Tuesday, December 29, 2015

Wholes and Parts (10)

As we have seen, the various roots of 1 provide the means of - indirectly - translating the true holistic nature of the natural numbers (that necessarily entail "higher" dimensions > 1) in the standard 1-dimensional manner.

In this way one is enabled to convert from  the (holistic) Type 2 to the (analytic) Type 1 nature of the number system.

So once again - to give the simplest example - the holistic nature of 2 consists in the intuitive recognition that the two opposite directions (+ 1 and  – 1 respectively) - representing polar reference frames - are identical, when simultaneously considered in a complementary fashion.  Such intuitive recognition thereby represents a pure energy state!  

We have already illustrated on many occasions this notion of 2 with respect to left and right turns at a crossroads.  Thus given the polar direction of approaching the crossroads (N or S) what is left (with respect to one polar reference frame is right (with respect to the other) and also what is right (with respect to the first frame) is left (with respect to the second).

Thus when considered in complementary fashion (with respect to both N and S directions simultaneously) left is right and right is left. In other words + 1 (representing the left direction) = + 1 (now representing the right) and + 1 (now representing the right) = + 1 (representing the left).

So from a rational dualiistic perspective, this interdependence of both directions represents pure paradox. However such paradox is directly appreciated in a nondual fashion (through holistic intuition).

So once again 2-dimensional appreciation is directly intuitive, entailing the simultaneous recognition of two (opposite) polar reference frames.

However when we reduce this analytically (i.e. in a 1-dimensional manner) we now recognise the two poles in a relatively separate dualistic fashion, whereby they are + 1 and  – 1 with respect to each other.

Thus in any actual situation, a turn can be unambiguously designated as + 1 or – 1 respectively. Thus if a left turn is denoted as + 1 , then in relative terms a right turn is – 1 (i.e. not a left turn).

However if the right turn is now designated as + 1, then the left turn is, in a  relative manner,  – 1 (i.e. not a right turn). 

Thus the relative independence of each direction is denoted by + 1 and  – 1 respectively through the circular number system (based on the n roots of 1 in the complex plane). And again in this case n = 2.

The corresponding interdependence is then denoted through the sum of the n roots = 0.

So when n = 2, the sum of the two roots = + 1 – 1 = 0. So the pure holistic notion of interdependence is thereby without quantitative significance! 

Therefore, we can now quickly generalise to state that the holistic notion of n (in Type 2  terms) can be indirectly reduced in a Type 1 manner through obtaining the n roots of 1.

Thus when  xn = 1, x– 1 = 0 and the n solutions of his equation provide the n roots of 1.

However one of these roots will always = 1, where holistic meaning reduces to its analytic (1-dimensional) counterpart.

Thus to obtain the non-trivial roots (with true holistic meaning) we divide, 

x– 1 = 0 by x – 1 = 0 to obtain,

x+ x+ x+ x+ .........+ xn – 1 = 0.

This is what I refer to as the Zeta 2 function which complements Riemann's Zeta function (which in this context represents the Zeta 1 function).

The true importance of the Zeta 2 solutions is that they provide the means of switching as between qualitative notions (of relative interdependence) and quantitative notions (of relative independence) respectively.

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