## Wednesday, October 19, 2011

### Number as Dynamic Interaction

When properly appreciated all numbers represent a dynamic interactive process that necessarily entails both quantitative and qualitative aspects (which provide the very basis for such interaction).

A number perception representing a quantity is strictly speaking meaningless in the absence of its corresponding number concept (which relatively is of a qualitative nature).

So for example the number "2" representing a specific number perception has no meaning in the absence of the universal concept of number.

And this concept of number is strictly qualitative relating to what is potential and infinite. So in this qualitative sense the general concept of number applies potentially to all (as yet unspecified) numbers in an infinite manner.

However any specific number is necessarily of an actual finite nature. And this relationship of actual number perceptions (as quantities) and the potential number concept (as qualitative) provides the very dynamic for subsequent number interaction in experience.

Of course just as a left turn off a road becomes a right (and a right turn becomes a left) when one switches the direction of movement along the road, likewise through dynamic switching in experience number perceptions equally can attain a qualitative and the number concept - relatively - a quantitative aspect.

So from this latter perspective the number perception is seen to embody, as it were, the general property of number (that is qualitative in nature) while the concept is understood to apply to all actual numbers (in quantitative terms).

This thereby represents the reverse direction of the relationship as between quantitative and qualitative in number experience.

From a psychological perspective the dynamic interaction by which both the quantitative and qualitative aspects of number appreciation occur, requires in turn both rational (conscious) and intuitive (unconscious) understanding. In this interaction the intuitive aspect points to the holistic potential aspect of qualitative recognition (that switches as between concept and perception). The rational then points to corresponding actual quantitative recognition (that again switches - in relative fashion - as between perception and concept).

In the conventional (Type 1) approach, Mathematics is formally interpreted in a rational manner. This entails in turn that a merely reduced quantitative notion of number is given.

The starting point for a true interactive (Type 3) approach is the recognition that every specific number quantity implicitly implies a corresponding dimensional number concept that is - relatively - of a qualitative nature (and vice versa).

Thus for example "2", which conventionally in Type 1 terms is interpreted (solely) as a number quantity, equally has an implicit qualitative aspect (as representing a number dimension).

So in more complete terms, the natural number quantity "2" is defined with respect to a (default) number dimension 1 which, in experiential terms, is - relatively - of a qualitative nature.

So in this context 2 is properly 2^1.

Now, the very nature of 1, when used as a qualitative dimension is that qualitative is necessarily reduced to quantitative type meaning. Therefore though the number concept properly refers here to a potential - as opposed to actual - meaning, from a linear (1-dimensional) rational perspective, this is then interpreted in a merely reduced manner (as applying to all actual numbers).

However just as we can define the extreme quantitative (Type 1) approach in terms of a default dimensional number (i.e. 1), likewise we can define an extreme qualitative (Type 2) approach in terms of a default base quantity that is also "1".

So "2" in this approach representing a dimensional number is written as 1^2.
The significance of this latter approach is that each number represents a unique manner of qualitative interpretation of mathematical symbols.
Thus from this perspective we have a - potentially - infinite set of ways for the logical interpretation of mathematical relationships. And the qualitative structure of each logical system of interpretation exactly matches the corresponding root structure (in quantitative terms)! So just as the two roots of 1 - for example - in quantitative terms are + 1 and - 1, equally the qualitative logical corresponding to 2 (as dimensional number) is based on the complementarity of opposite poles (i.e. + 1 and - 1, taken as interdependent. So in quantitative terms we interpret the roots in a linear either/or logic (based on the independence of opposites) in corresponding qualitative terms we interpret the dimensional number in both/and terms (based on their interdependence).

Extending this realisation, every mathematical symbol that is given a specific quantitative interpretation in Type 1 Mathematics, can equally be given a holistic qualitative interpretation in Type 2 terms.

In Type 3 Mathematics we then attempt to interactively combine both Type 1 and Type 2 understanding.

Thus the simple mathematical expression 2^2, in Type 3 terms represents a dynamic interaction as between 2 as base number and 2 as dimensional number (with each number having both quantitative and qualitative aspects always in opposite relationship to each other).