The conventional approach to the number system - as we have seen - is based on the attempt to view it in absolute terms as containing abstract independent entities with a static identity.
However if indeed numbers were independent in this sense, then strictly it would be impossible to establish their relationship with other numbers (which clearly implies interdependence).
So once again Conventional Mathematics can only deal with the interdependent notion through a gross form of reductionism which distorts its very nature.
Therefore the one viable way of addressing this issue is to fully accept the inherently dynamic relative nature of the number system in an interactive manner.
Thus from this perspective, numbers do indeed possess a quantitative aspect as independent. However this is now interpreted in a relative - rather than absolute - manner.
Equally numbers possess an (unrecognised) complementary qualitative aspect as interdependent. Whereas the quantitative aspect is associated directly with an analytic, the qualitative aspect is - by contrast - associated with a holistic type interpretation.
Thus the dynamic nature of number arises from the continual interaction of both analytic and holistic type aspects (which is in keeping with the authentic nature of mathematical experience).
So the very paradigm of Conventional Mathematics, which formally seeks to define number relationships in a merely quantitative manner, thereby considerably distorts interpretation of its true nature.
Thus properly conceived in dynamic terms we have both Type 1 (quantitative) and Type 2 (qualitative) aspects with respect to the interpretation of all mathematical symbols.
However this very recognition of two distinctive aspects to mathematical interpretation (with equal importance) then raises the question of consistency with respect to their combined interaction.
It is in this context that the solutions to the Riemann Zeta Function play a crucial role, for properly understood the non-trivial zeros that emerge, represent an unlimited set of numbers (as point singularities on the imaginary axis) where both the Type 1 and Type 2 aspects of the number system approach mutual identity.
So this set of numbers - representing the Type 3 aspect - relates to values where both quantitative (analytic) and qualitative (holistic) interpretations approach identity. Put another way - as befits a dynamic treatment - the non-trivial zeros represent points where both the quantitative (independent) and qualitative (interdependent) aspects of number are identical in a relative fashion.
Thus the non-trivial zeros (in the generation of this new Type 3 system of numbers) serve as the fundamental requirement for ensuring subsequent consistency with respect to all quantitative and qualitative interpretations (when considered in relative separation from each other).
When properly understood therefore, the non-trivial zeros are necessary to ensure the consistent use of axioms at a conventional mathematical level.
Of course this is not properly understood by practitioners at this level, precisely because of the formal adoption of a merely quantitative approach to interpretation!
So when the qualitative aspect is unrecognised, the very question of its consistent relationship with the quantitative does not arise.
So the quest to “prove” the Riemann Hypothesis continues unabated even though an enlarged more authentic framework of interpretation can quickly show why this in fact is not possible!
As we have seen, the relationship between the primes and the natural numbers (and the natural numbers and the primes) is just another way of expressing this ultimately important relationship as between both the quantitative and qualitative aspects (and the qualitative and quantitative aspects) of the number system respectively.
However when our very intuitions are firmly based solely on independent quantitative notions (even when attempting - as with the primes - to establish their relationship with the natural numbers) it is well nigh impossible to adequately convey in an intuitively accessible manner what the non-trivial zeros convey (from such a perspective).
Once again conventional notions are based on clear separation of the quantitative from the qualitative aspect. For conventional mathematicians - especially where serious research work is concerned - even the admittance of any unexplained qualitative aspect would be considered anathema!
Now this clearly poses a massive problem when the very nature of the non-trivial zeros lies at the other extreme of understanding, where both aspects are understood as identical (which again from a phenomenal perspective is necessarily in a relative sense).
So this requires forming an appropriate knowledge of the quantitative (analytic) aspect of number in a standard linear, also forming an utterly distinctive qualitative (holistic) notion of number in a circular manner (that equally applies circular logical notions) and then simultaneously relating the two forms of understanding so closely in understanding as to become identical.
There is even the further complication here, arising from the linear nature of the standard representation of number. This entails the conversion of the qualitative aspect, relating to a circular type understanding of the ordinal nature of number, indirectly in a linear fashion (i.e. on an imaginary scale).
And when you can do all this seamlessly, then you can perhaps appreciate what the non-trivial zeros truly represent!
Now I have demonstrated in many blog entries how this requirement relates to the complementary appreciation of two – relatively distinct – interpretations of the relationship of the primes to the natural numbers.
Again in the Type 1 approach, we interpret the primes in cardinal terms as the building blocks of the natural number system with all (except 1) expressed as a unique combination of prime factors.
So this relates directly to the quantitative notion of numbers as independent entities.
However this relationship is reversed in the Type 2 approach, whereby we interpret the natural numbers in ordinal terms as the building blocks of the primes, with these natural numbers (except 1) - expressed through its corresponding prime number of roots - representing a unique combination of ordinal components.
Thus in terms of the two-way understanding of each approach (Type 1 and Type 2) we face a critical paradox with respect to the fundamental nature of number, which cannot be reconciled in terms of either as considered separate in a relative manner.
So this paradox is only finally resolved in the Type 3 system where both the prime and the natural numbers (and the natural numbers and the primes) are understood as identical with each other (though strictly this position can only be approximated in phenomenal terms).
And once again the Type 3 system - or more correctly the Type 3 aspect of the number system - is represented through the non-trivial solutions for the Riemann Zeta Function.
What this implies is this! When properly realised, the truly complementary nature of the relationship between primes and natural numbers, leads to a directly opposite sequence of causation in the Type 1 and Type 2 interpretations respectively. Then like matter and anti-matter in physical terms, when both are combined, they fuse in a common psycho-spiritual energy (relating to greatly enhanced intuitive appreciation).
So ultimately rational understanding - in the separate Type 1 and Type 2 interpretations respectively of the relationship between the primes and natural numbers - becomes so refined and fleeting that it no longer even appears to arise in experience (through interpenetrating so seamlessly with intuition).
Therefore in truth, the fullest appreciation of the Type 3 aspect of the number system (relating to the non-trivial zeros) will require this new more comprehensive mathematical context, where the most advanced level of (holistic) contemplative awareness can seamlessly combine with a highly refined degree of (analytic) rational understanding.
In this new appreciation of the number system, numbers will no longer be viewed as abstract entities but rather as dynamic entities charged with enormous potential energy.
So at the one extreme, in the Type 1 aspect, numbers certainly appear as static and inanimate - merely quantitative - objects; however at the other extreme of the Type 3 aspect (again represented through the non-trivial zeros) they now appear as dynamic - and indeed - living entities charged with potentially unlimited energy.
Of course the consequent corollary in complementary terms is that the same zeros likewise represent potentially unlimited physical energy states! So, in this sense our very understanding of what is deeply implicit in earliest physical evolution, intimately depends on corresponding advanced psychological understanding, so that what was always deeply implicit in physical phenomena can now be made fully explicit through psychological appreciation of its original nature!
One key implication of this new perspective is that number can no longer be seen as divorced from the physical and psychological domains of reality but rather as necessarily deeply inherent within them as the ultimate explanation of their phenomenal existence.
And it is when this realisation starts to slowly dawn, that we will then begin to properly appreciate the truly enormous potential significance of this new enlarged appreciation of the number system with the non-trivial zeros lying there at its very centre.