## Friday, June 15, 2018

### Relationship of Primes to Natural Numbers: internal and external

We saw yesterday how every prime can be given both an independent quantitative identity (as a “building block”) of the natural number system and a shared qualitative identity (as a unique constituent factor of composite natural numbers).

And once again this key distinction as between the analytic and holistic nature of primes is unrecognised in conventional mathematical terms with the latter holistic meaning (in every relevant context) reduced in a mere analytic fashion.

And this problem by its very nature cannot be remedied within the present accepted mathematical paradigm, which is not geared to deal properly with holistic meaning that is inherently of an unconscious nature (though indirectly capable of expression in a paradoxical rational manner).

So again from the analytic perspective, a prime such as 2 is given an absolute unambiguous meaning in quantitative terms.

However when 2 is used as a factor of an even composite number, it then enjoys a relative shared meaning, that is holistic and qualitative in nature.

In other words, through the fact that 2 is thereby shared with other prime factors, it acquires a unique qualitative resonance (through this shared relationship).

So from this relative context, 2 resonates in a distinctive qualitative manner whenever it is used with any other - or combination of other - prime factors.

Thus in the simplest case, 2 for example attains a unique qualitative shared meaning when combined again with 2 (to derive the composite natural number 4).

However it then attains a distinctly unique shared meaning when combined with 3 (to derive the composite natural number 6). And of course this can be continued on indefinitely with 2 - and by extension - every prime number.

Therefore when one properly accepts this new dynamic interactive manner of understanding the primes, the very way one looks on their relationship with the natural numbers is fundamentally changed.

From the reduced analytic perspective, one starts with the primes as pre-given quantitative entities in an absolute manner.

Then one attempts to explain the derivation of the natural number system in one-way fashion as resulting from the unique relationship of prime factors (that are still misleadingly viewed in an absolute quantitative manner).
Thus the natural numbers are themselves then viewed as absolute entities in a merely quantitative manner.

However from the dynamic interactive perspective - where both analytic and qualitative aspects of number are explicitly recognised - it is all somewhat different.

Thus internally each prime is viewed in quantitative terms as composed of independent homogeneous units (that thereby lack qualitative distinction); however equally each prime is viewed in a qualitative manner as composed of uniquely distinct natural number ordinal members that are fully interdependent - and thereby interchangeable - with each other.
So in this sense each unit lacks quantitative distinction.

However from a dynamic perspective, these aspects (quantitative and qualitative) are viewed as complementary in a relative manner.

From the quantitative perspective, a prime is seen as a “building block” of the natural numbers.
However from the corresponding qualitative perspective, a prime is seen as composed of a unique set of ordinal natural number members.

Thus because of this inherent complementarity, both the quantitative and qualitative aspects of the primes can only find their appropriate interpretation within a dynamic relative framework.

Then when we extend this thinking externally to the relationship as between all the primes and the natural number system, again there are two aspects which interact with each other in a dynamic relative manner.

So from the quantitative perspective, we see the primes as the “building blocks” of the entire natural number system (with each composite natural number composed of a unique combination of prime factors).

However from the qualitative perspective it looks very different with the unique spacing as between each prime determined through the combined relationship of the primes with the natural numbers.

And when one reflects on the matter both of these aspects are necessarily interdependent.
Thus we cannot give an exact location to each prime in quantitative terms, without establishing the overall relationship of the primes to the natural numbers (in a quantitative manner).

Likewise we cannot establish an overall relationship of the primes to the natural numbers (in qualitative terms) without knowledge of the individual identity of each prime (in a quantitative manner).

Therefore the clear conclusion from this paradox - which only becomes properly apparent when viewed dynamically - is that we can neither pre-determine the individual quantitative identity of each prime nor the collective qualitative relationship of all the primes with the natural numbers.

Rather both of these features simultaneously arise in a synchronistic fashion (which is ultimately ineffable).

So the highest knowledge of the relationship between the primes and natural numbers is the clear realisation that their ultimate nature is indivisible with both mirroring each other in a perfect manner.