Paradoxical Nature of the Primes

We have already looked in detail at the number “2” distinguishing clearly both its analytic (quantitative) and holistic (qualitative) aspects of interpretation.

So from the former perspective, 2 can be viewed in quantitative terms as composed of independent homogeneous units.

However from the latter perspective 2 can equally be viewed in qualitative terms as uniquely composed of 1^st and 2^nd units which are interdependent (i.e. interchangeable with each other).

And to properly preserve both distinctions, 2 must be viewed in a dynamic interactive manner where quantitative and qualitative aspects are understood as complementary.

And what is true above in relation to the number “2” is equally true of all numbers.

So rather than being understood in abstract terms as absolute quantitative entities, as for example with the natural numbers - all numbers rightly represent dynamic interacting patterns, entailing the complementary action of both analytic (quantitative) and holistic (qualitative) aspects of behaviour.

This of course intimately applies therefore to the fundamental nature of the primes (with 2 the first member).

However when one looks at the conventional mathematical interpretation of the primes the analytic (quantitative) aspect is solely emphasised.

So the primes are thereby viewed in quantitative linear terms as the fundamental “building blocks” of the natural number system.

In other words, every natural number (except 1) represents a unique combination of one or more prime factors and all these natural numbers are then viewed in reduced linear terms (i.e. as points on the real line).

However each prime equally has an important circular identity, representing its (unrecognised) qualitative nature.

We can see this perhaps most simply in relation to the number 2.

As already stated the holistic (qualitative) appreciation of this number relates to recognition of 1^st and 2^nd members that are fully interchangeable with each other.

Indeed, once again this is the very recognition that is implicitly is involved when one recognises that both left and right turns at a crossroads are interchangeable depending on which direction (N or S) the crossroads is approached.

However the question that then occupied my mind for some time was how to give this holistic type recognition a satisfactory mathematical interpretation.

Returning to the crossroads left and right turns are necessarily opposites of each other.

So if approaching the crossroads (heading N) we denote a left turn as + 1, then a right turn (in this context) is – 1.

However if now approaching the crossroads (heading S) we now denote the right turn as + 1, then a left turn (in this alternative context) is – 1.

So the pure holistic recognition resides in the paradoxical appreciation that what is + 1 can equally be – 1 and what is – 1 can be + 1.

And this recognition, which is directly intuitive in origin, represents a psycho-spiritual energy state.

So far from intuitive insight directly corresponding with rational understanding it is quite the opposite in that it can only be indirectly expressed in rational terms in a paradoxical (i.e. circular) manner.

If we take the two roots of 1, we once again get + 1 and – 1. However on this occasion both results are clearly separated in a linear manner.

So for example when one approaches a crossroads from just one direction - say heading N - left and right turns have an unambiguous meaning. So if one designates a left turn as + 1, then a right turn is clearly –  1  (in this actual context).

However if one attempts to designate positive and negative signs to left and right turns without specifying the direction from which the crossroads is approached then clearly paradox is involved (in this potential context).

And this in a nutshell raises the key distinction as between analytic and holistic type recognition.

Analytic recognition relating to actual reality is unambiguous (amenable to linear rational interpretation) , whereas holistic recognition relating to potential reality is clearly paradoxical when conveyed in rational terms. So psychological intuition can only be indirectly conveyed in rational terms in a circular (i.e. paradoxical) manner.

However this raises enormous issues for conventional mathematical interpretation.

In other words professional mathematicians may indeed recognise the importance of intuitive insight (especially for creative new findings).

However there is then the mistaken belief that such intuition can be assumed to directly correspond - and thereby be formally reduced - in a strictly linear rational manner.

The key issue with respect to all mathematical interpretation is thereby completely overlooked i.e. to establish a consistent correspondence as between rational type understanding (of a linear nature) and intuitive recognition (that is paradoxical in rational terms).

And as we shall see this is the key issue relating to appreciation of the nature of the original Riemann Hypothesis and the generalised Riemann Hypothesis (applying to all L-functions).

A parallel way of expressing the dual nature of both the analytic and holistic aspects of number is to say that all numbers can be given interpretations as both structures of form and energy states respectively.

Thus the quantitative extreme, which is emphasised in conventional mathematical interpretation, views a prime (such as 2) as an absolute number form in an analytic manner. This corresponds directly with linear reason. And then the primes as structural forms are viewed as the “building blocks” of the natural number system. So the natural numbers are likewise viewed as structures of form uniquely comprising prime components as factors.

However the corresponding qualitative extreme is to view a prime (again such as 2) as a purely relative energy state in a holistic manner (with complementary psychological and physical attributes). This corresponds directly in psychological terms with intuitive insight (that indirectly can be conveyed in a circular i.e. paradoxical rational manner).   

Therefore in dynamic experiential terms, the understanding of number entails the interaction of notions of form with corresponding energy states resulting in a ceaseless transformation with respect to its inherent nature.   

So again from the analytic perspective a prime is viewed in quantitative terms as an independent “building block” of the natural number system in linear terms. So all composite natural numbers are assumed to lie on the same real line as the primes from which they are derived.

However from the corresponding holistic perspective a prime is viewed in qualitative terms as representing a unique circle of interdependence with respect to its group of natural number members (in ordinal terms).

So for example from this perspective, 5 as a prime is viewed as uniquely composed of 1^st, 2^nd, 3^rd, 4^th and 5^th members.

This uniqueness is then indirectly expressed in a circular number fashion through obtaining the corresponding 5 roots of 1 (which lie as equidistant points on the circle of unit radius in the complex plane).

Now while it is true that mathematicians have long been aware of this circular number system, once again their interpretation of its nature has been solely with respect to its analytic (quantitative) features.

However the true holistic significance in this context of the 5 roots of 1 is that they indirectly express the notions of 1^st, 2^nd, 3^rd, 4^th and 5^th (with respect to a group of 5 members) where positions are interchangeable.

And in this context the uniqueness of each prime is revealed through the fact that all its prime roots (with the exception of the default root 1), cannot be repeated with respect to any other prime.  

So from the analytic (quantitative) perspective, each prime is unique as it contains no factors (other than itself and 1).

However from the corresponding (qualitative) perspective, each prime is unique because each of its natural number ordinal members is unique (with the exception of the default last member).

And again this is highly revealing for the very interpretation of ordinal numbers in conventional mathematical terms is to treat them in fixed terms as the last member of its number group.

So 1^st is the last - and indeed only - member of  a group of 1; 2^nd is then the last member of a group of 2; 3^rd is the last of  a group of 3; 4^th is the last of a group of 4 and so on.

In this way the ordinal notion, which is inherently of a qualitative nature can be successfully reduced in a merely quantitative manner.  

However even brief reflection on the dual nature of a prime i.e. with both analytic and holistic properties, reveals its paradoxical nature.

Thus again from the quantitative perspective, each prime is unambiguously viewed in a one-way manner as a “building block” of the natural numbers.

However from the corresponding qualitative perspective, each prime is already seen to be composed uniquely of a group of natural number members in an ordinal manner.

So from the former cardinal perspective the natural numbers appear to uniquely depend on the primes. However from the corresponding ordinal perspective, each prime is already uniquely composed of a group of natural number members.

The clear implication of this paradoxical situation is that the primes and natural numbers are completely interdependent in two-way fashion with each other - ultimately in an ineffable manner - with respect to both quantitative and qualitative characteristics.