Wednesday, July 22, 2015

Zeta Zeros Made Simple (12)

We have seen that in cardinal (Type 1) terms, each natural number can be uniquely expressed as a product of the primes.

Then in complementary ordinal (Type 2) terms, each prime can itself be uniquely expressed as an ordered sequence of natural numbers.

Thus again from the cardinal perspective, 3 is viewed as prime building block with respect to the natural number system.

However, when viewed from the corresponding ordinal perspective, 3 is now viewed as uniquely composed of its natural number ordinal members i.e. 1st, 2nd and 3rd that indirectly can be represented in a quantitative fashion. (Now again the final member which in general terms is the pth of p members is not unique as it is always equal to 1).

However all other expressions for 1st, 2nd, 3rd, 4th,......(p – 1)th members are indeed unique!.

Thus, when we consider both the (Type 1) cardinal and (Type 2) ordinal nature of number, then the relationship between the primes and natural numbers is clearly seen as circular (in a two-way fashion).

Thus again from the cardinal perspective (based on the multiplicative approach), the primes are seen as the unique building blocks of the natural numbers; however when seen from the corresponding ordinal perspective (based on the additive approach)  the natural numbers are seen as the unique building blocks of each prime!

Thus the very nature of the number system, though which both the primes and natural numbers are mutually generated, is of a dynamic nature based on the bi-directional interaction of complementary opposites (that necessarily are of a relative nature).

And as this system approaches perfect synchronicity, the primes and natural numbers are increasingly seen as perfect mirrors of each other in a formless ineffable manner.

Thus if we initially conceive - as is the standard practice - of the primes and natural numbers in an analytic fashion, then the two sets of zeros (Zeta 1 and Zeta 2) can only be properly understood in a corresponding holistic fashion (where the interaction of both quantitative and qualitative aspects are mutually harmonised).

So the very essence of analytic interpretation, once again, is that polar opposites such as quantitative and qualitative are clearly separated in an absolute type manner.

However the corresponding essence of holistic interpretation is that that these same opposites are now fully harmonised in a relative type manner.

So once again in the standard cardinal interpretation, approaching from the additive perspective, the natural numbers 1, 2 and 3, for example, are clearly identified in analytic fashion as absolute quantities (that are independent of each other).

However properly speaking in corresponding ordinal interpretation 1st, 2nd and 3rd are identified in holistic fashion as relative notions (and thereby interdependent with each other).

Thus we can give these a relatively independent quantitative identity (as represented by the 3 roots of 1). However the qualitative interdependence of these roots is then expressed through their collective sum = 0.
In other words, though individually each ordinal position can be given  a relatively independent quantitative identity, collectively all of these positions have but a qualitative identity = 0 (in quantitative terms).

Thus we can see that very interpretation of roots is here of a holistic nature.
And then the Zeta 2 zeros simply represent the same roots with the one common (i.e. trivial root of 1) temporarily excluded.

Then again in the standard cardinal interpretation, this time approaching from the multiplicative perspective, the primes 2, 3 and 5 for example are again clearly identified in analytic fashion as absolute quantities (that are independent of each other).  

However properly speaking the corresponding "ordinal" interpretation (whereby the primes are uniquely combined to form the composite natural numbers) are again identified in holistic fashion as relative notions (and thereby interdependent with each other).

Thus from one perspective, we have the individual primes, as independent of each other; on the other hand we have the collective relationship of the primes that uniquely generates each composite natural number throughout the entire system.

Thus from the analytic perspective, the primes again appear as absolutely independent of the composite natural numbers in a quantitative manner. However from the holistic perspective, the combined relationship of primes is now viewed as fully interdependent with the natural numbers in a qualitative manner.

So now in complementary fashion each Riemann (Zeta 1) zero wonderfully expresses a location (on an imaginary axis) where both the primes and natural numbers are relatively identical with each other in a qualitative fashion. Here now the collective sum of zeros relatively yields an independent quantitative identity that can then be used to completely unravel the distortion in the general (continuous) estimate of prime number frequency.

Therefore the very role of the zeta zeros (Zeta 1 and Zeta 2) is to dynamically enable in a fully coherent and consistent manner, the continual switching in the number system as between both quantitative and qualitative aspects, In other words, the zeta zeros enable this two way switching as between Type 1 and Type 2 aspects, from the notion of number as absolutely independent (of other numbers) to the corresponding notion of number as consistently related to all other numbers in the system (and vice versa).

And as we have seen this switching as between quantitative and qualitative aspects (and qualitative and quantitative) is mediated through the corresponding two-way relationship as between the primes and the natural numbers (and the natural numbers and primes).

In conclusion, I cannot stress strongly enough how reduced - and thereby distorted - is the conventional mathematical approach to the number system.

At its very heart lies the totally unwarranted attempt to exclude all qualitative type considerations from quantitative type relationships. So there is no explicit notion of a holistic aspect to Mathematics that is distinct from the analytic! Therefore there is likewise no realisation that ultimately both analytic and holistic aspects must be combined in an integrated fashion to provide the appropriate framework for either aspect (in isolation).

In truth, the quantitative obsession in Conventional Mathematics is utter madness and yet this this is the "rock" on which such Mathematics is built.

Some time in the future, it will be clearly realised that as quantitative and qualitative notions are fully complementary in mathematical terms, that any attempt to coherently understand the quantitative, must thereby likewise include the qualitative!

However this conversion urgently needs to start now, which can thereby eventually lead to an unimaginable transformation in the entire scientific and intellectual landscape.

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