Sunday, November 27, 2011

A “Purely-Quantitative" Movement of “Meta-Monadization"?

A "Purely-Quantitative" Movement of "Meta-Monadization"?

Dear Readers,

The purely-qualitative[self-]movements, and the qualo-quantitative” [self-]movements, of “[self-]meta-unit-ization, or of “[self-]meta-<<monad>>-ization, that we have recounted in this blog are, as we have seen, key to grasping both the mental and the physical [self-]movements, or [self-]progressions of our experience -- from predecessor <<arithmos>>/category to successor [meta-]<<arithmos>>/category -- as [self-]<<aufheben>> -- i.e., as dialectical -- [self-]movements; as dialectical [self-]progressions.

Out of curiosity – even if out of nothing else – let us ask the question: 

Is there a purely-quantitative movement of “meta-unit-ization, or of “meta-<<monad>>-ization?

My answer:  Yes!

And this movement is key to the notations of all of the ancient arithmetics, as well as to that of our modern, Hindu-Arabic – place-value/zero-as-empty-place-holder – arithmetical notation.

This movement is fundamental to all purely-quantitative arithmetics. 

It is the very ordinal/cardinal progression of arithmetical numerals itself!

I hold that this universal <<aufheben>> progression of numeration was adopted by our ancestors in response to both their goal of brevity of written expression, and to their goal of cognitive-advantage.

The only one of the ancient arithmetical notations for which we can readily illustrate this purely-quantitative progression of “meta-unit-ization, given the typographical resources ready-to-hand, is that of the “Roman numerals”.

We can describe this purely-quantitative <<aufheben>> progression thusly --

I ---> I, I & I = II ---> I, II, I & I & I = III --->

I, II, III, I & I & I & I = IIII = IV ---> I, II, III, IV, IIIII = IVI = V;

...V ---> ...V,...V & V = X ---> ...V,...X;

...X ---> X,...X & X = XX ---> ...X,...XX,...X & X & X = XXX --->

...X,...XX,...XXX,...X & X & X & X = XL --->

...X,...XX,...XXX,...XL,...XXXXX = XLX = L;   

...L ---> ...L,...L & L = LL = C;

...C ---> ...C,...C & C = CC ---> ...C,...CC,...C & C & C = CCC --->

...C,...CC,...CCC,...C & C & C & C = CCCC --->

...C,...CC,...CC,...CCC,...CCCC,... C & C & C & C & C = D;

...D ---> ...D,...D & D = M, = ...D,...M,

= I,...V,...X,...L,...C,...D,...M, ---> .....

-- wherein the new numeral – the new unitV is a five-fold meta-unit of the <<arche’>>-unit I [IIIII = V], wherein the meta-unit X is a two-fold second-degree meta-unit of the first-degree meta-unit V [VV = X], wherein the meta-unit L is a five-fold third-degree meta-unit of the second-degree meta-unit X [XXXXX = L], wherein C is a two-fold fourth-degree meta-unit of the third-degree meta-unit L [LL = C], wherein D is a two-fold fifth-degree meta-unit of the fourth-degree meta-unit C [CC = D], and wherein M is a two-fold sixth-degree meta-unit of the fifth-degree meta-unit D [DD = MV].

Our modern Hindu-Arabic numerals regularize this purely-quantitative, meta-fractal, meta-unit-ic, <<aufheben>>-progression thusly --

1&1&1&1&1&1&1&1&1&1 = 10;

10&10&10&10&10&10&10&10&10&10 = 100;

100&100&100&100&100&100&100&100&100&100 = 1,000, etc., etc.

-- with 10 as meta-1”, with 100 as meta-10”, and with 1,000 as meta-100”, etc., etc.



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