Of course
this requires recognition of the fact - which I have continually sought to convey
over the past year or so - that there are two complementary aspects to the
number system (Type 1 and Type 2) which dynamically interact; that
corresponding to these two aspects we likewise have two complementary Zeta
Functions (Zeta 1 and Zeta 2) which likewise dynamically interact; and finally
that corresponding to these two Zeta functions we have two sets of
(non-trivial) Zeta zeros (also in dynamic interaction with each other).

In both the
Type 1 and Type 2 approaches, quantitative (cardinal) and qualitative (ordinal)
aspects of the number system are clearly separated.

So with the
Type 1, number is initially defined with respect to the former aspect
with each number representing a base quantity that can vary, which is defined
with respect to the fixed dimensional number 1 (serving as the default qualitative
value). So, for example in the number expression 2

^{1}, 2 represents the base quantity and 1 the default dimensional number (which - relatively - is of a qualitative nature).
Again the
customary quantitative bias of Conventional Mathematics is indicated by the
fact that this default dimensional value is ignored altogether!

So for
example the natural numbers are thereby merely defined in terms of their
quantitative base values i.e. 1, 2, 3, 4,… and not as 1

^{1}, 2^{1}, 3^{1}, 4^{1},…...
The Type 2
number approach is initially defined in inverse fashion with respect to the
latter aspect, with each number representing a dimensional value that can vary,
which is defined with respect to a fixed dimensional number 1 (serving as the
default quantitative value). So, this time in the inverted number expression 1

^{2}, 2 represents the dimensional value and 1 the default base number (which - relatively - is of a quantitative nature).
So in this
latter number approach, the natural numbers 1, 2, 3, 4,… (this time
representing dimensional values) are defined as 1

^{1}, 1^{2}, 1^{3}, 1^{4},…..
However
when we allow base and dimensional values to vary in the Type 3 approach, both
quantitative (cardinal) and qualitative (ordinal) aspects continually
interchange with each other.

So
initially we start by considering each quantitative base value as relatively
independent (in cardinal terms) and each qualitative dimensional value as
relatively interdependent (in ordinal terms). But relative independence
likewise implies relative interdependence and relative interdependence,
relative independence respectively.

This
therefore implies that both base and dimensional values alternate as between
quantitative and qualitative interpretations respectively.

Now this
all seems remarkably similar to Quantum Mechanics with respect to the particle
and wave features of matter. And indeed properly understood the quantum
mechanical features of matter spring directly from this prior dynamic nature of
number (from which they derive).

Now when we
look at the Zeta 1 and Zeta 2 Functions we can see that they both involve the
natural numbers in an inverse manner.

In the Zeta
1 the natural numbers are defined as the base quantities (defined with respect
to a negative dimensional value s, which can vary).

ζ

_{1}(s) = 1^{– s }+ 2^{– s}+ 3^{– s }+ 4^{– s }+…..
By contrast
in the Zeta 2 , the natural numbers are defined as the dimensional qualities
(defined with respect to base quantities s, which can vary)

ζ

_{2}(s) = 1 + s^{1 }+ s^{2}+ s^{3}+ s^{t – 1 }
Now initially
the Zeta 2 Function is defined in a finite manner.

However by
combining in regular groups of t terms, it can be extended in an infinite
manner

i.e. 1 + s

^{1 }+ s^{2}+ s^{3}+ s^{4 }+…..
The
(non-trivial zeros) for both of these Functions relates to solutions of s where

ζ

_{1}(s) and ζ_{2}(s) respectively = 0.
Now the key
philosophical significance of these zeta zeros is that they provide (in each
case) values for s, where both quantitative and qualitative interpretations of
s are reconciled i.e. where the cardinal and ordinal aspects of the number
system are simultaneously related. In other words because in dynamic terms,
continual interaction now takes place as between both quantitative and
qualitative aspects, a meaningful solution to the equations requires that a
dynamic identity be achieved as between both aspects.

The
importance of the prime numbers in this context is that serve as the means
through which the quantitative (cardinal) and qualitative (ordinal) aspects of
the number system are mediated in two-way fashion with respect to the natural
numbers.

So from the
Type 1 perspective the prime numbers serve as the cardinal building blocks of
the natural number system (in quantitative terms).

Then in
inverse fashion from the Type 2 perspective, the natural numbers serve as the
ordinal building blocks of each prime number (in qualitative terms).

So again, this
key relationship as between this two-way relationship as between the primes and
natural numbers (and natural numbers and primes) serves as the means by which
both the quantitative (cardinal) and qualitative (ordinal) aspects of the
number system are reconciled (in a bi-directional fashion).

And the
importance of (non-trivial) zeta zeros,
in both Type 1 and Type 2 terms, resides in the fact that they provide the solution
to this key issue of the ultimate identification of both quantitative and
qualitative aspects (with respect to the number system) from both perspectives.

It is
easier to demonstrate the inherent dynamic holistic nature of the zeta zeros
initially from the Zeta 2 perspective.

What does
it precisely mean to reconcile (or identify) quantitative and qualitative
aspects of number interpretation?

Well let’s
consider the simplest possible Zeta 2 solution which arises in the context of
the two roots of 1.

As 1 will
always be one of the t roots of 1 we can deem this as the trivial root.

The other
non-trivial root then arises in the context of the finite Zeta 2 expression

1 + s = 0,
i.e. s = – 1.

Now, as
explained before if we consistently combine terms in groups of 2, this also
serves as the solution of the infinite Zeta 2 expression,

1 + s

^{1 }+ s^{2}+ s^{3}+ s^{4 }+…. = 0
Now the
deeper significance of the two roots of 1 is that they serve as the means of
expressing in an indirect quantitative manner the true ordinal (i.e.
qualitative) significance of 1 and 2.

In other
words the very recognition of the 2

^{nd}member (of a group of 2) implies that we can (temporarily) negate exclusive identification with the 1^{st}member. So this dynamic negation of the 1^{st}member now enables us to posit recognition of a (new) 2^{nd}member.
Therefore
we express this ordinal relationship (as between 2 members) in an indirect
circular quantitative manner as + 1 and – 1. So, a continual process of
conscious positing (+ 1) and unconscious negation (– 1) is involved in the
dynamic interchange as between 1

^{st}and 2^{nd}units (in this context of 2).
Now by
extension we can provide an indirect quantitative means of translating the
ordinal relationships for any finite sized group t, through the corresponding
roots t roots of 1.

The deeper implications
here imply that the ordinal appreciation of number properly relates to
unconscious - rather than conscious - recognition. Therefore ordinal notions
can only be dealt with in a grossly reduced manner within the current
mathematical paradigm (as it is formally based on merely conscious rational notions).

So
indirectly, 1

^{st}and 2^{nd}(in the context of a group of 2) can be given quantitative expression as + 1 and – 1 respectively. The holistic qualitative appreciation of the interdependence of these numbers is then obtained through combining both (through addition). So (+) 1 – 1 = 0.
So each
member (in isolation) enjoys a partial quantitative existence (as independent);
yet when combined, a merely qualitative exists (as interdependent). This is why
the quantitative result = 0. So when opposite polarities are successfully
combined in a fully interdependent manner, no quantitative independence
remains.

One
interesting physical example of this phenomenon relates to the process whereby
matter and anti-matter particles combine fusing in pure energy. So what has a
distinct material existence (as independent particles) is transformed to pure
energy (through the interdependence of both).

This also
implies of course that number in its pure interdependent state represents a
qualitative energy state.

So the zeta
zeros therefore provide a means of reconciling both the partial independent
quantitative existence of number (as form) with their holistic interdependent
qualitative existence (as a pure energy state).

In a sense
all the Zeta 2 zeros can ultimately be expressed in terms of (+) 1 – 1 (as in all cases the sum of non-trivial t – 1 roots of
1 = – 1, with the trivial root = + 1).

Thus we can
give expression to the individual natural ordinal number members of any prime
number group p (i.e. 1

^{st}, 2^{nd}, 3^{rd},…..pth members) in an indirect quantitative circular manner (i.e. as equidistant points on the unit circle in the complex plane).
The
holistic qualitative interdependence of these members is then obtained through
combining the individual members (through addition).

So the sum
of roots = 0 demonstrating the qualitative interdependence of the combined p
members.

Now the
uniqueness of the prime numbers in this context is that each of its non-trivial
members (i.e. all roots except 1) are uniquely defined.

When the
number group is not prime, this will not be the case. For example – 1 is one of
the non-trivial roots of the 4 roots of 1. However it is not unique as – 1 is
also one of the two roots of 1.

Therefore
through the Type 2 aspect of the number system (with its associated Zeta 2
Function) we obtain precisely the opposite interpretation of a prime number
from the Type 1 aspect (associated with the Zeta 1 Function).

In the Type
1 (cardinal) approach (and its related Zeta 1 function), the prime numbers are
seen as the unique independent buildings blocks of the natural number system.

However in
the Type 2 (ordinal) approach (and its related Zeta 2 Function), the natural
numbers are seen as the unique independent building blocks of each prime
number. Thus from this context each prime number enjoys a distinctive holistic
identity through the collective qualitative interdependence of its individual
(natural number) members.

By contrast
in the Type 1 (cardinal) approach (with its related Zeta 1 Function), the prime
numbers are seen as the unique independent building blocks of each natural
number.

However as
we have multiplication of two numbers (> 1) entails a qualitative as well as
quantitative transformation.

So when for
example in conventional mathematical terms we say that 6 is uniquely expressed
through its prime factors 2 and 3, we are referring to this relationship in a
merely (reduced) quantitative manner.

The important
fact is however that a dynamic type transformation is equally involved, with
once again a reconciliation or identity existing as between individual isolated
numbers (with an individual independent identity) and an overall holistic
identity (as interdependent). Thus from this opposite perspective each natural number enjoys a distinctive holistic identity through the collective quantitative interdependence of its individual (prime) members,

And this is
what the famed zeta zeros (i.e. Zeta 1 non-trivial zeros) obtain i.e. an
identity as between independence with respect to individual numbers and an
interdependence with respect to these numbers taken as a collective
whole.

The Zeta 1
zeros are however of a more indirect nature than the Zeta 2 (and therefore more
difficult to express).

Once
again the Zeta 2 zeros provide an
indirect quantitative means of translating the qualitative (ordinal) nature of
each member of a group (with the overall additive relationship expressing their
combined qualitative interdependence).

However,
individual non-trivial Zeta 1 zero provides a - seemingly - independent quantitative
numerical measurement (with a fixed real part = ½ and an imaginary part that
varies).

Thus the
imaginary scale in this context provides an indirect way of representing, in a
linear analytic manner, what is inherently of a dynamic holistic nature (where quantitative
and qualitative aspects are reconciled as identical).

This indeed
is why these zeros bear such a close relationship with quantum chaotic energy
states. So in a sense the trivial zeros represent point singularities (as pure
energy states) on the imaginary number line.

So the
distinct quantitative nature of these zeros then results from their combined collective nature.

Therefore
the local independent quantitative nature of each individual prime number (as
distinct from the natural numbers) can be obtained through wave deviations associated
with the addition of the combined group of trivial zeros to a continuous general function,
expressing the general frequency of primes among the natural numbers.

Just as the
refined use of the unconscious in psychological terms can be used to correct rigid
identification of phenomena relating to conscious experience, likewise in reverse
fashion, in a physical mathematical sense, the refined use of the Zeta 1
(non-trivial) zeros can be used to obtain precisely the local rigid identity of
prime numbers as opposed to their general (interdependent) nature with the natural
numbers.

This is why
– in a very precise sense – these Zeta 1 zeros represent the perfect shadow
system to the primes.

So once
again because the conventional mathematical paradigm is based on a merely rational
conscious interpretation of its symbols, the primes are exclusively viewed with
respect to their (quantitative) existence in an independent analytical manner.

However the
Zeta 1 zeros properly represent the perfect shadow complement to this view of
the primes. Though these can indeed be also given a quantitative existence in
an analytical manner, they are of an imaginary (rather than real) nature,
serving is the indirect expression of their inherent dynamic holistic nature
(where separate independent elements are perfectly reconciled in an interdependent
group manner).

So the very
definition of “analytic” as I use the term, is that the quantitative and qualitative meaning of mathematical symbols
can be clearly separated from each other. In the conventional mathematical terms
this is used in an absolute sense i.e. where interpretation is formally of a
merely rational nature (which thereby totally excludes all genuine holistic
meaning of symbols).

The
corresponding meaning of “holistic” is the other extreme whereby the
quantitative and qualitative aspects of mathematical symbols are fully interdependent
with each
other.

This also
has an absolute interpretation (corresponding to pure intuition) which is of an
ineffable nature.

The proper
activity of Mathematics allows for both analytic and holistic interpretation
in a dynamic relative manner (with respect to both quantitative and qualitative
aspects).

The true
dynamic nature of both Zeta 1 and Zeta 2 zeros I derives from the fact that that they provide - from two
complementary directions - the mysterious identification of both the
quantitative and qualitative aspect of mathematical symbols i.e. where analytic
and holistic meaning are reconciled.

However
appreciation of their true nature will require the most radical revolution yet in scientific history whereby both (conscious) reason and (unconscious) intuition
need to be explicitly incorporated with each other in all mathematical interpretation. (And as we have seen the indirect incorporation of intuition in rational terms requires circular paradoxical - as opposed to strict linear - understanding).

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