We saw yesterday how in fact the (natural) number system can be given two complementary aspects representing quantitative and qualitative interpretation respectively.
The standard quantitative aspect is expressed in terms of the sequence of natural numbers (in each case raised to a default dimensional power = 1).
The (unrecognised) qualitative aspect is expressed in reverse fashion in terms of a default quantitative value = 1 that is raised to the successive natural numbers (as dimensional powers).
Now clearly from the reduced quantitative perspective - which characterises (Type 1) conventional interpretation - the second aspect seems of little interest with the value of each term = 1.
However the significance of the second qualitative aspect of the number system only becomes apparent when it is converted from linear to circular expression and it is here that the relationship of the integer power with its corresponding root form expression is vital for comprehension.
For the purposes of this exposition, we will confine ourselves to the qualitative meaning of 2 as a dimensional power. Fortunately, this in fact is sufficient to ultimately convey the true significance of the Riemann Hypothesis.
As we have seen, Conventional (Type 1) Mathematics can be accurately defined in terms of sole use of 1 a dimensional means of interpretation. So this leads - literally - to rational interpretation of a linear kind!
As we shall see, the deeper meaning of dimension relates to the number of polar directions that are involved with respect to interpretation.
And it is precisely because Type 1 Mathematics represents just one direction that seemingly it leads to absolute type truth.
So let us look a little more deeply at this matter!
Once again, actual mathematical experience necessarily entails the interaction of polar opposites in experience that are dynamically relative to each other.
For example it would be impossible to view the external nature of mathematical "objects", in the absence of corresponding mental constructs, which - relatively - are of an internal subjective nature.
So the notion of a number as objectively existing in external terms is strictly meaningless in experience, in the absence of the corresponding notion of a number perception which - relatively - is of an internal subjective nature.
Thus using simple mathematical language these two poles are positive and negative (and negative and positive) with respect to each other.
Another crucial example of such fundamental polarities in experience relates to the notions of whole and part (and is central to the mystery of the primes).
Thus in dynamic relative terms, any collective notion of general wholeness is necessarily of a dynamic relative nature with respect to its corresponding parts (and vice versa).
So again in dynamic relative terms these two polarities - which necessarily condition all mathematical experience - are positive and negative (and negative and positive) with respect to each other.
However it is the very nature of standard mathematical interpretation to ignore the dynamic relative nature of actual experience to fix these poles in an absolute manner.
Now as we know an absolute number refers to a situation where the sign of the number is ignored.
It is exactly similar with respect to standard mathematical interpretation. Thus, through ignoring the dynamic relative context of experience (with alternating positive and negative polarities) we thereby fix interpretation with the positive direction. In this way mathematical "objects" are given an unambiguous absolute identity with mathematical proof viewed in the same - ultimately misleading - fashion.
Therefore, in a very precise manner we perhaps now better appreciate what is meant by linear (i.e. 1-dimensional) interpretation from a qualitative perspective. And "1" in this context has now taken on a distinctive holistic meaning (which cannot be confused with the quantitative notion of 1)!
However, I am here offering the important insight that potentially an unlimited number of qualitative dimensions can be used for mathematical interpretation and that dimension corresponding to the number "2" is especially relevant in understanding the true nature of the Riemann Hypothesis.
Thus each dimensional number (in qualitative terms) bears an inverse relationship with its corresponding root (from a quantitative perspective).
Thus 2 as a dimensional number (in qualitative terms) is directly related to 1/2 (from a quantitative perspective).
So when we replace 2 in the second aspect of our number system with 1/2 we obtain 1^(1/2).
And this immediately leads to a quantitative result that is circular (i.e. that lies on the circle of unit radius in the complex plane).
Now there is huge confusion over this issue in standard (Type 1) mathematical interpretation, which directly arises from the lack of a distinctive qualitative aspect to number.
So the square root of 2 is given as either + 1 or - 1. Worse still, + 1 is then designated as the principle root (and subsequently used in numerical calculations). So when you take out a calculator input 1, the press the power function button and input 1/2 you get the answer 1 (i.e + 1).
However appropriately understood + 1 and - 1 pertain correspondingly to two different numbers in the qualitative system. So + 1 is properly the square root of 1^2 and - 1 is thereby the square root of 1 (1^1) as 1^(1/2).
So - 1 is then the appropriate number in the circular quantitative system that corresponds directly, in qualitative terms, with the dimensional number 2 (as a means of interpretation).
This implies that - 1 therefore relates to a circular type interpretation (from a logical perspective).
It is important to understand that we are now talking about a dynamic notion of negation (which necessarily entails its positive aspect).
In fact, psychological language is directly relevant here. Absolute type understanding pertains to the rational conscious.
Now the movement to qualitative understanding, entails a corresponding movement to the holistic unconscious, whereby dynamic negation (of what has been posited in conscious terms) takes place.
So in this dynamic context - 1 properly entails (+ 1 - 1). And when perfect negation takes place, in a similar manner to the annihilation of a matter by a corresponding anti-matter particle, pure energy results. However, in this psychological context we are referring to pure spiritual energy of an intuitive kind!
Such 2-dimensional interpretation is often referred to as the dynamic complementarity of opposites and is of a circular - rather than a linear - nature.
It can conveniently be referred to as paradoxical both/and logic, in contrast to unambiguous either/or logic, that typifies linear (1-dimensional) interpretation.
Once again, though directly it is of an intuitive holistic nature (that simultaneously embraces polar reference frames that are in opposition to each), indirectly it can be given rational expression in a paradoxical manner.
In fact, strictly the pure intuitive expression (which is nothing in phenomenal terms) refers to - 2 as a dimensional number with + 2 referring to its rational expression. So, to obtain pure intuitive appreciation of what the complementarity of opposites entails, we must dynamically negate any remaining identification with a rigid rational component! The Spanish mystic St. John of the Cross deals especially well with this type of negative dimensional understanding in a spiritual context emphasising its goal as - literally - nada (nothing). My point is that this qualitative type understanding has a direct holistic mathematical significance for interpretation of the first - and then by simple extension - all the remaining trivial zeros!
And from yet another angle, this again establishes that the Riemann Functional Equation establishes a clear relationship as between quantitative and qualitative type interpretation as between the RHS (for values of s > 1) and on the LHS (for values of s < 0).
I mentioned in an earlier example the apparently simple example of road directions at a crossroads.
However properly interpreted, this is far subtler than it might first appear (entailing the use of both 1-dimensional and 2-dimensional interpretation).
And when you properly understand this, you understand the true nature of the Riemann Hypothesis!
Once again in linear (1-dimensional) terms we can seek to obtain an unambiguous notion of direction by fixing reference frames (in an isolated independent manner).
So when I fix direction with movement "up" a straight road, I can for example, unambiguously identify a left turn at a crossroads .
Then, if I continue "up" the road past the crossroads and decide to turn back, now again - in an isolated independent manner - I will fix the reference frame with movement "down" the road. So when I come to the crossroads once again, I can unambiguously identify a "left" turn.
So using independent reference frames which befits the 1-dimensional approach, both turns at the crossroads are identified unambiguously as left.
However if we now switch from independent to interdependent reference frames, we realise that both turns are necessarily left and right (and right and left) in relation to each other.
In fact from this perspective what is left and right depends purely on convention, so that when both options are simultaneously considered, what is left (from one direction) is also "right" from the other and what is "right" is also "left" (from the other direction).
Indeed about 2,500 years ago the Greek philosopher Heraclitus had already dealt with this circular type of reasoning (that befits the 2-dimensional approach) when he stated,
"the way up is the way down;
the way down is the way up"
or in terms of my own illustration,
"a left turn is a right turn;
a right turn is a left turn"
So we can see here that the nature of linear reason is to operate in an unambiguous sequential fashion (within independently fixed polar frames of reference).
The nature by contrast of intuition - which then indirectly is translated in a paradoxical rational manner - is to simultaneously embrace interdependent reference frames in a holistic manner. And both types are necessarily involved in all understanding, whereby we continually strive to differentiate and then in turn integrate phenomena.
However it is a shocking indictment of conventional mathematical procedures that in formal terms, it ignores altogether this vital integral holistic aspect of understanding!
So, as we have seen the approach of Conventional Mathematics, in attempting to understand the nature of the primes, is to adopt isolated independent reference frames with respect to both their individual and collective nature.
So like the person who labels the two turns at a crossroads as left, mathematicians likewise attempt to identify both the individual and collective nature of the primes in a merely quantitative manner.
However as we well know, the turns at a crossroads are necessarily left and right (and right and left) in relation to each other.
The now accepted finding that the non-trivial zeros are encoded in the primes (and the primes equally encoded in the non-trivial zeros) clearly establishes their truly interdependent nature.
Properly understood therefore, this should immediately suggest that the relationship is as quantitative to qualitative (and qualitative to quantitative). But because of a marked inability to look outside the limitations of the conventional mathematical approach, this somewhat obvious conclusion is completely missed!
Indeed we can here suggest what precisely is the significance of the Riemann Hypothesis (where all the non-trivial zeros are postulated to lie on the real line = 1/2).
This numerical value of 1/2 actually relates to the real part of the dimensional power of s.
And as I have been stating here, in qualitative terms this directly points to the need for a 2-dimensional manner of interpreting these zeros.
Furthermore we have seen that 2-dimensional interpretation entails that two polar reference frames are considered in an interdependent manner.
As no one can now really doubt that the evidence already available establishes a two-way interdependence as between their individual and holistic nature, therefore the Riemann Hypothesis cannot be properly understood in the absence of appropriate 2-dimensional interpretation.
This also clearly establishes that the Hypothesis cannot be proved (or disproved) within the conventional (Type 1) mathematical paradigm (based on merely 1-dimensional quantitative interpretation).
So what the Hypothesis implies (in its requirement that all non-trivial zeros lie on the real line = 1/2) is that inherent in the nature of prime numbers are two diametrically opposed tendencies, individual and holistic respectively, which ultimately are totally interdependent with each other (in an ineffable manner).
This requirement therefore embraces both the quantitative and qualitative nature of the primes which ultimately is inseparable from the mystery of both the origins of life and all subsequent created evolution.