Addendum

Yesterday I had a vivid insight into the nature of the Riemann Hypothesis showing me clearly why from the conventional mathematical perspective it can seem as if about to yield up its secrets while always remaining tantalisingly out of reach.

And from the redefinition of the Riemann Hypothesis that I have suggested as an intimate relationship as between quantitative and qualitative interpretation this is exactly what one would expect.

The very difficulty that the Riemann Hypothesis raises, points directly to a central unresolved problem with the nature of mathematical proof.

Once again the proof of a general proposition (such as the Pythagorean Theorem) strictly is of a qualitative nature that potentially applies to an infinite (unspecified) number of cases; however the quantitative application of such a proof is of a different nature applying in actual terms to a finite number of cases (that can be specified).

Now because of the reduced rational bias of conventional mathematical interpretation the qualitative distinct nature of the general proof is thereby reduced to mere quantitative interpretation leading to the characteristic - unjustifiable - absolute nature of conventional truth.

However the Riemann Hypothesis is altogether more subtle and points to the necessary condition for proper reconciliation of both infinite (general) and finite (specific) notions.

As we have seen this condition (on which subsequent conventional mathematical appreciation properly depends) both predates and postdates as it were all phenomenal (quantitative) manifestations and corresponding (qualitative) interpretations of such reality. So the Riemann Hypothesis - which establishes this mysterious fundamental correspondence as between quantitative and qualitative reality - is already implicit in the very axioms that are used in Conventional Mathematics while ultimately transcending any (phenomenal) attempt to understand its very nature.

So quite clearly - once we appreciate its true nature - the Riemann Hypothesis cannot be proven (or disproven) from within conventional mathematical axioms!

And we can see how this problem of attempted proof is manifesting itself. From one perspective at the general level theorists have seemingly been closing on the ultimate target of proof e.g. by demonstrating that an infinite no. of non-trivial zeros exist on the critical line (with real part .5), and also by slowly showing that a higher and higher percentage of possible zeros must lie on this line. However even if 99.9999...% of possible zeros could be demonstrated to lie on the critical line this would not constitute an acceptable proof!

Meanwhile from the quantitative empirical perspective all valid zeros (now exceeding countless billions) have been found to exist on the critical line. However once again now matter how much further we go in this direction (with no exceptions showing) this will never establish a proof of the Riemann Hypothesis.

And this is the very point as the Riemann Hypothesis indicates clearly that there is is fact no phenomenal identity as between the qualitative area of general proof and the corresponding quantitative area of specific examples!

So the very notion of mathematical proof - though still immensely valuable - needs to be redefined dynamically in the light of the implications of the Riemann Hypothesis, whereby it is understood to be of a merely relative nature and necessarily subject to uncertainty.