The corollary of this from the psychological perspective is that both conscious and unconscious aspects of understanding are necessarily involved in all interpretation of such relationships (again in a dynamic interactive manner).
So the qualitative aspect is directly related to the unconscious aspect of interpretation!
However as we know Conventional Mathematics (that is qualitatively 1-dimensional in nature) is based on a further rational illusion that its meaning can be successfully conveyed in a linear rational manner.
Though the importance of the unconscious e.g. through supporting intuition may well be informally recognised (especially where creative insight is required) in formal terms this is allowed to play no part in accepted mathematical interpretation.
So once again the standard approach is based on a striking limitation (where unconscious intuition - though of a qualitatively distinct nature - is reduced in a merely rational manner.
This issue again is of the most fundamental possible and is glossed over completely within the mathematics community. This is why there such a great need for “outsiders” to identify key issues that remain so steadfastly ignored by “respected” mathematicians.
Once one accepts the equal importance of the qualitative aspect with the quantitative, and the corresponding equal importance of the (unconscious) intuitive with the (conscious) rational aspect of interpretation, then a key problem of the first magnitude arises with respect to the relationship as between the quantitative and qualitative aspects of meaning.
In a truly profound manner the non-trivial zeta zeros (both Zeta 1 and Zeta 2) provide the answer to this problem. They show us from two opposite directions i) the precise numerical relationship as between quantitative and quantitative aspects in an (external) physical fashion and ii) the complementary precise relationship - through interpretation - as between conscious and unconscious aspects in an (internal) psychological fashion.
Indeed once one appreciates the dynamic nature of number (from both external and internal perspectives) then one readily accepts that number is inherent in all physical and psychological processes (as the most intrinsic means of their encoding).
This entails for example that the history of the universe in the first instant of its phenomenal evolution is inseparable from the nature of number (with respect to its quantitative and qualitative aspects).
This also entails that the ultimate final realisation of the meaning of this universe is likewise inseparable from a full appreciation of this nature of number (which of course entails the full appreciation of the zeta zeros).
Admittedly there has been growing recognition in recent years of striking parallels as between the Zeta 1 zeros and certain quantum chaotic physical processes.
However - because of the lack of a dynamic paradigm - mathematicians are approaching this relationship largely from the wrong direction.
In other words they are wondering what the physical processes reveal about the zeta zeros when really it should relate to what the zeta zeros reveal about the nature of these physical processes. Thus the quantum behaviour of nature is already inherent in the dynamic nature of the number system! However because they are accustomed to looking at numbers in absolute terms i.e. as static unchanging entities they are unable to readily make this connection.
However the dynamic nature of the number system equally entails, that the physical aspect of its external behaviour with respect to nature is fully complementary with the psychological aspect of corresponding internal interpretation.
This immediately implies that the zeta zeros (both Zeta 1 and Zeta 2) have immense potential relevance in the human contemplative quest of attaining full enlightenment.
Once again from a dynamic perspective comprehensive mathematical understanding entails the equal specialisation of both reason and intuition. Such specialised intuitive attainment thereby ultimately requires the most advanced contemplative state (where reason can dynamically interpenetrate with intuition - without rigidity - in a highly transparent fashion).
So this situation where pure contemplation can be married with extremely refined rational structures of a dynamic nature will eventually be necessary for the most comprehensive understanding of mathematical relationships (which I refer to as Type 3).
Indeed when one thinks about it from the external objective perspective, the zeta zeros (from two directions) are associated with the ultimate relationship of the primes to the natural numbers.
Now there is a remarkable parallel - which can only be made through holistic terms - with the goal of human evolution.
Anyone for example familiar with Jungian psychology would be able to see this readily in terms of the unification of both conscious and unconscious aspects of the personality.
Now the untrained unconscious expresses itself - especially in earliest childhood - through a mass of uncontrolled primitive impulses.
We all can perhaps accept easily enough how our conscious behaviour with respect to the natural world can be readily hijacked through unconscious impulsive projections.
So the very task of properly recognising the intricate relationship of the primes to the natural numbers (in external physical terms) is ultimately inseparable from the corresponding psychological task of successfully reconciling the unconscious (and its primitive desires) with the rational conscious mind.
This immediately entails that all unconscious impulses are in fact encoded with respect to the qualitative aspect of prime number behaviour. Thus the unravelling of such primitive impulses is inseparable from directly unravelling this prime number code (with respect to its qualitative aspect).
Therefore the full attainment of spiritual contemplative development is inseparable from this task of gradually unravelling all primitive impulses, so that finally the unconscious can then be fully married with the conscious mind.
Thus the ultimate nature of number in external physical terms (at the earliest stages of evolution) with respect to the identity of its holistic and analytic aspects, is inseparable from the ultimate understanding of the nature of number (at the most advanced stages of evolution) where the holistic unconscious and analytic conscious aspects of personality can be finally fully merged with each other.
So when Hilbert maintained that the problem of the zeta zeros was not only the most important problem in Mathematics but absolutely the most important, in this respect he was fully right!
When one accepts that all phenomenal creation is encoded in number (with respect to both its quantitative and qualitative aspects), then the very purpose of evolution can be seen as the attempt to realise its most intrinsic secret (which is ultimately ineffable in origin).
I will finish up this blog entry with reference to a striking feature of the Zeta 2 and Zeta 1 zeros respectively (with immense psychological implications).
All of the (non-trivial) Zeta 2 zeros lie on the circle of unit radius in the complex plane.
As we have seen conventional mathematical reason relates to the conscious aspect of understanding (and is linear in nature). It is directly associated in turn with quantitative interpretation
Now the fact that all the zeta zeros here lie on a circle indicates that we have now switched to the unconscious aspect of understanding (and thereby circular in nature). Now to be precise, pure intuition is ineffable. However when we attempt to express its nature (as the interdependence of opposite reference poles) indirectly in a rational manner, it creates paradox in terms of standard (linear) reason.
So the Zeta 2 zeros are therefore directly associated with the qualitative aspect of understanding (that is indirectly translated in a quantitative manner).
The holistic nature of these zeros then arises through the inevitable dynamic interplay of both independent aspects (as quantitative) and interdependent aspects (as qualitative).
So the Zeta 2 zeros relate properly to the holistic (qualitative) aspect of understanding with respect to the number system. This is then translated indirectly in a circular rational manner. And properly understood the numerical symbols thereby generated are translated accordingly in this manner.
Though I have found the Zeta 2 zeros to be of equal importance to the Zeta 1 (and fully complementary with them) they remain unrecognised. This is due to the fact that Conventional Mathematics is completely lacking a holistic (qualitative) dimension!
Now the Zeta 1 zeros are all postulated to lie on a straight line through ½. However this straight line is of an imaginary - rather than real - nature.
This likewise has remarkable psychological connotations.
If we simplify psychological development the first task is to successfully differentiate the conscious mind (thereby attaining mastery with respect to linear reason).
And this capacity has reached a highly specialised level in our present culture.
However the next task (which occasionally unfolds with true spiritual aspirants) is to now successfully integrate the unconscious mind (thereby attaining mastery with respect to intuitive capacity). These then in a mathematical context would be translated in a circular rational fashion.
So this contemplative extreme relates directly to the qualitative holistic aspect of mathematical development (to which the Zeta 2 zeros directly relate)
However the final task relates to the task of then releasing all this unconscious intuitive capacity in terms of everyday activity.
So when one traces development of the great religious leaders the final stage of their lives often is remarkably active. So they have reached a sufficient stage of mastery as to be able to engage with everyday practical concerns (now transformed through an enlightened spiritual perspective).
The Zeta 1 zeros in fact represent the mathematical equivalent of the same process.
What they entail is the ability to bring the qualitative aspect of holistic unconscious awareness to bear within a standard analytic setting (based on linear reason).
With religious heroes this would be identified as the ability to integrate successfully the (unconscious) contemplative aspect of specialised intuitive enlightenment with the many demands of (conscious) everyday activities. This is sometimes referred for as the marriage of contemplation and activity and generally recognised as the most advanced stage of spiritual attainment!
The corresponding mathematical equivalent would be the ability to integrate the qualitative holistic aspect of mathematical appreciation with its (recognised) quantitative analytic aspect.
So a huge amount of attention in recent years has been given to interpretation of the Riemann Zeta Function (and associated Riemann Function) from the standard analytic aspect.
However once again proper interpretation of the (non-trivial) Zeta 1 zeros requires that holistic qualitative appreciation be properly integrated with its quantitative counterpart.
I have already explained in some detail in a previous blog entry the holistic significance of these zeros. (So I will be brief here)!
½ signifies an equal balance as between quantitative and qualitative aspects (with associated equal balance as between holistic and analytic interpretation of symbols.
Now the fact that the points lie on an imaginary line is very interesting.
In holistic terms the imaginary represents the indirect analytic means of expressing meaning that is directly of a holistic nature.
So for example the first zeros on the imaginary line are 14.134725 (and also – 14.134725).
However this represents an indirect analytic means of providing values that inherently are of a qualitative (interdependent) nature. This explains the puzzle of why such numbers in parallel quantum terms are associated with energy states.
An energy state simply represents the dynamic qualitative nature of number!
Now it also appears that all these non-trivial imaginary parts are transcendental in nature.
I wrote an on-line book some 20 years ago explaining the holistic meaning of the various number types.
I concluded then that the most refined state possible in the phenomenal realm relates to what is both transcendental and imaginary!
Transcendental in a holistic context relates to what is understood as neither quantitative nor qualitative (separately) but as the relationship between both aspects.
So understanding of phenomena needs to be extremely dynamic and refined to operate at this level.
The fact that they are imaginary, entails that they relate to unconscious projections. Now again an extreme mastery would be required to be able to spontaneously recognise all projections immediately in experience as expressing the balanced relationship as between both quantitative and qualitative aspects of understanding. This would entail that they would instantly dissolve and pass from memory as soon as they arise in experience.
Because the Zeta 1 zeros relate to the most intrinsic nature of matter we would not be able to identify them with measurable phenomena. Therefore they would serve as the final bridge as between phenomenal and ineffable reality in physical terms.
This would entail that their full understanding would equally require the most advanced stage of enlightenment possible (consistent with remaining in the phenomenal realm).
We are an awful long way from such realisation at our present stage of evolution.
What is truly remarkable however is that we have come far enough to at least begin to appreciate their true nature.
The Riemann Hypothesis (that all the non-trivial zeros lie on the imaginary line through ½), in holistic terms entails that both quantitative and qualitative aspects of understanding can ultimately be fully identified with each other.
Acceptance of this postulate properly belongs to faith and not to reason.
And quite clearly this postulate cannot be proved (or disproved) with reference to merely the quantitative aspect of mathematical interpretation.