Monday, October 29, 2012

Incredible Nature of the Zeta Zeros (19)

In order to appreciate what is involved, we need to keep placing the non-trivial zeros of the Riemann Zeta Function in a greatly enlarged mathematical context.

As we have seen conventional mathematical appreciation is based on the abstract notion of number as independent entities.

This is especially the case with respect to the treatment of the prime and natural numbers which are customarily represented on the same number line.

However this appearance of independence hides a deep-rooted problem.

For - as it is well known - the natural numbers (except 1) are uniquely derived in cardinal terms as the product of prime number factors. Therefore from this perspective, there is an important sense in which the natural numbers thereby depend for their identity on the primes.

Equally - though unfortunately not yet properly recognised - each prime number in an important manner is derived from its (internal) natural number members in an ordinal fashion.

So for example the prime number 5 necessarily is composed of a 1st, 2nd, 3rd, 4th and 5th member in ordinal terms.
Furthermore whereas the cardinal notion of number lends itself to quantitative type interpretation, strictly the ordinal notion relates directly to a qualitative - rather than quantitative - meaning.

So from one arbitrary perspective, the natural numbers depend on the prime numbers (for their quantitative identity).

Then from the equally valid alternative perspective, the prime numbers likewise depend on the natural numbers (for their qualitative identity).

Therefore in a crucial manner, both the primes and natural numbers (and the natural numbers and the primes) are mutually interdependent.

So, if we are to give a coherent interpretation of the relationship between both (i.e. primes and natural numbers) we must adapt from the onset a dynamic interactive approach.

In this context it is certainly true that both the primes and natural numbers can be given a relative independent identity (within two distinctive reference systems).

However their mutual interdependence then requires the complementary identity of both these systems.

Thus from the onset, I define three aspects to the number system, which necessarily interact with each other in a dynamic relative manner.

First we have the Type 1 aspect geared directly to the quantitative interpretation of number in cardinal terms.

Secondly we have the Type 2 aspect geared directly to the qualitative interpretation of number in ordinal terms.

Thirdly we have the Type 3 aspect geared directly to interpretation of the mutual interdependence of both the qualitative and quantitative aspects of number.

Therefore again in Type 1 (cardinal) terms, the primes appear as the building blocks of the natural number system in quantitative terms, where each natural number (except 1) can be uniquely expressed as the product of prime number factors.

Then in Type 2 (ordinal) terms, the opposite relationship now applies with the natural numbers appearing as the (internal) building blocks of the primes in qualitative terms, so that each prime number is uniquely expressed by its natural number members (i.e. through obtaining its respective roots).

Finally in Type 3 terms, the mutual interdependence of both the primes and the natural numbers (and natural numbers and the primes) is now recognised in both quantitative and qualitative terms. So in this mutual identity - which is of a necessarily relative approximate nature - no distinction remains with respect to the quantitative and qualitative interpretation of number.

Next, with respect to the Riemann Zeta Function we likewise have three matching interpretations in accordance with the Type 1, Type 2 and Type 3 aspects of number respectively.

The Type 1 aspect can be identified with the standard quantitative mathematical approach to interpretation of the Function. However there is one very important distinction in that, properly understood in dynamic interactive terms, such interpretation is now of a relative - rather than absolute - nature.

I refer to this modified Type 1 aspect of the Riemann Zeta Function,

1^(– s) + 2^(– s) + 3^ (– s) + 4^ (– s) + …….,

as the Zeta 1 function (or more briefly Zeta 1).

To be more precise the standard Type 1 approach is identified with an approach where quantitative and qualitative aspects of interpretation are (formally) separated from each other in an absolute manner. This in effect leads to the reduction of qualitative to quantitative meaning!

The new dynamic manner therefore that I propose for Type 1 is based on separation of these two aspects as relatively independent. This thereby enables a more balanced focus on quantitative meaning (without undue reductionism with respect to the qualitative aspect).

The Type 2 aspect is then identified with a qualitative mathematical approach to interpretation of the Function.

More precisely it is identified with recognition of the necessary interaction of both quantitative and qualitative aspects (with the main focus on the holistic implications of such interaction).

Therefore associated with Zeta 1, we have complex analytic interpretation in quantitative terms (with indirect recognition of a complementary holistic aspect of interpretation).

Then associated with Zeta 2, we have complex holistic interpretation in qualitative terms (with indirect recognition of a complementary analytic aspect
of interpretation).

Zeta 2 is in fact associated with another related Function,

s^1 + s^2 + s^3 + s^4 +……+ s^t.

This is similar to Zeta 1 (turned inside out), so that dimensional values (s) become base quantity values with respect to Zeta 2. Likewise the base values (1, 2, 3,…) with respect to Zeta 1, become the dimensional values with respect to Zeta 2.

Further complementarity also exists in that whereas Zeta 1 is defined as an infinite series, Zeta 2 is defined in finite terms. Also s in Zeta 1 complies with – s in Zeta 2.

Zeta 3 then entails the mutual interaction in interpretation with respect to both Zeta 1 and Zeta 2. Therefore whereas both quantitative and qualitative aspects enjoy a relative degree of separation with respect to Zeta 1 and Zeta 2 respectively, with Zeta 3, they becomes so closely intertwined in understanding as to become identical.

This also highlights the other key aspect of this new enlarged dynamic interactive approach.

Basically it applies to the two key polarity sets.

Therefore from one key perspective we now see all mathematical understanding as necessarily entailing the interaction of both quantitative and qualitative aspects (in relative terms).

Equally from the other key perspective, we see all such understanding as likewise necessarily entailing the interaction of both internal (mental) and external (objective) aspects.

In other words the objective mathematical reality, we wish to portray (in this context the non-trivial zeros) has no strict meaning in the absence of the corresponding mental lenses through which they are viewed.

So to objectively view the non-trivial zeros in an appropriate fashion, we must ensure the complementary nature of their subjective means of interpretation.

Thus the absolute notion of a static mathematical universe existing out there in some unchanging space is but a mistaken illusion that must now be fully discarded, for strictly, mathematical truth has no meaning in the absence of the manner through which it is interpreted!

And by employing a radical new interpretation - that better accords with the experiential dynamics of understanding - the very nature of Mathematics utterly changes.

Finally with respect to the non-trivial zeros in accordance with the three aspects of number interpretation and corresponding three Zeta Functions, we again have three sets of non-trivial zeros arising from solution to the respective equations for s when the value = 0.

So in Zeta 1 the non-trivial solutions represent the infinite set of solutions for s,

ζ(s) = ζ(1 - s) = 1^(– s) + 2^(– s) + 3^ (– s) + 4^ (– s) + ……. = 0.

In Zeta 2, the non-trivial solutions represent the finite set of solutions for s (except 1),

s^1 + s^2 + s^3 + s^4 +……+ s^t = 0 (where t can be any integer, regardless of how large).

Then in Zeta 3, both sets of solutions are so clearly understood as two necessary sides of the same phenomenon, that they approach mutual identity.

To put it simply, the non-trivial zeros relate directly to the interdependent nature of the number system.

And because conventional mathematical interpretation recognises solely (in formal terms) the independent notion of number, we cannot properly interpret their significance from within this restricted context.

However once we accept the dynamic nature of number (containing necessary aspects that are relatively independent and also interdependent with respect to each other) then it becomes obvious that we can focus – in the customary manner – on number as form with the appearance of material independence; equally we can at the other extreme, focus on number as energy where any lingering notions of a (solely) independent identity lose their meaning.

Put even more simply, just as conventional (independent) notions of number are based on an (unchanging) material form, the non-trivial zeros - representing the corresponding (interdependent) notion of number - relate directly to energy states.

As Einstein demonstrated, mass and energy are equivalent; so we have a new form of equivalence in the relationship of the conventional notion of number (as independent form) with the opposite notion of the non-trivial zeros (in their interdependent fusion) as energy states.

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