And the
crucial point here is that from the appropriate dynamic perspective that a
complementary relationship exists so that what is interpreted in an quantitative manner with respect to each individual term (related to a
specific prime), is then interpreted in a qualitative manner with
respect to the collection of terms (where this specific prime operates as a
shared common factor).

However as
always with such dynamic complementary relationships reference frames can be
switched, so that equally each individual term (with respect to the product over
primes expression) can be viewed in a qualitative manner. Then when
this is the case the corresponding interpretation with respect to the
collection of natural numbers where the prime operates as a factor is in an
quantitative manner.

So let me
now illustrate this point at greater length with respect to the example
yesterday of ζ(2), where we focussed on the first individual term in the RHS
product over primes expression i.e. 4/3, which is based on the prime number
“2”.

Now as we
have seen in other entries, 2 as a prime can be given both analytic
(quantitative) and holistic (qualitative) interpretations, which in the
dynamics of understanding keep interchanging with each other.

Thus when we
define 2 in cardinal terms as composed of independent homogeneous units (that
are thereby devoid of qualitative distinction) i.e. 2 = 1 + 1, we
thereby interpret this prime in an analytic (quantitative) manner.

However
when we equally recognise that 2 is necessarily composed of both a 1

^{st}and 2^{nd}unit, which are interchangeable with each other i.e. where each unit now is given a distinct qualitative identity in ordinal terms, then we interpret this prime in a corresponding holistic (qualitative) manner.
Once again,
from the dynamic interactive perspective, both of these interpretations are
relative, so that the analytic (quantitative) aspect of “2” cannot have any
strict meaning in the absence of the
corresponding holistic (qualitative) aspect; equally the holistic (qualitative)
aspect of “2” cannot have any strict meaning in the absence of the
corresponding analytic (quantitative) aspect.

And of
course this dynamic interactive nature equally applies to all other primes,
where both analytic (quantitative) and holistic (qualitative) aspects are
necessarily related to each other in a relative fashion.

So again
referring to the RHS product over primes expression, we can start by
defining each prime with respect to its holistic (qualitative) aspect where
component units are understood as interdependent (and fully interchangeable)
with each other.

So from
this perspective the prime number “2” (to which the first individual term
relates) can be given a holistic (qualitative) interpretation.

Then with
respect to the corresponding sum over the integers expression, 2 as a factor of
all even terms is given a complementary analytic
(quantitative) interpretation.

Clearly
therefore, though in the earlier case “2” is viewed in holistic terms as a
common factor (of all even numbers) it equally enjoys a relatively independent
identity (where it can be viewed as separate from other factors). And it is
this latter quantitative interpretation of a factor that is now emphasised in
this context.

So overall
each individual term (with respect to the product over primes expression) related to a specific prime, can be given both an analytic (quantitative)
and holistic (qualitative) interpretation.

Likewise
the collection of terms (with respect to the sum over integers expression)
which includes a specific prime as factor, can be given both analytic
(quantitative) and holistic (qualitative) interpretations.

But once
again the key requirement in correct dynamic interactive terms is that the
relationship between both expressions is of a complementary nature (with
analytic complementing holistic and holistic complementing analytic
interpretations respectively).

And in this
latter case where each individual term (with respect to the product over primes
expression) is given a holistic (qualitative) interpretation, then the
collective product of these terms is now given a complementary analytic
(quantitative) interpretation and vice versa so that when the terms (with respect to
the sum over integers expression) that share a specific prime as factor is
given a holistic (qualitative) interpretation, then the additive sum of terms
is - relatively - of an analytic (quantitative) nature.

So in
dynamic interactive terms, as we move from collective (whole) to an individual
(part) identity or in reverse from an individual (part) to a collective (whole)
identity, the very nature of number likewise keeps switching as between both its
analytic (quantitative) and holistic (qualitative) aspects, in a relative
complementary manner.

Thus the
notion that number somehow can be properly interpreted in an absolute - merely
quantitative - manner is utterly misleading and only valid as an important special case in a restricted
limited context.

And it is
this misleading notion of number - which ultimately is not fit for purpose -
that has dominated the development of Mathematics over several
millennia.

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