The primes are often referred to as the atoms of the number system.
However once again, this simply reflects a reduced Type 1 (i.e. quantitative) interpretation whereby all natural numbers externally considered can be expressed as the unique product of primes in cardinal terms.
However as we have seen there is an equally important unrecognised Type 2 (i.e. qualitative) interpretation whereby each prime is necessarily comprised internally of a set of natural number members.
Therefore if we take the prime number “5” to illustrate, from an internal perspective this is necessarily composed of a 1st, 2nd, 3rd, 4th and 5th member respectively. And the very essence of ordinal – as opposed to cardinal – interpretation is that it involves qualitative type distinctions!
And because the qualitative aspect strictly relates to the (interdependent) relationship as between numbers, without such distinctions it would not be possible to order number coherently in any relevant context.
Once again because of the extreme quantitative bias of Conventional Mathematics, this key issue is avoided and thereby grossly misinterpreted.
Thereby insofar as an internal structure is recognised it will be portrayed in a merely quantitative manner.
So from this perspective 5 is internally comprised of 5 homogenous units (understood in a merely quantitative manner).
So 5 = 1 + 1 + 1 + 1 + 1.
However the problem in defining number in such a manner is that it provides no means of providing a coherent order in qualitative terms. In other words the ability to rank numbers in an ordinal fashion reflects the qualitative aspect of number and cannot be explained in a quantitative manner!
Therefore properly understood, all mathematical understanding reflects the interaction of quantitative with qualitative type notions. However in formal terms this vitally important qualitative side remains completely unrecognised in the accepted interpretation of number.
The issue as I have repeatedly stated on these blogs could not be more fundamental in that through completely ignoring this qualitative aspect of mathematical understanding, present accepted interpretation is simply not fit for purpose.
We are now reaching a stage where Mathematics is in need of the most fundamental radical overhaul in its history intimately affecting every notion.
I have seen this clearly for many years now with a conviction that has steadily increased. Therefore I suspect that there are many others in our culture ready to reach the same conclusion. Sadly however, the greatest resistance to such fundamental change is likely to be experienced from within the Mathematics profession itself. For it requires a willingness to look at mathematical notions from a completely different perspective, before the reduced and misleading nature of present interpretation becomes fully apparent.
So returning to the primes, when we look at these numbers in an internal manner, their qualitative nature (as it were) is revealed in a highly dynamic interactive manner that closely resembles the quantum nature of physical reality.
And just as we cannot hope to understand the nature of quantum physical reality with reference to Newtonian type physics, even more we cannot hope to understand the internal nature of number with respect to standard quantitative mathematical notions.
The very essence of the conventional approach is that the major poles of understanding i.e. external (objective) and internal (subjective); individual (part) and collective (whole) are abstracted from each other leading to reduced absolute type interpretation. So in effect the internal subjective is reduced to the external objective aspect; likewise in any context the whole is reduced to part notions i.e. where the whole is interpreted as merely a collection of its individual parts in a merely quantitative manner.
However to properly embrace the subatomic nature of number we need to accept as our starting point the dynamic interaction in experience of both external and internal aspects of understanding and likewise whole and part aspects. So in this new understanding both poles maintain a – relatively - independent identity, while also sharing a common interdependence.
So the notion of any truth as absolute in objective terms is surrendered here with a new appreciation that all objective results necessarily reflect a certain type of mental interpretation; equally all quantitative mathematical notions equally can be given a coherent qualitative meaning with a proper integrated understanding thereby equally combining both aspects.
So in this entry I will demonstrate again this new thinking with respect to the number “2”, which in many ways serves as the blueprint for dynamic interpretation of all other numbers.
We are well accustomed to the conventional notion of 2 as a cardinal number (defined merely in quantitative terms).
Now this actually reflects just one limited type of interpretation which is accurately defined as linear (1-dimensional).
So the very essence of such linear appreciation is that in explicit formal terms we treat the objective pole as independent of mental interpretation (thereby creating the illusion of numbers possessing an abstract identity in absolute terms).
Likewise we treat the quantitative aspect as independent of the qualitative creating the further illusion that numbers possess a mere quantitative identity.
So the nature of number in conventional terms simply reflects the 1-dimensional type of (mental) interpretation that is employed.
However when we employ a distinctly different appreciation, the very nature of number changes in line with this new understanding.
So the key characteristic of 2-dimensional interpretation is that external and internal poles are now considered as dynamically interdependent.
This leads to what is often referred to as the complementarity of opposites. So instead of an unambiguous either/or logic suited to linear understanding, we now by contrast employ a paradoxical both/and logic in circular terms.
I have used the example of a crossroads on many occasions to illustrate the subtleties associated with this new interpretation.
We can give two – relatively - independent interpretations to movement along a (vertical) road i.e. “up” or “down”.
So if we are moving up the road an encounter a crossroads then the left turn will have an unambiguous meaning. Then having gone through the crossroads, if we now define direction as “down” the road when we again encounter the crossroads, a left turn will have an unambiguous meaning. However when we simultaneously attempt to embrace both directions “up” and “down” as interdependent, we are faced with paradox as the turns at the crossroad are now necessarily right and left (and left and right) with respect to each other.
So quite simply, the logic associated with independence is unambiguous and linear, whereas the logic associated with interdependence is circular and paradoxical.
Now the huge unrecognised problem with Conventional Mathematics is that it attempts to interpret, in any relevant context, the holistic notion of interdependence in a reduced analytic manner. In other words it attempts to explain what is qualitative (i.e. interdependence) in a merely reduced quantitative manner (as befits independence).
Once again let me demonstrate this problem with respect to the conventional treatment of the primes.
Now as we have seen Conventional Mathematics is based on linear interpretation (where polar reference frames are treated as independent).
So when we fix this frame with the individual nature of primes, unambiguous quantitative results can be obtained. This is directly analogous to fixing the movement along our vertical road so as to unambiguously identify a left turn at the crossroads!
Then when the reference frame is switched to study of the collective behaviour of the primes, again unambiguous quantitative type results can be derived. This is now analogous to again unambiguously identifying a left turn at the crossroads when we approach it from the opposite direction.
However as we can see when we consider “up” and “down” simultaneously as interdependent then the turns at the crossroads must necessarily be “left” and “right” (and “right” and “left”) in relationship to each other.
In like manner when we consider both individual (part) and collective (whole) notions in relationship to each other, such prime notions of behaviour must be both quantitative and qualitative (and qualitative and quantitative) respectively.
In other words the behaviour of the primes in terms of their individual and collective identity necessarily entails the two-way interplay of analytic (quantitative) and holistic (qualitative) interpretation with respect to each other.
However quite remarkably, Conventional Mathematics allows no formal recognition for this holistic (qualitative) dimension of understanding!
Putting it bluntly therefore, the conventional approach to understanding prime behaviour is similar to the attempt to understand two intersecting turns at a crossroads as both having a left direction!
Indeed it is truly futile therefore to attempt to understand the primes merely in a quantitative manner for intrinsic to their behaviour is a qualitative (holistic) aspect that cannot be successfully reduced in a quantitative manner.
Now in mathematical terms, the complementarity of opposites simply represents the Type 2 interpretation of the number “2”.
Here we obtain the 2 roots of 1 to quantitatively express the two ordinal members of 2 in a circular fashion as + 1 and – 1 respectively.
So we posit movement up the road as unambiguous (+ 1) by implicitly negating the corresponding down direction (as – 1) . Equally, we then unambiguously posit the movement down the road ( + 1) by implicitly negating the corresponding up direction (as – 1). So crucially we now define independence in a relative – rather than – absolute manner.
Then we understand the holistic interdependence of these two directions by combining (adding) + 1 and – 1 simultaneously = 0 (which has no quantitative significance).
So the Type 2 (circular) interpretation of each prime number, representing its internal nature, combines the relative quantitative independence of each of its natural number ordinal members (indirectly expressed in a circular quantitative manner) with their overall collective holistic interdependence. And this qualitative interdependent aspect is indirectly represented as the sum of all the individual members = 0!
Such understanding requires the interplay of refined reason with intuition with the independent aspect provided directly by reason and the holistic component by intuition.
As we move on to larger numbers, this interplay of recognition of independent members becomes so seamlessly combined with their collective interdependence that they no longer even appear to arise.
So the full integration of both the quantitative and qualitative aspects leads ultimately to the experience of number in Type 2 terms as representing pure (spiritual) energy states, which complements their existence as equally representing pure energy states in physical terms.
Thus we can now perhaps see here with reference to the simplest example of 2, how both distinctive linear (analytic) and circular (holistic) interpretations can be given to every number.
Thus from the standard linear perspective, prime numbers such as 2 are viewed externally somewhat like indivisible unique atoms from which all natural numbers are quantitatively derived.
However from the new circular perspective, each prime number is viewed internally as a group whose ordinal members (except 1) are uniquely defined in natural number terms (which then can be indirectly expressed in a circular fashion).
When one appreciates this fact, then the ultimate relationship as between the primes and the natural numbers can be clearly seen as paradoxical.
Put another way – when properly understood in a bi-directional fashion - ultimately the primes and natural numbers are fully interdependent with each other i.e. mutually contained in each other in an indivisible manner.
With this realisation the true nature of the Riemann Zeta Function becomes clear as the relationship between analytic (quantitative) and holistic (qualitative) type meaning with the Riemann Hypothesis as the condition required for the mutual identity of both aspects.
Therefore from this perspective, as the Riemann Hypothesis entails the condition
for ultimate identity of both the quantitative and qualitative aspects of number, it cannot be proved or disproved from the standpoint of Conventional Mathematics (as it allows no formal recognition for the qualitative aspect).