We will briefly return to a key issue that I raised earlier, i.e. that the ordinal notion of a number has a purely relative meaning (depending on context).
In a grouping of two objects, the 2nd member might seem unambiguous in meaning, in the sense that no other interpretation of 2nd can be given in this context. And indeed this is precisely how the dimensional number 2 is defined in the circular number system.
So 1^2 in this context (in qualitative terms) corresponds with the quantitative notion of 1^(1/2) = - 1.
And if we are to preserve the true qualitative meaning of the ordinal nature of a number (associated with interdependent relationships with other members) then we must thereby use the qualitative meaning of this circular result (i.e. - 1) in interpretation of the 2nd member (of a grouping of two members).
So what does this mean?
Well! Obviously if we have two objects in a group (in cardinal terms) we must first recognise these objects as relatively independent in a quantitative linear manner(using 1-dimensional interpretation).
However the ordinal recognition of the 2nd member (in relation to the first) requires establishing an interdependence as between the two objects (in qualitative terms).
So quite literally this entails the temporary negation in understanding of the 2nd object as independent (i.e. - 1).
In other words we start with two independent units that are posited in rational conscious terms i.e. + 1 and + 1. At this stage the very notion of 2 has no strict meaning in the recognition of each separate individual unit.
However once we can then negate the separate existence of one of these units (thereby establishing interdependence with the other unit), the notions of 1st and 2nd are thereby involved. So 2 in this ordinal relationship context of interdependence implies the negation of 1 (as independent).
Now all of this might seem initially strange to anyone approaching number in conventional mathematical terms!
However this is precisely the point for as remarkable as it might seem, no coherent interpretation of the ordinal nature of number is possible within this conventional framework. Because of its merely linear basis of interpretation, qualitative notions of interdependence are necessarily reduced in a quantitative manner. And when one investigates the implications of this approach, deep confusion arises!
So whereas we can give an unambiguous interpretation to the ordinal meaning of 2 (within the context of 2 objects) this inevitably changes when we switch to 3 or more objects.
In other words the meaning of 2nd in the context of 3 objects is necessarily distinct from the meaning of 2 within the context of 2 objects!
And of course we can can extend this indefinitely so that 2 (in an ordinal sense) can be given an unlimited number of possible interpretations. We could equally express this point by saying that the 2 has an unlimited set of possible qualitative interpretations (depending on the group context in which the notion of 2nd arises)!
Now to solve this problem of unlimited ordinal interpretations (with respect to number) we need to recognise a corresponding unlimited set of interpretations associated with each number (as representing a dimension). This then enables us, through the reciprocal nature of its root, to obtain a numerical value in the circular number system.
So to solve the problem of the ordinal nature of 2 (in the context of a group of 3 objects) we obtain 1^(2/3) to obtain the 2nd root of 3 (i.e. - .5 - .866i).
And we can see that this involves both real and imaginary aspects, which likewise entails a unique configuration with respect to independence and interdependence.
So the ordinal and cardinal nature of number is inevitably mixed in a dynamic interactive manner which explains the complex nature of results that generally arise.
To sum up this section therefore, the circular system of numbers (corresponding in quantitative terms to the various roots of 1), provides in direct complementary fashion the appropriate means for the ordinal interpretation of number.
So underlying the natural number system 1, 2, 3, 4, 5,.... (when interpreted in an ordinal manner) is a unique harmonic system of complex numbers. And these numbers are derived with respect to corresponding roots of 1 (derived from these numbers in quantitative terms). However then associated with the quantitative expression of each of these numbers is a corresponding qualitative meaning, that allows for their appropriate ordinal interpretation.
The importance of the prime numbers in this context is that they provide (among the natural number set of its ordinal members) unique circles of interdependent relationships!
So once again 5 for example is a prime number. Therefore with respect to the 5 roots of 1, a unique set of complex numbers (on the circle of unit radius) can be generated which define the appropriate qualitative interrelationship of these 5 members!
Therefore we can perhaps see, that once we allow for a distinctive ordinal interpretation of number (in dynamic relative terms) that we necessarily must recognise a dual structure to the number system (representing - in quantum mechanical terms - both its particle and wave aspects respectively).
However since the ordinal nature of number is necessarily related to its cardinal nature, once we accept the dynamic relative nature of the number system (from an ordinal perspective) we must do likewise from a cardinal perspective.
This therefore entails that the cardinal number system must likewise possess two aspects i.e. a particle like system of the recognised counting numbers and an accompanying wave system (representing the interdependence of primes among the natural numbers).
So the non-trivial zeros of Riemann Zeta function are the direct counterpart (from the cardinal perspective) of the various prime numbered roots of 1 (on the ordinal side).
The clear implication of all this is that the Riemann Zeta Function in fact establishes the precise relationship as between both the cardinal and ordinal aspects of number. Or to put it in the terms that I customarily use, it establishes the precise relationship as between the quantitative and qualitative aspects of number! And the Riemann Hypothesis is the essential condition required to establish the identity of these two aspects!
And once again I must state bluntly, that as the conventional mathematical approach allows for no distinctive ordinal treatment of number, it cannot thereby properly convey either the true nature of the Riemann Zeta Function or its associated Riemann Hypothesis!