Friday, August 9, 2013

The Holistic Nature of the Number System (4)

As we have seen the ordinal interpretation of number relates to number interdependence, as opposed to the corresponding cardinal notion of number as independent.

Whereas analytic interpretation is indeed appropriate with respect to the cardinal notion of independence, by contrast holistic appreciation is required for the ordinal interpretation of number as interdependent.

Once again analytic interpretation corresponds with rational understanding (of a linear nature).

However holistic interpretation corresponds directly with intuitive understanding (that is indirectly rationally expressed in a circular i.e. paradoxical manner).

Again the essence of analytic interpretation is that it is 1-dimensional i.e. based in any context on just one independent pole of reference.

The essence of holistic interpretation is that any number but 1 serves as the dimension. In the simplest case, which in many ways serves as the blueprint for all holistic interpretation, two interdependent poles of reference are used.

I have explained on many occasions the relationship as between these two approaches in the context of a crossroads.

Once again if we adopt independent polar frames of reference, we can give unambiguous interpretation to left and right turns at a crossroads.

So if one approaches the crossroads (travelling North), one can give an unambiguous definition of left and right to each turn; then when one approaches the crossroads travelling South, again both turns are unambiguous.

The reason for this lack of ambiguity is that the reference frame is independent, referring to just one pole (which again is the essence of 1-dimensional interpretation).

However, if one now simultaneously attempts to include both reference frames North and South, then definition of left and right is paradoxical. So each turn at the crossroads can be both left and right depending on context.

The reason for the paradox is now due to the fact that the reference frames are interdependent (defined in a 2-dimensional manner).

So the very essence of paradox is that it is nondual (and thereby cannot be grasped in an analytic fashion).

And the appreciation of such paradox relates to a distinctive type of holistic understanding (where the two reference frames are united in a directly intuitive manner).

Therefore, though we indirectly can express such nondual appreciation in a circular (paradoxical) rational manner, in direct terms it is of a holistic intuitive nature.

So the holistic appreciation of 2, is the recognition of  + 1 and – 1 as complementary opposite poles of recognition.

Therefore, the very basis of intuitive insight in experience is the recognition of the complementary - and ultimately identical - nature of external (i) (objective) and internal (subjective) polarities and (ii) part (quantitative) and whole (qualitative)  polarities.

Indeed these four polarities, representing the 4-dimensional qualitative nature of dynamic interaction in experience, can be conveniently represented - like the directions on a compass - by four equidistant points on the circle of unit radius (in the complex plane).

In like manner the n-directional qualitative nature of dynamic interaction between polarities can be represented by n equidistant points on the circle!

As I have said, in many ways 2-dimensional interpretation serves as the simple blueprint for all holistic recognition.

The important point here is that with holistic recognition we must always start with (1-dimensional) analytic interpretation.

So before we can recognise the two turns at a crossroads in holistic interdependent terms as both left and right, we must initially recognise them in independent terms as unambiguously either left or right. So we must apply two independent reference frames separately (as independent) before combining both (as interdependent).

Now in qualitative terms "+" simply implies positing in a conscious manner and + 1 implies such positing with respect to one independent reference frame.

Thus once again, Conventional Mathematics in this qualitative sense, is 1-dimensional with the one root of 1 = + 1.

In qualitative terms,  "– " implies (dynamic) negation in an unconscious manner. Thus the very means by which the unconscious is activated in experience is through the dynamic negation of consciously posited reference frames.

Without the unconscious it would be thereby impossible to switch reference frames. Thus the all important process (in any context) of relating part to whole (and whole to part) implicitly implies unconscious recognition. And as we have seen, intuitive appreciation in experience relates directly to unconscious type recognition!

So the holistic appreciation of interdependence is directly of an unconscious nature.

Once again Conventional Mathematics cannot deal with this notion (except in a grossly reduced sense) as it formally defines relationships in a merely conscious manner.

Thus the Type 1 aspect of the number system relates directly to the analytic (conscious) aspect of mathematical experience.

The Type 2 aspect relates - by contrast - to its holistic (unconscious) aspect.

So again from a Type 1 perspective, 2 is defined as 21. From a Type 2 perspective, 2 is defined as 12.

Then associated with the Type 1 system is the standard Riemann Zeta Function (which I refer to as Zeta 1),

i.e. 1–s  + 2–s  + 3–s  + 4–s  +……..

Here each of the natural numbers is raised to the negative of the same dimensional value (i.e. s).

However equally associated with the Type 2 aspect is an alternative Zeta Function (which I refer to as Zeta 2),

i.e. 1 + s1  + s2  + s3  +….. + st – 1
Here, in reverse s as the quantitative value is raised to each of the natural values (up to t – 1).

So whereas Zeta 1 is conventionally defined as an infinite series, Zeta 2 is of a finite nature (though it can be given a qualified meaning in infinite terms).

Now the derivation of the non-trivial zeros for Zeta 1 relates to the equation

1–s  + 2–s  + 3–s  + 4–s  +……..  = 0

The corresponding derivation of non-trivial zeros (whose role is as yet unrecognised) relates to the equation,

1 + s1  + s2  + s3  +….. + st – 1  = 0 (where t is a prime number).

The simplest example is where t = 2, and

1 + s1  = 0,

so that s =  – 1.

As we know  + 1 will always constitute one of the roots of 1.

In this sense it is a trivial root (as it is repeated in every case).

The deeper qualitative significance of this root is that it implies that we must always start with analytic interpretation (within independent reference frames) before we can achieve their holistic interdependence.

However the fascinating feature of the Zeta 2 zeros is that they already have eliminated the common root of 1).

Thus where t is prime the solutions of the Zeta 2 equation relate to the t – 1 non trivial roots (which are directly of a holistic nature geared to the interpretation of interdependence).

Now the unconscious holistic nature of these solutions is indicated by the fact that their sum = – 1.

Thus in one sense, the higher dimensional interpretations (where t is prime > 2), relate to ever more refined unconscious (i.e. holistic) type appreciation.

The uniqueness of the prime numbers in this regard is that there non-trivial roots are unique (i.e. never duplicated in other prime root solutions).

Thus in Type 1 terms a prime number is unique because of its independence in cardinal terms i.e. has no natural number factors.

In Type 2 terms a prime number is unique because of its interdependence in ordinal terms i.e. all members (apart from the 1st) are defined in a non-repetitive manner.  

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