Just to recap where we have reached!
When one recognises that the number system has in fact two aspects that dynamically interact with each other, the relationship between the primes and the natural numbers (and the numbers and the primes) can be expressed in two ways that are complementary.
Form the Type 1 (cardinal) perspective the natural numbers (excluding 1) can be expressed as the unique product of prime number factors. So here the cardinal primes are seen as the building blocks of the natural number system.
From the Type 2 (ordinal) perspective, the primes can be expressed as the unique sum of (circular) numbers that represent each natural number in ordinal terms. So for example the prime number 3 would be represented as the sum of its 3 roots (1st, 2nd and 3rd, from this perspective).
In relation to the Type 1 approach – as is well known – we have for example the Prime Number Theorem and the Riemann Hypothesis.
So on its simplest expression the Prime Number Theorem, representing the general frequency of the primes among the natural numbers, can be expressed as n/log n (with the relative accuracy continually increasing as n increases).
What is fascinating is that a parallel prime number theorem can be expressed in relation to the Type 2 approach. Once again a remarkable form of complementarity is involved.
As I have stated the Type 1 approach offers but a reduced (1-dimensional) approach in qualitative terms of number relationships.
Now we can use a reduced 1-dimensional quantitative approach of the prime roots of 1 to establish an amazing parallel prime number theorem in Type 2 terms.
In other words we simply treat negative signs as positive and imaginary as real values.
When one then obtains the mean average of the sum of real parts (or imaginary parts) of these roots (now expressed in a positive real manner) the answer converges ever closer to 2/π (as p increases).
And 2/π = i/log i.
So for the Type 1 system we have a Prime Number Theorem (referring to the cardinal linear measurement of the distribution of prime numbers) = n/log n. Likewise we have an alternative Prime Number Theorem (referring to the indirect ordinal measurement of prime number distribution in circular terms = i/log i).
We have also a parallel Riemann Hypothesis with respect to the Type 2 system in the behaviour of actual deviations of the mean average values of roots from 2/ π (= i/log i). In other words the ratio of the difference of the mean of cos values (real part) divided by the corresponding difference of the mean of sin values (imaginary part) approaches .5.
The unexplained deviation, in a manner that parallels the non-trivial zeros, can be eliminated through the use of the ratio of prime numbers.
For example, I have demonstrated how using just the (1-dimensional values in quantitative terms) of the 3 roots of 1, how one can predict – for example – the corresponding deviation of the mean average of both real and imaginary parts of the 127 roots of 1 from i/log i, to a surprisingly high degree of accuracy.
Before moving directly on to consideration of the nature of the non-trivial zeros, I wish to explain here the deeper nature of the qualitative (ordinal) interpretation of number.
In dynamic terms, all experience is governed by the interaction of two fundamental polar sets of opposites. In other words what is viewed as external and objective is necessarily related to what is internal and subjective (and vice versa); likewise what is whole in any context is necessarily related to corresponding parts (and vice versa).
Now in geometrical terms we could arrange these polarities – like as on a compass – as 4 equidistant points on the circle of unit radius. The first set can be arranged as points on the real axis and the second set as points on the imaginary axis.
So from a conscious (real) perspective, reality is viewed as composed of holons (or whole-parts) in the standard reduced manner (where every whole is composed of constituent parts).
However to view the true relationship between whole and part in an unreduced manner, we need to include unconscious understanding (the indirect representation of which is imaginary in qualitative mathematical terms). So when we recognise wholes (in Jungian terms as archetypes) this represent an unconscious recognition of phenomena that are imaginary in nature (which cannot be confused with corresponding conscious recognition of a local kind).
Likewise each part can also reflect the (universal) whole in an intuitive unconscious manner that is imaginary. So in the dynamics of experience real (conscious) recognition continually interacts with imaginary (unconscious) awareness.
So from a psychological perspective, when appropriately understood in a qualitative mathematical fashion, all reality is necessarily complex, reflecting the interaction of (real) conscious and (imaginary) unconscious aspects.
From the corresponding physical perspective, all reality likewise is necessarily complex in qualitative mathematical terms, reflecting the interaction of phenomena in a real (analytic) and imaginary (holistic) fashion.
In fact, another way of saying the same things is that all physical and psychological processes entail both differentiated (analytic) and integrated (holistic) elements.
Now the very reason why we think we live in a “real” world is that we are conditioned that way by the limited (1-dimensional) paradigm, which largely defines scientific and mathematical interpretation in a merely conscious (rational) manner.
So what is solely conscious is understood as solely “real”.
However we know that despite our perceptions of the world as "real" in qualitative terms, that quantum reality for example corresponds to complex interpretation in quantitative terms.
However to properly appreciate such reality, we clearly need a scientific paradigm that is also complex in qualitative terms! Even more we need a paradigm in Mathematics that is complex in qualitative terms! And of course this is the very process I am engaged with in these blogs. Therefore to understand quantum reality in an appropriate qualitative philosophical manner, we need to recognise that it represents the dynamic interaction of particle behaviour in both an analytic (real) and holistic (imaginary) fashion.
After more than 40 years now in the development of the qualitative aspect of Mathematics – what I refer to as “Holistic Mathematics” - I have come to marvel at the manner in which every mathematical symbol, with an accepted Type 1 quantitative type interpretation, can likewise be given a remarkably coherent qualitative meaning in Type 2 terms.
Therefore, not only are numbers, for example, so important with respect to quantitative notions of order in analytic terms, they are equally important with respect to qualitative notions of order of a distinctive holistic kind. And sadly as yet we have precious little idea of what this greatly neglected qualitative aspect entails. Quite incredibly, its existence is not even formally recognised by the Mathematics profession!
So clearly, the enormous changes that are required with respect to appreciation of the true nature of Mathematics cannot be effected from within the recognised mathematical community.
Once again that is why I believe there is an important need for a blog such as this that actually addresses the truly fundamental issues that are completely ignored by recognised practitioners.
So we have a coherent (holistic) qualitative – as well as recognised (analytic) quantitative – notion of number.
Put simply, this qualitative notion relates to the manner in which the fundamental dynamic polarities that I have mentioned - that necessarily condition all experience - are configured.
Thus each number in this sense represents a distinctive manner of configuring the dynamic relationship between these polarities.
Jung is especially interesting in this respect. Though not a mathematician in any formal sense, his concepts do lend themselves especially well to qualitative mathematical interpretation.
The number 4 especially engaged his attention. Indeed his great follower Marie Louise Franz has said “Jung devoted practically the whole of his life's work to demonstrating the vast psychological significance of the number four”.
So she is referring here to the no. 4 in a Type 2 (circular) qualitative manner. Now when one considers that the qualitative notion of number equally applies to the holistic nature of physical reality and that every other number has both a physical and psychological relevance in qualitative terms, we can perhaps begin to appreciate the important relevance of the neglected Type 2 aspect of number.
In this qualitative respect the prime numbers are again especially important as each one constitutes an interdependent group (of natural numbers in ordinal terms) that is unique.
So ultimately, all qualitative features of phenomenal reality are derived from such prime number constituents.
In my own career, I have especially concentrated on the numbers 2, 4, and 8 with respect to their qualitative dimensional significance (which possess an especially important significance as an integral scientific means of integration).
Putting it simply, higher number interpretations reflect an increasing refinement with respect to the ability to configure the fundamental polar opposites in experience.
It is far too early in human evolution to obtain any real insight into the experiential nature of the qualitative aspect of large numbers. However we certainly have come far enough to at least begin to recognise its existence!