Though it
is not quite as clear-cut as this, the analytic - for convenience - can be
identified with the quantitative and the holistic with the qualitative aspect
of interpretation respectively.

Indeed
right away we have a problem with the very use of the word “analytic” which is
a specialised more limited interpretation within Conventional Mathematics.

Here
analytic relates to the study of infinite series (real and complex), limits,
the use calculus notions with respect to such series etc.

However in
the wider more universal scientific use of the term, analytic - in any context
- applies to the breaking down of a whole into its component parts.

And because
of the very nature of present scientific method this implies a
reduced (i.e. quantitative) notion of a whole which is seen merely as the
sum of its constituent parts.

So this is
the sense in which I use the word analytic in a mathematical context (which
would include all analysis in the narrow more specialised sense in which the
terms is used).

Indeed the
analytic aspect of understanding concurs perfectly with what I characterise as
the 1-dimensional approach.

So once
again what this precisely means is that where the interpretation of any
relationship in concerned that only one polar frame of reference is used.

Therefore all
of current Mathematics - at least what is formally accepted as valid
Mathematics - is one dimensional (in this qualitative interpretative sense).

Thus in
relation to the first key polarity set i.e. external and internal, mathematical
symbols and relationship are given a mere external (i.e. objective) identity in
absolute terms that is not influenced through internal (subjective)
interpretation.

Therefore
though in dynamic interactive terms external and internal polarities of
experience are positive (+) and negative (–) with respect
to each other. However the very essence of absolute interpretation is to ignore
this distinction in effect treating meaning in a merely positive (+) fashion.

So this is
a perfect example of what is meant by the 1-dimensional approach where rational
interpretation is linear (and unambiguous) in just one positive direction!

This
1-dimensional approach equally characterises treatment in conventional terms
with respect to the second key polarity set i.e. quantitative and qualitative.

Here symbols
and relationships are given a mere quantitative identity, again in absolute
terms that is not influenced through qualitative interaction.

There is
huge confusion in present Mathematics with respect to this fundamental point.

For example
the natural numbers 1, 2, 3, 4, etc are
defined as independent with respect to their mere cardinal identity in
quantitative terms.

However the
very ordering of numbers (whereby we meaningfully can place numbers in relation
to each other) requires a distinctive ordinal identity (of a qualitative
nature). In other words in ordinal terms a number only has meaning in relation
to other members of a number group.

So the
meaning of 3

^{rd}for example only has meaning in the context of a number group (which can arbitrarily vary in size). Therefore, 3^{rd}in the context of 3 numbers clearly carries a very distinct meaning from 3^{rd}in the context of 300!
Thus
ordinal identity therefore relates to the notion of an interdependent - rather
than independent - number identity.

So the huge
reductionist assumption that is made in Conventional numbers is that we can
order the cardinal numbers in a merely quantitative manner.

However once we depict the cardinal numbers e.g. as successive points on a number line,
we are thereby assuming a dimensional context that properly relates to a
qualitative ordinal identity!

Thus the very
essence of 1-dimensional interpretation is that it directly confuses the
quantitative notion of individual numbers (as independent units) with the
qualitative dimensional notion of the overall general order or collective
interdependence as between various numbers.

And let’s
be utterly frank here. Our cherished notions of the number system (that have
developed now over several millennia) are thereby based on a fundamental
confusion (i.e. where qualitative is reduced to quantitative interpretation).

So the
qualitative nature of number is not just something vague, as I have often read,
such as number personalities or even number archetypes, but as referring
directly to the fundamental issue of the means by which we are enabled to
achieve an overall order with respect to the number system.

Therefore
without properly recognising a qualitative dimension we cannot strictly derive
meaningful quantitative notions of number.

Put another
way the reduced (i.e. merely quantitative) notions of number we have inherited
are defined in a merely 1-dimensional analytic manner.

Now if
pressed sufficiently professional mathematicians may eventually concede that
the mental constructs we use to interpret mathematical reality are strictly of
a subjective rather than objective nature; with much greatly difficulty they
may even concede that ordinal notions strictly refer to qualitative rather than
quantitative meaning.

However
they will then go on to happily assume a direct absolute correspondence as
between (i) objective mathematical reality and subjective mental interpretation
and (ii) quantitative objects of an independent nature and a overall qualitative
dimensional context (that implies interdependence between these objects).

So in both
cases a basic reduction in meaning is involved. This does not entail that no
useful benefit can be achieved through such reductionism. Clearly the
development of conventional Mathematics proves otherwise. However it does entail
that mathematical edifice has been built on a limited and - ultimately - faulty
foundation.

Therefore
the essence of the analytic approach is to clearly attempt to separate on the
one hand

(i)
objective truth from subjective mental interpretation and

(ii) quantitative
(independent) objects from a qualitative (relational) context.

Then having
attempted this clear separation a direct correspondence is assumed as between
both in absolute terms.

So once
again such analysis entails the implicit belief that mental constructs directly
correspond with the mathematical reality (thereby interpreted) and also that general
relationships between objects correspond with the (assumed) independent identities
of these objects.

Now if one
thinks clearly about it such assumptions are untenable and even farcical.

Surely it
offends common sense to maintain for example that numbers are independent (in
an absolute sense) when clearly numbers can be placed in relationship with
other numbers!

The essence
of the holistic approach by contrast is that it is inherently dynamic and interactive
in nature.

Now the
remarkable fact that immediately arises is that all numbers (and indeed all
mathematical symbols) can be given a holistic - as well as analytic - identity.

Thus associated
with each number for example can be defined a unique mathematical reality in
holistic terms. Therefore for example, when the lens of interpretation keeps
changing the corresponding mathematical reality to which it corresponds
likewise keeps changing (in a relative manner).

So all
numbers in holistic terms (except 1) are associated with unique holistic interpretations
of mathematical reality (defined in a merely relative manner).

The significance of the number 1 in this context is that it represents the
special limiting case where mathematical reality is defined in an absolute
manner.

And this is
the number that defines all conventional mathematical interpretation. So once
again Conventional Mathematics can be precisely characterised as 1-dimensional (in
qualitative terms).

So with all
other dimensional numbers (other than 1) we have a unique dynamic interaction as
between opposite polarities ,constituting in turn unique holistic interpretations.

Now the
significance of this for interpretation of the Riemann Zeta Function is
immense.

As we know
, the one value for which the Function remains undefined in
analytic (quantitative) terms is where s (the dimensional value) = 1.

However we
now have a corresponding qualitative interpretation where the Function likewise
remains undefined in holistic (qualitative) terms for s = 1.

What this
means quite remarkably is that the Riemann Zeta Function remains uniquely
undefined when we seek to interpret it in a mere analytic (i.e. conventional mathematical) manner.

This therefore
implies that the Riemann Zeta Function - when appropriately interpreted in a
dynamic interactive fashion - provides an ingenious means of reconciling
analytic and holistic interpretation of the number system i.e. its cardinal and
ordinal aspects.

And clearly
again this cannot be done in an absolute analytic manner!

Now to keep
it simple, I will illustrate briefly the holistic approach with reference to the (qualitative) dimensional notion of 2
(which in a very special manner is the most important).

Now the
essence of 2 in this sense is that it is defined by a 1

^{st}and 2^{nd}dimension.
I have
explained before how indirectly we give quantitative expression to such ordinal
notions through obtaining the corresponding two roots of 1 i.e. (of 1

^{1 }and 1^{2}).
So writing
these roots (in reverse order) we obtain + 1 and – 1.

Now the 1

^{st}dimension relates to analytic interpretation. So we interpret both numbers in an independent quantitative manner.
However the
2

^{nd}(new) dimension relates directly to qualitative interpretation (of an interdependent nature).
So + 1 and – 1 now represent the dynamic manner in which we switch as between the
opposite poles of experience (with 1 representing each pole).

For
example with relation to the first polarity set ,we posit the external pole
(i.e. by making it conscious) to get objective knowledge of mathematical
relationships.

However in
order to switch to the internal pole of mental interpretation (of these
objects) we must negate the external pole (thereby rendering it unconscious).

If in
reverse, we start by positing the internal pole, we then switch to the external (by
negating the internal pole).

So there is
a dynamic interdependence as between both poles with - what is labelled as
- positive or negative depending purely on context.

And as all
mathematical experience necessarily entails both external objects and internal interpretation,
then the dynamic interpretation of this experience requires 2-dimensional - rather
than 1-dimensional – interpretation.

So in this
context the very number 2 takes on a remarkable new holistic meaning.

Thus once
again in analytic (cardinal) terms 2 = 1 + 1 (with no qualitative distinction
as between units).

However in holistic
(ordinal) terms 2 = 1

^{st}+ 2^{nd}which indirectly is represented as (+) 1 – 1 = 0. Thus the qualitative interdependent nature of the relationship is exemplified by the absence of a quantitative result!
In actual
experience the analytic (cardinal) understanding of numbers relates directly to
rational interpretation (of a linear kind).

The holistic
(ordinal) understanding relates however directly to intuitive appreciation
(which indirectly is expressed in a circular i.e. paradoxical rational manner).

Therefore
in actual experience (including of course mathematical) there is always a sense
in which opposite poles remain separate. However there equally is an important
sense in which they overlap as interdependent.

Whereas
rational understanding relates directly to their separation, intuitive appreciation
arises from realisation of their mutual interdependence.

So the
analytic aspect of understanding is directly of a linear rational nature
(though indirectly requiring, especially in creative work, a degree of
intuition).

However the
holistic aspect of understanding is directly of an intuitive nature (though
indirectly requiring rational expression in a circular paradoxical manner).

Once again
we can see the highly reduced nature of Conventional Mathematics, which is formally
defined merely in terms of linear rational notions. So intuition - which is
directly associated with qualitative type understanding - is formally reduced
to (linear) reason. And this again is why qualitative meaning is reduced to
quantitative!

From this reduced
perspective, mathematical symbols and relationships are given but one unambiguous
interpretation.

However
when we move - at a minimum - to 2-dimensional interpretation, all mathematical
symbols and relationships are given two unique interpretations (in both
analytic and holistic terms).

I have illustrated,
in this entry, the nature of both interpretations - analytic and holistic - for
the number 2.

However in principle
every mathematical symbol and relationship can equally be defined for both,
with comprehensive mathematical understanding arising from the interaction of
twin aspects.

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