I have already suggested that one revealing way of appreciating the nature of the non-trivial zeros (both for the Zeta 1 and Zeta 2 Functions) is as the shadow to the conventional natural system of number (in both cardinal and ordinal terms).
In Jungian psychological terms this would thereby imply that these two sets of non-trivial zeros represent the unconscious complement of what we conventionally understand as number (in conscious terms).
Likewise in physical terms it equally implies that the two sets of non-trivial zeros represent the holistic complement of what is conventionally interpreted in analytic terms with respect to the natural world.
Again from a Jungian perspective, true psychological integration requires the combined integration of both the (revealed) conscious and (hidden) unconscious aspects of personality.
Complementarity exists in the relationship between the natural number system (cardinal and ordinal) and the two sets of non-trivial zeros.
So from the conscious extreme the natural counting numbers seem fully accessible to understanding (even from a very early age).
However from the corresponding unconscious extreme, the two sets of non-trivial zeros – certainly in terms of what they inherently represent – seem almost entirely inaccessible to conventional understanding (at any age of development).
So what appears as completely obvious in terms of (revealed) conscious appears completely inaccessible (in terms of (hidden) unconscious behaviour. So identification with the conscious thereby blots out entire recognition with respect to the opposite unconscious extreme (relating to the hidden nature of the number system).
This thereby entails that ultimately true psychological integration requires appropriating this enormous shadow with respect to the nature of both sets of non-trivial zeros, so that eventually they can become fully accessible in intuitive terms to understanding.
The deeper significance is that Conventional Mathematics thereby is greatly lacking a true integral dimension with respect to overall understanding.
Once again this is due to a total failure in formal terms to recognise its hidden unconscious dimension.
Alternatively it represents the complete domination (in formal terms) of quantitative over qualitative type interpretation of mathematical symbols.
Expressed in yet another way it represents sole recognition of the analytic dimension of mathematical interpretation through the exclusion of its equally important holistic aspect.
Yet again it entails the remarkable fact that Mathematics as it currently stands is entirely lacking in any genuine notion of the nature of interdependent type relationships.
And the saddest part – though his is easily appreciated in Jungian terms – is that because of its extreme one-sided emphasis, it thereby is greatly lacking any insight into the nature of its own considerable shadow.
Thus what are understood as rigorous procedures (within its limited terms of reference) can equally be seen from an outside perspective as representing highly reduced understanding at every turn.
And by definition when one accepts such reductionism – as a recognised member of the mathematical profession - without question, one thereby becomes unable to tolerate or even appreciate any outside criticism of its inherent nature.
Though the Riemann Hypothesis is not necessary to identify the fundamental problem with present mathematical interpretation, it does however – when appropriately interpreted – provide a wonderful illustration of the precise nature of this problem.
So let me once more state it simply! Properly understood Mathematics is inherently dynamic in nature representing the two-way relative interaction of aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.
However Conventional Mathematics is based on the special case where such interaction is effectively completely ignored. So the qualitative – though of equal importance - is thereby reduced to the quantitative.
Mathematics is then misleadingly viewed as relating merely to quantitative relationships in absolute terms.
The Riemann Hypothesis relates intrinsically to the fundamental nature of the number system and cannot thereby be properly understood – not alone solved – in conventional mathematical terms.
It thus reduces and thereby gravely distorts its inherent dynamic nature as the interaction of twin elements that are quantitative and qualitative (and equally physical and psychological) with respect to each other.
So number inherently represents the fundamental encoding of all physical and psychological phenomena in quantitative and qualitative terms.
Conventional Mathematics is thereby gravely in need of addressing its quite enormous shadow.
In physical terms this requires the recognition that all mathematical processes entail a qualitative holistic dimension that cannot be reduced in mere quantitative terms.
In complementary psychological terms it implies that that the understanding of such processes entails an unconscious (intuitive) aspect that cannot be reduced in a rational (linear) manner!
As I have repeatedly stated, because of the very nature of dynamic interaction, all interpretation that is directly analytic (i.e. quantitative) has a corresponding shadow – initially unrecognised - that is indirectly holistic and qualitative in nature.
Likewise all interpretation that is directly holistic (i.e. qualitative) has a corresponding unrecognised shadow that is indirectly of an analytic quantitative nature.
When we initially look at the number system it appears directly in a linear rational manner revealing its quantitative aspect. So here we literally view all numbers in quantitative terms as lying on a straight line!
However this number system has a corresponding shadow that is indirectly holistic and qualitative in nature.
Bringing this aspect to light depends much more on intuitive (unconscious) development rather than on linear rational understanding.
In fact indirectly it can be then given a paradoxical rational interpretation in a circular manner.
I have frequently over the past year or so in my blogs drawn direct attention to the nature of this holistic aspect identifying firstly the Type 2 aspect (of the number system) and then the corresponding Zeta 2 Function.
Not surprisingly, as Conventional Mathematics totally lacks in formal terms a holistic dimension, no recognition exists as yet of the enormous importance of what – I refer to as – as the Zeta 2 non-trivial zeros. These in fact are fully complementary with the Type 1 zeros.
The Type 2 non-trivial zeros essentially relate to the holistic nature of numbers (as dimensions).
As understanding becomes increasingly dynamic the nature of higher dimensions (the structure of which is related to the corresponding roots of the number in question) unfold in understanding.
Ultimately however as holistic appreciation deepens, remaining rigid elements of a linear rational level gradually are completely dissolved.
In psycho-spiritual terms this is identified with the continual deepening of a contemplative (intuitive) state that ultimately transcends all analytic interpretation (of a rigid linear kind). Increasingly however, dynamic rational structures of a circular (paradoxical) kind are required to mediate such understanding.
This represents the unconscious – as opposed to the conscious – extreme of interpretation.
So we start in development with the understanding of number with the direct linear rational understanding of number (i.e. its quantitative aspect).
Then when development proceeds to higher stages we gradually can uncover the shadow of this aspect of number (i.e. its initially unrecognised holistic qualitative dimension).
The full unfolding of this alternative holistic aspect leads to an increasingly dynamic appreciating of number that is directly intuitive but indirectly is conveyed in a refined circular rational fashion.
This understanding ultimately culminates in a spiritually contemplative appreciation of the transcendent nature of reality (i.e. the purely holistic appreciation of the infinite nature of the number system).
And the Type 2 zeros relate directly to this circular holistic aspect!
However the holistic also has a shadow aspect (in its unrecognised analytic aspect).
What this entails from a psycho spiritual perspective is that the pure holistic appreciation of the infinite nature of reality (as transcending all finite notions) must now be made immanent in all finite phenomena.
This means in effect that one now gradually learns to integrate the new qualitative aspect of appreciation with all scientific type phenomena (formerly understood in mere quantitative terms).
Let me briefly illustrate. From a conventional scientific perspective (i.e. quantitative) the recognition of object phenomena e.g. roses is of a reduced collective nature where we can impersonally classify all individual members as belonging to the same general class of rose!
Now the corresponding qualitative recognition requires that each rose be understood uniquely with respect to its distinctive attributes.
Remarkably it is the same with the number system.
The quantitative recognition of number is impersonal in nature where the qualitative distinction as between numbers is not taken into account.
Indeed the failure to appreciate the true nature of multiplication in Conventional Mathematics relates to this lack of qualitative distinction.
So 2^2 (represents a number expressed with respect to the second dimension). However the quantitative result = 4 (i.e. 4^1) represents a number expressed with respect to the (default) 1st dimension.
Thus the qualitative change in the nature of the number (which always results through multiplication) is thereby ignored. In other words the qualitative aspect of interpretation is reduced to the quantitative!
The very point about the Riemann Hypothesis is that it relates to this key problem of reconciling the quantitative with the qualitative nature of number.
The non-trivial zeros (i.e. Type 1) therefore relate to the holistic aspect of the number system now interpreted in an immanent manner, whereby the qualitative aspect can be properly incorporated with each finite member of the number system.
Put another way just as the conventional linear number system represents the interpretation of number with respect to its quantitative aspect, the non-trivial zeros (Type 1) represent the corresponding interpretation with respect to the qualitative aspect.
Now the conventional system (esp. primes and natural nos.) are understood as points on a real line. In qualitative terms “real” is synonymous with rational interpretation of a quantitative kind!
From a holistic perspective, “imaginary” relates to the indirect rational way of conveying meaning that is of an unconscious intuitive nature. So the recognition of “qualities” as associated with consciously recognised objects, always comes directly from the unconscious mind (in its interaction with the conscious).
In this sense such qualitative appreciation is of an imaginary nature. So whereas the quantitative number system is of a real, the corresponding qualitative appreciation of its nature is imaginary! So therefore the (Type 1) non-trivial zeros lie on a straight line that is imaginary.
If the unconscious remains undeveloped we thereby cannot properly identify true quality in any context!
Unfortunately this is a damning criticism that can be made of Conventional Mathematics (which fails to recognise its qualitative dimension)!
This therefore poses considerable difficulties in appreciating the nature of the non-trivial zeros (Type 1) which relate to the qualitative aspect of number (with respect to their finite nature). By contrast, the Type 2 non-trivial zeros relate to the qualitative aspect of number (with respect to their infinite nature).
Now I have already stated that all phenomena fundamentally represent the original dynamic interaction of the number system with respect to its quantitative and qualitative characteristics.
What is truly remarkable is that though remaining inaccessible to conventional understanding, the non-trivial zeros (Type 1 and Type 2) must necessarily implicitly exist in the unconscious mind of all human beings as the very means through which we are enabled to make qualitative distinctions with respect to objects.
What is even more remarkable is that these same zeros must innately exist with respect to all natural phenomena providing their very capacity to become manifest in nature (exhibiting both quantitative and qualitative characteristics).
In an important sense one key goal of evolution requires fully uncovering in understanding these hidden – though currently largely inaccessible - original aspects of our number system!
Indeed the very nature of both types of zeros is pointing clearly to the present inadequate nature of Conventional Mathematics.
Mathematics is not strictly about the quantitative but rather the dynamic interaction as between its quantitative and qualitative aspects. Therefore to properly understand this interaction we must face its persistent shadow in a continual failure to recognise the qualitative dimension.