Friday, June 26, 2015

Zeta Zeros Made Simple (5)

We now come to the discuss the crucially important role of the primes with respect to the natural number system.

In the conventional mathematical approach - that is solely geared to absolute type appreciation of the quantitative (Type 1) aspect of the number system, the primes are customarily viewed as the essential building blocks of the natural numbers.

Thus - apart from 1 - every natural number is either prime or composed as a unique product of primes.

So for example 6 = 2 * 3 (and cannot be derived from any other combination of prime factors).

Therefore from this quantitative perspective the direction of causation with respect to the primes and natural numbers is strictly one-way, with the natural numbers necessarily derived from the primes.

However, even very early on, I could see a big problem with this way of thinking, as the natural numbers are already implicit in the very use of the primes!

For example, if I refer to 7 as a prime (in a cardinal sense) then this implies that I can meaningfully distinguish its 1st, 2nd, 3rd, 4th, 5th, 6th and 7th terms.

Thus the very definition of a prime in cardinal terms, implicitly implies an ordered set of natural numbers (in ordinal terms).

So we insist on viewing 7 as an independent prime building block of the (cardinal) natural numbers; however it is already implicitly defined in terms of a set of (ordinal) natural numbers.

This therefore suggests that once we incorporate ordinal - as well as cardinal - meaning, that the relationship as between primes and natural numbers is necessarily bi-directional in a circular paradoxical manner.

Seen from another angle, implicit in the very listing of the primes (in cardinal terms) as 2, 3, 5, 7, a corresponding set of natural numbers (in ordinal terms) as 1st, 2nd, 3rd, 4th,...

We could equally say from the opposite perspective, that implicit in the very listing of the ordinal natural numbers 1st, 2nd, 3rd, 4th,..., are the corresponding prime numbers 2, 3, 5, 7,...., (which are necessary to drive the natural numbers).

Therefore from this more comprehensive number perspective (incorporating both cardinal and ordinal notions) the relationship between the primes and natural numbers is necessarily one of two-way interdependence.

In fact this is just another way of stating once again that the number system necessarily entails both quantitative and qualitative aspects in dynamic relationship with each other. Thus, if we designate the cardinal numbers in a quantitative manner, then the ordinal numbers are thereby - relatively - of a qualitative nature.

When on reflects for a moment on this matter, it should appear somewhat obvious.

So again for example when refers to "3" in a cardinal manner, it thereby is interpreted in quantitative terms as independent of other numbers. However when one then refers to 3rd (in an ordinal fashion) it necessarily entails a qualitative relationship (with respect to a corresponding group of numbers).

However, when one looks at the conventional treatment of ordinal numbers, this qualitative aspect is rarely if ever referred to in explanation. Rather the more neutral term of "rankings" is used, thereby subtly reducing the qualitative aspect in quantitative terms.

However when one properly grasps the true qualitative nature of ordinal numbers with respect to cardinal, then one realises that - rather than rigid absolute entities defined in a merely quantitative manner - the number system by contrast necessarily represents a two-way dynamic interactive process entailing both cardinal and ordinal aspects (that are - relatively - quantitative and qualitative with respect to each other).

This then leads to a new enhanced appreciation of the true relationship of the primes and natural numbers, which is now seen to ultimately operate in a two-way synchronous manner.

So with respect to the origins of the number system, the primes and natural numbers ultimately "cause" each other. In other words, they mutually arise in a dynamic synchronous manner.

And the importance of this relationship is that through the interaction of the primes and natural numbers, both the quantitative and qualitative aspects of the number system are mutually transmitted (in a two-way interactive fashion).

One can hardly over state the significance of this realisation, for it opens up the way for a vastly enhanced appreciation of the true significance of the number system.

So far we have sought to understand the number system in a merely quantitative manner (in a greatly distorted reduced manner).

However we are now on the verge of appreciating that the true role of number equally embraces the quantitative as well as quantitative realms. And we are in a position to show how these are truly related (which will greatly enhance appreciation of both aspects).

So from this new dynamic perspective. the number system is now seen as inseparable from the course of all phenomenal evolution (in quantitative and qualitative terms). Indeed, quite simply, all phenomenal form (in space and time) is ultimately encoded in a dynamic number pattern as its most fundamental "genes".  In this sense, manifest evolution simply represents the decoded nature of number!

Another related insight regarding the nature of primes, was equally to lead to this new dynamic appreciation of their nature.

As I have stated before, every mathematical symbol with a standard analytic interpretation can equally be given a corresponding (unrecognised) holistic interpretation.

Some years ago, when I was seriously developing my holistic mathematical understanding, I spent a considerable time establishing all the holistic equivalents to the recognised number types e.g. prime, natural, positive and negative, rational, irrational (algebraic and transcendental), imaginary, complex etc.

It struck me forcibly that the holistic notion of "prime" bore a very close relationship with the nature of what we identify as "primitive" in psychological terms.

Thus a primitive instinct represents a direct confusion of conscious with unconscious type meaning i.e. where the holistic desire for overall meaning is confused with immediate localised phenomena in experience.

From a mathematical perspective, this could be expressed as the confusion of holistic and analytic (or likewise quantitative and qualitative).

Therefore unraveling such confusion in coming to realise the true nature of the primes, would be inseparable from the psychological process, whereby both conscious and unconscious become properly differentiated from each other (before being properly integrated).

Thus in mathematical terms, the first stage of conscious differentiation relates to Type 1 (analytic) mathematical interpretation.

The second stage of unconscious differentiation relates to Type 2 (holistic) mathematical interpretation.

The third stage (by which conscious and unconscious can be maturely integrated relates to Type 3 (radial) mathematical interpretation.

And this is the stage that we are now at, in attempting to unravel the mystery of the primes and zeta zeros.

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