Wednesday, May 2, 2012

Nature of Number System (3)

To understand number in an appropriate dynamic interactive manner, we need to move to a circular rather than strictly linear appreciation. Now to be precise, such circular likewise entails linear appreciation in a refined manner.

In other words in such understanding all numbers arise from the interactions of various poles of understanding (which comprise the use of numbers in qualitative terms as dimensions).

Now as we have seen the essence of linear understanding is that it is - by definition - based on uni-polar reference frames. So as interpretation - in any context - is based on just one pole, it is - literally - therefore 1-dimensional in nature.

This thereby excludes any genuine notion of interaction and therefore any genuine notion of interdependence (which requires at a minimum two poles).

So for example in conventional mathematical terms the objective status of a number is unaffected by our subjective interaction with that number. In other words through linear interpretation, the objective pole is isolated in an absolute independent manner.

So the inherently qualitative notion of interdependence (arising from the dynamic interaction as between distinct poles) is thereby inevitably reduced in a misleading quantitative manner.

Therefore, in failing to recognise a distinctive qualitative aspect to understanding of its symbols, Conventional Mathematics cannot properly deal with the notion of interdependence (which once again is directly of a qualitative nature).

Now the starting point for the recognition of this refined circular/linear approach (representing the dynamic interactive nature of number) is – what I have referred to as – the Type 2 number system.

Now once again the Type 1 system with respect to the natural numbers is defined as
1^1, 2^1, 3^1, 4^1,….

So in this system each natural number (as base quantity) is defined with respect to an invariant dimensional number i.e. 1 (which is - relatively - of a qualitative nature).

From another perspective we can express this by saying that the base quantities represent cardinal numbers whereas the dimensional number - by contrast - is of an ordinal qualitative nature.

So 1, in this ordinal context, relates to the 1st dimension.

Now in the Type 1 system the pure nature of addition (as positing) can be isolated.
Thus when we add numbers in the Type 1 system as quantities, the dimensional number remains unchanged (in qualitative terms).

Thus for example 2^1 + 3^1 = 5^1.

In the Type 2 system we have a complementary number system where each natural number (as power) now represents an ordinal dimensional while the base quantity remains fixed as 1.

Therefore in this system, the natural numbers are represented as,

1^1, 1^2, 1^3, 1^4,…..

Now in the Type 2 system the pure nature of multiplication can be isolated
So when we multiply numbers, the base number remains unchanged (in quantitative terms).

So 1^2 * 1^3 = 1^5.

This demonstrates the very important fact that whereas addition (with respect to the Type 1 system) is of a quantitative nature, multiplication (with respect to the Type 2 system) is - relatively - qualitative in nature.

Thus the key problem of reconciling the nature of addition and multiplication with respect to the primes points to the fact that – properly understood – they relate to the quantitative and qualitative aspects of mathematical understanding respectively.

So once again Conventional Mathematics must necessarily deal with the operation of multiplication in a reduced quantitative manner.

Strictly speaking therefore when we multiply two numbers (with base number other than 1) both a quantitative and qualitative transformation is involved.

So, 3^1 * 4^1 = 12 in quantitative terms.

However the dimensional nature of units strictly has now changed from 1 to 2 (which we can easily visualise in dimensional terms as a rectangle with sides 3 and 4 units respectively).

However in conventional terms this dimensional change is ignored.

So from this linear perspective

3^1 * 4^1 = 12^1.

Thus - quite literally - in the linear approach when carrying out quantitative calculations, the dimensional nature of numbers is ultimately reduced to 1.

However the key to unlocking the truly circular nature of the Type 2 system requires an important transformation whereby each power or exponent of 1 (as representing a dimensional number that is qualitative) is directly related in complementary terms with its quantitative root!

So for example to establish the circular nature of 1^2 we obtain the second root of
1, i.e. 1^(1/2).

So more generally the number D (as qualitative dimension) is inversely related with 1/D (as quantitative root).

Therefore when the root (1/D) is interpreted in a quantitative numerical manner, the corresponding dimension (D) is interpreted directly in a qualitative manner!
This implies that associated with each number (representing a dimension) is a distinctive qualitative means of interpretation.

The implications of this are enormous!

Once again Conventional (Type 1) Mathematics corresponds to the qualitative interpretation of 1 as dimension (whereby in effect qualitative notions are reduced to quantitative)!

However corresponding to every other number as a dimension is a distinctive means of overall qualitative interpretation (that entails a unique configuration of both quantitative and qualitative type appreciation).

Now the only dimensional number for which the Riemann Zeta Function is undefined is where D = 1.

This therefore implies - that when appropriately understood - this Function establishes at all other defined points complementary relationships as between quantitative and qualitative type meaning. Thus the mystery inherent in the primes relates directly to the key relationship as between quantitative and qualitative type meaning inherent in number symbols.

So once again, this demonstrates why it is ultimately futile trying to approach this mystery merely in conventional mathematical terms (which is solely of a 1-dimensional nature)!

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