Saturday, January 21, 2012

Three Mathematical Interpretations

I will now attempt to demonstrate how the the result of the simple expression,
2^2 would be interpreted in Type 1 (Conventional), Type 2 (Holistic) and Type 3 (Comprehensive) mathematical terms.

Now in Type 1 terms 2^2 = 4 (i.e. 4^1).

So what we obtain here is a numerical result in a reduced quantitative manner.

Now if we were to represent this in geometrical terms, 2^2 would represent a (2-dimensional) square of side 2 units. Therefore its area would = 4 square units.
In other words, though a qualitative change in the nature of the units involved, results from squaring (i.e. in moving from a 1-dimensional to a 2-dimensional format, this is simply ignored in Type 1 terms. Thus from a reduced quantitative perspective the result of 2^2 is indeed 4.

So correctly we could write the result 4, 2 with 4 representing the quantitative and 2 the qualitative dimensional aspect respectively the answer is given as 4, 1 (where the 1 which is merely a default value can be ignored).

In Type 2 terms 2^2 = 2 (i.e. 1^2).

Here - in reverse fashion - we look at the qualitative change that has taken place (while ignoring the quantitative value).
Now though we used a square to illustrate 2 dimensional reality in the previous part, this itself is but a reduced notion based on linear extension of what corresponds to 1-dimensional interpretation.

Now as we have no direct interest in the quantitative transformation brought about from 2^2 but rather in the qualitative transformation involved we can replace the quantity by 1 to obtain 1^2. So whereas Type 1 Mathematics is 1-dimensional in nature, Type 2 Mathematics is by contrast 1-quantitative so that the dimensional number involved is expressed with respect to the default base quantity 1!

Now the key to mathematically expressing the significance of the dimensional number D is to express 1 with respect to its inverse i.e. 1/D.

Thus when we replace in this case D with 1/D we obtain 1^(1/2) = - 1. Now this represents a quantitative value on the circle of unit radius (in the complex plane).

Now this same number - 1 can be given a coherent qualitative meaning that corresponds to 2 as a dimension (i.e. 2-dimensional interpretation).

In holistic mathematical terms + 1 implies the positing of conscious form; conventional mathematical reason implies such positing!
However - 1 holistically implies the negation of unitary form and relates directly to the unconscious intuitive aspect of understanding.

So 2-dimensional understanding relates directly to the intuitive aspect of understanding by which qualitative holistic connections are made in understanding. However this is inevitably reduced in Type 1 Mathematics to linear (1-dimensional) rational format.

In Type 3 terms 2^2 = 4, 2

In other words Type 3 Mathematics combines both Type 1 and Type 2 understanding.

Therefore form one perspective we recognise the quantitative transformation implied by the expression 2^2 i.e. 4 (which is expressed with respect to the default dimensional number of 1). However equally we recognise the qualitative transformation implied by 2^2 i.e. 2 as dimensional number (which is expressed with respect to the default quantitative number of 1).

So understanding here becomes of an increasingly interactive nature whereby one continually moves from quantitative to qualitative (and qualitative to quantitative understanding). In psychological terms this would imply the balanced mix of reason and (higher-level) intuition. Remember again that in Type 1 Mathematics intuition - though present - is inevitably reduced to reason!

The implications here are truly enormous!

Type 1 Mathematics is entirely defined in terms of just one logical system of interpretation (which in precise holistic mathematical terms is 1-dimensional).

However as a unique system of interpretation is associated with every number (as dimension), this implies that Mathematics can be given an unlimited number of logical interpretations (with each interpretation implying a unique manner of configuring reason and intuition).

So in the expression 2^2 this implies 2-dimensional interpretation. Now as conscious rational (1-dimensional) implies positing (+), corresponding unconscious intuitive understanding comes from the negation (of what has been posited). So correctly 2-dimensional interpretation implies - in paradoxical rational terms - the complementarity of opposites (where what is true is merely relative, as positive and negative at the same time.

2-dimensional mathematical interpretation thereby implies a more refined appreciation of mathematical symbols where for example a number has now two directions in understanding (which are positive and negative with respect to each other).

In other words the number 2 can be looked on as an external object (which is posited in experience) or as an internal construct (which arises through negation of the external aspect). In other words with 2-dimensional understanding one keeps switching as between external object and internal perception in a dynamic interactive manner.

So a number - by definition - at this level of understanding does not statically exist (as an independent object) but rather has a merely relative existence through the dynamic interaction of both positive and negative polarities of experience.

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