Saturday, November 21, 2009

Significance of Riemann Hypothesis

I had an intuition last night regarding the significance of the Riemann Hypothesis which was of a faint - yet truly incredible - beauty.

Following intense reflection on the Hypothesis over the past two years, I have come to the firm conclusion that its true relevance has been missed.

Let me attempt to put this once again into perspective.

A complete approach to Mathematics entails three distinct aspects:

1) Conventional (Analytic) Mathematics that is geared directly to the quantitative interpretation of mathematical symbols.

2) Holistic Mathematics that is geared directly to the qualitative interpretation of these same symbols.

3) Comprehensive Mathematics - what I term Radial Mathematics - which simultaneously combines both the analytic and holistic aspects.

Though the Riemann Hypothesis does not serve as a necessary condition for the existence of these three aspects, it does however provide a remarkable demonstration of their relevance.

Putting it bluntly, the Riemann Hypothesis cannot be resolved i.e. either proven or disproven with sole reference to the first aspect i.e. Conventional Mathematics. Even less can it be fully appreciated from this perspective.

My interest in the Riemann Hypothesis initially stemmed from a desire to reveal the signficance of the - so called - "trivial zeros" of the zeta function.

Then in a wonderful moment of insight, it became clear to me that the true explanation for these zeros could be provided not through conventional quantitative, but rather through corresponding holistic qualitative interpretation.

As often demonstated, conventional understanding of mathematical symbols is based on rational interpretation using linear (either/or) logic entailing the clear separation of opposite poles of understanding (such as internal and external).

However holistic understanding is based on paradoxical interpretation, using circular (both/and) logic of the complementarity - and ultimate identity - of these same poles.

Thus the interpretation of the "trivial zeros" requires the use of circular - rather than linear - logic for satisfactory resolution.

This qualitative mathematical interpretation then has immediate important applications in both psychological and physical terms (that are complementary).

So in psychological terms, the zeros (corresponding to negative even integers) relate to the higher dimensions (representing pure spiritual emptiness). These advanced stages of integration can be fruitfully equated with the higher contemplative states of development.

In complementary physical fashion - though I have less intuitive grasp of this aspect - the zeros would correspond to successive dimensions representing pure states of material energy.

Therefore in a marvellous manner the Riemann Zeta Function requires the twin use of both analytic (quantitative) and holistic (qualitative) aspects for adequate interpretation.

One aspect corresponds directly with standard analytic understanding i.e. for values of s > 1 in the zeta function (where s represents the power to which each of the natural numbers in the series is raised).

Another aspect corresponds directly with qualitative holistic interpretation i.e. for values of s < 0.

Finally the third aspect corresponds directly with radial interpretation i.e. for complex values of s.

Once again the significance of the non-trivial zeros in standard analytic terms, has been extensively examined with reference to the general distribution of prime numbers.

However, what is not properly recognised is that a corresponding holistic interpretation exists for the non-trivial zeros. This then has immense potential significance for understanding of the most fundamental aspects of both physical and psychological reality.

In my work on the Riemann Hypothesis available at "The Riemann Hypothesis",
I have investigated in some detail the psychological significance of the non trivial zeros relating to both quantitative and qualitative aspects of understanding. These represent dynamic equilibrium states, whereby contemplation can be fully reconciled with intense engagement in practical affairs. Put another way, phenomena are here clearly understood as but secondary appearances, thereby enabling their total reconciliation with underlying spirit (as the primary reality) Thus any undue attachment to phenomena (arising through mistakenly identifing appearances as reality) is thereby avoided.

At a deeper level this entails that primitive instinctive phenomena (as the roots of all conscious experience) are immediately eroded as they arise in experience, through continual relationship to an underlying ineffable holistic state. This serves therefore as the qualitative counterpart to the quantitative understanding of the distribution of prime numbers!

However because physical and psychological aspects of reality are dynamically complementary, a qualitative explanation also exists with direct application to to physical reality.

I have already suggested that the underlying structure of string reality bears an intimate connection with prime numbers.

Now it has already been demonstrated that the non-trivial zeros are intimately related with certain quantum chaotic energy states of sub-atomic particles. However I would suggest that their connection with physical reality is much deeper.

Thus the qualitative aspect of understanding with relation to the non-trivial zeros ultimately is required to explain the relationship of strings to the holistic void that underlies all physical reality.

In other words such understanding is needed to consistently explain how strings keep emerging from and likewise continually return to an empty underlying state (representing the continuing potential for existence of phenomenal physical reality).

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