Wednesday, October 7, 2009

Reflections on Riemann Hypothesis Article

It is now some months since I completed the article A Deeper Significance: Resolving The Riemann Hypothesis.

This is turn followed on more detailed work on the topic at my website Radial Development.
What I was attempting t0 do was indeed very ambitious i.e. resolving the Riemann Hypothesis.

Now from one perspective it might seem absolutely presumptuous that an amateur with no special ability in - what is conventionally called - mathematics should even attempt to tackle such a problem.
However through Holistic Mathematics - which I have been developing now for some 40 years - I was confident that I could approach the issue from an entirely new perspective.

So in the end I indeed resolved the issue to my own satisfaction, with the Riemann Hypothesis in the context of this new perspective having a remarkably simple resolution.

Just to briefly summarise my findings!

The Riemann Hypothesis is true! however this cannot be proven (or disproven) within the axioms of conventional mathematics .

Rather the Riemann Hypothesis arises as the direct consequence of the fundamental axiom of a more comprehensive mathematical approach - which I term Radial Mathematics - that combines twin aspects (relating to distinctive logical systems).

The first aspect is provided by Conventional Mathematics which is based on linear either/or logic (requiring the separation of opposites).
The second aspect is provided by Holistic Mathematics which is based on an alternative circular both/and logic (requiring the dynamic complementarity of opposites).

So a comprehensive mathematical understanding really requires both the analytic and holistic interpretation of symbols (represented by Conventional Mathematics and Holistic Mathematics respectively).

Just as in geometrical terms the line and circle are reconciled at the central point which is common to both, likewise linear and circular understanding are ultimately reconciled at a common point which is ineffable.

This provides the appropriate context for understanding the true nature of prime numbers. They inherently combine two logical systems the ultimate nature of which is ineffable.

The significance of the real part of 1/2 (to which all non-trivial solutions to the zeta function are intimately related) can best be seen as representing a golden mean as between opposite extremes. Reconciling linear with circular logic requires maintaining complete harmony as between dualistic opposites.

So inherent in the resolution of the Riemann Hypothesis - and also inherent in the fundamental nature of prime numbers - is the consistent reconciliation of the two distinctive types of logic that are themselves necessary for proper comprehension of the issue.

Thus the most fundamental axiom of Radial Mathematics (entailing the twin use of both linear and circular logic) entails the truth of the Riemann Hypothesis.

However as this truth relates to the relationship as between two distinctive types of logic it cannot be resolved with reference to just one!

Thus the Riemann Hypothesis - though necessarily true from a radial perspective - cannot be resolved (i.e. either proved or disproved) within the axioms of Conventional Mathematics.

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