The
quantitative aspect in the linear approach is expressed through addition.

So for
example 2

^{1}= 1^{1 }+ 1^{1}^{}^{ }

However the
qualitative aspect - indirectly expressed in a quantitative manner - in the
circular approach is expressed through addition.

So 0 = 1

^{st}+ 2^{nd}i.e. indirectly expressed in quantitative terms as 0 = + 1 – 1.
Then the
qualitative aspect in the linear approach is expressed through multiplication.

So for
example 1

^{3}= 1^{1 }* 1^{1 }* 1^{1}
However when
this is expressed in a circular manner - in an indirect quantitative fashion
through the 3 roots of 1 - we arrive back at the quantitative aspect of number,

i.e. 1 * (– .5 + .866i) * (1 (– .5 – .866i) = 1

Thus for any
prime number p (except 2) the product of its p roots = 1, while the sum of its
p roots = 0.

Now the
analytic interpretation of this is of course well known. However there is a
much deeper corresponding holistic significance in demonstrating the
complementary nature of addition and multiplication (when reflected through
linear and circular frameworks respectively).

So whereas
Conventional Mathematics can indeed analyse circular relationships in
quantitative terms, it necessarily does so analytically from within a rational
linear framework.

However to
properly interpret these relationships from a corresponding holistic
perspective, we must do so within a rational circular framework (where
paradoxical complementary relationships indirectly reflect holistic
understanding that is directly of an intuitive nature).

Once again
this holistic aspect is entirely missing from Conventional Mathematics (as
formally understood).

So when we
appreciate mathematical relationships in an appropriate holistic
manner, we are then enabled to seamlessly convert from quantitative to
qualitative aspects respectively through continually switching in turn as
between linear and circular modes of interpretation.

Perhaps
the most important holistic mathematical appreciation of all relates to the qualitative
significance of the imaginary notion.

Imaginary numbers are indeed used mow extensively in quantitative analytic
terms within Conventional Mathematics.

However the
corresponding qualitative holistic interpretation of imaginary numbers - which is
equally important - is entirely missing from conventional mathematical interpretation.

I have found my acquaintance with Jungian notions extremely helpful in arriving at
the holistic mathematical meaning of the imaginary.

Again in
Jungian terms, when a conscious function such as thinking is unduly dominant,
its unconscious (shadow) complement is projected in a somewhat blind manner on to
conscious phenomena.

Now in
scientific terms, “reality” is defined in a merely conscious rational manner
which is properly geared to analytic type interpretation.

However we
are all perhaps aware that our actual experience is
conditioned to a degree by unconscious type projections (which relate to a distinctive
holistic meaning).

So when we
experience objects there is indeed a specific local aspect (according with
conscious recognition).

Likewise however
there is a universal holistic aspect (according with unconscious meaning).

For example
one might speak of a “dream house”. Now the house indeed can be consciously
verified as an object; however there is also a holistic aspect here in the
experience that serves a deeper (unconscious) holistic desire for
meaning.

Now the “imaginary”
in qualitative terms, simply refers to objects as indirectly representing this
holistic aspect of meaning.

In this
sense all reality is complex with objects having a - relatively separate -
local identity (as real) and a whole relational identity - as imaginary.

As we have seen, this is
equally true of all mathematical objects (such as numbers).

Indeed the
ordinal nature of number properly relates to its qualitative holistic identity.

In
direct terms this relates directly to an unconscious - rather than
conscious - recognition that is then indirectly expressed rationally in a circular
(paradoxical) fashion. It then can be converted, as it were, into linear type interpretation
through giving it an “imaginary” rather than “real” identity.

As we have seen, in quantitative terms, the imaginary number i is given as the square root of – 1.

Now, in the context of the Zeta 2 Function, – 1 is all important, serving in
qualitative terms as the expression of unconscious meaning. In other words, in experiential
terms, the very way in which the unconscious is activated in experience, is
through the dynamic negation of conscious type meaning.

So once
again, in Jungian terms, if the emphasis on conscious experience is unduly
dominant (as it most certainly is in Conventional Mathematics), then the unrecognised
unconscious aspect will be blindly projected onto consciously understood phenomena
(and then directly confused with them).

And of
course this is precisely what has happened in Mathematics. Though ordinal
appreciation is qualitative - inherently relating to the unconscious - this
is blindly projected on to the conscious recognition of mathematical objects
(as quantitative) and directly confused with them.

So we are
still labouring under the huge fallacy that the number system can be understood as
quantitative in a merely reduced manner, though in truth
we can give no meaningful order to this system without qualitative recognition.

And without
such qualitative order in an overall context for relating mathematical objects,
the quantitative aspect itself can have no strict meaning!

So again the
definition in quantitative terms of i is the square root of – 1;

In holistic terms, – 1 defines the 2

In holistic terms, – 1 defines the 2

^{nd}dimension. So to indirectly express this in terms of the 1st we again obtain the square root.
What this
means in effect is that from a holistic perspective, the imaginary notion relates to the
indirect expression (in a linear quantitative manner) of meaning that is
properly of a qualitative nature.

If we then
apply this to the Zeta 1 zeros, they are all postulated to lie on an imaginary
line (through .5).

What this
holistically implies, is that all these zeros directly represent a qualitative -
rather than quantitative -meaning.

In other words
the very nature of the zeta zeros is that they represent an indirect
quantitative way of expressing the qualitative (ordinal) nature of the number system
as a whole.

Now the
Zeta 2 zeros express this holistic nature more directly (in a circular manner)
thus demonstrating the unique qualitative nature of each prime number
group (representing a individual natural number members in an ordinal relationship with
each other).

The Zeta 1
zeros express this holistic nature more indirectly (in an imaginary linear
manner) and demonstrate the unique qualitative nature of the
(composite) natural numbers (representing a unique grouping of prime numbers in
ordinal relationship with each other).

So put most
simply, the zeta zeros represent, from two distinct perspectives, the (hidden)
qualitative nature of our number system.

Though this
representation is necessarily given an indirect quantitative expression, as
its true nature is holistic, it cannot be successfully interpreted in a
conventional mathematical manner (that is exclusively concerned with
quantitative meaning).

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