Thursday, November 12, 2015

Clarifying Analytic and Holistic Interpretations of "2" (19)

To put it simply, the Riemann (Zeta 1) zeros represent the (hidden) holistic basis of the cardinal natural number system (as explicitly understood in the standard analytic fashion).

From a corresponding psychological perspective, these zeros represent the (hidden) unconscious basis of the same natural number system (as explicitly understood in a conscious rational manner).

Then in a parallel fashion, the alternative (Zeta 2) zeros represent the (hidden) holistic basis of the ordinal natural number system (as again explicitly understood in the standard analytic fashion).

And from the corresponding psychological perspective, these zeros represent the (hidden) unconscious basis of the same ordinal natural number system (as explicitly understood in a conscious  rational manner).

However because these two sets of zeros (Zeta 1 and Zeta 2) are dynamically complementary with the analytic aspects in a two-way interactive fashion, they are in truth both simultaneously involved as the hidden holistic basis with respect to the natural number system (in cardinal and ordinal terms).

All of this leads to the need for the most fundamental change possible with respect to the customary interpretation of number.

Rather than the number system existing in an unchanging absolute fashion in some universal "mathematical heaven", it has an inherently dynamic interactive nature, entailing both quantitative and qualitative aspects in relative fashion.

The ultimate nature of this system is synchronistic in a holistic manner, approaching a pure ineffable identity. It is then only at the opposite extreme of analytic type interpretation (within isolated polar reference frames) that it approaches the appearance of being composed of rigid number objects of form!

Crucially, as the true interpretation of number, with respect to its dynamic interactive nature, entails the balanced interaction of both conscious (rational) and unconscious (intuitive) aspects of understanding, it is simply not possible to appreciate its nature within the accepted confined of present Mathematics. As such Mathematics is formally interpreted within a merely (conscious) rational type framework, it inevitably reduces the qualitative aspect in a quantitative manner.

It thereby reduces the inherently dynamic nature of the number system in a limited static manner!

As the nature of number is so fundamental, underlying everything else in phenomenal creation, the greatest revolution yet in our intellectual history now awaits, where the unconscious aspect of understanding needs to be explicitly united with its conscious counterpart.

This revolution, which will require a great deal of time to unfold, will then not have only profound implications for Mathematics but also most importantly for all related sciences and for society generally, gradually leading to a much more comprehensive world view that at present would be impossible to envisage.

One may validly enquire as to the status of the Riemann Hypothesis from this new dynamic perspective on number.

In fact, its very nature in now transformed in a manner that cannot be appreciated within the conventional mathematical framework.

I have commented before many times on the important fact that from the analytic quantitative perspective, the Riemann zeta function is undefined at just one point, i.e. where s (representing a power or dimensional number) = 1.

The holistic counterpart of this is that the Riemann zeta function likewise is undefined for s = 1, where it now defines in qualitative terms the 1-dimensional approach to mathematical understanding.

And as Conventional Mathematics is formally defined in a 1-dimensional manner (i.e. within isolated polar reference frames) then the true significance of the Riemann zeta function simply cannot be appreciated from this standpoint.

In other words, properly understood, the Riemann zeta function maps out numerical values that correspond to both analytic (quantitative) and holistic (qualitative) interpretations of number respectively.

Basically values for the function on the RHS for values of s > 1 correspond to analytic interpretation that are mapped with corresponding holistic values on the LHS for values of s (< 0).

So through the Riemann functional equation, values of  ζ(s) can be mapped with corresponding values of  ζ(1 –  s).

This means for example that ζ(2) is mapped with ζ(– 1).

Now  ζ(2) = 1 + 1/4 + 1/9 + 1/16 + ..... which in the standard logic of numerical interpretation, does indeed converge to a finite value (π2/6).

Then  ζ(– 1) = 1 + 2 + 3 + 4 +.... which again in the standard logic of numerical interpretation diverges (to an infinite value).

However in terms of the Riemann zeta function,  ζ(– 1) = – 1/12!

So clearly, a different form of interpretation is required to give meaning to this value, which in fact now corresponds to a holistic - rather than analytic - value!

And because analytic and holistic are complementary opposites, this entails that where a series diverges (in the standard analytic manner) it converges (in a corresponding holistic manner) and vice versa.

Now the value for s = .5, assumes a central importance here, for it entails that both ζ(s) and ζ(1 –  s) are now identical.

Put another way, this entails that both the analytic (quantitative) and holistic (qualitative) interpretations of number coincide.

The additional requirement that both ζ(s) and ζ(1 –  s) = 0 requires that all of the non-trivial zeros lie on an imaginary line through .5.

Therefore we can now reinterpret the Riemann Hypothesis as the central requirement for ensuring the coincidence of both the quantitative and qualitative aspects of number. In other words it serves as the fundamental requirement for ensuring the consistency of the natural number system with respect to both cardinal and ordinal interpretation!

Now, when one reflects on this for a moment, it should be obvious that there is no way that this proposition can be proved (or disproved) in the conventional mathematical manner.

Because the qualitative aspect is reduced (in every context) to quantitative interpretation, within conventional mathematical axioms, this means that the consistency of quantitative with qualitative aspects is in effect already assumed through the use of these axioms.

Therefore one cannot hope to prove (or disprove) a prior assumption, using an axiomatic approach that already implicitly contains that prior assumption (literally as an act of faith).

In fact, properly understood, the true significance of the Riemann Hypothesis is much greater than what is presently realised, for it in fact starkly reveals the enormous limitations imposed by the present conventional mathematical approach (which attempt to formally exclude qualitative considerations from all its procedures).

Thus the Riemann Hypothesis - when properly interpreted - expresses the mysterious requirement for the coincidence of quantitative with qualitative type interpretation of mathematical symbols.

From a psychological perspective, this relates to the requirement that both conscious (rational) and unconscious (intuitive) aspects of understanding can be synchronised in a consistent manner with respect to all mathematical interpretation.

This cannot be proved for conventional proof is already implicitly based on the assumption (that such synchronisation of meaning is already assured).

Thus correctly understood, all mathematical interpretation is implicitly based on an initial massive act of faith in the consistent correspondence of both the quantitative and qualitative meaning of its symbols !

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