Wednesday, December 28, 2011

Positive and Negative Dimensions

Actual experience always entails a dynamic interaction of both conscious and unconscious. Putting it simply, the conscious relates directly to what is finite and actual, whereas the unconscious relates to what is potential and infinite.

As these two aspects are qualitatively distinct from each other, equally they require distinctive methods of intellectual translation.

At its most basic, I have long argued for the need for a binary digital means of interpreting development (which represents an ongoing transformation process). In this system, 1 represents unitary form and relates to a linear (1-dimensional) method of interpretation. By contrast 0 represents emptiness i.e. nothingness as the potential for all form. Though in direct terms in experience this is strictly of an intuitive nature, indirectly it can be translated rationally through the use of complementary opposites.

So in a comprehensive approach to development, we must combine both linear (either/or) logic that is unambiguous with circular (both/and) logic of a paradoxical nature.

All phenomena entail the interaction of perceptions and concepts which are - relatively - quantitative and qualitative with respect to each other.

From one perspective, a specific perception points to what is actual in experience. For example if I notice a cat running around my back garden, the specific perception of the cat points to an actual object in experience.
However this perception of a cat would not be possible in the absence of the general concept of "cat" which potentially has an infinite range. Thus the concept of "cat" which provides the unique quality of "catness" in experience applies potentially to all cats.

However with merely linear understanding, the unique notion of the concept (as potential and infinite) is immediately reduced to finite actual interpretation. So then the concept of "cat" is identified as having a merely finite actual meaning. In this manner the dynamic interaction of (holistic) intuition and (specific) reason that are necessary for the identification of a cat to take place is subsequently interpreted in a merely reduced rational manner.

This problem is of the first magnitude when it comes to mathematical understanding!

So the identification of a particular number (as representing a distinct mathematical object) arises out of a dynamic experience that entails the interaction of both intuition and reason.

Thus intuition is necessary in formulating the general concept of number (as potentially applying to an infinite set of - as yet unspecified - numbers).

Reason is likewise necessary in formulating the specific perception of an actual number (that is finite in nature). So crucially in experience there is always this intersection of finite with infinite meaning, whereby both the general concept and specific perception (relating to that concept) can be identified with each other.

And of course this is equally true of mathematical experience, whereby for example the recognition of number always entails the identification of infinite with finite notions of meaning.

However crucially in Conventional (Type 1) Mathematics holistic notions - that are potentially of an infinite nature and relating directly to intuitive recognition - are formally reduced in a merely rational manner.

In this way the potential nature of the number concept is quickly lost and is identified with all actual numbers (of a finite nature). So inevitably with Conventional Mathematics the infinite notion is reduced in a merely actual manner.

For example such reductionism underlies all mathematical proof.

The general proof of a proposition properly applies potentially to all - as yet unspecified - examples within its class.

So the Pythagorean Theorem for example where in every right angled triangle the square on the hypotenuse equals the sum of squares on the other two sides, applies potentially to the infinite class of unspecified actual right angled triangles.

However with mathematical proof, the assumption is then made that there is a direct correspondence as between infinite and finite notions. In other words the applicability of the general proof to potentially all triangles is misleadingly identified as synonymous with the limited set of actual triangles (where the theorem can be demonstrated to hold). The problem however is that an indeterminate set of actual triangles will always remain for which the proof can never be actually verified! So properly understood an inevitable uncertainty principle necessarily applies to all mathematical proof.

The holistic nature of the number concept properly relates to the dimensional aspect of number.

What struck me forcibly - even as a child - is the reduced manner in which the dimensional aspect of number is treated.

For example, I remember an school how an acre was defined as 4840 square yards!
This measurement is clearly of a 2-dimensional format. However conventionally when a number is squared the qualitative transformation in the dimensional nature of the units is ignored.

So for example 70^2 = 4900 (i.e. 4900^1). However, here the qualitative nature of the units has clearly changed (from 1-dimensional to 2-dimensional format). However this qualitative transformation is then simply reduced in quantitative terms. So we can see here in a very graphic manner how the conventional approach of Type 1 Mathematics is - literally - of a linear nature i.e. where all number transformations are ultimately expressed in reduced (1-dimensional) quantitative terms.

So in moving from the reduced approach of Conventional (Type 1) Mathematics to a more dynamic interactive approach (which properly incorporates intuition), the first thing to understand is that unconscious appreciation requires the negation of what is conscious.

Thus if we identify Conventional Mathematics with the linear rational approach (i.e. 1-dimensional) then the unconscious intuitive counterpart of such understanding is represented by the negation of the 1st dimension.

Now consider the following integer number 4 (i.e. 4^1).

If we raise this to the dimensional power of - 1 i.e. 4^(- 1) we obtain 1/4.

So the very means by which in experience whole moves to part recognition implicitly entails experience with respect to the negative 1st dimension.

Thus the recognition of 4 as a number requires the rational recognition that 4 belongs to the number concept (that is then understood as applying to all actual numbers).

Imagine we have a cake made up of 4 slices! Such recognition requires linear (1-dimensional) understanding that is rational and positive. Thus the cake is identified here with its four slices.

However the opposite part recognition of each slice (as 1/4 of the whole cake) requires the temporary negation of the whole cake (as 4 slices). This then enables a switch to part recognition of each slice (in the context of the cake).

So without implicitly being able to negate rational understanding of the whole, it would not be possible to appreciate the unique identity of each slice (in the context of the whole).

Thus in holistic (Type 2 terms) rational understanding with respect to qualitative dimensions relates to the positive sign of these dimensions.
Intuitive understanding is then directly identified with the negative expression of these dimensions.

This applies to all higher dimensions (other than 1).

For example the rational expression of (positive) 2-dimensional understanding can be described as the logic of complementary (real) opposites in experience i.e. where all conscious experience is paradoxically conditioned by the interaction of polar opposites.

For example Hegelian philosophy attempts to give rational expression to such 2-dimensional understanding.
However true intuitive recognition, compatible with such understanding, properly applies to corresponding negative 2-dimensional intuitive understanding (which is ineffable in nature).

Indeed appreciation of the qualitative significance of the first of the trivial zeros in the Riemann Zeta Function springs directly from such understanding.

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