Monday, October 6, 2014

Holistic Synchronous Nature of Number System (4)

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.

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.

Radial Mathematics is then explicitly identified with both Type 1 and Type 2 (in mutual conjunction with each other) i.e. the complex aspect of mathematical appreciation

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