We saw
how every number can be equally given a holistic (qualitative) as
well as analytic (quantitative) meaning.

In
direct terms, the latter holistic meaning is equivalent in psycho-spiritual
terms to an intuitive energy state.

And as
psychological and physical aspects of reality are dynamically complementary,
this entails that number can equally be given a holistic meaning in terms of
physical energy states.

Because
of the close correspondence as between the Riemann zeros and certain data
representing the excited energy states of atomic particles, this clearly
suggests that the zeros relate directly to the holistic - rather than analytic
- aspect of number.

However
speaking still somewhat in general terms, a fascinating and crucially important
point can be made regarding the holistic nature of number.

As we know
modern science, with its strong quantitative bias is rooted in analytic
interpretation.

However
what is not all yet realized in our culture is that likewise all of the various arts are
intrinsically rooted in the holistic interpretation of number.

In other
words all the qualitative attributes universally manifest in created phenomena are
encoded in the notion of number (when given its appropriate
holistic interpretation).

So again
every number, as it were, has its own unique holistic signature, which forms the
fundamental basis for the qualitative aspects of all phenomena.

Therefore
sometime in the future for example it will be readily acknowledged that that
true aesthetic appreciation (in all its forms) is rooted in enhanced holistic
mathematical appreciation.

And sadly
we still live in an age where the holistic aspect of mathematics is not even
formally recognised!

It must
be emphatically stress however that holistic mathematical appreciation cannot be
considered as an optional add-on to present analytic understanding, as it
requires a radically distinct mind-set, where increasing specialisation in a
contemplative type vision of reality is required.

In the past,
such specialisation in contemplative training was related to advanced spiritual
practice (associated with the various religious movements East and West).

However
rarely was sustained attention given - with the possible exception of the Pythagoreans - to the implication of advanced
contemplative states for mathematical interpretation. So perhaps it is only now
that the true need is emerging in our culture for a radically new mathematical
approach.

Perhaps in
making this point I could usefully relate my own experience where holistic
understanding naturally emerged while attending University following on profound disillusionment
with the conventional mathematical approach.

Then as the
holistic aspect underwent considerable development, for many years I suffered a
sharp decline in ability to follow the established abstract approach to
mathematical problems.

And it is
now only in recent years - following several decades of holistic
training - that I have been gradually able to look at a fundamental problem
such as the Riemann Hypothesis from a dynamic perspective (entailing both
analytic and holistic aspects) where it now appears in an entirely different
light.

In some
future golden age, I suggest that mathematics by its very nature will entail
the balanced integration of both its analytic and holistic aspects. However we
are still very far away from that day due to a completely blindness at present to
the need for true holistic appreciation.

So to
return to the primes, we have seen that each prime can be given both an analytic
(quantitative) and holistic (qualitative) interpretation.

Again from
the former perspective, each prime is unique it that it has no constituent
factors (other than itself and 1). Thus it thereby serves as a quantitative
“building block” of the natural number system.

However
from the latter perspective, each prime is unique as it is composed in ordinal
terms of a group of natural number members, which are themselves (apart from the
last) unique.

Thus again
for example 5 as a prime is unique from the former quantitative perspective in
that it has no factors (other than 5 and 1).

Then from
the latter qualitative perspective, 5 is unique in that its 1

^{st}, 2^{nd}, 3^{rd}, and 4^{th}members are distinct. Indirectly this is demonstrated through the corresponding four (of 5) roots of 1, i.e. 1^{1/5}, 1^{2/5}, 1^{3/5}, 1^{4/5}which cannot repeat for any other prime.
The final
root i.e. 1

^{5/5}, by definition = 1, and this then provides the means by which the ordinal notion is reduced in conventional mathematical terms.
So in conventional
usage the ordinal positions are not treated as interchangeable, but rather
fixed with the last unit member of each number group.

Now
conventionally, as we know the natural numbers (apart from 1) are obtained
through a unique combination of prime factors.

So again
from this perspective 6 = 2 * 3.

6 therefore
is uniquely composed in quantitative terms of its two component prime “building
blocks” i.e. 2 and 3.

So from the
analytic perspective 6 is now likewise considered as an independent quantity in
an absolute manner.

However
though 2 and 3 initially can indeed be considered in isolation as independent
“building blocks”, the very fact of combining them creates a new unique holistic
interdependent identity. And this is a vitally important point.

Once again
in isolation 2 and 3 have an independent identity as prime quantities.

However
when combined together as factors of a new composite natural number i.e. 6,
they now attain a shared status as prime factors of that number.

Thus there
is a clear distinction to be made as between the analytic (quantitative) notion
of a prime as an independent “building block” (in isolation from other primes)
and the corresponding holistic (qualitative) notion of that same prime as a
shared factor of a composite natural number.

We
have already seen in a previous blog entry the distinction as between the
analytic and holistic aspects of number intimately relates to the complementary
nature of the operations of addition and multiplication respectively.

And if one
is to understand the true nature of dual sum over natural numbers and product
over primes expressions (which are common to all L-functions) then this
distinction is vital.

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