And in this
entry, I wish to probe deeply the precise nature of the latter two
complementary expressions.

The starting
point here is remarkably simple, though quickly becomes much more intricate.

Let us
start with the simplest of all expression viz. x = 1.

Then by the
standard laws of conventional algebra, x – 1 = 0.

However if
we now square both sides of each expression, something strange happens.

For in the
first case, x

^{2}= 1, so that x^{2}– 1 = 0; however in the second case, (x – 1)^{2 }= 0.
And the
latter equation, when expanded is x

^{2}– 2x + 1 = 0.
So though we have started with two similar expressions i.e.
x = 1 and x – 1 = 0, to quickly find through
squaring both that two new distinct expressions emerge.

And,
remarkably what has really happened is that the two final expressions i.e. x

^{2}– 1 = 0 and x^{2}– 2x + 1 = 0, really relate to two distinctive notions of number that are circular and linear with respect to each other. Using a fruitful analogy from quantum physics, they represent thereby the wave and particle aspects respectively of number.
However the deeper implications here is that the
understanding of number itself can only be properly understand in a truly
interactive manner entailing both analytic (quantitative) and holistic
(qualitative) appreciation that keep switching in the dynamics of experience.

And Conventional Mathematics is completely unsuited to this
new form of understanding as it reduces (in every context) the holistic
(qualitative) aspect of understanding in a merely analytic (quantitative)
manner.

Therefore to understand properly what happens when we square
our original expressions i.e. x = 1 and x – 1 = 0 we
must allow for two distinct aspect to the number system, I refer to as Type 1
and Type 2 respectively.

Thus
expressed in Type 1 (analytic) terms the natural numbers are defined as,

1

^{1}, 2^{1}, 3^{1}, 4^{1}, …
In other
words, they are defined in linear (1-dimensional) cardinal terms as fixed
independent quantitative entities i.e. as points on the real number line.

So the
natural numbers are defined with respect to a base number that varies against a
fixed (default) dimensional number = 1.

However in
Type 2 (holistic) terms the natural numbers are defined in circular
(multi-dimensional) ordinal terms as relatively interdependent qualitative
relationships entailing the unique sub-units of each number,

1

^{1}, 1^{2}, 1^{3}, 1^{4}, …
So for
example from the Type 1 perspective 2, i.e. 2

^{1 }= 1 + 1, where the units are considered in quantitative terms as independent and homogeneous, thereby lacking any qualitative distinction.
However
from the Type 2 perspective 2, i.e. 1

^{2}= 1^{st}+ 2^{nd}where the units are considered in qualitative terms as interdependent (i.e. interchangeable) and unique, thereby lacking any quantitative distinction. So what is 1^{st}in one context can be 2^{nd}in another related context (and vice versa).
Now the
clue to what truly happens when we square the expression x = 1 (i.e. x

^{1 }= 1^{1}) is that we now switch directly from the Type 1 to the Type 2 system.
So again in
conventional terms, when we square both sides x

^{2 }= 1 (i.e. as a number still interpreted in Type 1 quantitative terms). So as this conventional mathematical interpretation ignores the qualitative aspect, 1^{2 }is thereby reduced in a quantitative (1-dimensional) manner as 1^{1}.
However,
properly understood x

^{2}= 1^{2 }(i.e. as a number now interpreted in a Type 2 qualitative manner).
Then when we
square x – 1 = 0, we
now interpret in complementary fashion this relation in a Type 1 quantitative
manner.

So (x –
1)

^{2}= 0, i.e. x^{2}– 2x + 1, has two linear roots i.e. + 1 and + 1 respectively (as the same two points on the real number line).
However x

^{2}= 1^{2}, has two circular roots i.e. + 1 and – 1 respectively (as two points on the unit circle).
In fact what we have here are two distinct mathematic
notions of dimension that are analytic and holistic with respect to each other.

The 1st linear notion of dimension is the one that is
conventionally recognised in mathematics.

So 1

^{2}, can be geometrically represented in 2-dimensional terms as a square (with side 1 unit).
So if one side represents the length, the other side
represents the width.

So the two roots of the equation (x – 1)

^{2}= 0, i.e. + 1 and + 1 represent thereby both the length and width respectively of the square of 1 unit.
And by
extension the three roots of the equation of (x – 1)

^{3}= 0, i.e. + 1, + 1 and + 1 represent the length, width and height respectively of a cube of 1 unit.
And though
we cannot envisage this in pictorial terms the n roots of the equation (x –
1)

^{n}= 0 represent the n sides respectively of a hypercube of I unit.
However the
true nature of the 2

^{nd}circular notion remarkably, is not properly understood in conventional mathematical terms as it is in fact directly associated with an entirely distinctive holistic form of dimension.
Of course,
circular notions e.g. with respect to the various roots of 1 are indeed
recognised, but invariably in a merely analytic manner (where they are
considered as separate from each other).

However the
true holistic notion of dimension requires that the various roots of 1 - as
indirect quantitative representations of qualitative notions - be interpreted
in an interdependent manner (where they are understood as interchangeable with
each other).

I will
illustrate this again briefly with an oft-quoted example regarding the
interpretation of a crossroads.

Now when
one approaches a crossroads along a straight road - say heading N - then a left
turn for example has an absolute unambiguous meaning.

This is
because the frame of reference i.e. the direction of movement, is
one-dimensional. So there is only one direction considered here in terms of
approaching the crossroads i.e. N.

So we can
unambiguously identify the left turn in this context as + 1 with the other
right turn (which by definition is not a left turn) thereby as – 1.

Thus + 1 (a left turn) and – 1, as the two conventional
roots of 1, carry here a strictly analytic meaning.

Now, if alternatively we were to approach the crossroads from
the other direction (heading S) then again left and right turns can be given an
unambiguous meaning represented as + 1 and – 1 respectively (as the
interpretation is again 1-dimensional with only one direction of approach to
the crossroads considered).

However if now consider the approach to the crossroads
simultaneously from both N and S directions, then circular paradox is clearly
involved for what is left from one directionis right from the other; and what
is right from one direction is left from the other.

So in numerical turns what is + 1 from one direction (e.g. a
left turn) continually switches to – 1 from the other (i.e. right) turn and
vice versa.

Thus what we have here is a holistic 2-dimensional
interpretation of left and right (i.e. + 1 and – 1) which are fully relative
and thereby interchangeable with each other.

Now whereas
1-dimensional interpretation from one fixed reference frame is absolute and
analytic, 2-dimensional interpretation (from two polar reference frames simultaneously)
is by contrast relative and holistic in nature.

We could
validly equate then 1-dimensional with (linear) rational and 2-dimensional with (circular) intuitive
interpretation respectively.

However
indirectly we can give intuitive appreciation an indirect rational
interpretation in a paradoxical logical fashion.

So whereas
with linear logic opposite polarities such as + and – are clearly separated, with circular logic, +
and – are understood as
fully interdependent (and thereby interchangeable) with each other.

Now the
importance of this is that all experience is conditioned by polarities such as
external and internal and whole and part that continually interact in dynamic
fashion with each other.

This
intimately applies also to mathematical understanding.

Therefore
through conventional mathematical interpretation is based on the assumption of
the abstract independent existence of “objects” (such as number), strictly
these have no meaning apart from subjective mental constructs that are used in
their interpretation. And both objective and subjective aspects are thereby
external and internal with respect to each other.

However
once again conventional mathematical interpretation is based on the misguided
belief that “objects” such as numbers can be properly understood in an external
(1-dimensional) manner possessing an absolute quantitative identity.

And in
general, n-dimensional interpretation from the holistic perspective entails highly
refined interdependent relationships entailing n interchanging reference
frames.

However,
though not strictly valid, the corresponding analytic approach attempt to give
independently viewed objects a succession of higher dimensions in space.

Though in
experiential terms, it is not possible to go beyond 3 space dimensions in this
manner, the extension of the linear notion of dimensions can then be abstractly
extended to n
dimensions.

So the key
point again regarding the two complementary expressions of the Zeta 2 function
is that they relate to the analytic and holistic notion of dimension
respectively.

Thus the infinite
sum of reciprocals of the unique numbers associated with (x – 1)

^{ n}corresponds to the analytic (linear) notion of dimension (envisaged as n independent linear directions in space).
The
complementary approach relates to x

^{n}– 1 or 1 – x^{n}= 0. Then to get rid of the one linear dimension, we divide by 1 – x to obtain 1 + x^{1}+ x^{2}+ x^{3}+ … + x^{n – 1 }= 0.
This then represent the finite expression of the Zeta 2
function.

And then for the geometric series expressions used to define
each term of the Zeta 1 (Riemann) function, we use the infinite version of this
function, i.e.

1 + x

^{1}+ x^{2}+ x^{3}+ …