Whereas
analytic interpretation is indeed appropriate with respect to the cardinal
notion of independence, by contrast holistic appreciation is required for the
ordinal interpretation of number as interdependent.

Once again
analytic interpretation corresponds with rational understanding (of a linear
nature).

However
holistic interpretation corresponds directly with intuitive understanding (that
is indirectly rationally expressed in a circular i.e. paradoxical manner).

Again the
essence of analytic interpretation is that it is 1-dimensional i.e. based in
any context on just one independent pole of reference.

The essence
of holistic interpretation is that any number but 1 serves as the dimension. In the simplest case,
which in many ways serves as the blueprint for all holistic interpretation, two
interdependent poles of reference are used.

I have
explained on many occasions the relationship as between these two approaches in
the context of a crossroads.

Once again
if we adopt independent polar frames of reference, we can give unambiguous
interpretation to left and right turns at a crossroads.

So if one approaches
the crossroads (travelling North), one can give an unambiguous definition of left
and right to each turn; then when one approaches the crossroads travelling South, again both
turns are unambiguous.

The reason
for this lack of ambiguity is that the reference frame is
independent, referring to just one pole (which again is the essence of
1-dimensional interpretation).

However, if
one now simultaneously attempts to include both reference frames North and South,
then definition of left and right is paradoxical. So each turn at the
crossroads can be both left and right depending on context.

The reason
for the paradox is now due to the fact that the reference frames are
interdependent (defined in a 2-dimensional manner).

So the very
essence of paradox is that it is nondual (and thereby cannot be grasped in an
analytic fashion).

And the
appreciation of such paradox relates to a distinctive type of holistic
understanding (where the two reference frames are united in a directly
intuitive manner).

Therefore,
though we indirectly can express such nondual appreciation in a
circular (paradoxical) rational manner, in direct terms it
is of a holistic intuitive nature.

So the
holistic appreciation of 2, is the recognition of + 1 and – 1 as complementary opposite poles of
recognition.

Therefore, the very
basis of intuitive insight in experience is the recognition of the
complementary - and ultimately identical - nature of external (i) (objective)
and internal (subjective) polarities and (ii) part (quantitative) and whole
(qualitative) polarities.

Indeed
these four polarities, representing the 4-dimensional qualitative nature of
dynamic interaction in experience, can be conveniently represented - like the
directions on a compass - by four equidistant points on the circle of unit
radius (in the complex plane).

In like
manner the n-directional qualitative nature of dynamic interaction between
polarities can be represented by n equidistant points on the circle!

As I have
said, in many ways 2-dimensional interpretation serves as the simple blueprint
for all holistic recognition.

The
important point here is that with holistic recognition we must always start
with (1-dimensional) analytic interpretation.

So before
we can recognise the two turns at a crossroads in holistic interdependent terms
as both left and right, we must initially recognise them in independent terms as
unambiguously either left or right. So we must apply two independent reference
frames separately (as independent) before combining both (as interdependent).

Now in
qualitative terms "+" simply implies positing in a conscious manner and + 1
implies such positing with respect to one independent reference frame.

Thus once
again, Conventional Mathematics in this qualitative sense, is 1-dimensional with the
one root of 1 = + 1.

In
qualitative terms, "– " implies (dynamic)
negation in an unconscious manner. Thus the very means by which the
unconscious is activated in experience is through the dynamic negation of
consciously posited reference frames.

Without the unconscious it would be thereby impossible to switch reference
frames. Thus the all important process (in any context) of relating part to
whole (and whole to part) implicitly implies unconscious recognition. And as we
have seen, intuitive appreciation in experience relates directly to unconscious
type recognition!

So the
holistic appreciation of interdependence is directly of an unconscious nature.

Once again
Conventional Mathematics cannot deal with this notion (except in a grossly
reduced sense) as it formally defines relationships in a merely conscious manner.

Thus the
Type 1 aspect of the number system relates directly to the analytic (conscious)
aspect of mathematical experience.

The Type 2
aspect relates - by contrast - to its holistic (unconscious) aspect.

So again
from a Type 1 perspective, 2 is defined as 2

^{1}. From a Type 2 perspective, 2 is defined as 1^{2}.
Then associated with the Type 1 system is the standard Riemann Zeta Function (which
I refer to as Zeta 1),

i.e. 1

^{–s }+ 2^{–s }+ 3^{–s }+ 4^{–s }+……..
Here each
of the natural numbers is raised to the negative of the same dimensional value (i.e. s).

However
equally associated with the Type 2 aspect is an alternative Zeta Function
(which I refer to as Zeta 2),

i.e. 1 + s

Here, in reverse s as the quantitative value is raised to each of the natural values (up to t – 1).^{1 }+ s^{2 }+ s^{3 }+….. + s^{t – 1}^{}

So whereas Zeta
1 is conventionally defined as an infinite series, Zeta 2 is of a finite nature
(though it can be given a qualified meaning in infinite terms).

Now the
derivation of the non-trivial zeros for Zeta 1 relates to the equation

1

^{–s }+ 2^{–s }+ 3^{–s }+ 4^{–s }+…….. = 0
The
corresponding derivation of non-trivial zeros (whose role is as yet
unrecognised) relates to the equation,

1 + s

^{1 }+ s^{2 }+ s^{3 }+….. + s^{t – 1 }= 0 (where t is a prime number).
The
simplest example is where t = 2, and

1 + s

^{1 }= 0,
so that s
= – 1.

As we
know + 1 will always constitute one of
the roots of 1.

In this
sense it is a trivial root (as it is repeated in every case).

The deeper
qualitative significance of this root is that it implies that we must always
start with analytic interpretation (within independent reference frames) before
we can achieve their holistic interdependence.

However the
fascinating feature of the Zeta 2 zeros is that they already have eliminated the
common root of 1).

Thus where
t is prime the solutions of the Zeta 2 equation relate to the t – 1 non trivial
roots (which are directly of a holistic nature geared to the interpretation of
interdependence).

Now the
unconscious holistic nature of these solutions is indicated by the fact that
their sum = – 1.

Thus in one
sense, the higher dimensional interpretations (where t is prime > 2), relate
to ever more refined unconscious (i.e. holistic) type appreciation.

The
uniqueness of the prime numbers in this regard is that there non-trivial roots
are unique (i.e. never duplicated in other prime root solutions).

Thus in
Type 1 terms a prime number is unique because of its independence in cardinal
terms i.e. has no natural number factors.

In Type 2
terms a prime number is unique because of its interdependence in ordinal terms i.e.
all members (apart from the 1

^{st}) are defined in a non-repetitive manner.
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