Thursday, December 2, 2010

True Significance of Riemann Hypothesis (2)

We have already explained the Riemann Hypothesis in terms of that intimate connection enabling consistency as between reality as viewed objectively in quantitative terms and the corresponding psychological constructs necessary for overall qualitative interpretation of its nature.

An even deeper appreciation of this relationship entails the incorporation of both (holistic) intuitive and (analytic) rational type interpretation.

In a direct sense, (holistic) intuition relates to the empty spiritual aspect of reality (that is infinite in potential terms); by contrast (analytic) reason relates to the formal material aspect (that is finite in an actual manner). So the dynamic relationship of reason and intuition in experience pertains to the central relationship of finite to infinite.

And in a crucially important sense, the Riemann Hypothesis again can be expressed as the fundamental requirement for consistency with respect to this relationship.

For example in strict terms - though this is overlooked with respect to conventional mathematical interpretation - a general proof of a theorem applies to an infinite no. of potential cases. However any specific examples, illustrating the general proof, relate to a finite no. of actual cases.

So the proof of the Pythagorean Theorem for example applies potentially to all possible right hand triangles (in infinite terms). However the applicability of the theorem necessarily relates to a finite no. of actual right hand triangles.
Thus inherent in the belief that we can apply a general (potential) proof to specific (actual) examples is an implicit acceptance of the fundamental consistency as between finite and infinite realms.
And this is exactly the same consistency that is implied by the Riemann Hypothesis!

Unfortunately, as Conventional Mathematics is formally based merely on rational interpretation, the significance of the Riemann Hypothesis will always remain out of reach (when approached from this perspective).

Looked at in yet another equivalent manner, the fundamental axiom - to which the Riemann Hypothesis relates - enables consistent dynamic switching as between the opposite polarities of experience (such as external and internal).

I will explain here the basic nature of this dynamic interaction before explaining its precise mathematical significance in the next contribution.

All experience - indeed all development processes - entail twin processes of differentiation and integration respectively (based on two distinct logical systems).

Differentiation essentially entails linear logic whereby opposite polarities are clearly separated.
Integration by contrast entails circular logic whereby these same opposites are viewed as complementary (and ultimately identical).

In human experience differentiation and integration relate to the conscious and unconscious respectively. Reason is directly identified with conscious understanding and best suited to detailed analytic understanding of specific aspects of reality (using linear logic); intuition is directly identified with unconscious understanding and correspondingly suited to holistic appreciation of the overall nature of reality (using circular logic).

Conventional mathematical appreciation - as stated - is based on sole recognition of reason (using linear logic). And central to this appreciation that objective truth can be clearly separated from subjective (in absolute type fashion).

Now in order to switch from conscious recognition with respect to external (objective) reality to corresponding conscious recognition with respect to internal (psychological) constructs (and then back again), the unconscious is always implicitly involved. The unconscious essentially is based on intuitive recognition that external and internal (as opposite poles of experience) are dynamically complementary (and indeed ultimately identical). So in a sense to view objective reality in an absolute manner is very unbalanced (from this unconscious perspective). So in an attempt to establish balance, experience switches direction (to the internal psychological pole).

However when mathematical reality is viewed in a merely rational manner, the internal pole is also given a somewhat absolute identity. This leads to the view that there is thereby a direct correspondence (in absolute terms) as between the mental constructs that Mathematics uses for interpretation and the objective reality to which these relate.

This belief in turn leads to a somewhat rigid relationship between polarities whereby experience continues to confirm the same rigid assumptions (on which interpretation is based).

Indeed this rigidity has been so great that it has blotted out recognition altogether of the equally important holistic aspect of mathematical understanding. Again this is directly based on a special type of intuitive understanding (conveyed through coherent qualitative interpretation of mathematical symbols).

So true qualitative appreciation of reality - relative to quantitative - is of a holistic intuitive nature. So when we attempt to express such appreciation in merely rational linear terms, we simply reduce the qualitative aspect to the quantitative.

So not surprisingly the very nature of Holistic Mathematics cannot be understood from this perspective.

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