Saturday, August 10, 2013

The Holistic Nature of the Number System (5)

Yesterday, I introduced again the two Zeta Functions (Zeta 1 and Zeta 2) which correspond in turn to the quantitative and qualitative aspects of the number system (i.e. Type 1 and Type 2) respectively.

Now properly understood - in dynamic interactive terms – these two Functions are complementary with each other.

Once again the Zeta 1 Function (i.e. the standard Riemann Function) is given in conventional terms as the infinite series,

1–s  + 2–s  + 3–s  + 4–s  +……..,

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

By contrast the Zeta 2 Function is given as the finite series,

1 + s+ s+ s+….. + st – 1 (with t prime),

where in reverse, s as the quantitative value is raised to each of the natural values (up to t – 1).

The value of s will of course be distinct for each Function!

So through each Function we have the expression of the natural number system (both in finite and infinite terms) alternating in each case as between dimensional values (that are qualitative) and - relatively - base number values (of a quantitative nature).

Now the zeta zeros (Zeta 1 and Zeta 2 respectively) represent the solutions of each Function, where the sum of terms in the series expression = 0.

Thus when we look at these solutions in a dynamic interactive manner, the crucial key as to their nature is revealed.

Putting it simply, the zeta zeros represent the solutions for which both the quantitative and qualitative aspects of the number system are perfectly reconciled (from two complementary perspectives).

In fact it is easier to initially appreciate the precise nature of such holistic solutions from the Zeta 2 perspective (which we will return to shortly).

So the Zeta 2 solutions show how the internal nature of each prime number is comprised of a series of individual natural numbers in ordinal terms, with the (indirect) quantitative nature of each of these natural numbers (in isolation) perfectly reconciled in a holistic manner with the overall qualitative interdependence of this prime number grouping.

Now the uniqueness of each prime number in this context is that all its natural number ordinal members (except 1) are uniquely defined in an (indirect) quantitative manner. In other words these prime number roots are unique for each prime number group.

Thus for example 5 is a prime number. Now by definition 5 as a group is composed of its 5 individual members which are 1st, 2nd 3rd, 4th and 5th  in natural number ordinal  terms. Each of these ordinal members can then be indirectly defined in a quantitative (relatively independent) manner through the corresponding 5 roots of 1.

Then when we combine these roots to demonstrate their corresponding relative interdependence (in a qualitative manner) their sum = 0.

So in this way we can demonstrate a unique holistic balance for each prime number as between its individual members (in quantitative terms) and their overall collective interdependence (in a qualitative manner).

We are accustomed to interpreting prime numbers in an absolute analytic fashion as the independent quantitative building blocks of the natural number system.

So from this perspective, the qualitative relational aspect of the primes become fully separated and then fully reduced to the quantitative aspect.

And once again the analytic aspect of the number system is defined by such absolute reductionism (whereby it is formally interpreted in a merely quantitative manner)!

However the holistic aspect is altogether distinctive in nature (and lying at the other extreme) where quantitative elements, while maintaining their independent identity in a relative manner, are yet seen - relatively - as fully interdependent with each other in qualitative collective terms.

And it is vital to grasp that zeta zeros directly represent the latter holistic interpretation of number!

And so once again this is exactly why the very nature of these zeros is - necessarily - completely missed through the conventional mathematical approach.

We cannot therefore hope to appreciate the holistic nature of number (entailing the balanced interaction of its quantitative and qualitative aspects) in an analytic manner (that is formally based on mere quantitative interpretation).

Now I have spelt out in some detail here the holistic nature of the Zeta 2 zeros.

Though we will return to this in future entries in this “holistic” series on my blog, the general nature of the - greater recognised - Zeta 1 zeros can easily be expressed.

Just as the Zeta 2 zeros are defined for the internal ordinal natural members of each individual prime, the Zeta 1 zeros are defined externally for the number system as a whole with respect to the collective nature of the primes with respect to the (composite) cardinal natural numbers.

So the non-trivial zeros of the Zeta 1 function ensure a complementary type of holistic harmony as between the qualitative interaction of primes and the quantitative natural numbers resulting.
Thus the Zeta 1 and Zeta 2 serve as complementary mirrors to each other.

In the case of the Zeta 2 each cardinal prime is defined in an ordinal manner through a finite set of natural number members (1, 2, 3…p).

In the case of Zeta 2, the cardinal natural numbers are defined in terms of the ordinal activity of the prime number groupings that uniquely define their nature.  

When one can dynamically interpret these relationships simultaneously in both Zeta 1 and Zeta 2 terms, the primes and natural numbers are revealed as perfect mirrors of each other (in an ultimately ineffable manner). And this is because the Zeta zeros likewise serve as perfect holistic mirrors of each other.

Another interesting feature worth emphasising here is that the true holistic nature of the zeta zeros (Zeta 1 and Zeta 2) implies that they in fact represent energy states.

Once again the conventional analytic interpretation of number is based on the apparent existence of numbers absolutely existing as rigid immutable forms.

So in analytic quantitative terms “2” for example is interpreted as a fixed number symbol.

However just as matter can be transformed into energy (and energy into matter), it is exactly similar in mathematical terms.

So the understanding of numbers existing as absolute forms represents the extreme analytic interpretation (where the qualitative aspect is reduced to the quantitative).

However when, at the other extreme, we understand numbers in pure holistic terms, they are understood as representing energy states.

Of course both of these aspects necessarily interact to a degree in experience, representing the inherent dynamic nature of number.

So here number forms in rational (analytic) terms interact with number energy states in a (holistic) intuitive manner leading to a continual dynamic transformation in the experience of number.

Of course in formal terms such experience is greatly misrepresented in a mere rational (analytic) manner.

Recent strong correlations as between the Zeta 1 zeros and energy states in quantum physics suggest that in some sense the zeros do indeed represent energy states!

However none of this can have intuitive resonance within the restricted conventional manner in which number is presently interpreted.

However it makes perfect sense when one interprets the zeros in an appropriate holistic manner.

However what is still being completely missed is that from a dynamic perspective, physical and psychological aspects are complementary.

So not alone do the zeta zeros represent physical energy states, they equally represent psychological - or more correctly - psycho-spiritual energy states.

Therefore, correctly understand the zeta zeros (in appropriate terms)  represent psycho-spiritual energy states that have complementary correlates in nature (in a psycho-physical manner).

And of course in dynamic interactive terms, physical nature is necessarily understood in a psycho-physical manner!

The spiritual traditions often make the distinction as between dual and nondual reality.

Dual reality - literally - relating to “twoness” refers to analytic type understanding, where the poles of understanding (such as internal/external and quantitative/qualitative are clearly separated.

Nondual reality - literally not-two - refers to holistic type understanding, where the opposite poles of understanding (that necessarily condition all experience) are understood as complementary and ultimately identical in a spiritual manner.

So in these terms, the zeta zeros relate directly to a pure nondual intuitive type appreciation at the deepest possible level of the ultimate interdependence of both the qualitative and quantitative (and indeed internal and external) aspects of the number system.

Now as with all phenomenal experience, holistic appreciation is necessarily of a relative approximate nature (which entails a degree of supporting analytic activity).

So the zeta zeros can indeed be encoded in a phenomenal numerical manner.

However it is important to remember that their inherent nature is holistic and thereby they relate to the most dynamically interactive experience that is humanly possible, while remaining in the world of phenomena. In this sense they are like the most fleeting of virtual particles.

We are a long, long way in our present evolution from being able to appreciate their inherent nature truly in the appropriate interactive manner (which will entail the most advanced states of spiritual contemplative development).

However we have come far enough to begin to understand their true nature, which is as far away as is possible from the rigid analytic notion of number that dominates conventional mathematical interpretation.        

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