Monday, August 1, 2016

Riemann Hypothesis: New Perspective (8)

We have looked at the apparent fact that all the Riemann zeros lie on an imaginary line (drawn through .5 on the real axis).

Therefore in dynamic interactive terms, the "real" nature of the natural number line i.e. where all real numbers are viewed in linear rational terms (as lying on the 1-dimensional line) is complemented by the "imaginary" nature of a corresponding number line on which all the Riemann zeros are postulated to lie.

An as we have seen - again in dynamic interactive terms - when the interpretation of the "real" number line takes place in the conventional analytic manner (based on the assumed independence of number), then the interpretation of the "imaginary" number line (containing the Riemann zeros) should then rightly take place in a holistic manner (where the corresponding interdependence of number - which appears paradoxical in analytic terms - is now equally emphasised).

Looked at from a psychological experiential perspective, this implies that both conscious (analytic) and (unconscious) holistic appreciation of number be brought into a dynamic equilibrium with each other (the ultimate nature of which is truly ineffable).
Expressed more simply, this implies the balanced recognition of both reason and intuition with respect to all number relationships.

Because of the reduced nature of accepted mathematical interpretation, number is treated solely with respect to its quantitative (independent) nature that is viewed in an absolute manner.

However there is always - inescapably - an unrecognised qualitative aspect to recognition, where one accepts that numbers can be consistently related with each other (i.e. as interdependent with each other).

In conventional mathematical terms this qualitative aspect is blindly assumed to be consistent with the quantitative aspect (in a static absolute manner) which strictly speaking is a completely untenable position.

So once we recognise the equal importance of both independent and interdependent aspects, we must then treat number in a dynamic interactive fashion with complementary aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.

So in this dynamic context, the key issue for Mathematics is that consistency can be maintained as between both quantitative and qualitative aspects.

And this consistency requires than a complementary holistic linear formulation of number exists that  complements the accepted analytic linear interpretation.

And this again is the statement of the Riemann Hypothesis with the additional requirement that the imaginary line - containing the Riemann zeros - passes through .5 on the real axis!

Once again this clearly cannot be proven through conventional mathematical methods (as its axioms already blindly assume consistency).

So its truth depends on acceptance of the twin complementary nature of the number system, which ultimately reflects an initial act of faith in the subsequent consistency of the whole mathematical enterprise.

We also know that for every "positive" expression of a Riemann zero i.e. a + it, a corresponding "negative" expression equally exists i.e. a - it and again the assumption of the Riemann Hypothesis is that a = .5.

Now one might query as to what the second "negative" version of each zero refers!

To briefly recap, I have suggested that the the frequency of the Riemann zeros coincides very closely to the manner in which the natural factors of numbers  accumulate as we move up the number scale. And just as these factors progressively increase (as we move higher up the linear scale to n), likewise it is similar with the trivial zeros moving on a corresponding "circular" scale up to t where n = t/2π.

However, we are referring solely here to the "positive" zeros!

One must remember however that in dynamic interactive terms, switching of reference frames continually takes place.

Therefore when we associate the "positive" zeros with the holistic interpretation of the zeros (that complement the analytic interpretation of the natural number system) then the "negative zeros coincide with the corresponding analytic interpretation of the zeros (that complement the corresponding holistic interpretation of the natural number system).

And of course what is "positive" and "negative" in this dynamic interpretation is merely relative depending on context.

Therefore the fact that all Riemann zeros are postulated to have both a positive and negative equal identity on the imaginary line simply reflects the fact that all these zeros can be given - in dynamic interactive terms - both a holistic and analytic interpretation respectively.

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