In the last blog entry, I contrasted the (static) analytic notion of independence with the corresponding (dynamic) holistic notion of interdependence.
Once again the analytic notion is 1-dimensional in nature, corresponding to interpretation with respect to just one polar frame of reference; the holistic notion by contrast is multidimensional corresponding to interpretation with respect to a number of reference frames simultaneously.
The simplest - and most important - case of holistic interpretation is then with respect to 2 complementary reference frames that are positive and negative in terms of each other. (These 2 dimensions are thereby represented in quantitative terms by the corresponding 2 roots of unity i.e. + 1 and – 1).
Therefore when we holistically view both the random and ordered nature of the primes through the complementary cardinal and ordinal aspects of number, we then can truly realise the ultimate dynamic interdependence of both these notions.
Now such interdependence is directly qualitative, entailing the (unconscious) negation of what has already been (consciously) posited in experience. So, again, at a crossroads initially within single reference frames, one can posit both L and R turns in an unambiguous manner. However when one then simultaneously attempts to combine approaches to the crossroads from both N and S directions, this leads to the realisation of the mutual interdependence of L and R (as both L and R). This then enables us to (unconsciously) negate in experience the unambiguous conscious identification of each turn as either L or R (separately).
Now this latter holistic recognition arising from dynamic negation (of the single posited pole) of recognition is properly of a 2-dimensional nature.
However customary mathematical understanding takes place in an analytic manner within single (posited) poles of reference.
Therefore to indirectly express the qualitative holistic recognition, that is 2-dimensional, in a 1-dimensional manner, we take the equivalent of the square root of – 1.
In other words, we have here the vitally important point here that the imaginary notion itself represents the indirect quantitative means of expressing the holistic qualitative notion of interdependence in a reduced analytic manner.
So all the Riemann (Zeta 1) zeros are postulated to lie on the imaginary line (through 1/2).
This in fact points clearly to the fact that the numbers representing these zeros intrinsically represent the holistic notion of interdependence with respect to both the random and ordered nature of the primes throughout the number system. Now if we were to postulate that a zero could lie off this imaginary line, then we would in effect be maintaining that such a zero was of a qualitatively different nature (than those lying on the imaginary line)!
Once again we saw that when approached from the cardinal perspective in analytic (1-dimensional) terms, both the random and ordered nature of the primes lie at two extremes from each other, with individual primes highly random and the collective relationship of the primes (with the natural numbers) highly ordered respectively.
Then when approached from the ordinal perspective, in reverse manner, both the ordered and random nature of the primes again lie at two extremes from each other with now the individual natural number members of each prime highly ordered and the overall collective nature of the primes highly random.
So when we combine both reference frames (cardinal and ordinal) both random and order notions are rendered paradoxical, for what is random from one perspective is ordered from the other, and what is ordered from one perspective is random from the other.
Thus quite remarkably, the Riemann non-trivial zeros, in a direct sense represent holistic points approaching pure interdependence with respect to both the random and ordered nature of the primes. These points can then be indirectly expressed in an analytic manner on the imaginary axis (through 1/2).
Now it may help to appreciate this more clearly by once again going back to the number system that contains both primes and composite natural numbers.
Each prime (from the cardinal perspective) is of a random nature (with no component factors). However each composite number by contrast is of an ordered nature (with multiple factors).
I showed already how there is an extremely close link as between the frequency of the Riemann (Zeta 1) non trivial zeros and the corresponding frequency of the natural factors of the composite natural numbers.
However we keep moving between two extremes. So we start with 1 and then the two primes 2 and 3 (with no factors). Then suddenly we hit 4 (with 2 factors). We then have another prime 5 (with no factors) and then 6 (with 3 factors).
Therefore with respect to factors (up to 6) we have 2 factor points at 4 and another 3 factor points at 6).
So these factor points occur in a discontinuous manner due to the existence of primes (with no factors).
Thus the Riemann non-trivial zeros can be fruitfully seen as an attempt to smooth out these factor points so as to take account of the random nature of the primes.
Therefore though the overall sum of factors and non-trivial zeros will closely match each other (up to a given number) the factors will occur as discontinuous blocks of numbers (where order is separated from randomness) whereas the non-trivial zeros will occur as single points (where at each location order is closely balanced with randomness).
Put another way, remarkably at the location of each Riemann zero, the primes and natural numbers closely approximate the situation where they can be seen to be identical with each other in a dynamic relative manner. And such identity equates to pure energy states (in both physical and psychological terms).
Expressed again in an alternative manner, at these points the cardinal and ordinal aspects of the number system, likewise approach full identity with each other (again in a dynamic relative manner).
Indeed this situation can be viewed even more clearly from the ordinal perspective.
So if we take for example the prime 3, this has 3 individual members (in a natural number fashion) i.e. 1st, 2nd and 3rd members respectively. Now remember the various prime roots of 1 (except for the trivial case of 1), represent the Zeta 2 zeros!
Indirectly we can give these ordinal members an individual quantitative identity, through the 3 roots of 1 i.e. – .5 + .866...i, – .5 – .866...i and 1.
However the interdependent identity of these roots is expressed through their sum = 0.
So the prime 3, expressed in this sense, is inseparable from its 3 natural number members in ordinal terms.
Thus each member can preserve a certain random individual identity, while remaining closely ordered with the remaining members of its group.
In this way the zeta zeros (Zeta 1 and Zeta 2) can approach perfect balance as between order and randomness, both with respect to the cardinal behaviour of the primes in terms of the number system as a whole and their corresponding ordinal behaviour in terms of each individual prime.