Wednesday, February 4, 2015

Intricacies of Addition and Multiplication (7)

As stated on several occasions every natural number enjoys both a quantitative (analytic) and qualitative (holistic) meaning that dynamically interact in experience.

So again for example the quantitative notion of 3 (as representing independent units) in cardinal terms is dynamically inseparable from the corresponding qualitative notion of 3  (as representing the interdependence of uniquely distinct units) in an ordinal manner.

And this applies by extension to all the primes and natural numbers.

However we can likewise give both quantitative and qualitative meanings to all rational fractions (which likewise interact in dynamic fashion).

Again using the reference point of 3 to illustrate let us look closely at the quantitative meaning associated with the fractions 1/3, 2/3 and 3/3.

Now 1/3 represents the reciprocal of 3.

In a more complete Type 1 manner 1/3 represents the reciprocal 31.

Now 3 represents an integer, which is - literally - a whole number.

However 1/3 as reciprocal represents its corresponding part which can be written as 3– 1.

So implicitly the change from whole to part entails the negation of the (default) 1st dimension which is the basis of all rational linear understanding.

This negation then causes to a degree an unconscious fusion of positive and negative polarities in experience which is the direct basis of all intuitive type understanding.

So the very means by which we are enabled to switch from whole to part (with respect to number) and indeed with respect to phenomenal recognition in any context implicitly requires a switch from conscious (analytic) to unconscious (holistic) type understanding.

However once this transition from whole to part has been achieved, in explicit terms the result will be interpreted in a merely (reduced) quantitative manner.

So 1/3 = (1/3)1.

Thus is we imagine a small cake divided into 3 slices, 1/3, 2/3 and 3/3 i.e. (1/3)1, (2/3)1 and (3/3),represent the various options in terms of expressing the constituent parts with respect to the whole cake (which is equally recognised as composed of 3 parts).

So the whole cake (i.e. 1 unit) = 3/3. However once again this properly entails the simultaneous recognition of the whole cake (as 1 unit) and 3 part units which entails both rational and intuitive type recognition.

Of course we can equally have 2/3 (as 2 part slices with respect to the whole) and 1/3 (as 1 part slice in relation to the whole).

Now even though in abstract terms we may use integers and fractions without respect to concrete phenomena, the basic rationale remains the same in that implicitly the change from whole to (reciprocal) part recognition implicitly entails both intuitive (unconscious) as well as (rational) conscious recognition.

However the corresponding qualitative interpretation of these same fractions is perhaps even more surprising.

In Type 2 terms, these will now be represented as 11/3, 12/3 and 13/3 .

These can then be readily identified as representing the 3 roots of unity which are
– .5 +.866i, .5   .866i and 1 respectively.

What is remarkable about these roots (though clearly not properly recognised) is that they express in a reduced quantitative manner, ordinal relations (that are directly of a qualitative nature).

So 11/3 relates in this context  of the small cake relates to the 1st of  the 3 slices.

12/3 relates to the 2nd of the 3 slices, while 13/3 relates to the 3rd of the 3 slices.

Thus once again, what is remarkable is that we can then give these ordinal notions (of a qualitative nature) an indirect quantitative interpretation.

In other words we have here a ready means of converting as between the Type 2 and Type 1 aspects of the number system (from qualitative to quantitative format).

In more general terms, in order to express the ordinal rankings with respect to any number n in an indirect quantitative manner, we simply obtain the corresponding n roots of 1.

Once again through this process, these qualitative rankings, which are unique for each cardinal value of n, can be given a unique quantitative value!  

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