One of the fundamental implications which the Riemann Hypothesis has for Mathematics relates to the nature of proof.
As I have suggested, when the true nature of such proof is properly recognised i.e. in dynamic experiential terms, then it is necessarily subject to the Uncertainty Principle.
However the full implications of this are very revealing for what this actually implies that every mathematical proof has in fact two distinct aspects (that are both quantitative and qualitative in relation to each other). So the Riemann Hypothesis establishes the axiomatic condition for consistency as between these two aspects. Thus the immediate corollary that follows is that every proposition with an (established) quantitative meaning equally can be given a coherent qualitative interpretation. This also implies that every theorem with an established proof in conventional terms equally can be given a coherent qualitative proof in holistic mathematical terms.
Indeed from a comprehensive mathematical perspective this would then entail that a theorem has not been properly proved until both aspects of proof have been fully established.
And this immediately points to the nature of the Uncertainty Principle that is involved.
So continual focus on just one aspect of proof (quantitative) thereby blots out recognition of the equally important alternative aspect (qualitative). This in fact explains the current nature of mathematics whereby total focus on the mere quantitative aspect of proof has completely blotted out recognition of the equally important alternative aspect.
Using an analogy from quantum physics it is as if we have concentrated so much on the mere particle existence of light, that we have no conception that light equally has a valid existence as a waveform!
The implications of what is being stated here are extremely far reaching in scope for once again it implies that strictly speaking the notion that Mathematics can have a mere abstract meaning (with no direct relevance to reality) is quite untenable.
So the very reason why this so often appears the case is precisely because of the lack of any developed - or even undeveloped - qualitative aspect to current mathematical thinking.
However when the qualitative aspect is properly integrated, then every mathematical proposition has a potential relevance to physical reality (in both quantitative and qualitative terms). And because physical and psychological aspects are complementary, this implies that every mathematical proposition has an equal potential relevance for psychological reality also!
Now we are still - to coin a phrase - a million light years from having any true conception of this comprehensive nature of Mathematics. However I can say with confidence, that as this is in fact the case, it will eventually be recognised.
It has to be stated however that the two aspects of proof (that I have outlined) are of a different nature (requiring uniquely distinct types of understanding). The quantitative aspect conforms to linear logic in the sequential establishing of unambiguous type rational connections between variables; the qualitative aspect by contrast conforms to circular logic in the simultaneous holistic recognition of complementary intuitive type connections (that are indirectly given a rational form).
One fascinating possibility is that a certain common pattern necessarily applies to both aspects. In other words with the appropriate type of proof for a proposition, one would be able to suggest the necessary structure of the qualitative aspect (from corresponding knowledge of the quantitative). Alternatively, one would be able to equally suggest the necessary structure of the quantitative aspect of proof (from corresponding knowledge of the qualitative).
So for example an established quantitative proof that did not lend itself readily to its qualitative partner would be deemed in some sense inefficient with a better proof still to be established.
In my own work I have given some - though necessarily limited - consideration to this issue. For example one of earlier "successes" was to resolve - what I call - the Pythagorean Dilemma, by providing the corresponding qualitative aspect of proof as to why the square root of 2 is irrational!
Now the Pythagoreans would have already established a quantitative proof as to why this root is irrational. However they implicitly recognised that it also required a qualitative aspect of proof (which they could not provide).
So a future stage of my own investigations will now relate to searching for this common pattern as between the two aspects of proof.
In other words the quantitative aspect really shows how the square root of 2 is irrational; the qualitative aspect then explains the deeper philosophical reason of why this root is irrational.
However there has to be a common structure to both aspects of an appropriate proof so that same symbols can be equally read in accordance with both linear establishing the quantitative and circular logic establishing the quantitative aspect of proof respectively.
Once again this is implied by the Riemann Hypothesis.