I have already redefined the Riemann Hypothesis demonstrating how it relates to an intimate correspondence as between both the quantitative and qualitative interpretation of mathematical symbols.
Personally I find this connection to be of enormous significance. For many years now I have been developing an alternative type of Mathematics where each mathematical symbol is given a holistic as opposed to a strictly analytic interpretation as in Conventional Mathematics. So this has continually raised the question of this mysterious relationship as between quantitative and qualitative.
So like two blades of a scissors, properly understood we have two aspects of Mathematics that - in a more balanced appreciation - would be recognised as of equal importance i.e. standard and holistic.
Then in a comprehensive approach to Mathematics both of these aspects would increasingly interact in a dynamic manner that would be both immensely productive and highly creative (representing the full expression of both rational and intuitive type capacities).
However this very interaction already assumes an intimate correspondence as between both aspects. And this in fact is the very same axiom or assumption that underlines the truth of the Riemann Hypothesis.
So in strict terms we can never prove that the Riemann Hypothesis is true (or indeed false). The reason for this is that the axiom - to which the Hypothesis relates - is already implicit in the very axioms that are used in Conventional Mathematics.
Put another way if we wish to doubt the validity of the Riemann Hypothesis, then we must necessarily doubt the validity of all the axioms that we conventionally use, which would in turn undermine belief in the truth of any mathematical proposition.
Thus in the deepest sense the truth implied by the Riemann Hypothesis is not implied by reason but rather by faith! For without such faith the whole mathematical edifice constructed so painstakingly over the past few thousand years would be without any foundation.
However it is indeed possible to probe a little more into what the Riemann Hypothesis truly implies.
Central to this truth is that we can never divorce objective knowledge of reality from the psychological constructs that we must necessarily use to interpret this reality. In other words we cannot divorce quantitative type results from qualitative type interpretation.
Now the great appeal of Mathematics to so many is a belief in its pure objective nature. So for example when a proposition is proven as true the belief is that this proof possesses an "objective" validity that is absolute.
Strictly speaking however this absolute view of mathematical truth is not warranted. For the very "objective" truth that we demonstrate through a mathematical proof is itself but a reflection of the mental constructs that we deem appropriate in arriving at such a conclusion.
So expressed now in more refined terms, that directly relates to experience, the faith that we place in mathematical truth (such as a proof of a theorem) in dynamic interactive terms expresses the belief that an automatic correspondence exists as between what is objectively demonstrated to be true and the subjective mental constructs that we must necessarily use in reaching such a conclusion.
Now in static terms there are two equally valid ways that we could express such proof:
(i) as pertaining to objective reality (as independent of psychological interaction). In other words the proposition is thereby absolutely true in quantitative terms.
(ii) as pertaining to psychological mental reality (as independent of any physical interaction). So again the proposition is absolutely true in qualitative terms.
Now in effect - when we interpret in this static absolute manner as in Conventional Mathematics - both possible interpretations will directly correspond with each other. So it does not matter in effect which polar aspect of explanation we might give. In other words the qualitative can thereby be reduced to the quantitative aspect.
However once we view mathematical understanding in a dynamic interactive manner, the relationship as between both aspects is of a merely relative nature. So quite literally we now accept that both external (objective) and internal (psychological) aspects are (dynamically) related to each other in experience.
Though interpretation of a mathematical proposition cannot now be strictly of an absolute nature, a high level of trust can still be placed in proof (by accepting this intimate correspondence as between both aspects). And once again this intimate correspondence - enabling us to place such great value in what (objectively) corresponds to our (psychological) interpretations of reality - is the very same axiom to which the Riemann Hypothesis applies.
So the key significance of the Riemann Hypothesis is completely missed through conventional mathematical interpretation.
And it is this very fact which makes it so central. For once we understand what it really is implying, then we must accept that it is exposing the most fundamental weakness possible with respect to Conventional Mathematics. In other words - properly understood - there are two aspects of equal importance to Mathematics. Yet one of these aspects has been effectively ignored almost entirely throughout our mathematical history!
And the Riemann Hypothesis - when appropriately appreciated - is there as that essential axiom enabling the proper integration of both branches (on which is based - what I refer to as - Radial Mathematics).
Yes! the Riemann Hypothesis has indeed considerable implications for the true nature of prime numbers!
More importantly however it has even greater implications for the true nature of Mathematics.