Wednesday, May 24, 2017

Number and Development (9)

As we have seen, there are two aspects with respect to the understanding of all numbers - and indeed by extension all mathematical relationships - that are quantitative (analytic) and qualitative (holistic) with respect to each other.

Both of these aspects can only be properly understood in a dynamic relative context, where each type of understanding  implies the other in a complementary manner.

Now, from a psychological perspective, the analytic aspect is directly related to rational type appreciation (of a conscious kind); by contrast the holistic aspect is directly related to intuitive type appreciation (of an unconscious nature).

Therefore, when understood appropriately, the role of intuition with respect to mathematical understanding is utterly distinct and cannot be confused with reason.

However, because in effect conventional mathematical understanding entails the reduction of holistic type meaning (in an analytic manner), equally this entails the corresponding reduction of intuitive  (directly related to the unconscious) with rational type appreciation (of a distinctive conscious nature).

So once again the truly central issue with respect to all mathematical interpretation is thereby missed.

From the external physical perspective, this relates to consistency with respect to both the quantitative and qualitative interpretation of its symbols; from the corresponding psychological perspective - which in dynamic terms complements the physical - this equally relates to consistency with respect to the rational and intuitive interpretation of these same symbols.

In fact as I have repeatedly stated in my blog entries, from the appropriate dynamic interactive perspective, the Riemann Hypothesis can be seen as a key statement with respect to this central issue.

And by the same token, because in conventional mathematical terms the qualitative aspect is not formally recognised (as distinct from the quantitative) this implies that such attempted "proofs" of the Riemann Hypothesis are rendered futile!

When one looks at the nature of the primes from this dynamic interactive perspective, appreciation of their very nature is thereby transformed.

Once again using "3" to illustrate this of course represents a prime number!

Now from the conventional analytic perspective in cardinal terms, this prime thereby represents a constituent "building block" of the natural number system.

However from the (unrecognised) holistic perspective, in ordinal terms, the position is reversed with each prime group representing a unique configuration of its constituent individual members.

So "3" for example, thereby represents a unique configuration with respect to its 1st, 2nd and 3rd members.

The next prime "5" would then represent a unique configuration with respect to its 1st, 2nd, 3rd, 4th and 5th members.

However this begs the significant question as to the derivation of the 4th member (which implies the number "4"). Therefore though from the cardinal perspective "5" is already viewed as an independent "building block", clearly from the ordinal perspective the composite natural number "4" is directly implied with respect to its 4th member!

In other words, from a dynamic interactive perspective it is quite untenable to maintain this absolute stance with respect to the primes as representing the independent "building blocks" of the natural number system!

Certainly from a relative perspective, the primes appear as the "building blocks" of the natural number system (in cardinal terms). However from an equally valid alternative relative perspective, each constituent prime appears as representing a unique configuration of its individual natural number members (in an ordinal manner).

So again in cardinal terms, the natural numbers appear to be determined by the primes; however from the ordinal perspective, each prime appears to be determined by its natural number members.

The key implication therefore is that from a dynamic interactive perspective - which represents the true nature of the number system - both the primes and natural numbers are co-determined in a synchronistic manner (that is ultimately ineffable).So the primes and natural numbers ultimately mirror each other in a perfect manner.
And through right understanding one can experientially approach, to an ever closer degree, true appreciation of this perfect mirroring.

So in terms of Band 5 development, a new appreciation of "dimensional" numbers started to open up, whereby they could now become fully grounded in the linear (1-dimensional) levels of Band 2.

So I now came to the clear realisation that the very means of "converting" the qualitative notions of 1st, 2nd, 3rd, ... entailed the holistic appreciation of the corresponding roots of 1.

So again for example in Type 2 terms, we can represent 3 as 13.

Thus is holistic terms 3 represents the notion of 3 dimensions as fully interdependent with each other (which is directly grasped in an intuitive manner).

Thus with each number is associated a distinctive "quality" of intuition. Thus associated with 2 is the quality of appreciating the interdependence of 2 related dimensions, with 3, the quality of appreciating the interdependence of 3 related dimensions, with 4 the quality of appreciating the interdependence of 4 related dimesnions and so on.

However we can indirectly convert such qualitative notions, in a quantitative (1-dimensional) rational manner, by taking the corresponding roots of 1 (associated with each number).

Thus for example the 2 roots of 1, i.e. + 1 and – 1 express - in a necessarily paradoxical "circular"  manner - the linear rational notion of the interdependence of two objects.

As I have repeated many times before this naturally arises in our appreciation of the paradoxical nature of turns at a crossroads.

So in approaching the crossroads from one direction one can unambiguously define left and right turns. So if "left" is designated as + 1 (as 1st), then "right" in this context is designated as – 1 (as 2nd).

However when the crossroads is approached approached from the opposite direction, what was formerly a left turn is now right and what was a right turn is now left. So what was + 1 (as 1st) is now – 1 (as 2nd), and what was  – 1 (as 2nd) is now + 1 (as 1st).

So in this context of mutual relative interdependence,  + 1 and – 1 can switch between each other (with each possessing a merely relative validity). In fact the interdependence of the two numbers is expressed through the requirement that their sum = 0.

Likewise the 3 roots of 1 i.e. + 1,.5 + .866i and .5 –.866i express (in an indirect linear rational manner) the interdependence of 3 numbers with respect to 1st, 2nd and 3rd positions (which can mutually switch in a relative manner as between each other).

And in more general terms the n roots of 1 likewise express (in an indirect linear rational manner) the interdependence of n numbers with respect to 1st, 2nd, 3rd,..., nth positions (which can all mutually switch in a relative manner as between each other)..

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