We have seen that – appropriately understood – all mathematical understanding necessarily entails dynamic interaction patterns governed by the two fundamental sets of polarities (that underlie phenomenal reality).
So an external aspect is always related to an internal aspect (and an internal to an external). Likewise wholes are always related to parts (and parts to wholes) in a manner that entails the necessary interaction of quantitative and qualitative aspects (in relative terms).
Therefore, a great reductionism pervades present conventional interpretation of Science (and especially Mathematics).
Here the attempt is made to view:
(a) the external aspect of reality in abstraction from the internal in a merely objective type appreciation:
(b) the relationship between wholes and parts in a grossly reduced fashion amenable to mere quantitative interpretation.
However I have been at pains to show in these blog entries how this approach unravels completely in the quest to understand the fundamental mathematical problem with respect to the relationship as between the primes and the natural numbers.
In other words the coherent interpretation of this relationship (which ultimately will always remain shrouded in mystery) requires placing Mathematics within its true dynamic context.
Indeed this then reveals – apart from the long recognised Type 1 quantitative aspects - a hitherto unrecognised shadow interpretation (i.e. Type 2) which is directly based on interpretation of the dynamic interaction as between these fundamental polarities.
So the marvellous revelation can then start to dawn that very mathematical symbol can be given two distinctive interpretations in both an analytic (Type 1) and holistic (Type 2) manner.
Then the true nature of mathematical activity can be understood as the continual interplay of both Type 1 and Type 2 aspects. Through this interaction the Type 1 aspect (as quantitative) is also revealed to possess a Type 2 aspect (as qualitative); likewise the Type 2 aspect (as qualitative) is also revealed to possess a Type 1 aspect (as quantitative). Indeed I used this appreciation to show how an alternative Prime Number Theorem, Riemann Hypothesis amd Zeta (2) zeros can be shown to exist in Type 2 terms!
Now this might appear remarkably reminiscent of the nature of Quantum Mechanics in Physics.
So formerly matter was understood as composed of constituent particles. Then Quantum Mechanics revealed that at the sub-atomic level matter exists in both particle and wave form (in complementary manner).
Then even more strangely it was revealed that the particle aspect is also wave-like and the wave form also particle-like.
However the even deeper connection that has not properly understood is that this behaviour of matter at a sub-atomic level is itself rooted in the prior nature of mathematical activity (which is thereby inherent in such behaviour).
So the complementary nature of sub-atomic matter (indeed strictly all matter) as possessing complementary particle and wave expressions itself reflects the deeper complementarity of both quantitative and qualitative aspects with respect to all phenomena. And this is rooted in the very nature of mathematical activity when appropriately understood in dynamic terms.
Therefore though Physics has been forced to accept at an experimental level the strange reality of quantum behaviour, it still mistakenly attempts to view this through a mere quantitative lens which is deeply misleading. In fact this is the key reason why such findings are counter intuitive in terms of the prevailing paradigm.
So both quantitative and qualitative aspects necessarily interact with respect to all mathematical (and indeed extended scientific) behaviour.
This means that we need to redefine the way we interpret reality as containing both real (analytic) and imaginary (holistic) aspects.
So from this dynamic perspective all physical (and of course mathematical) reality is necessarily of a complex nature. It only appears as (solely) real when we attempt to understand it from a merely reduced quantitative perspective.
So we have so far learnt only to give complex notions (with real and imaginary parts) mere quantitative expression. However they equally can be given qualitative expression and in this context the imaginary part relates to the holistic aspect of interpretation.
Now when we switch briefly to the Riemann Zeta Function we can perhaps appreciate a key problem with respect to its interpretation.
As Conventional Mathematics is explicitly based on mere quantitative interpretation, it thereby can only attempt to understand the complex plane (on which the Zeta function is defined) from this limited (quantitative) perspective. However the qualitative interpretation of the mathematical complex notion implies that both real (analytic) and imaginary (holistic) aspects be both combined.
And as we have seen the remarkable fact is that Conventional Mathematics is powerless to deal with holistic notions (relating to interdependence) in an authentic manner. As its very rationale is based on abstract notions of independently existing entities, it can only approach holistic notions (requiring a genuine appreciation of the nature of interdependence) in a grossly reduced manner.
As is well known the only value for which the Riemann Zeta Function remains undefined in quantitative terms for the dimensional value (where s = 1).
Now as I have stated on so many occasions (though the penny may yet have to drop as it were), the only value for which the Riemann Zeta Function remains undefined in qualitative terms is for the same dimensional value (where s = 1).
Now in this qualitative context 1 here relates to linear (1-dimensional) method of interpretation which defines the very nature of Conventional Mathematics.
Thus the clear implication of this is that the Riemann Zeta Function – when appropriately understood – remains uniquely undefined when approached from the conventional mathematical perspective. And the very reason for this this is that it lacks any holistic notions of interdependence (which is fundamental to understanding the key relationship between the primes and natural numbers).
Now in qualitative terms for any dimensional value of s (≠ 1) a dynamic relationship as between quantitative (analytic) and qualitative (holistic) notions is implied.
Therefore once again the one type of mathematical approach that misses the target completely in this crucial respect is that of Conventional Mathematics (which is so wrongly identified in our culture as the only valid form of mathematical interpretation!)
I have emphasised a great deal the relationship between quantitative and qualitative (which is required to avoid merely reducing wholes - in any context - to constituent parts. This therefore represents one of the fundamental polar sets of relationships (governing all phenomenal reality).
However, the other set relating to internal and external is equally important!
Though this seems to be quickly forgotten when doing Mathematics, any relationship we identify in external (objective) terms necessarily reflects a certain psychological interpretation that is - relatively - of an internal nature.
So, in dynamic terms, we do not have just mathematical symbols, hypotheses, and relationships etc. existing in (absolute) objective terms. Rather we always have a dynamic relationship as between objects and mental interpretations that are relative in nature. And once we recognise that the standard conventional interpretation with respect to mathematical “objects” represents but one of many possible interpretations, then perhaps we can better appreciate how relative in fact is mathematical truth!
Thus once again we have created the mistaken impression that Mathematics represents a somewhat absolute version of “objective” truth through misleading adopting the assumption that only one valid means of interpretation can exist! And this assumption is utterly without foundation.
So an extremely important implication of setting Mathematics in its correct dynamic perspective is the realisation that not alone is it necessarily inherent in all physical phenomena (as an essential means of their encoding) but also that it is likewise necessarily inherent in all psychological phenomena (as a similar means of encoding).
In other words – properly understood in dynamic terms – both the physical and psychological aspects of phenomenal reality are necessarily of a complementary nature.
Therefore Mathematics – as what is most essential to both aspects – likewise necessarily applies to both the physical and psychological aspects of reality in equal measure.
I can say this with considerable confidence. My own applications of Holistic Mathematics initially related more to psychological – rather than physical – reality.
Indeed some 20 years ago I attempted to portray human development (including the advanced contemplative stages) from a holistic mathematical perspective.
So I set out to show how the basic structure of every stage of psychological development is defined uniquely in holistic mathematical terms. It was only later that I began to properly discover how all these stages necessarily had a complementary interpretation in physical terms.
Some of this work indeed has a deep relevance for greater appreciation of the nature of the non-trivial zeros.
Though it is now being accepted that the Riemann zeros do indeed have important implications for certain physical systems, I can see no recognition of their equal importance for the understanding of advanced psycho-spiritual states of development.
And once again because in dynamic terms the physical and psychological aspects of reality are complementary, we will never properly appreciate the physical nature of the non-trivial zeros (without equal recognition of their corresponding psycho spiritual significance).