponding negative to the positive direction associated with conscious phenomena. So when one realises that the posited world of object phenomena is in relationship with a subjective self, then the negative direction is brought into play. And depending on the clarity of such recognition, a certain fusion (of both polarities) takes place in the form of psycho spiritual energy (i.e. intuition).
As we have seen, intuition - which is directly qualitative in nature - belongs to a distinct holistic realm of experience that should not be confused with what is rational and analytic.
However it is then possible to indirectly express this holistic aspect in an analytic manner (whereby its distinct nature is preserved. And this is done through the imaginary notion.
So form a holistic perspective, the imaginary represents an indirect analytic means of expressing the qualitative aspect (relating to the unconscious appreciation of interdependence).
In this way, both real and imaginary objects (such as numbers) can be treated as independent. However the independence associated with the imaginary in quantitative terms, represents but an indirect means of expressing the true interdependence of these objects (in a qualitative manner).
So to express the unconscious negative (–) direction - which in dynamic terms entails a necessary fusion with the corresponding positive as intuitive energy - indirectly in 1-dimensional fashion (as merely positive), we - literally - take its square root .
We have now an exact corresponding holistic interpretation of the imaginary notion (representing the square root of – 1).
The analytic (quantitative) and holistic (qualitative) aspects of experience are thereby themselves real and imaginary with respect to each other.
Though of course both real and imaginary numbers are now used in conventional mathematical terms, from a qualitative perspective, only the real (i.e. quantitative) aspect is recognised.
However a comprehensive approach to Mathematics must include real and imaginary aspects in both quantitative and qualitative terms. And this requires recognition of the neglected holistic aspect, which is totally excluded from consideration in formal terms.
So the four roots of 1, + 1, – 1, + i and – i, as the linear (1-dimensional) "conversion" of the 4 dimensions, are extremely significant, as they bring together the holistic relationship of the two fundamental sets of polar opposites (that condition all phenomenal experience).
So external and internal are - relatively - positive (+) and negative (–) with respect to each other.
Likewise quantitative and qualitative are real and imaginary with respect to each other.
And real and imaginary aspects, have both positive and negative directions associated with them.
So we can posit objects, including mathematical, in both real terms (as directly conscious) or in imaginary terms as indirectly conscious (representing the unconscious holistic aspect of experience).
And then we can likewise negate these objects in both real and imaginary terms as we move towards a purer holistic (unconscious intuitive) experience.