The second
of the two key polarity sets relates to whole and part (collective and individual) which is
vitally important with respect to all mathematical understanding.

Again
however conventional interpretation leads to basic reductionism whereby whole
and part are both understood in a merely quantitative manner with the whole
thereby indistinguishable from the sum of its parts.

However
when we allow for the complementary interaction as between the poles, both quantitative
and qualitative aspects of interpretation are involved. Then once again like
the two turns at a crossroads, when one turn is designated left, the other -
relatively - is necessarily right (and vice versa), likewise, in any dynamic
context, if the whole is viewed in a quantitative manner, then the parts -
relatively - are qualitative in nature; likewise when the whole is viewed in a qualitative
manner, the parts are - relatively - quantitative in nature.

Thus whole
and part interactions entail both quantitative and qualitative aspects that keep alternating
between each other in a dynamic interactive manner.

Again it
may be instructive to probe the psychological dynamics through which both quantitative
and qualitative aspects arise in experience.

We have
already dealt with the external and internal polarity set.

Therefore
for example when we form knowledge of a number as object (in external terms)
this is necessarily balanced to a degree by a corresponding mental perception
that is - relatively - of an internal nature.

Though in
formal terms these aspects are considered as separate and reduced in terms of
each other in an absolute conscious manner, implicitly some degree of (unconscious)
awareness necessarily must also exist of the complementarity of both poles (thereby
enabling the mental interaction with number objects to take place).

This then
leads to an intuitive fusion of opposites whereby the (independent) specific actual
nature of the number (both as object and mental perception is thereby cancelled
out). So what happens - though we explicitly awareness of the
nature of this process may well be absent, we move to a potential - rather than actual - knowledge of interdependence that
carries an infinite meaning.

So in
effect with respect to our example of the number “2”, we switch from the actual
specific experience of this number (both as separate object and perception
respectively) to a new holistic appreciation of the interdependent notion of “twoness” that
potentially applies in any context to “2”.

Without such implicit appreciation of this notion of “twoness” (that is potentially infinite in scope) we would be unable to recognise specific examples of “2” in an actual context.

Without such implicit appreciation of this notion of “twoness” (that is potentially infinite in scope) we would be unable to recognise specific examples of “2” in an actual context.

Thus in the
understanding of any number - and indeed every mathematical phenomenon - the
dynamic relationship between actual (conscious) and potential (unconscious)
notions necessarily takes place.

The conscious
appreciation relates directly to rational (analytic) type interpretation; the
unconscious appreciation, by contrast relates directly to intuitive (holistic)
type interpretation.

Though mathematicians
may informally recognise the importance of intuitive understanding (especially for
creative work) explicitly Conventional Mathematicians is presented
as a set of absolute rational type connections which gravely distorts its true
nature.

So we have
moved from the actual notions of external and internal polarities (relating
to the number “2” as object and mental perception respectively) to the potential
appreciation of the notion of “twoness” as holistically applying to all
possible specific cases of “2”.

In the
dynamics of understanding this causes a decisive shift from the notion of
number as a specific object (and corresponding perception) to the conceptual notion of the holistic nature of number
as applying in all possible cases.

However this intuitive holistic understanding then becomes quickly reduced in an actual manner. So
one quickly moves from the potential (intuitive) holistic notion of number (and corresponding mental concept) as
applying in an infinite manner, to the rational notion of the general notion of number as specifically applying in all actual cases.

Indeed the conventional
notion of mathematical proof is based on this merely reduced interpretation. Thus
when a theorem e.g. is generally proved (e.g. the Pythagorean Theorem) strictly
this has an infinite - merely potential - application.

Then when
we identify a specific application in an actual finite context it is assumed
that this directly corresponds with the general proof.

Now I am
not arguing that conventional mathematical proof has no value! However I am saying is that
it is based on the direct reduced assumption whereby what is potentially
true is assumed to be equally true in an actual manner.

Thus the
mighty assumption of a direct coherence as between quantitative (finite) and qualitative
(infinite) notions is thereby made.

However
this assumption of coherence which underlies all conventional mathematical
relationships, cannot be itself addressed within this framework (which simply
reduces one to the other).

This is why
it is imperative to move to a dynamic interactive interpretation of mathematical relationships that can embrace both quantitative (analytic) and
qualitative (holistic) aspects as equal partners.

So in all
mathematical understanding we have the dynamic process through which specific objects
(and mental perceptions) are related to whole objects (and mental concepts).

And in the
very dynamics of such understanding the unconscious (based on complementary
recognition of both poles) is vitally necessary in enabling the switching as
between specific and whole objects (on the one hand) and perceptions and
concepts on the other.

Now the
very nature of conventional mathematical interpretation leads to an extreme
degree of rigidity in the nature of such interaction whereby actual and
potential objects (and corresponding perceptions and concepts) are assumed to
confirm each other in an absolute manner.

However we understand
more appropriately carefully distinguishing rational (analytic) from intuitive
(holistic) type appreciation, mathematical interpretation is revealed to be
strictly of a - merely - arbitrary relative understanding which paradoxically
leads to a greatly enhanced appreciation of its true unlimited nature.

Now once the
true holistic nature of mathematical understanding (as qualitatively distinct
from its analytic counterpart) is properly realised a key problem that relates
to how this can be effectively translated in a phenomenal manner.

In everday experience
when we experience phenomena both conscious (analytic) and unconscious
(holistic) aspects are involved.

So If I am
looking to buy a house, clearly the house has an actual existence that can be
consciously identified. However the house will equally embody the more holistic
(unconscious) desire for meaning. So in a sense I will be searching for my “dream”
house than embodies these deeper holistic desires.

Strictly
this is true of all phenomena and indeed of all mathematical phenomena.

Insofar as
the object (and corresponding perception) has an analytic identification this
corresponds to “real” conscious meaning.

However
insofar as the object has a holistic identification this corresponds to an “imaginary”
unconscious meaning.

So just as
in quantitative terms we now recognise that complex numbers have both “real” and “imaginary” aspects, likewise
in qualitative terms we realise that the enhanced complex interpretation
of mathematical relationships (combining both analytic and holistic aspects)
equally contains both “real” and “imaginary” aspects.

So the “imaginary”
aspect of mathematical interpretation strictly represents an indirect rational
way of communicating holistic meaning.

Thus in the
very dynamics of understanding in the very way that mathematical perceptions and
concepts interact, we keep switching as between “real” actual and “imaginary”
potential meaning.

However in
formal terms this process in interpreted in a merely reduced rational
manner as solely “real” from a qualitative perspective.

So Conventional
Mathematics is formally confined to merely Type 1 appreciation (of a quantitative
analytic kind) i.e. the real aspect of mathematical appreciation.

Holistic
Mathematics is formally identified with merely Type 2 appreciation (of a qualitative
holistic kind) i.e. the imaginary aspect of mathematical appreciation.

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