As we have seen natural numbers can be given two distinct definitions (in relative isolation from each other):
1. In cardinal terms as a collective homogeneous group (i.e. where individual units have no unique identity). This relates to the analytic (quantitative) aspect of number which I refer to as the Type 1 interpretation.
2. In ordinal terms as the relationship between unique individual members of a number group (where the collective group - from this perspective - has no distinct identity). This relates to the holistic (qualitative) aspect of number which I refer to as the Type 2 interpretation.
In actual experience, both of these aspects of number continually interact. Strictly speaking therefore, we cannot form an analytic appreciation of the nature of number without its corresponding holistic aspect; likewise we cannot form a holistic appreciation of number without its corresponding analytic aspect.
Ultimately both analytic and holistic aspects are of a purely relative nature.
Though such pure relativity relates to an absolute ineffable state, in the phenomenal experience of number it can only be approximated.
And properly understood this is what the Riemann Hypothesis relates to as the closest approximation in phenomenal terms to the ultimate ineffable state where both the analytic (quantitative) and holistic (qualitative) aspects of number are fully identical.
I have already explained in an earlier blog how the non-trivial zeros - corresponding to the Type 1 interpretation - represent in fact the perfect shadow number system to the primes (that comprise the natural number system - except 1 - in cardinal terms).
Equally the non-trivial zeros – corresponding to the Type 2 interpretation – represent the perfect shadow number system to the natural numbers (that comprise each prime group - except 1 - in ordinal terms).
Properly understood therefore, both the cardinal and ordinal numbers have no meaning in the absence of their corresponding shadow (wave-like) number systems as represented by the non-trivial zeros in both cases.
So both these particle-like features of number (as independent) and their corresponding wave-like properties (as interdependent) are simultaneously co-determined in a manner that is ultimately ineffable.
Put another way both the Type 1 and Type 2 aspects of the number system have a particle and wave-like identity reflecting their analytic and holistic aspects respectively.
From a psychological perspective this implies that all numbers have both conscious aspects (as analytic) and unconscious aspects (as holistic) respectively.
Now just as in quantum physics through interaction, every particle possesses wave-like properties and every wave particle-like properties, likewise with respect to number, the analytic (Type 1) interpretation of number possesses holistic (Type 2) properties and the holistic (Type 2) interpretation analytic properties respectively.
This means - as I have repeatedly stated – that we cannot possibly hope to understand the nature of the Type 1 (quantitative) non-trivial zeros – to which the Riemann Hypothesis directly relates - in the absence of Type 2 (qualitative) interpretation.
Equally, we cannot possibly hope to understand the nature of the Type 2 (qualitative) non-trivial zeros - to which an unrecognised counterpart Riemann Type Hypothesis relates - in the absence of Type 1 (quantitative) interpretation.
And I have indicated in several of my blogs what such a quantitative interpretation entails.
The basic message is that the inherent nature of the number system (and indeed of all mathematical activity) is of a dynamic relative nature entailing the interaction of both its quantitative (analytic) and qualitative (holistic) aspects.
It therefore cannot possibly be understood within the current mathematical paradigm, which is geared formally to - mere - quantitative interpretation!
In a nutshell this is the essential message underlying all my blog entries and urgently needs to be grasped!
So every mathematical symbol with a standard (Type 1) quantitative interpretation in conventional mathematical terms possesses an equally important (Type 2) qualitative interpretation from a holistic perspective.
Properly understood therefore mathematical meaning necessarily represents the combined interaction of both perspectives.
Thus once again to illustrate the number 2 has a standard quantitative interpretation in linear mathematical terms (representing the separate collective whole identity of a homogeneous set of units in a cardinal manner).
However the number 2 equally has a qualitative interpretation in circular terms (representing the interdependence of the two unique individual members of a group in an ordinal manner).
Likewise the operations + and – have a recognised quantitative meaning in standard mathematical terms; however they equally have a - largely unrecognised - qualitative meaning! So + in this context relates to the direct positing of meaning in a (conscious) rational fashion. – relates to the corresponding negation of such rational meaning in an (unconscious) intuitive manner.
So deeply implicit in the holistic notion of 2 is the interaction of two polar directions of experience that are positive and negative with respect to each other.
Indeed deeply implicit in the holistic notion of 1 is the existence of just one direction of experience that is positive (in an absolute manner).
So linear (1-dimensional) rational understanding with just one direction, necessarily leads to the reduction of holistic to analytic (and unconscious to conscious) interpretation respectively.
Once again though indeed it represents an important special case, in holistic terms, 1 represents the only dimension that interprets mathematical meaning in an absolute manner.
For all other numbers (as dimensions) a dynamic relationship as between analytic and holistic (conscious and unconscious) is implied.
Once again this is the Type 2 explanation as to why the Riemann Zeta Function remains uniquely undefined for s = 1!
As the Function - when properly interpreted - relates to the relationship as between analytic and holistic type meaning, it thereby remains undefined in conventional (1-dimensional) terms, where analytic type meaning is solely recognised in an absolute manner!
Therefore it is ultimately futile - not only in trying to prove - but more importantly in even trying to understand the Riemann Hypothesis from this limited perspective.