Yesterday we explained the nature of the non-trivial zeros of the Zeta 1 Function as a set of numbers that represent the interdependence of both its quantitative and qualitative aspects in a linear manner.
Earlier we saw that the corresponding non-trivial zeros of the Zeta 2 Function is likewise a set of numbers, which represents the interdependence of both its quantitative and qualitative aspects in a circular manner (where each integer representing a dimension is associated with its own circular group of individual members).
And just as the size of these circular groups (of non-trivial zeros) steadily increases in Type 2 terms with larger integers, likewise the frequency of non-trivial zeros steadily increases as we ascend the linear number scale in Type 1 terms.
So to put it more simply, whereas the Type 2 zeros express the interdependence of the primes and natural numbers (internally) from a (circular) holistic, the Type 1 zeros express this interdependence (externally) in terms of a linear quantitative perspective.
I now wish to probe more deeply into the philosophical significance of the manner in which the non-trivial zeros are represented.
Firstly - which gives rise to the Riemann Hypothesis - all the Type 1 zeros appear as complex numbers of the form ½ + it and (½ - it) respectively.
½ has a crucial significance in this context.
It must be remembered that the non-trivial zeros relate to numbers (as dimensional powers or exponents).
Now the qualitative significance of ½ can be expressed in terms of the two roots of unity i.e. + 1 and – 1 respectively. These two points lie at opposite ends of the diameter of this unit circle in the complex plane with the midpoint at 0. However if we represent this diameter in a linear fashion (on a positive number scale) with length 1, the midpoint that divides the line will lie at ½.
So the significance of this is that the non-trivial zeros arise only in the context of the opposite polarities (internal and external) which govern all reality, remaining perfectly balanced with each other. Indeed this is required in order to fully preserve the independence of individual quantitative elements with corresponding interdependence with respect to their collective qualitative nature.
Without this equality, an unbalanced emphasis - either on the quantitative or qualitative aspects of the relationship - would arise.
So in the simplest manner, the requirement that all non-trivial zeros lie on the real line through ½, is necessary to ensure the ultimate identity of both the quantitative and qualitative aspects of number.
Once again it is futile trying to prove this in a conventional mathematical manner as this serves as a necessary prior condition for the subsequent consistent use of its very axioms.
At a deeper level, this also implies that appropriate interpretation of the nature of the non-trivial zeros requires the realisation that ultimately the objective nature of mathematical objects (such as numbers) cannot be separated from their subjective means of mental interpretation.
Now this could not be more damaging in terms of the conventional paradigm, which is based on the complete separation of objective from subjective poles leading to the reduced (and ultimately untenable) interpretation of mathematical objects existing in independent terms as abstract unchanging entities!
So we are now at the other extreme, where understanding of mathematical phenomena is so dynamic that they barely seem to appear in experience. So this experience of the pure interdependence of number requires that objective data be fully integrated with their subjective means of interpretation.
This implies a highly subtle appreciation, where rational structures interact in such a refined manner with holistic intuition that they can be seamlessly integrated with each other.
The next interesting fact is that the non-trivial zeros always have an imaginary component.
Now the importance of the imaginary notion can hardly be overstated.
Basically it relates to the means of interpreting - what is properly of a holistic circular nature - indirectly in a linear manner.
So from this perspective, the imaginary aspect relates to the unrecognised qualitative side of Mathematics.
This is obscured in current interpretation by the fact that all imaginary notions can equally be given a merely reduced quantitative interpretation.
Thus the Riemann Zeta Function for example is defined with respect to the complex plane, which necessarily entails real and imaginary number variables.
However in conventional mathematical terms, this is interpreted in a merely quantitative manner!
However, equally the complex plane can be given a qualitative definition, where it relates to the need for both quantitative (analytic) and qualitative (holistic) interpretation.
And once we recognise that all reality (physical and psychological) necessarily entails both differentiated (quantitative) and (integrated) holistic elements, then it is easy to appreciate that we necessarily live in a complex world (from a mathematical perspective). The reason again why we think that it is simply “real”, reflects a scientific paradigm that reduces the qualitative to the quantitative aspect!
However this qualitative appreciation of the imaginary notion is completely overlooked in conventional terms, with no recognition of the subtle vital dynamics that underlie the number system through the interaction of its quantitative and qualitative aspects.
In fact putting it bluntly, conventional mathematical interpretation completely misrepresents the true relationship of the primes to the natural numbers (and the natural numbers to the primes) as this relationship cannot be properly approached in a merely (reduced) quantitative manner!
So the fact that all the non-trivial zeros lie on an imaginary line (through ½), points to the fact that these zeros directly relate to holistic notions of circular type interdependence (that are then indirectly represented on an imaginary number scale).
And earlier in explaining the basic formula for calculating the average gap as between the non-trivial zeros (as we ascend this number scale) I employed this very fact!
Next, we see that the non-trivial zeros always occur in pairs.
Actually there is a very close relationship here with the nature of virtual particles in physics.
As sub-atomic reality becomes increasingly dynamic, extremely short-lived virtual particles tend to arise spontaneously as matter and anti-matter pairs (which then quickly annihilate each other turning into energy).
Virtual in this context is synonymous with imaginary, as these particles serve as but an indirect expression of the ultimate interdependent nature of matter (in what we could refer to as the holistic ground of reality).
We have a similar situation in psychological terms – especially in earliest infancy – before stable phenomenal perception emerges. So any phenomena that arise do so in a spontaneous short-lived virtual manner, as the most primitive expression of the unconscious (which is the corresponding holistic ground of all psychological phenomena).
However these occurrences are of a merely implicit nature in earliest physical and psychological life.
Thus it requires the other extreme of full maturity (where conscious and unconscious are seamlessly integrated in experience) before what is most fundamentally implicit with respect to nature, can finally be made fully explicit with respect to human understanding.
Once again complete experience of the nature of the non-trivial zeros will require a profound level of contemplative awareness (together with the most refined degree of rational understanding). And we are a long way from that yet in our evolution, though remarkably have already reached a position, whereby perhaps we can now at least glimpse the nature of its wonderful secrets.
A further intriguing factor regarding the nature of the imaginary part of these zeros is that they represent transcendental numbers.
Some 20 years ago - long before I seriously looked at the Riemann Hypothesis – I set out to show how all the structures of all various stages of psychological (and complementary physical) evolution can be precisely encoded in a holistic mathematical manner in number terms.
What I concluded at that time was that the most advanced contemplative structures (before full union in mystical contemplative awareness) would have an imaginary transcendental structure.
So it is no surprise therefore in a complementary fashion, that the most fundamental numbers (representing the interdependence of primes and natural) would also implcitly contain the same imaginary transcendental structure.
We have already seen that imaginary in this context represents the indirect linear expression of what is directly of a holistic (circular) nature. In psychological terms this refers to the indirect conscious expression of what is directly of a holistic unconscious nature.
The nature of transcendental in this qualitative context can best be appreciated with respect to the most famous transcendental number π.
Now π in quantitative terms represents the perfect relationship as between the circumference of a circle and its line diameter.
In corresponding qualitative terms π would represent the perfect relationship as between circular (holistic) and linear (analytic) type understanding.
All transcendental numbers entail in dynamic terms the relationship of circular and linear type notions.
Thus to understand in transcendental terms is to interpret relationships in neither an analytic or holistic manner (as separate) but rather as the interdependent relationship of both aspects!.
So imaginary transcendental understanding goes a step further in being able to understand a linear presentation of meaning as the indirect representation of a relationship entailing the interaction of both analytic and holistic aspects.
And this precisely is what the non-trivial zeros represent i.e. the indirect linear presentation of the relationship between the primes and the natural numbers that are understood as analytic (quantitative) and holistic (qualitative) with respect to each other.
The inclusion of ½ as the real part entails the further requirement that both the objective nature and mental interpretation of this reality be perfectly identified with each other (so that no separation exists).
So therefore we can now give a full qualitative explanation of the nature of the non-trivial zeros in this fashion!
The non-trivial zeros represent the simultaneous identity of numbers both as objects and mental interpretation, that express in an indirect linear form, the ultimate complete interdependence as between the primes and natural numbers from a dynamic relative perspective, both of which are understood as possessing analytic (quantitative) and holistic (qualitative) aspects with respect to each other.
So the full explicit understanding of the non-trivial zeros mirrors precisely the corresponding - merely implicit - nature of the non-trivial zeros in mathematical terms, where the most evolved understanding (from a psycho spiritual perspective) is necessary to interpret the least evolved - which is thereby equally the most involved - nature of reality (in a corresponding physical fashion).
The non-trivial zeros therefore represent in number form, complementary facets of the deepest intrinsic and extrinsic nature of both the physical and psychological aspects respectively of the dynamically interactive system, which lies at the very core of all created phenomena.
And this as close as one can get in the phenomenal realm to formless ineffable reality.
So now one can perhaps appreciate a little better what the non-trivial zeros truly represent and why they are so significant!