It is my intention in these blogs - for anyone who cares to read - to combine elements of the teacher, preacher, prophet, iconoclast, radical, antagonist - indeed whatever it takes - to challenge existing mathematical perceptions and in some measure awaken readers to a wonderfully enlarged mathematical universe that still remains totally undiscovered.
Though the discussion on this blog - as is the intention - necessarily centres around the Riemann Hypothesis, this really is being used to serve a much greater purpose relating to the true (unrecognised) nature of Mathematics.
The insights that I offer here do not come from idle speculation, but rather arise from a deep struggle in my late teens with the nature of Mathematics leading to a significant personal transformation, the implications of which I have continued to explore for some 45 years.
We are accustomed to viewing electromagnetic radiation in physical terms as a spectrum comprising many different bands; likewise this is true from a psycho spiritual perspective.
And just as natural light comprises one narrow band with respect to the physical spectrum, equally the natural intuitions that inform the vast bulk of intellectual discourse in Western society likewise come from one narrow band (around the middle of this psychological spectrum).
And Mathematics - as conventionally understood - represents par excellence, an extreme specialisation with respect to the thinking associated with this narrow band.
So as I have stated so many time before, in a very precise qualitative manner, Conventional Mathematics is based on a 1-dimensional logical approach (where isolated uni-polar reference frames of enquiry are adopted).
However on the full spectrum, representing the latent possibilities for psycho-spiritual development, many further bands of ever more refined forms of intuitive energy exist.
In the past such bands were always universally associated with spiritual contemplative development, which tended to be strongly wedded to the belief systems of the various mystical traditions. Though many detailed accounts have been left by courageous voyagers who successively traversed these realms, invariably they are couched in a religious manner (employing the symbols of their respective traditions).
So, throughout history the deep implications of the contemplative vision for Mathematics and the Sciences have therefore been largely ignored.
Thus, my own particular concern with respect to contemplative type development has been significantly related to its implications for scientific type understanding and especially for mathematical interpretation.
So from the perspective I now customarily adopt, I can see clearly that what we conventionally term Mathematics represents in fact just one narrow band on the overall mathematical spectrum (where qualitative type meaning is directly reduced in quantitative terms).
However potentially an unlimited number (i.e. dimensions) for other mathematical systems exist, that represent - precisely defined - dynamic configurations with respect to both quantitative and qualitative meaning.
Putting it another way, Conventional Mathematics is formally based on exclusive recognition of the conscious aspect of recognition (which gives it an absolute type appearance); however in all other mathematical systems both conscious and unconscious are involved in a dynamic relative manner.
Stating it yet another way, Conventional Mathematics necessarily entails the confusion of potential (infinite) with actual (finite) notions of meaning, whereas in all other systems a careful distinction between both is maintained.
Or finally, to emphatically make the point here, no formal recognition is given to the role of intuition in Conventional Mathematics (where again its use is reduced to rational type interpretation). However in all other systems, intuition is recognised as operating in a vitally distinct holistic manner, and cannot be encapsulated in a linear (i.e. 1-dimensional) rational fashion.
It might be instructive in this regard to explain how my holistic appreciation of the nature of prime numbers emerged.
When I was initially engaged in in attempting to map out the psychological spectrum, not surprisingly I spent some considerable time dealing with earliest infant development and the processes by which phenomenal understanding emerges.
I then began to appreciate that there were close connections as between the meaning of the words "primitive" in a psychological context and the corresponding holistic notion of prime numbers.
In earliest development, phenomena enjoy but a fleeting momentary existence, and the reasons for this are very interesting to probe.
Because neither conscious nor unconscious aspects of personality have yet been properly differentiated, holistic notions (relating to the unconscious) are directly confused with specific phenomena. In other words the neonate continually confuses the general dimensions (of space and time) in experience with specific objects. Therefore because - quite literally - phenomenal objects cannot yet be placed in a proper dimensional context - they enjoy but an immediate transient existence.
I hasten to add that this insight is also very relevant to the nature of sub-atomic particles, which likewise enjoy but a fleeting momentary existence. This clearly implies that at this level of material organisation, the natural dimensions of space and time have not yet properly formed. Of course we do not properly recognise this fact when we insist on imposing perceptions of dimensions appropriate to the macro level on sub-atomic reality!
So we can accurately characterise - in both physical and psychological holistic terms - earliest reality as existing in a - mathematical - prime framework, where both the individual (object) and collective (dimensional) identity of the primes is confused.
Putting it more precisely from a physical perspective, the prime numbers contain two aspects relating to an infinite potential ground (as their dimensional attributes) and to finite actual entities (where they can be identified as individual numbers).
And in corresponding psychological terms with respect to their manner of interpretation, the prime numbers again contain two corresponding aspects relating to an unconscious (with respect to their holistic) and a conscious (with respect to their specific) nature respectively.
Incidentally when I read some years ago that a quantum physical basis had been discovered for the Riemann non-trivial zeros, I was not at all surprised; in fact due to the holistic nature of my approach, I had already formed the conclusion several years earlier that this was necessarily the case!
So with reference to the whole spectrum of development, we can see clearly that there are necessarily two aspects to the primes (with matching physical and psychological correspondents).
And this means in turn that from a dynamic interactive perspective - which is the proper context of experience - prime numbers have no strict reality apart from the interpretations that are brought to bear on them!
Now mathematicians such as Hardy would have recoiled in horror at such a revelation.
From their perspective, the identity of numbers was as absolute as one could get (eternally existing - as it were - in some sort of immutable Universe).
And of course the reason they could think this, relates to the specialised nature of rational development that only unfolds at a later stage of human development.
So the supposed absolute nature of numbers, simply reflects the absolute nature of the rational paradigm from which they are interpreted!
Or to put in an equivalent manner, reality inevitably reflects the manner we look at it; so when we formally exclude the role of the unconscious altogether, numbers do indeed appear as absolute fixed entities!
However once we begin to incorporate the unconscious appropriately with conscious interpretation, this comforting worldview irretrievably breaks down.
Thus, stating it in more nuanced language (incorporating both conscious and unconscious aspects) there is an inherent potential in prime numbers which only then later in human history can become actualised in a finite manner. And as such actualisation remains a continual on-going process, the prime numbers can be accurately seen - without hyperbole - as embracing the entire course of created evolution!
So for example, since Riemann's path breaking discoveries in 1859, it is known that underlying the actual identity of specific primes (that had long been observed) is a - hitherto unknown - harmonic structure.
Therefore though this potential was always inherent in the nature of the primes, we had to wait till 1859 for its very first revelation in actual terms.
Furthermore we can safely say that the full implications - which are truly mind boggling - of this revelation, have not yet evolved.
If we revert to Quantum Mechanics, we can perhaps appreciate that though the foundations were laid in the 1920's, the continued reverberations with respect to the implications of this new theory lasted for a considerable period afterwards (and have in no way yet been fully realised).
For example, one fascinating issue related to the apparent non-local effects of sub-atomic particles, so that a particle such as a light photon seemingly could communicate with another photon (even at a great distance).
Einstein in his (unconscious) desire to maintain a strict deterministic approach railed against such new findings. However in the 80's, though controversy necessarily remains, it was experimentally demonstrated that non-local effects with respect to particles do in fact occur.
Now properly understood, this is also true in an even more profound manner with respect to the nature of prime numbers.
Though prime numbers in their (actual) individual identities, seem the most "local" of all numbers; yet with respect to their (infinite) potential identity (in relation to all other primes) they are equally the most "non-local" of numbers. So they combine, in their very nature, extremes with respect to two opposing modes of identity!
Thus again, at the level of (actual) individual primes they are the most independent of all numbers. However equally at their collective (potential) level they possess the ability to communicate with all other primes, thereby preserving a perfect synchronicity with respect to overall behaviour.
However the deeper implication of this finding requires the surrender of an exclusive rational approach to Mathematics!
Quite simply the present paradigm is geared to deal merely with local effects; by its very nature it cannot deal with non-local effects, which correspond to a distinctive holistic mode of behaviour (and matching interpretation).
Put another way, we are now truly at the crossroads in Mathematics with respect to the prime numbers. The unconscious now urgently needs to be explicitly incorporated (with conscious understanding) in the recognition of a distinctive qualitative aspect to their behaviour. And of course by extension this recognition necessarily applies to all Mathematics!
Then, once again, the Riemann Hypothesis points to the condition for the mutual identity of both the quantitative (local) and qualitative (non-local) aspects of the primes.
And eventually, it will be realised that this new feature of the primes had always been potentially inherent in their very nature (which however could only be actualised when appropriate human evolution with respect to corresponding unconscious understanding had taken place)!