Wednesday, September 2, 2015

Zeta Zeros and the Changing Nature of Number (5)

We have seen how in fact there are two notions of a prime.

1) The Type 1 (quantitative) notion where each prime is viewed as an independent building block of the natural number system in cardinal terms.

2) The Type 2 (qualitative) notion where each prime is viewed - by contrast - as uniquely defined by its natural number members in ordinal terms.

So 5 for example as a prime is expressed in Type 1 terms as 51 (where it relates to the base number that is raised to the default dimensional number of 1).

Then in Type 2 terms it is expressed as 1(where it relates to the dimensional  number that is expressed with respect to the default base number of 1).

Though we may initially attempt to isolate these two interpretations (of quantitative and qualitative) in truth they are fully complementary with each other, so that Type 1 and Type 2 meanings arise through the mutual dynamic interaction of both aspects (which I refer to as Type 3).

Again in conventional terms, there is just one interpretation of the relationship between the primes and natural number system with all natural numbers expressed as unique combinations of prime factors.

So for example in conventional terms 6 (as a composite natural number) is uniquely expressed as the product of 2 and 3 i.e. 2 * 3.

However it should now be apparent that there are in fact two complementary interpretations in Type 1 and Type 2 terms.

So from a Type 1 perspective, 6 i.e. 6= (2 * 3)1.

However from a Type 2 perspective 6 i.e. 1= 1(2 * 3)

This entails again that from the Type 1 perspective, the number 6 (as a composite natural number) is uniquely defined by its prime factors in cardinal terms.

However from the complementary Type 2 perspective, 6 (as a combination of primes) is uniquely defined by its unique natural number members (1st, 2nd, 3rd, 4th, 5th and 6th) in an ordinal manner.

Thus when one properly appreciates the complementary nature of both the Type 1 and Type 2 aspects of the number system (relating to quantitative independence and qualitative interdependence respectively), then it becomes quite apparent that both the primes and natural numbers ultimately approach full identity with each other in an ineffable manner!

As we have seen the Zeta 2 zeros (as the non-trivial roots of 1) express the ordinal notion of number that is unique for each prime.
So once again using the prime number "5" to illustrate, 1st, 2nd, 3rd and 4th are uniquely defined in a Type 1 manner by the Zeta zeros in this case as the solutions to

1 +  s+ s2  +  s3  +  s4  = 0. The remaining "trivial" notion of 5th (in the context of 5) reduces to the cardinal notion of 1 (i.e. in cardinal terms 5 is understood as composed of 5 independent units).

So the Zeta 2 zeros therefore express the truly relative (holistic) identity of the ordinal notions!

Now what is astounding - when one comes to clearly realise its significance that the ordinal notions themselves initially derive from the attempt to reduce (in a 1-dimensional manner) what in fact belong to"higher" dimensions (based on the holistic interdependence of each unit).

In psychological terms this means that the ordinal notions, relate directly to the unconscious aspect of understanding which is them made amenable to conscious interpretation through reduction in a linear (1-dimensional) manner.

Therefore though we assume that the ordinal notions directly express the conscious aspect through rational interpretation, this in fact is not the case!

Put another way, the number system - and by extension all Mathematics and related sciences - cannot be properly interpreted in a merely rational (i.e conscious) manner.
So what we have in fact at present with Conventional Mathematics is but a grossly reduced interpretation of the true reality.

So coherent understanding will entail the full incorporation of both conscious and unconscious aspects (in the incorporation of both Type 1 and Type 2 modes). And as we have seen this will incorporate both the analytic and holistic interpretation of all mathematical symbols!

We have in fact two interrelated approaches to the number system.

First we have the Peano system where each number is expresses as through the addition of 1.

Now in my approach I started with each prime expressed in this manner. So again for example,

2 = 1 + 1 and 3 = 1 + 1 + 1

However the second approach then expresses each (composite) natural number as a product of primes.

So in this approach 6 = 2 * 3

Therefore the two approaches quickly overlap with the clue to their reconciliation that - as we have seen, both can be given Type 1 and Type 2 formulations.

So therefore, though we initially confined the Zeta 2 zeros to the n solutions of s

 1 +  s+  s2  + .... + st – 1 = 0, where t is prime,

we can now extend this (through the second formulation) where t is any natural number.

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