However it
is vital to appreciate from the outset that this significance cannot be
properly understood within the accepted confines of Conventional Mathematics, which
solely recognises, in formal terms, the quantitative (analytic) interpretation of
its symbols.

Indeed from
one important perspective, the Riemann Hypothesis - when adequately interpreted
- points directly to a critical limitation in the overall accepted framework
of Mathematics.

The number
system is fundamental to the reality we call Mathematics. And what in turn is
central to the number system is the relationship between the primes and the
natural numbers.

Now a prime
has no factors (other than itself and 1). So 7 for example is an early example
of a prime. All other natural numbers (i.e. composites) can then be uniquely
expressed as the product of primes. So for example 30 ( = 2 * 3 * 5) is
therefore composed of 3 prime factors and does not allow for any alternative factor combination.

Therefore the
primes have been long accepted in conventional wisdom as supremely important,
which - rather like atoms in physics - serve as the basic “building blocks” of the
natural number system.

However it
has also been keenly recognised that the individual behaviour of the primes is
highly unpredictable with no regular pattern occurring.

Then at the
beginning of the 19

^{th}century it was discovered that their overall frequency did indeed correspond to a simple log pattern.
Thus an
overall dichotomy became apparent as between the - apparent - random behaviour
of each individual prime and an increasingly regular pattern that characterised
their overall frequency.

It was then
Bernhard Riemann who on 1859, made immense strides in reconciling this discrepancy
as between whole and part (i.e. the regular collective order of the primes with
their random individual behaviour).

However it
is my strong contention that the overall philosophical significance of his seminal
paper on the primes, has never been properly grasped.

For to put
it simply, Riemann's findings point to the fact that underlying the conventionally
accepted natural number system of a discrete “particle” (i.e. part) is a highly
intricate continuous “wave” like system of equal importance, that in truth is
dynamically inseparable from the manifest nature of natural numbers.

This discovery took place long before the quantum revolution that shook the physical world in the
1920’s. Here paradoxical features of behaviour were revealed as the inherent
nature of sub-atomic matter. So for example at this level it was now understood that physical behaviour manifested itself in a complementary dual fashion with both particle and wave like features.

One might
ask why parallels were not then drawn as between this behaviour at a physical
level and the inherent nature of number revealed by Riemann more than
half a century earlier!

However
quite simply, the accepted framework of Mathematics is so rigid that no
proper accommodation of these findings could be made within its accepted
confines.

Then starting
from the 1970’s evidence began to emerge suggesting striking parallels as
between the “Riemann zeros” (to which the wave like features number relate) and
findings regarding energy levels in atomic physics.

However
though mathematicians can no longer ignore these findings, they still lack the means of properly explaining their nature.

In fact the
real position is so revolutionary that it undermines the very paradigm that
Mathematics has been built on now stretching back over several millennia!

So what has
wrongly emerged in our culture is the view that numbers - and indeed all mathematical relationships -
can be interpreted abstractly in an objective manner with respect to their mere quantitative
(analytic) characteristics.

However in
truth numbers - and by extension all mathematical relationships – are inherently
experiential in nature, comprising the dynamic interaction of both quantitative
(analytic) and qualitative (holistic) characteristics.

And it is
only in this relative interactive context (comprising both analytic and
holistic aspects) that the true relationships as between the particle and wave features
of number can be properly interpreted.

Putting it
simply, one overriding confusion lies at the heart of all accepted mathematical interpretation.
This relates to the attempted reduction - in every context - of qualitative
notions of interdependence in a merely quantitative manner (i.e. as
independent).

So for
example we can indeed attempt to look at the individual primes in a
quantitative analytic manner (as independent).

However the
overall collective relationship of the primes (to the natural numbers) properly relates
to their corresponding qualitative nature (as interdependent). However we cannot then also attempt -
without resorting to gross reductionism - to express this qualitative nature solely in an analytic manner! Rather we must now switch - in complementary relative terms -
to a corresponding holistic interpretation of symbols.

In psychological terms this clearly entails incorporation of both conscious (rational) and unconscious (intuitive) modes of understanding, with the unconscious aspect indirectly conveyed in a subtle circular (paradoxical) rational manner.

However no
recognition whatsoever exists - at a formal level - in present Mathematics
of the qualitative holistic interpretation of mathematical symbols
(which in truth is of equal importance to the analytic).

This is why
I stress once again without a hint of hyperbole that the greatest revolution
yet in our mathematical history - indeed in our intellectual history - is now
required where both quantitative (analytic) and qualitative (holistic) aspects
of interpretation are accepted as equal interacting partners, both in relation
to the fundamental nature of the number system and by extension to all
mathematical and scientific relationships.

So in this
context the present accepted analytic approach represents but a limited special
case of an altogether much more comprehensive understanding.

The great
German mathematician Hilbert in responding to what he considered the greatest
problem in Mathematics once replied,

“The
problem of the zeta zeros, not only in Mathematics but absolutely (the most
important)”.

Now once
again the “zeta zeros” relate to the variety of solutions to a certain key equation - the Riemann zeta function - from
which the wave pattern of number is constructed.

Over the
past 10 years or so, I have come to agree with Hilbert, though for reasons that
he would have been loath to consider!

The famed Riemann
Hypothesis is also based on a certain assumption regarding these zeros i.e.
that they all lie on an imaginary line drawn through .5 on the real axis.

In fact,
when properly interpreted this assumption serves as the central condition for
the ultimate identity of both the quantitative (analytic) and qualitative (holistic)
aspects of number.

In other words
the Riemann Hypothesis serves as the key requirement for the mutual
consistency of mathematical symbols in both quantitative and qualitative terms
(which is fundamental for all subsequent mathematical interpretation).

However once again this clearly cannot be properly appreciated through the present mathematical paradigm (which recognises solely the quantitative aspect).

However once again this clearly cannot be properly appreciated through the present mathematical paradigm (which recognises solely the quantitative aspect).

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