Wednesday, July 20, 2011

Holistic Mathematical Dimensions (1)

As we have seen dimensional numbers as used in conventional Type 1 Mathematics are given a reduced quantitative meaning (that conceals their true nature).

Therefore to unravel the true qualitative meaning of dimensions requires the use of a holistic Type 2 mathematical approach.

Furthermore this holistic approach has direct implications for appreciation of the true nature of space and time in both physical and psychological terms (which from this qualitative perspective are inherently complementary).

The starting basis for this new appreciation of dimensions is the realisation that all phenomenal reality is conditioned by 3 sets of polarities.

The first of these relates to the fact that in experience phenomena necessarily entail the interaction of polarities that are - relatively - external and internal with respect to each other. Thus to experience any phenomenon (e.g. mathematical symbol objectively as external requires a corresponding subjective mental construct that is thereby - relatively - of an internal nature.

Now if we denote conscious phenomena in holistic mathematical terms as "real" then all phenomena are conditioned by opposite polarities that are - relatively - positive and negative (with respect to each other).

The second set of polarities relates to the further fact that in experience all phenomena switch as between opposite aspects that are whole and part with respect to each other.

Now in conscious experience a basic reductionism takes place whereby wholes and parts are reduced in quantitative terms (with the whole in any context thereby representing the sum of its constituent parts).
This indeed leads to the experience of holons i.e. whole/parts where the qualitative distinction of whole and part is not maintained.

To properly distinguish whole and part in qualitative terms thereby requires unconscious (as well as conscious) recognition.
Ultimately qualitative recognition entails a spiritual element to understanding. Then with pure intuitive type recognition from one perspective each individual part can uniquely mediate Spirit (as an archetypal individual symbol). Likewise from the other perspective the collective whole can likewise mediate spirit (as an archetypal collective symbol).

In holistic mathematical terms the relationship as between conscious and unconscious is real to imaginary.

So now when we combine the two sets of polarities with respect to external/internal and individual/collective aspects we have - relatively - both real and imaginary distinctions in a positive and negative manner.

The third set of polarities relates directly to the distinction as between form and emptiness in recognition of the fact that - properly understood - all phenomenal experience entails both material and spiritual aspects.

This third set actually entails a special case of the two former sets which can be explained as follows.

The geometrical way of representing these polarity sets is with respect to a circle of unit radius (in the complex plane) .

Now to represent the first set we draw a straight horizontal line through the centre to meet the circumference on both sides. So if the right hand radius is + 1, then the left hand radius is - 1.

To represent the second set we now draw a straight line vertically through the centre so as to again meet the circumference of the circle on both sides.

So if the upper (vertical) radius is + i, then - relatively - the bottom radius is - i.

Finally to represent the third set we draw a straight line diagonally (from left to right) through the centre (at an equal distance from both horizontal and vertical lines).

Now if the upper diagonal line (or radius) is 1/k (1 + i) where k = square root of 2, then the lower part (as radius) is 1/k (- 1 - i).

Now if we choose the other direction for the diagonal line from bottom right to upper left, then the corresponding representations will be 1/k(1 - i) and 1/k (- 1 + i) respectively.

So the holistic mathematical representation of the 3 polarity sets corresponds in structural terms to the 2nd, 4th and 8th roots of 1 respectively.

The key to qualitative dimensional appreciation is the recognition that these quantitative roots bear an inverse relationship with their corresponding dimensions in qualitative terms.

So in this sense all experience is fundamentally structured according to the holistic qualitative meaning of mathematical dimensions.

And as such experience relates to both physical and psychological aspects of understanding, this thereby entails both physical and psychological reality (which are complementary) are fundamentally structured in accordance with Type 2 mathematical appreciation of dimensional numbers.

No comments:

Post a Comment