At first glance the specific prime numbers 2, 3, 5, 7,... appear to display no obvious pattern.
And till the beginning of the 19th century, this situation remained. Then largely due to Gauss this situation changed when he discovered a striking collective pattern to the primes based on the natural log function.
So from an individual perspective the primes numbers formerly seemed the most independent of numbers (exhibiting no obvious pattern), Yet from the new general perspective it could now be seen that they displayed an amazing regularity.
It was the genius of Riemann to try and connect these two aspects. He made amazing strides in analysing the complex nature of the primes so that he was able to suggest a formula that ultimately could eliminate deviations arising from the general estimate of prime number frequency.
He made the wonderful discovery that associated with the primes are a - potentially - unlimited number of solutions to the Riemann Zeta Function (defined in the complex plane). These solutions are commonly referred to as the non-trivial zeros (of the Zeta Function) or more simply Riemann's zeros. And associated with these zeros are elaborate wave forms which can - magically as it were - be used to correct any remaining deviations arising from his general estimate of prime number frequency.
So underlying the prime number system, seemingly is composed of discrete individual members, is a subtle wave form pattern. And it is in the relationship as between the two aspects that the mystery of the prime numbers resides.
Thus from one perspective we can use the waveforms associated with the non-trivial zeros in a systematic pattern to eliminate deviations with respect to the general estimate of prime number frequency (up to a given natural number).
In corresponding reverse manner we can also use knowledge of the individual prime numbers to eliminate any remaining deviations in the general estimate of the frequency of the non-trivial zeros (up to a given number on the imaginary number scale).
So this displays admirably the two-way interdependence as between the individual primes on the one hand and the corresponding wave patterns associated with these primes.
Then coming from either direction, mediated through the natural numbers, is a means of moving from individual primes to their collective wave patterns on the one hand and likewise in reverse manner from these wave patterns to the individual primes.
Right away this new newly discovered nature of the primes (by Riemann) suggests a direct correspondence with quantum physics.
Again it is truly remarkable in this regard that Riemann can be directly linked with the two great developments in physics of the 20th century. His developments in Geometry (Riemannian Geometry) was to prove invaluable to Einstein in formulating his Theory of General Relativity. However equally - though not yet sufficiently recognised - his work on the prime numbers provides a direct mathematical link with Quantum Mechanics.
As is now commonly known at the sub-atomic level of matter, particles possess a dual complementary existence exhibiting - depending on the context of observation - both wave and particle effects. Indeed even at the everyday macro level, objects exhibit both particle and wave aspects (though in practical terms the wave aspect can be effectively ignored)!
We can now say exactly the same thing in relation to prime numbers. At a deep complex level of investigation, prime numbers likewise exhibit both particle and wave aspects. However once again at the macro level of "real" interpretation these wave aspects do not reveal themselves and are effectively ignored. So once again it requires the complex analytic techniques first used by Riemann to reveal - as it were - the subatomic structure of the primes.
It is revealing that the primes are often referred to as the "atoms" of the number system which then serve as the "building blocks" for the natural numbers. However though indeed valid from one partial perspective, this is unduly reductionist for the primes - as we now know - contain a remarkable wave pattern system. And because of the two-way interdependence as between the particle and wave aspects, we can no longer hope to understand primes as mere "building blocks".
So as with subatomic particles we now recognise that the particle aspect of the primes also contain a wave aspect and the wave likewise a particle aspect.
However now we get to the crucial point! Just as it is now understood that the behaviour of particles at the sub-atomic level is governed by the Uncertainty Principle, equally we can assert that the prime numbers - at a deeper level of investigation - are governed by a corresponding mathematical Uncertainty Principle.
In physics if we try to precisely fix the position of a particle we blot out recognition of its corresponding momentum. And in turn, if we then try to precisely measure momentum it tends to block out any precise knowledge of its position.
It is similar in relation to the primes. When we try to precisely track the individual location of the primes, it tends to block out corresponding knowledge of their general distribution. And mathematical history testifies to this point! For though the primes were known and studied for well over 2000 years, it was not until the 19th century that this customary stance with respect to prime number analysis was abandoned.
And in turn, when we then concentrate on the general distribution of the primes this tends to block out recognition of the precise location of each individual prime!
So let me put it bluntly so that the point cannot be easily ignored. Uncertainty - by its very nature - lies at the heart of the number system and by extension at the heart of all mathematical relationships!
Furthermore the Uncertainty Principle that is now accepted with respect to the physical world - though not unfortunately its many implications - has its deeper roots in the nature of the prime numbers!
In other words the Uncertainty Principle that governs the physical world is ultimated rooted in the uncertainty that lies at the heart of mathematics.
The wave forms associated with prime numbers were discovered by Riemann long before the breakthroughs in Quantum Physics. However just as Einstein's General Theory of Relativity has its mathematical roots in Riemannian Geometry, I am confident it will be accepted in the future that Quantum Mechanics - especially in relation to the Uncertainty Principle - has its roots in Riemann's amazing discoveries with respect to the complex Zeta Function.
Then, even deeper implications of what I am saying here, have not yet been addressed by either the physics or mathematical community.
Indeed it is again ironic that the Uncertainty Principle itself can be effectively used in a related manner to highlight the key dilemma!
In truth all mathematical and scientific understanding entails two aspects of understanding which are interrelated in experience.
One the one hand we have reason - pertaining directly to the conscious - which in the current mathematical and scientific means simply linear reason. Such reason properly relates to differentiated interpretation with respect to specific actual phenomena (that are necessarily of a finite nature).
However equally we have intuition, pertaining directly to the unconscious. Though the importance of such intuition might well be informally recognised especially for creative work, in formal terms it simply is reduced through the conventional paradigm to reason. And in contrast to reason, intuition relates to integral holistic interpretation with respect to general phenomena of a universal potential nature (that are necessarily infinite in origin).
So a great price that has been paid for the admittedly enormous advances using the present paradigm. Thus we have learnt to automatically reduce - in any context - what is truly qualitative and distinct to mere quantitative interpretation.
Whereas reason tends to be linear, intuition is inherently of a circular nature (when indirectly interpreted in rational fashion). In other words linear reason is based on the clear unambiguous separation of opposite polarities (such as external and internal) whereas intuition is based on their complementary - and apparently paradoxical - identity.
So the key implication for mathematics and science is that a more comprehensive paradigm requires the incorporation of both linear and circular modes of logical interpretation. And remember these circular modes serve as the indirect expression of ever more refined forms of reason! Put another way, in qualitative terms we need a paradigm that contains both real and imaginary aspects!
We have seen that an attempt to precisely fix the position of a particle blocks out recognition of its complementary (unrecognised) aspect.
Thus the obsession with a paradigm that is geared to mere quantitative interpretation has all but blotted out recognition of its corresponding qualitative aspect.
And this problem cannot be solved through ever more sophisticated attempts to adapt scientific interpretation of reality to the conventional linear rational approach!
Since my own "conversion" over 40 years ago, I have consistently proposed that a more comprehensive Mathematics entails at least three aspects.
The first is what I now term Type 1 Mathematics which relates to the standard conventional approach based on mere quantitative interpretation. This does not mean of course that qualitative considerations can play no part, but rather that they inevitably become reduced, due to the limited nature of an approach adopted, to quantitative interpretation!
The second is what I now term Type 2 Mathematics (which formerly I referred to as Holistic Mathematics). This is directly based on training in ever more refined states of intuition (that often are associated with the spiritual contemplative vision). In more formal terms, it relates to ever more intricate circular logical systems which have a precise mathematical basis (in holistic terms). Putting it in a Type 2 manner, every number for example has a direct qualitative significance as representative of a dimensional meaning.
So In Type 2 terms, conventional mathematical interpretation is based on a linear (1-dimensional) approach. However potentially an unlimited number of possible other interpretations exist, according to numbers qualitatively used as alternative dimensions.
When seen from this perspective, what is currently known as Mathematics simply refers to 1-dimensional interpretation. However this leaves a potentially unlimited set of alternative dimensions of interpretation (that have yet to be explored). The general nature of all these alternative interpretations is that each dimensional number relates to a unique means of configuring the relationship between reason and intuition (that now is understood in merely relative terms). Put another way, in Type 2 terms all mathematical symbols have both particle and wave aspects (with the wave aspect corresponding to a unique type of holistic appreciation that depends on the quality of intuition that inspires it).
My own mathematical journey has largely consisted of developing the implications of this type of appreciation which - even from my necessarily limited perspective - I can see will be truly breath-taking in scope.
The third aspect of Mathematics, which promises to be by far the most comprehensive, is what I now refer to as Type 3 (formerly Radial) Mathematics. It is designed to combine both Type 1 and Type Mathematics in ways that can be both extraordinarily productive and creative.
Properly understood all Mathematics is implicitly of a Type 3 nature. However - as I have stated - because one aspect (i.e. the holistic qualitative) has been entirely blotted out from formal interpretation, in effect it is reduced to Type 1.
The insights offered here are highly unlikely to arise from within the conventional mathematical community as existing practitioners are still strongly wedded to conventional interpretation. So I offer them as an outsider who from an early stage could see deep cracks in the mathematical edifice.
So coming back to Riemann and the famous legacy of The Riemann Hypothesis, it can now be restated in an especially cogent form as relating to the central condition necessary for the reconciliation of both Type 1 and Type 2 Mathematics.
When seen in this light, not alone can it not be proved (or disproved) in Type 1 terms, it cannot even be properly appreciated in this manner.
In other words there is a quantitative and qualitative meaning that can be given to every mathematical symbol which cannot be successfully reduced in terms of each other.
So the wonderful mystery then arises as to how - despite these distinctive meanings - a remarkable overall coherence is preserved. And this coherence lies deep in the nature of prime numbers themselves that preserve a unique quantitative identity yet display in collective terms remarkable holistic qualitative characteristics.
It is as if prime numbers are able to communicate with each other so effectively, that while contributing distinct individual notes, no discordance arises with respect to their collective symphony. Indeed this is precisely what is happening as - properly understood - in dynamic interactive terms, the primes are living entities (that are necessarily mediated though phenomena of nature). And once again, here we have the connection with subatomic particles which can communicate with each other (even at a great distance).
And this mystery cannot be interpreted in mere quantitative terms! Rather it relates to a prior mystery that ultimately governs the entire pattern of evolution (in both a quantitative and qualitative manner).
So properly understood, the implications of the Riemann Hypothesis are already contained in the mathematical axioms we use (in Type 1 Mathematics). Its acceptance thereby represents a massive act of faith in the subsequent consistency of the whole mathematical enterprise.