Wednesday, July 27, 2016

Riemann Hypothesis: New Perspective (4)

To properly understand the nature of the Riemann zeros, one requires the ability the be able to look at the relationship between the primes and natural numbers simultaneously from two reference frames (which are paradoxical in terms of each other).

Now a simple example of where this ability is required related to the nature of a crossroads, where one can readily appreciate that left and right turns have a purely relative meaning (depending on context).

Thus when travelling N on a straight path, one can unambiguously define left and right turns at the crossroads; equally when travelling S in the opposite direction towards the crossroads, one can again unambiguously define left and right turns.

Thus within each single polar reference frame (i.e. either N and S considered separately) one can give an unambiguous meaning to L and R turns. However when we now consider these turns simultaneously from both N and S directions, paradox is involved. For what is L (from the N direction) is R (from the S); equally what is R (from the N direction) is L (from the S).

Therefore, in dynamic experiential terms, where both N and S directions (representing polar reference frames) are simultaneously combined, L and R turns then have a merely relative meaning (depending on arbitrary context).

Now with respect to this simple crossroads, most people will have little difficulty in intuitively appreciating the paradox that what is L (from one perspective) is R (from the opposite perspective); and what is R (from one perspective) is L (from the opposite).

However an altogether much greater difficulty is likely to arise, when applying the same kind of understanding to appreciation of the Riemann zeros.

Now here instead of N and S directions, we have quantitative and qualitative type interpretations of mathematical number symbols. Once again the quantitative aspect relates to their separate independence (from each other); the qualitative aspect then relates to their mutual interdependence (with each other). Then in place of L and R turns we have numbers representing base and dimensional values respectively.

So again in general terms with respect to an, a is the base and n the dimensional number respectively.

And both the Type 1 and Type 2 aspects of the natural number system are defined with respect to base and dimensional numbers..

With the Type 1 aspect, the base number varies over all the natural numbers, while the dimensional number remains fixed as 1.

With the Type 2 aspect, in inverse complementary terms, the dimensional number varies over the natural numbers, while the base number remains fixed as 1.

And in dynamic experiential terms - as with the crossroads - both quantitative  and qualitative appreciation of number symbols is involved with respect to base and dimensional values.

Now the zeros of Riemann zeta function entail that

1– s + 2– s + 3– s + 4– s + ..........  = 0

So The Riemann zeros establish a direct relationship as between the natural number system (as base number symbols) with complex numbers of the form a +/– it (as corresponding dimensional number symbols).

And when we start by viewing the base natural numbers in a quantitative manner, then the dimensional numbers (representing the Riemann zeros) must be viewed in a corresponding qualitative manner.

Going back to the crossroads example, analytic interpretation is unambiguous in an absolute linear rational manner. Therefore from this perspective a turn at a crossroads is either L or R.

However, corresponding holistic interpretation of the crossroads requires a circular (paradoxical) type logic, where both turns are simultaneously viewed as both L and R (depending on relative context).

Therefore the clear implication here is that when we initially view the natural number system in a customary analytic, the Riemann zeros must be interpreted in a complementary holistic fashion.

Now again in analytic terms, Type 1 and Type 2 aspects of the number system are clearly separated; however in holistic terms, Type 1 and Type 2 are simultaneously integrated with each other. 

So holistic appreciation of the Riemann zeros requires that one simultaneously views the number system from both its Type 1 and Type 2 aspects. Indeed, crucially the Riemann zeros entail those points in the number system where both Type 1 (quantitative) and Type 2 (qualitative) interpretations directly coincide with each other. 

We could equally say that they represent the points where randomness and order with respect to the number system coincide; we could also say that they represent points where the primes and natural numbers coincide; finally we could say that they represent points where both addition and multiplication coincide. 

Once again these statements have no meaning from the (linear) analytic perspective, which defines conventional mathematical interpretation. However they do they indeed have a very deep significance from the (circular) holistic perspective, which unfortunately is completely overlooked in conventional terms. And the holistic interpretation inherently requires a dynamic interactive mode of appreciating mathematical symbols that combines both quantitative (as independent) and qualitative aspects (as interdependent) respectively. So in this dynamic environment, both independence and interdependence acquire a merely relative meaning (depending on context).
So once more, we cannot hope in the present context to appreciate the nature of the Riemann zeros in a solely analytic fashion (employing independent reference frames).

Thus it is strictly speaking impossible to properly appreciate the Riemann zeros from a mere quantitative perspective without explicit recognition of the qualitative aspect of mathematical understanding!

Therefore the apparent absolute nature of the natural number system in accepted analytic terms,  where all natural numbers can be represented as unique combinations of primes, itself is intimately dependent on an equally important holistic number system (represented by the Riemann zeros) where the dynamic relative independence and interdependence of both its quantitative and qualitative aspects is given expression.

So from this perspective, the Riemann zeros simply represent the (hidden) holistic basis of the natural number system (as analytically interpreted in conventional mathematical terms).

However, as always, in dynamic interactive terms, we can switch reference frames.

So from an equally valid perspective, the Riemann zeros represent the (hidden) analytic basis of the natural number system (as holistically appreciated with respect to the interdependence of all such numbers).

Then from a more comprehensive perspective, it can be clearly seen that both the natural number system and the Riemann zeros are dynamically interdependent with each other with respect to both analytic and holistic aspects, ultimately approaching a state of pure synchronicity in an ineffable manner.

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