Once again,
if we take the number 3 to illustrate in cardinal terms this is treated
unambiguously in quantitative terms as representing a collective whole integer. Thus
if we wish to break it into constituent units i.e. 1 + 1 + 1 this must be rendered
in homogeneous terms (without qualitative distinction).

However if
we look on 3 in ordinal terms, it takes on a very distinctive qualitative type
meaning based on its relationship with other numbers in a group.

The
simplest case would involve a group of 3 members. So the ordinal notion of 3

^{rd}therefore implicitly entails setting up a relationship with the two other members in the group, which thereby can be designated in this context as 1^{st}and 2^{nd}.
To do this
we must initially fix the position of the 1

^{st}member. Now the fascinating thing about 1^{st}is that ordinal and cardinal meanings necessarily coincide! In other words with the 1^{st}member we have by definition no outside context yet (with other members). So in this sense the ordinal notion of 1^{st}must coincide with the cardinal notion of 1.
Once again
this is precisely why the 1-dimensional paradigm employed in Conventional
Mathematics ordinal and cardinal meanings coincide! So in effect ordinal
notions are effectively reduced in cardinal terms!

So the
fixing of position of the 1

^{st}member in a group implies the cardinal notion of 1.
Now this
can be done in any of 3 different ways (as there is no distinction as between
the three cardinal units of 3).

This means
that in true circular terms we can have three distinctive arrangements of 1

^{st}, 2^{nd}and 3^{rd}.
In other
words this ordinal approach demonstrates the true interdependence of the group
(in a merely relative manner).

Therefore, each of the 3 members of the group can be 1

^{st}, 2^{nd}and 3^{rd}(depending on context).
So we have
moved quickly here from one extreme to another.

In the
cardinal definition, 3 has an unambiguous (linear) quantitative meaning that is independent
of other numbers.

In the
ordinal definition 3 (as 3

^{rd}) has a merely (circular) qualitative meaning that is intimately dependent on its relative relationship with other numbers (as interdependent).
So
depending on context, any of the 3 individual members of the number group can
be designated as the 3

^{rd}!
In
1-dimensional interpretation, as the ordinal relationship is confined to
switching as between one member of a group, effectively it becomes
indistinguishable from its cardinal identity.

Thus in
1-dimensional interpretation the cardinal numbers are literally represented as
points (drawn at an equal distance from each other) with their corresponding
ordinal identities assumed to follow from their cardinal positions.

This
equally coincides with the fact that when the dimension of a number is 1, the
corresponding (one) root of the number is thereby identical. When the dimension (i.e. exponent) > 1, then the structure of the corresponding roots becomes more complex. So for example when we have 3 roots, these serve, in an indirect quantitative manner, to provide the ordinal relationships between the 3 members of a prime group!

Therefore,
once a number is defined with respect to any other dimensional number other than 1, we then have to draw a
clear distinction as between cardinal and ordinal type interpretations with
respect to number.

Once again
this is of supreme importance with respect to interpretation of the Riemann
Zeta Function which of course entails the natural numbers defined with respect
to varying dimensional powers (i.e. s).

Now the one
dimensional value for which the Riemann Zeta function is undefined is where s = 1.

From a
holistic perspective, this is precisely because no distinction can be drawn
here as between cardinal (quantitative) and ordinal (qualitative) type
interpretations.

So this
immediately suggest that - when appropriately defined - the Riemann Zeta
Function, via the Functional Equation, relates to the intimate connections as
between quantitative and qualitative type meaning. The significance of .5 in
the context of the Riemann Hypothesis thereby relates to the condition necessary for
the mutual identity of both cardinal and ordinal interpretations.

Therefore, the
importance of .5 is that it represents the dimensional value relating to the Zeta 1 Function (with all the non-trivial
zeta zeros presumed to lie on the imaginary line drawn through this point).

However we
can demonstrate an equal remarkable significance to .5 in the context of the
Zeta 2 Function (where it now represents a corresponding quantitative value).

As
mentioned in the two previous blog entries we can initially define the Zeta 2
Function initially as,

ζ

_{2}(s) = 1 + s^{1 }+ s^{2 }+ s^{3 }+….. + s^{t – 1 }(with t prime)
And for the
zeta zero solutions we set,

ζ

_{2}(s) = 1 + s^{1 }+ s^{2 }+ s^{3 }+….. + s^{t – 1 }= 0
The
question then arises as to what happens is we attempt to extend the Zeta 2
series in the(conventional) infinite manner!

1 + s

^{1 }+ s^{2 }+ s^{3 }+…..
As we
have seen in the simplest finite case case (the non-trivial second root of 1)

1 + s

^{1 }= 0 with s = – 1.
Thus if we
insert this value in the infinite series we get

1 – 1 + 1 –
1 + 1 –……

Now clearly there
are only two options here for the value of the infinite series!

If we have
an even number of terms then the sum = 0!

If we have
an odd number of terms the sum = 1.

As the
chance of an even or odd number of terms is similar, then we can say that the
probable value of the series = .5.

There
is a simple identity formula we can use to obtain this value

1/(1 – s) =
1 + s

^{1 }+ s^{2 }+ s^{3 }+…..
So when we
insert the value of s = – 1 on the LHS, the value on the RHS = ½ (i.e. .5)

What is
remarkable here, is that the result of the formula for the first of these zeta
zeros, strictly represents - not an actual - but rather a probable value!

What is
even more remarkable is that the probable value of the infinite series for all
Zeta 2 zeros is likewise .5.

One way of
expressing this is that the sum of all non-trivial roots of 1 = – 1 and when we
insert this value in the formula we get .5.

However I
will demonstrate it more fully for the two non-trivial roots of the 3 roots of 1 (correct to 3
decimal places),

i.e. the
solutions for 1 + s

^{1 }+ s^{2 }= 0 i.e. s = – .5 +.866i and – .5 +.866i.
Now there
three possibilities with respect to the number of terms in the series here here

3n, 3n + 1
and 3n + 2. (where n = 1, 2, 3, 4,....)

If we have
3n terms the value of series in both cases = 0 (for each root value used). .

However if
we have 3n + 1 terms there are two possibilities (for the two root values) with the
answer in each case = 1

If we have
3n + 2 terms there are again two possibilities.

For the 1

^{st}root value, the sum of series = 1 – .5 +.866i = .5 +.866i
For the 2

^{nd}root value the sum of series = 1 – .5 –.866i = .5 - .866i
Thus the
sum of these two possibilities = 1

Therefore
for 6 different options (i.e. 2 root values applies to 3 different terms of
series) the overall expected value = 2 + 1 = 3.

Therefore
the average expected value = .5.

So
remarkably for all Zeta 2 zeros (and indeed all non-trivial root values for the natural numbers), the expected value (or probable value) of the
infinite Zeta 2 series = .5.

And this is
what determines the real part of s for non-trivial zeros (in the complementary
Zeta 1 series) where,

ζ

So what constrains the non-trivial zeros of the Zeta 1 Function to lie on the imaginary line through .5 is the fact that the (probable) value of the Zeta 2 infinite Function for all its non-trivial zeros = .5._{1}(s) = 0.And of course there is a circular relationship involved here, as the very reason why this value of .5 holds with respect to the Zeta 2 Function is because of its application in turn to the Zeta 1 Function.

So correctly understood, in dynamic interactive terms the value of .5 is simultaneously co-determined with respect to both Functions!

However
what is important to point out is that this value of .5 cannot be interpreted in the
standard absolute analytic manner.

In truth it
represents a merely probable value. This in turn reflects an inherently dynamic
interpretation of the series where both quantitative and qualitative aspects
necessarily interact.

It
is very reminiscent of the nature of interpretation within Quantum Mechanics.

Indeed at a
deeper level, the very behaviour manifested by particles in quantum physical
terms is rooted in the (true) inherent dynamic nature of the number system.

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