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 nature, 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.

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.

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