Note 21 - Powers and Roots

We have concentrated on the most fundamental numbers 0, 1 and 2. In this manner we have illustrated the horizontal operations of addition and subtraction i.e. where quantitative change takes place with respect to the 1st dimension (which remains fixed).

So 1

^{1}+ 1^{1}= 2^{1}; also 1^{1}- 1^{1}= 0^{1}We then have given these operations a dynamic holistic interpretation, with addition relating to the (conscious) positing of (quantitative) phenomena (or perceptions), and subtraction to their corresponding (unconscious) negation.

Likewise we have illustrated the vertical operations of multiplication and division where now qualitative (or dimensional change) takes place with respect to a fixed unit quantity.

So 1

^{1}* 1^{1}= 1^{2}; also 1^{1}/ 1^{1}= 1^{0}We then likewise have given these operations a dynamic holistic interpretation with multiplication now relating to the (conscious) positing of qualitative dimensions (or concepts) and division to their corresponding (unconscious) negation.

So for example as a concept applies potentially to any (actual) perception within its class, conceptual understanding has the power to literally multiply perception (without limit).

However numbers representing quantities and numbers representing qualities (or dimensions) are inherently quite distinct. However in terms of conventional "real" (i.e. conscious) understanding this distinction is overlooked.

Therefore a double form of reductionism takes place with qualities reduced to quantities and quantities reduced to qualities. In other words - as with scientific empirical investigation - (qualitative) wholes are reduced to (quantitative) parts; likewise with scientific theory, (quantitative) parts are reduced to (qualitative) wholes.I have suggested therefore that the recognition of both the quantitative and qualitative aspects of number requires a mathematically complex formulation.

Therefore to convert a "real" number representing a qualitative dimension (indirectly) to quantitative format it must be expressed as "imaginary".

Likewise to convert a "real" number representing a quantity (within a dimension) to qualitative dimensional format again it must be expressed in "imaginary" terms.This is all deeply relevant to the dynamics of experience which continually entails the interaction of concepts (i.e. the qualitative dimensions for phenomena) and perceptions (i.e. the quantitative data to which the dimensions relate).

So if we are not to continually reduce concepts to perceptions and perceptions to concepts we must define all phenomena in mathematically complex terms i.e. with both "real" and "imaginary" aspects which continually keep switching between one another.

One example of this reductionism relates to the way in which space dimensions are continually confused with objects. So when we maintain that an object is 3-dimensional in space we are treating both the object and the dimensions in merely quantitative terms.

Strictly speaking therefore - in dynamic holistic terms - if the dimensions are "real", then the objects to which they relate are - relatively - "imaginary". Likewise if the objects are "real" then the dimensions to which they relate are "imaginary".

So in dynamic interactive terms, "reality" is in fact complex with both what - relatively - are (quantitative) objects and (qualitative) dimensions continually switching between each other (i.e. from "real" to "imaginary" and "imaginary" to "real" format).Of course the psychological experience of phenomena directly complements what is physically observed. Therefore the complex psychological interaction of perceptions and concepts (and concepts and perceptions) exactly mirrors the corresponding complex physical interaction of object phenomena with dimensions (and dimensions with objects).

So the dynamics of this activity - as we have seen - are deeply rooted in fundamental holistic mathematical notions.

The diagonal operations of powers and roots entail both horizontal and vertical

aspects.We defined the horizontal operations (of addition and subtraction) by holding the qualitative dimensional characteristic fixed as 1.

Likewise we defined the vertical operations (of multiplication and division) by holding the quantitative object characteristic fixed as 1.

In diagonal terms both the quantitative and qualitative aspect can vary leading therefore to both a quantitative and qualitative transformation.

So we illustrate this fundamental diagonal operation by raising 2 to the power of 2 which is

2^{2 }i.e. duality in both object (quantitative) and dimensional (qualitative) terms.Now 2

^{2}= 4 (square units).This can best be appreciated by visualising the operation in geometrical terms as a square with each side = 2 (i.e. 2

^{1 }units)

The area of the square is then 4 square units.

So both a (horizontal) quantitative and (vertical) qualitative transformation has been brought about.Now the reverse operation involves taking the square root.

So if we obtain the square root of 4 square units we obtain 2 linear units (i.e. 2^{1}) again demonstrating that both a quantitative and qualitative transformation has taken place.Again these dynamics are directly relevant to the very nature of experience (when appropriately understood).

For example to illustrate, let us take the three cognitive stages that make up the middle band of the Spectrum i.e. concrete operational (conop), formal operational (formop) and vision-logic.

Now the very nature of concrete operational understanding is that it largely relates to - mere - quantitative transformations (within embryonic conceptual structures that remain largely fixed).

So in this sense, concrete operational understanding involves the horizontal positing (and to a limited extent) negation of specific quantitative phenomena (within relatively fixed - undeveloped - conceptual qualitative structures).

Formal operational understanding however - by contrast - largely relates to qualitative transformations directly with respect to more generalised conceptual classes.

So (pure) formal operational understanding involves the vertical positing (and again to a limited extent) negation of more universal qualitative (i.e. dimensional) phenomena with respect to a fixed quantitative (concrete) base. In othere words it relates directly to significant development in (qualitative) conceptual understanding of reality.

Vision-logic however entails the combined interaction of both the horizontal (quantitative) and vertical (qualitative) aspects, leading - especially when well developed - to significant quantitative and qualitative transformations in phenomena.

So in dynamic holistic terms, vision-logic relates to the holistic mathematical equivalents of the operations of powers and roots entailing both (quantitative) phenomenal and (qualitative) dimensional transformation.

Fundamentally this entails diagonal transformation with respect to both form and emptiness (though necessarily at this comparative early stage of development with a significant degree of reductionism remaining).What is interesting here is that all the levels of the Spectrum can be sub-divided with respect to concrete, formal and vision-logic stages (what I refer to as sub-levels) and be given a precise holistic interpretation based on the dynamic interactive appreciation of the fundamental mathematical operations

i.e. (horizontal) addition and subtraction

(vertical) multiplication and division and

(diagonal) powers and roots