I am attempting in these blogs to use a variety of related insights to convey the true meaning of the famed zeta zeros and their relationship to the primes and natural number system.
My key contention all along is however that we cannot hope to properly understand the nature of these zeros from within the restricted current paradigm of Mathematics, geared as it is to the formal interpretation of meaning that is of a reduced – merely quantitative – nature.
So it is the interaction of polar aspects of understanding that are (1) internal and external with and (2) quantitative and qualitative with respect to each other that the true meaning of number (and the corresponding nature of prime and natural numbers) properly resides.
Once again, appropriately interpreted in a dynamic relative manner, all mathematical meaning entails the relationship as between what is objectively known (as external) with corresponding subjective constructs of mental interpretation (as – relatively – internal).
Numbers from this perspective inevitably reflect the manner by which they are interpreted.
Therefore when we interpret them in a reduced manner in 1-dimensional terms (as solely objective) they appear to have a fixed absolute identity.
However when we interpret them in a more balanced manner (as reflecting the inevitable interaction of objective data with subjective interpretation) they now appear in dynamic terms as possessing a merely relative identity.
Likewise all mathematical meaning entails the interaction of what is identified as independent in quantitative terms with an interdependent aspect through relationship that is strictly of a qualitative nature.
Once again when we seek to interpret numbers is a reduced manner as solely independent they appear to possess a merely quantitative identity.
However in a more balanced understanding, an inevitable interaction necessarily exists as between two polar aspects that are – relatively - quantitative and qualitative with respect to each other.
Expressed in an equivalent manner all numbers possess both analytic (quantitative) and holistic (qualitative) aspects in a dynamic relative manner.
So the physical identity of numbers (as objective) possesses a psychological aspect (as interpretation) with analytic and holistic aspects respectively.
Whereas the psychological counterpart of the analytic aspect relates to conscious (rational), the psychological aspect of the holistic aspect relates to unconscious (intuitive) understanding respectively.
Indirectly however this holistic aspect can be expressed in rational terms in a circular (paradoxical) fashion.
What is even more remarkable is that this (real) circular aspect can then be further indirectly conveyed in a linear rational fashion that is imaginary!
So once again not only can real and imaginary notions be given a merely quantitative (analytic) interpretation in mathematical terms; equally they can be given a qualitative (holistic) interpretation as imaginary.
Therefore to understand number relationships – and indeed all mathematical relationships – in a more comprehensive manner (with interacting analytic and holistic aspects) we require a complex rational paradigm (in qualitative terms) with both real and imaginary aspects.
Those familiar with Jungian notions – which lends itself very well to holistic mathematical interpretation – will readily recognise the notion of the shadow.
So when a conscious function of understanding is unduly dominant in experience, the shadow unconscious aspect is projected outwards as blind projection.
Therefore though the meaning of the projection is properly of an (unconscious) holistic nature, it is misleadingly confused with specific conscious object phenomena.
We see this for example in relation to spiritual beliefs which can be identified in an unduly narrow fashion with the symbols and rituals of the various religious traditions. Too often the projection of such limited interpretations has served to justify war and persecution on a grand scale.
Therefore, though projections are necessarily embodied in conscious symbols their deeper significance is of a holistic (unconscious) nature.
As Conventional Mathematics rigidly identifies itself in a merely conscious rational manner, this thereby betrays a deep unrecognised shadow whereby it remains steadfastly blind to holistic interpretation at all levels.
This is of paramount significance in relation to appreciation of the true nature of the famed non-trivial zeros of the number system.
Rightly understood these can fruitfully be interpreted as representing the perfect shadow counterpart to the prime numbers.
In other words when we view the prime numbers as separate (in an analytic rational manner) as the independent building blocks of the natural number system, the non-trivial zeros exist in relative terms as the perfect holistic counterpart to this system.
As we have seen this holistic aspect properly relates to the unconscious aspect of understanding that directly manifests itself in an intuitive manner. This can then indirectly be expressed in a circular rational fashion which in turn can be converted in a linear imaginary fashion.
Therefore right away here we have the simple qualitative explanation of why all the non-trivial zeros line up on the same imaginary line!
So the prime numbers as separate independent entities – and this can only properly be viewed in a relative manner – have their perfect shadow number counterpart in the non-trivial zeros, which through their overall collective behaviour represent the interdependent extreme of the relationship of the primes to the natural numbers.
Thus the one extreme of the prime numbers viewed - in relative terms - as analytic and independent, is necessarily counterbalanced by the opposite extreme of the non-trivial zeros as holistic and interdependent!
We could equally say that the discrete particle nature of the prime numbers is necessarily counterbalanced by the continual wave nature of the non-trivial zeros (which are – relatively – analytic and holistic with respect to each other).
However from the equally valid opposite perspective each non-trivial zero has a discrete independent identity in imaginary terms (i.e. as a point on the same imaginary line).
Therefore from this imaginary perspective the discrete independent identity of each non-trivial zero (in analytic terms) is perfectly counterbalanced by the collective interdependent identity of the prime numbers.
So just as we can explain the actual deviations of individual prime numbers from their overall general behaviour through the collective behaviour of the non-trivial zeros (in holistic terms), equally we can explain the actual deviations of individual trivial zeros from their overall general behaviour through the collective behaviour of the prime numbers (again in holistic terms).
So properly understood in a dynamic interactive perspective, both the primes and the non-trivial zeros can be given, in their relevant contexts, both extreme analytic and holistic interpretations respectively.
This points to the fact that ultimately the relationship between both is of a purely relative nature (in phenomenal terms) which equally implies their ineffable origin in an absolute manner.
Once again the great (unrecognised) limitation of Conventional Mathematics is that it cannot – by its very interpretations – view the relationship between the primes and the non-trivial zeta zeros (and the non-trivial zeros and the primes) in a satisfactory manner.
Though the relationship between both is inherently dynamic and interactive with twin analytic and holistic aspects, Conventional Mathematics can only attempt to view both in a fixed analytic fashion. This therefore ultimately only serves to misrepresent their very nature!