Skip to main content

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 on Riemann Hypothesis.
What I was attempting to 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.

Comments

Popular posts from this blog

The Number 137

The number 137 has raised considerable interest. Its reciprocal (1/137) approx. is referred to as the fine structure constant in physics and is related to the probability of electrons (or other particles) emitting or absorbing particles. Much has been written regarding the "mystical" properties of this number. Indeed some years ago my attention was drawn to its significance through correspondence relating to Jungian archetypes. And just recently an interesting article by Giorgio Piacenza has been published on Frank Visser's Integral World web-site. Without wanting to claim too much for the "mystical significance" of this number, I would like to initially broaden the topic to highlight some important general properties of prime numbers (of which 137 is a specific example). From one perspective prime numbers can be viewed as the basic building blocks of the natural number system (which we literally view in a linear manner as stretched out on a strai...

Integral Science - holistic mathematical nature

In Conventional Mathematics both real and imaginary numbers are used with respect to their (merely) quantitative interpretation. However the key starting point of Holistic Mathematics is the realisation that every mathematical symbol can also be given a corresponding qualitative meaning. The limitation therefore of Conventional Mathematics is that it is confined in qualitative terms to merely real understanding (corresponding to default one-dimensional interpretation). The role of Holistic Mathematics is to provide corresponding imaginary interpretation in qualitative terms. Whereas real interpretation corresponds directly with conscious, imaginary interpretation - by contrast - corresponds directly with unconscious understanding. Once again the real aspect relates to linear logic (where opposite polarities in experience are clearly separated) whereas the imaginary aspect relates in turn to circular logic (where such opposite polarities are treated as complementary). Thus when we separ...

Multidimensional Nature of Time and Space (10)

As we have seen, Conventional Science is based on rational understanding of a linear logical kind which directly conforms with the qualitative interpretation of 1 (as a number dimension). And of course it is the very nature of such interpretation that qualitative notions are thereby reduced (for any relevant context) in a merely quantitative manner! Also, as we have seen, directly associated with this approach is the interpretation of time also as 1-dimensional (where it moves in a single positive direction). However, once we recognise that associated with all numbers is a corresponding qualitative - as well as recognised - quantitative interpretation, then potentially we can have an unlimited number of mathematical interpretations (all of which assume a certain limited validity within their appropriate relative context). This likewise entails that time (and space) itself - when appropriately understood - possesses a potentially unlimited number of possible directions (associated...