Friday, February 17, 2012

2-Dimensional Interpretation

All experience of reality is necessarily conditioned by fundamental polar reference frames which thereby deternines the manner in which such reality is interpreted.

Two of these polar reference frames we have referred to before and are deeply relevant to the manner in which mathematical - as indeed all other phenomenal experience - takes place.

The first of these refers to the relationship as between external and internal (which can be equally expressed as objective and subjective, outer and inner etc.)

So when we look at mathematical experience, in a dynamic interactive manner, we must necessarily recognise the two-way relationship as between external and internal (and internal and external) aspects.

So for example we cannot form knowledge of a number "object" in the absence of a corresponding mental perception of such an "object". So a two-way relationship continually exists as between both the (physical) object and the (psychological)perception that are - relatively - external to internal and internal to external respectively.

Equally in mathematical experience a two-way dynamic relationship necessarily exists as between whole and part (and part and whole).

So, from a psychological perspective, concepts (as wholes) are necessarily in relation to perceptions (as parts) and vice versa. And because the psychological and physical aspects of reality are likewise dynamically complementary, in corresponding physical terms we have dimensions (as wholes) related to objects (as parts) and vice versa.

For example, we cannot form a particular perception of number in the absence of the corresponding concept of number; in corresponding physical terms, we cannot form a number object in the absence of the notion of a number dimension.

So, we have seen how in a precise manner, Conventional Mathematics is 1-dimensional in nature; and of course the standard representation of numbers in this approach uses a 1-dimensional straight line!

Thus, once again the very nature of the standard mathematical approach is to represent the dynamic relative nature of experience in a misleading absolute fashion.

And the essence of how this is achieved, requires - in any context - using just one polar direction, thereby creating an independent frame of reference.

So, in relation to external and internal, mathematical symbols are given an absolute objective existence. This in turn implies that (internal) interpretation is given an absolute identity. So in Conventional Mathematics, correct interpretation is assumed to correspond directly with the absolute nature of mathematical objects!

Thus, in a very literal sense with such absolute interpretation, we ignore the fact that external and internal are positive (+) and negative (-) in relation to each other by assuming that both are in fact positive (+).

Now, I do not question for a moment the great value of this 1-dimensional approach (where the focus is on merely reduced quantitative interpretation of mathematical symbols). The problem is that exclusive focus on this one aspect simply blinds us to the much more comprehensive nature of mathematical understanding that is truly possible.

By contrast, the essence of the 2-dimensional approach is that it is always based on the dynamic interaction of two complementary reference frames that are positive (+) and negative (-) in relation to each other.

The true direct nature of 2-dimensional appreciation is of a holistic intuitive kind i.e. where the two-way interdependence of opposite polar reference frames can be directly intuited.

By contrast as we have seen the direct nature of 1-dimensional interpretation is of a rational linear kind (based in any context on one isolated polar frame of reference).

However though the direct nature of 2-dimensional appreciation is intuitive, indirectly it can be given rational expression in a circular logical fashion.
So, in contrast to the unambiguous either/or logic of the 1-dimensional rational approach, the 2-dimensional is based on a paradoxical both/and logic (arising from the recognition of two complementary poles that are relative).

So, 2-dimensional appreciation is inherently of a qualitative holistic nature.

The mathematical nature of this understanding (Type 2) can be expressed through our second number system where 1 is raised successively to the natural numbers (representing dimensions).

So once again each number (as qualitative dimension) bears a close structural relationship with its reciprocal (in a quantitative manner).

So therefore in this number system 1^2 (in qualitative terms) corresponds closely with 1^(1/2) from a quantitative perspective.

And as we have seen 1^(1/2) is correctly interpreted quantitatively as - 1 (which lies as a point on the circle of unit radius in the complex plane).

Thus this circular number also directly corresponds with the qualitative mathematical interpretation of - 1.

And as this in qualitative terms, requires the negation of what is rational linear (and conscious) it thereby implies what is intuitive (and unconscious in origin).

And we can see directly that the dynamic process of reaching this state implies the negation of what is positive. Thus in a circular rational fashion, 2-dimensional understanding implies both positive and negative poles (in the complementarity of opposites).

We saw already - using the crossroads analogy - how in a very precise holistic mathematical manner, the correct statement of the nature of the Riemann Hypothesis requires both 1-dimensional and 2-dimensional interpretation.

Once again, in approaching the nature of the individual primes and their general frequency (in isolation from each other) we require the standard type of 1-dimensional interpretation on which Conventional Mathematics is based.

However like someone who can only recognise a left turn, when approaching the mathematical crossroads from the alternative individual and collective perspectives, where the central mystery of the Riemann Hypothesis lies, both turns will be interpreted in a merely quantitative manner.

However, just as we realise, that when we look simultaneously at the two turns at the crossroads as interdependent, they are necessarily left and right (and right and left) in relation to each other, in like manner when we view both the individual nature of the primes and their general frequency as interdependent, then the two aspects are necessarily quantitative and qualitative (and qualitative and quantitative) in relation to each other. And the Riemann Hypothesis is a statement of the ultimate condition necessary for this two-way identity of quantitative and qualitative aspects!

However such appreciation of this two-way interdependence with respect to the primes is based on 2-dimensional (rather than 1-dimensional) interpretation!

So, once again, not alone can the Riemann Hypothesis not be proved (or disproved) in conventional mathematical terms, it cannot even be properly interpreted in this fashion.

Of course the Riemann Hypothesis has indeed a key relevance for the nature of the primes. However its deeper significance is that it has an even greater relevance for the true nature of Mathematics.

In other words, it clearly points to the need for an altogether more comprehensive vision of Mathematics which can recognise - and then ultimately integrate - the (unrecognised) qualitative side with its established quantitative aspect.

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