This
confusion arises directly from attempting to deal with - what inherently
represent - dynamic interactive relationships (possessing a mere relative
validity) in a misleading static absolute fashion.

This is
especially true - in the most fundamental sense - with respect to the true
nature of the number system. Rightly interpreted, this entails in experience
the continual interaction of both quantitative (analytic) and qualitative
(holistic) aspects (with an arbitrary relative validity). However for several
millennia now we have committed ourselves to a reduced interpretation (in
absolute terms) with sole recognition given to the quantitative aspect.

Now I would
always recognise the important validity of this approach (in a limited partial
i.e. quantitative sense). However the
huge mistake is in elevating it as being synonymous with all valid Mathematics.

My
criticism could not be more fundamental for it embraces all accepted present
mathematical activity.

So it cannot
be countered through more contemporary developments such as p-adic number
systems for these too are interpreted within the same reduced quantitative
approach.

The precise
nature of the conventional mathematical approach can be accurately described as
1-dimensional.

To
understand this more clearly let me once more place this observation in
context.

All
phenomenal experience - including of course mathematical - is conditioned by
the interaction of important polarity sets (two of which are especially important).

So all
external (objective) recognition necessarily entails a corresponding internal
mental means of (subjective) interpretation.

Thus
external and internal polarities necessarily dynamically interact in a
continual manner in experience.

Secondly
all quantitative (analytic) recognition necessarily entails a corresponding
qualitative (holistic) aspect with both of these aspects again continually
interacting in experience.

Now a whole
new type of mathematical appreciation - operating according to uniquely
distinctive means of circular type interpretation - opens up when we consider
mathematical symbols and relationships explicitly in such a dynamic context.

So we here
recognise in a much more refined rational manner how interpretation keeps switching
as between opposite polar reference frames (in a dynamic interactive manner).

Thus for
example in the simplest dynamic situation we must use two complementary
reference frames. And such higher-dimensional interpretation (where each number
represents a certain dynamic configuration with respect to opposite polarities)
defines the qualitative (holistic) approach.

Now
Conventional Mathematics avoids all such dynamic considerations by simply
defining interpretation with respect to just one pole of reference. So
typically we portray number as possessing an objective identity (without
reference to internal interpretation). Equally we portray numbers as possessing
an independent quantitative identity without reference to their overall
relationship with other numbers (which is properly of a qualitative nature).

Now it
should be immediately obvious that it is not in fact possible to give a
cardinal (quantitative) identity to a number without an implied ordinal
(qualitative) identity and vice versa.

However the
quantitative confusion runs so deep in Conventional Mathematics that we have
somehow convinced ourselves that the ordinal nature of number likewise refers
to mere quantitative identity!

In other
words the reduced nature of our interpretation is so entrenched that we no
longer even recognise the huge confusion involved!

So the
vital key to appreciating the dynamic nature of number is that in experience,
we always have the interaction of both quantitative and qualitative aspects
(operating according to two uniquely distinct modes of interpretation)

So in
actual experience we necessarily switch as between cardinal and ordinal modes
in a dynamic manner.

However Conventional
Mathematics completely misrepresents the nature of such interaction by attempting
to interpret number with respect to its mere quantitative aspect.

And because
it is necessarily defined in a 1-dimensional (quantitative) manner, furthermore
it is entirely lacking any means of properly incorporating its neglected
qualitative aspect.

This is
precisely why an entirely new paradigm is now urgently needed in Mathematics.

Let us now
see how understanding of the relationship as between the primes and the natural
numbers is transformed through 2-dimensional (rather than 1-dimensional) interpretation.

In the
conventional mathematical approach, the primes are seen in cardinal terms as
the quantitative building blocks of the natural number system (the atoms as it
were of the system).

However
this approach is utterly misleading.

If we take
the prime number 3 for example, in cardinal terms it is given a collective
quantitative identity. If we were to attempt to break this number down in quantitative terms
through its constituent units, it would be represented as 1 + 1 + 1.

But there
remains a crucial problem here in that this quantitative definition (in terms of homogeneous units) leaves us with no means of making an
ordinal (qualitative) distinction as between the units.

In other
words in a number group of 3 members we thereby have no means of uniquely distinguishing
in any context its 1

^{st}, 2^{nd}and 3^{rd}members!
So the
ordinal (qualitative) recognition of 1

^{st}2^{nd}and 3^{rd}(which depends on the relational context of each member within the group) cannot be derived from the cardinal (quantitative) definition of number (viewed in collective whole terms as independent).
Though this
issue is truly of the first magnitude it is completely overlooked in
conventional terms.

So it needs
an utterly distinctive (Type 2) holistic interpretation of number to deal with
the ordinal aspect! So, as I have outlined repeatedly in various blog entries we can indirectly express in quantitative terms the ordinal nature of n units (with n a prime number). This is done in a circular - rather than linear manner - as points on the circle of unit radius, through obtaining the corresponding n roots of 1. Thus the relative independence of each member of the number group in an indirect quantitative manner is given through each root (in isolation). The overall qualitative interdependence - again of a relative nature - of the n members is then expressed through the sum of roots = 0.

Thus when
we adopt both the Type 1 (cardinal) and the Type 2 (ordinal) approaches to
number, the relationship between the primes and natural numbers operates in two
diametrically opposite ways.

From the
Type 1 approach the primes appear to serve as unique quantitative building
blocks of the natural numbers (in a cardinal manner).

However
from the Type 2 approach the natural numbers appear to serve - in reverse
manner - as the unique qualitative building blocks of the primes (in an ordinal
fashion). So again from this perspective the prime number 3 comprises 1

^{st}, 2^{nd}and 3^{rd}members (which as we have seen can then be indirectly given quantitative expression in a circular numerical fashion).
So the
quantitative (cardinal) perspective we see the natural numbers in terms of the primes;
then from the qualitative (ordinal) perspective we see the primes in terms of the natural
numbers!

Now the
holistic (2-dimensional) approach essentially consists in wedding these two complementary
perspectives (which in dualistic terms are paradoxical in terms of each other)
in a single nondual intuitive type realisation. And the very ability to “see”
in a unified manner through such intuition, is intimately related to the
corresponding ability to identify opposite characteristics accurately with
respect to the two frames of reference!

Therefore
the holistic (qualitative) is utterly distinct from the corresponding analytic
(quantitative) approach.

he latter is directly rational in a linear
manner; the former however is directly intuitive (which is then indirectly
expressed rationally in a circular paradoxical fashion).

So when we
are enabled to switch smoothly in an increasingly dynamic manner as between
both reference frames, we see the primes and natural numbers ever more closely
as perfect mirrors of each other (with the ultimate pure spiritual experience
of this relationship ineffable in nature).

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