Tuesday, February 23, 2016

Holistic Synchronicity (1)

I have mentioned repeatedly that properly understood, number is inherently of a dynamic nature, entailing the interaction of both Type 1 and Type 2 aspects that are - relatively - quantitative and qualitative with respect to each other. (It precisely mirrors therefore the wave/particle duality that applies to sub-atomic - indeed strictly all -  physical particles).

This entails that the number system is characterised by a series of remarkable synchronous type connections.
And as I have indicated, the true relationship of the primes (to the natural numbers) and in reverse fashion natural numbers to the primes, cannot be properly understood without recognition of these synchronous relationships.

For example, the distribution of the primes among the natural numbers (in Type 1 terms) is characterised by the significant fact that the average gap as between primes ~ log n.

However balancing this finding (in Type 2 terms) is the corresponding finding that the average frequency of natural factors for each number, also ~ log n.

Therefore in Type 1 terms, we are viewing the relationships of (individual) primes with respect to the (collective) natural numbers.

However in Type 2 terms we are viewing the relationship of (collective) primes - in that every combination of factors ultimately entails a relationship as between constituent primes - with respect to each (individual) number.

So the relationship here is clearly of a complementary nature.

Finally, whereas in Type 1 terms we are looking at numbers (as base quantities), in Type 2 terms we are now looking at numbers as representing (dimensional) factors. So again this relationship is of a complementary nature (i.e. as quantitative as to qualitative).

So properly understood, neither of these relationships can be given prior independence but arise in a dynamic synchronous manner with respect to each number (individually) and the number system (as a collective whole).

This again is the reason why the ultimate relationship of the primes to the natural numbers (and the natural numbers to the primes) cannot be successfully analysed in the conventional analytic manner. For this assumes the static independent identity of number, whereas in truth its inherent nature is dynamic and synchronous (entailing the complementarity of opposite reference frames).

In this regard, I suspected that another simple synchronous relationship was lying there waiting to be discovered.

As is well known (Hardy-Ramanujan Theorem) the average frequency of (distinct) prime factors comprising each natural number ~ log log n (with this approximation improving as n increases).

Therefore applying the notion of holistic complementarity as necessarily applying to this finding, this therefore implies that log log n should equally represent the spacing as between natural numbers (based on some appropriate criterion).

On reflection, I would interpret this (necessary) complementarity as follows.

The average frequency of natural factors per number ~ log n. Therefore the average gap (regarding this frequency) with respect to prime factors is given by the log of this number i.e. log log n assuming that the same behaviour governs the distribution of primes with respect to both base numbers (in Type 1 terms) and dimensional factors (in a Type 2 manner).

Thus on the one hand we have a complementary relationship as between the average gap between natural numbers representing primes and the average frequency of natural number factors (as log n).

On the other hand we have a further complementary relationship as between the average gap between natural numbers representing prime factors and the average frequency of (distinct) prime factors (as log log n).  

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