Thursday, October 22, 2015

Clarifying Analytic and Holistic Interpretations of "2" (9)

Before moving on, I will briefly here explain what the holistic interpretation of dimensional numbers relates to!

All experience is conditioned by two fundamental sets of polarities which can be referred to as internal/external and whole/part respectively.

So in both physical and psychological terms (which in dynamic terms are complementary) a ceaseless interaction takes place as between these two opposite sets of polarities.

In geometrical terms we can represent these as 4 equidistant points on the unit circle (in the complex plane) with the sets horizontal and vertical with respect to each other.

These 4 points (i.e. + 1, – 1, + i and – i) then represent the indirect quantitative interpretation corresponding to the 4 dimensions.
The startling implication of this is that the qualitative notion of "4" (i.e. "fourness") implicitly implies the unconscious recognition of both real and imaginary opposites in their unified state!

Then we attempt to express this state of "fourness" in an indirect quantitative manner, (conforming to 1-dimensional interpretation) it entails the circular paradox of opposites in both real and imaginary terms. 

Thus once again our intrinsic experience of  dimensions - as relating to the qualitative aspect of number - relates directly to the unconscious (intuitive) aspect of experience.

Furthermore - though this goes beyond our present purposes - the intrinsic notion of space and time in physical terms, when appropriately understood, conforms directly with this mathematical interpretation of dimension (as the holistic qualitative nature of number).  

However, once again, in standard mathematical terms, based on merely (conscious) rational notions, the qualitative aspect is invariably reduced in a merely quantitative manner!

Just as with a compass, we can then view 4 dimensions as representing four equidistant points on the unit circle (represented by the four roots of 1).

Then all other dimensions can be viewed as representing unique configurations with respect to these 4 basic co-ordinates.

Now we can express the n roots of 1 by the equation,

1 = st , i.e. 1 s =  0.

Now as we have seen, in all cases one of these roots i.e. 1 –  s = 0, (where s = 1) represents the 1-dimensional case. Here ordinal (qualitative) meaning of number is directly reduced in a cardinal (quantitative) manner.   

Therefore to isolate the remaining non-trivial roots (where a distinct ordinal meaning is preserved) we divide 1 s =  0, by 1 –  s = 0 so as to obtain,

1 + s+ s+ s+….. + st – 1    = 0

I refer to this as the (finite) Zeta 2 function which is designed to complement the well know Riemann zeta function (which I refer to as the Zeta 1 function).

Therefore using this terminology,

ζ2(s) = 1 + s+ s+ s+….. + st – 1    = 0.

Now initially I define this function for prime values of t (where solutions for s are unique).

Therefore in the simplest case where t = 2,

ζ2(s) = 1 + s1   = 0.

So s (i.e. s1)  =  – 1.

This therefore represents the first of - what I refer to as - the Zeta 2 zeros. 

Now, the crucial role that this zero plays is that it enables one to "convert" the inherently qualitative notion of the 2nd (of two dimensions) indirectly in a quantitative manner.

And as we have seen (with the example of the crossroads) the very way we recognise a right turn when the other turn is posited as left is through negation of this polar direction. 

So right (in this context) is  – 1 (with respect to the left turn as + 1).

So what is happening here - though in this case it seems intuitively obvious - is that we are recognising that in simultaneous two-dimensional terms (when both N and S directions of approach are considered) that the two turns are paradoxically both left and right.

Therefore when we isolate a direction unambiguously as left (when approached in a N direction) then - relatively in this context - the 2nd turn must be right.

However we can switch the frames of reference through approaching the crossroads from the alternative S direction. So now - what was formerly identified as right (the 2nd turn) is now + 1 and what was formerly left is now – 1.

So it is important to realise that + 1 and  – 1 in this context are merely relative and interchangeable.

In other words though we we are using  quantitative means of expression to represent left and right turns, the two numbers + 1 and  – 1 and are now understood in a relative interdependent manner.

So we can perhaps now sum up the crucial difference as between the quantitative and qualitative notions of "2".

With the quantitative notion of "2", the two units are independent. (So + 1 is absolutely separated from – 1).

By contrast with the qualitative notion of "2" (as "twoness") the two units are interdependent with each other (and thereby interchangeable). (So + 1 can only be relatively separated from – 1 in an arbitrary context, which can be reversed where + 1 and  – 1 now switch positions with each other.

Pu more simply, the quantitative notion of "2" concurs with linear (either/or) logic; the qualitative notion of "2" concurs with circular (both/and) logic.      

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