Monday, November 2, 2015

Clarifying Analytic and Holistic Interpretations of "2" (14)

We have seen that the quantitative independence of primes is expressed through their unique individual status as 1-dimensional numbers (i.e. prior to their combination in multiplication operations).

However the qualitative interdependence of primes thereby arises through such multiplication whereby they achieve a new dimensional status.

Therefore, for example, when in the simplest case 2 is multiplied by 2 i.e. 2 * 2, 2 now acquires, through this multiplication operation, a unique qualitative resonance arising from the dimensional transformation in the nature of units thereby involved.

Put more simply, this dimensional transformation relates directly to the fact that 2 is now a unique factor of the (composite) natural number "4".

Therefore the qualitative nature of the primes relates directly to their existence as factors of the (composite) natural numbers.

However the natural numbers themselves (representing combinations of primes) can likewise attain a new qualitative resonance through being factors of (composite) natural numbers.

So 4 for example which represents the unique prime combination 2 * 2, can itself exist as a factor of a (composite) natural number. For example 4 (i.e. 2 * 2) is clearly a factor of 8. So therefore a distinct qualitative resonance attaches to both 2 and 4 (as factors of 8) in this case.

Therefore again, whereas the quantitative independence of the primes is expressed through the isolated individual status of the primes, their qualitative interdependence is expressed through the natural factors (including primes and composite natural numbers) of the composite natural numbers.

The question then arises as to how precisely we should measure these natural factors.

Though this process is in some respects arbitrary, the various choices that can be validly made will not affect the long term behaviour of theses factors with respect to the natural numbers.

In my own approach, I exclude consideration of the factors of primes. As by definition each prime must contain the number itself and 1 as factors, these can be thereby be considered as trivial (and thereby excluded).

With respect to the (composite) natural numbers, once again I do not include 1 as a factor. However I do include the number itself (as it represents a unique combination of primes).

Therefore if we take for example, the (composite) natural number "12" to illustrate, though it has only 2 (distinct) prime factors (i.e. 2 and 3), it contains - according to these definitions - 5 natural number factors i.e. 2, 3, 4, 6 and 12.

We can now therefore define two complementary interpretations of the relationship of the primes to the natural numbers. The fact that they are complementary, once more testifies to the two-way dynamic interactive nature of both the primes and natural numbers!

The standard (Type 1) definition views the primes and (composite) natural numbers in their independent isolated status (i.e. in 1-dimensional terms as points on the real number line). It then attempts to measure the quantitative frequency of these primes with respect to all the natural numbers.

The alternative (Type 2) definition however views both the primes and (composite) natural numbers in their related (i.e. interdependent) collective status as factors of each (composite) natural number.

It then attempts to measure to quantitative frequency of the (distinct) prime factors with respect to the natural number factors (of each number).

What is crucial to appreciate however is that through quantitative measurements can be made in isolation (of prime frequency) that relative to each other, they are quantitative as to qualitative!

Again this can be illustrated with respect to turns at a crossroads. If one approaches the crossroads heading N, one can unambiguously identify a left turn. If one then changes the frame of reference and now approaches the crossroads in the opposite direction, heading S, one can again unambiguously  identify a left turn.

However both turns are now unambiguously identified as left, which makes no sense with respect to the simultaneous relationship between them.

Thus relative to each other (where both N and S directions are recognised in approaching the crossroads) the tuns must be relatively left and right in relation to each other.

Likewise in mathematical terms, one can within an isolated framework (representing numbers as individual entities) measure prime frequency in a quantitative manner.

Again in isolated terms, one can, now representing numbers as factors, measure prime frequency in a quantitative manner.

However when one simultaneously recognises both aspects, then the two measurements must be quantitative as to qualitative (and qualitative as to quantitative) respectively.

So if we arbitrarily fix (in Type 1 terms) the frequency of the primes in relation to the natural numbers in a quantitative manner, then - relatively - the corresponding frequency of (distinct) prime with respect to natural factors (of each number) is of a qualitative nature.

Now in Type 1 terms, a simple measurement of prime number frequency is given as n/log n.

Dirichlet proved that the average frequency of the natural factors (or divisors) of a number n approaches log n (strictly log n – 2γ + 1 =  log n – .15443.. . ).

And according to the Hardy-Ramanujan Theorem the average frequency of the (distinct) prime factors approaches log log n.

Therefore in Type 2 terms, the ratio of natural to (distinct) prime factors of n ~ log n/ log log n.

If we let log n = n1, then  log n/ log log n can be expressed as n1/log n1.
So remarkably the two formulae  of (1) the distribution of the primes with respect to the natural numbers and (2)  the distribution of (distinct) prime factors with respect to natural number factors (of each number) are of the same format.

According to the formula for (1) if n = 1,000,000 then we would expect approximately 72,382 primes.

Then according to the formula for (2) if a number has 1,000,000 natural number factors,
(i.e. e1,000,000), then then the ratio of natural factors to (distinct) prime factors would be again approximately 72,382.

From the Type 1 perspective we view the primes as the "building blocks" of the natural numbers.

However with respect to the Type 2 perspective, the prime factors are the "building blocks" of the natural number factors.

And because both of these are complementary (i.e. opposite) to each other, this implies that the direction of causation as between the primes and natural numbers (and natural numbers and primes) is bi-directional.

This in fact means that ultimately the behaviour of the primes with respect to the natural numbers (and natural numbers with respect to the primes) is of a purely synchronistic ineffable nature.  

There is one more deeply revealing observation that can be made!

From the Type 1 perspective the relationship between primes and natural numbers is of an additive nature.

From the Type 2 perspective, however, the relationship between prime and natural numbers factors is of a multiplicative nature.

So just as addition and multiplication - when appreciated in an appropriate dynamic manner - are quantitative as to qualitative (and qualitative as to quantitative), the relationship between the primes and (composite) natural numbers is likewise quantitative as to qualitative (and qualitative as to quantitative) respectively. 

No comments:

Post a Comment