Friday, June 3, 2011

Odd Numbered Integers (8)

It might help to summarise the rationale of what has been involved in making these qualitative connections with the Riemann Zeta Function (for negative odd integer values).

Once again it is vital to appreciate that standard (unambiguous) quantitative type interpretation of numbers is associated with a default dimensional value of 1. So for example when numbers are raised to a power (other than 1), the attempt is made to obtain a reduced numerical result (in terms of 1).

Thus in this context 2^2 = 4 (i.e. 4^1). Therefore, though a qualitative change in the nature of the number takes place (through raising to the power of 2) the result is expressed in a reduced merely quantitative manner.

However the Riemann Zeta Function diverges for negative odd integer values of the dimensional power s.

So when s = - 1, we obtain in this context the series 1 + 2 + 3 +..... which clearly in standard terms sums to infinity.

However through the process of analytic continuation the Riemann Zeta Function can be given an alternative finite interpretation for all negative odd integers of s as a rational number.

Now the fascinating explanation for this alternative behaviour is that the finite numerical result now obtained relates to a circular rather than linear interpretation.

For simplicity, I have referred to this result as qualitative (rather than quantitative). However strictly speaking this is not properly correct.

So in a more complete fashion we have distinguished two distinct types of explanations for numerical results of the Riemann Zeta Function.

For values of s > 1, they can be given the standard linear (absolute) type interpretation. This results from the assumed absolute separation here of qualitative interpretation from the objective numerical result obtained.

However for values of s < 0, they can only be given an alternative circular (relative) type interpretation inherently in terms of the actual value for s utilised.

Here we operate according to the different assumption that both qualitative interpretation with resulting numerical values are always in dynamic interaction with each other. Thus the rational values that result from the Riemann Zeta Function thereby reflect the circular interrelationship of aspects of understanding that are always qualitative and quantitative with respect to each other. Furthermore this relationship tends towards complementarity (whereby both aspects are perfect mirrors of each other).

Now strictly this perfect complementarity properly only applies for negative even integer values of s. Thus the resulting value of 0 reflects that we can no longer separate any phenomenal result (in merely quantitative terms). So we have here pure intuitive realisation (in psychological terms) which perfectly mirrors appreciation of the empty holistic ground (underlying all phenomenal awareness of physical type reality)

However in a provisional limited sense for the negative odd integers - even though interdependence again characterises the relationship between qualitative interpretation and quantitative type results - a certain relative degree of separation can take place.

Thus from the psychological perspective the numerical result of the Function has a qualitative interpretation (as a certain mode of rationality).

Meanwhile from the physical perspective the same numerical result can be given a quantitative interpretation (i.e. representing a certain quantum relationship).

However as the numerical magnitude of dimensions increases a progressively higher level of pure energy characterises all relationships.

Thus from the psychological perspective, rationality becomes so refined that it ultimately cannot be separated from the spiritual intuitive energy with which it interacts.

Likewise from the corresponding physical perspective, quantitative phenomena become so unstable and short-lived that they can no longer be distinguished from the pure physical energy with which they are associated.

And this pure energy itself ultimately becomes inseparable from an empty holistic ground of nature (as the source of all physical reality) that is complemented by a pure empty spiritual experience (as the goal or realisation of all reality).

I will draw attention to another crucial distinction.

From the standard linear perspective, all numerical values are given an abstract identity (as essentially independent of all physical and psychological behaviour).

However from the corresponding holistic circular perspective, all numerical values necessarily express fundamental phenomenal relationships (with both physical and psychological aspects) that ultimately tend to full complementarity.

This suggests therefore that the results of the Riemann Zeta Function - as I have been explaining - have a direct relevance to both physical and psychological reality.

Likewise this is true of the famed non-trivial zeros which now have dual interpretations (both in the standard abstract sense and the new holistic sense as intimately related to both physical and psychological reality).

Whereas there is now some recognition that these zeros may indeed have a direct physical interpretation, there is no recognition as yet of their corresponding psychological relevance (as representing various refined states of interpretation).

And ultimately - as I have repeatedly stated - the very message of the Riemann Hypothesis is that physical and psychological aspects (through both quantitative and qualitative type interpretation) are ultimately inseparable.