Wednesday, March 20, 2013

Two Complementary Zeta Functions (4)

It is now recognised that the non-trivial zeros (i.e. the Type 1 zeros) in all probability have a spectral interpretation relating to the frequencies of some dynamic system of a physical nature.

Now this comes as no surprise to me at all, as I would have long suspected that this was necessarily the case. Indeed I would go considerably further in emphasising that this spectral interpretation relates to a dynamic system with twin physical and psychological aspects that are complementary.

However my own interest in the Riemann Hypothesis initially arose from a prolonged interest in the nature of the (unrecognised) non-trivial zeros (Type 2).

And in this context I had already made use of a spectral interpretation as just one illuminating pathway towards appreciating their true nature.

Now the electromagnetic spectrum is perhaps the best-known physical spectrum.

So natural light comprises just one narrow band on this overall spectrum. Below this band lies electromagnetic energy of higher frequency and shorter wavelength than light (e.g. x-rays and gamma rays). Above this band lie corresponding energies of lower frequency and longer wavelength (e.g. microwave and radio).

Because in dynamic terms of complementarity, a psychological aspect necessarily exists to such a spectrum with fascinating implications for the spiritual contemplative journey.

The term “light” of course is equally used in a spiritual as well as physical sense where it refers to the holistic aspect of intuition or illumination which necessarily interacts with (analytic) reason.

Just as natural light inhabits the middle band of the physical spectrum likewise the intuition that characterises conventional (linear) reason likewise inhabits the middle of the psychological spectrum.

However it has been long known that through authentic contemplative development that higher bands on this spectrum are attainable where the intuitive light becomes ever more refined and thereby capable of interacting with paradoxical (i.e. circular) rational structures.

Indeed ultimately these structures become so unrestricted and transparent that they no longer appear to possess any remaining rigid structure becoming fully integrated with the very light through which they are mediated.

Now associated with the deepest level of contemplation is a quality of peace and utter stillness. This illumination can therefore be readily characterised as (intuitive) light of exceptionally long wavelength (and corresponding low frequency).

By contrast in the spiritual literature the other extreme (of short wavelength and high frequency) is often related to periods of purgation where an incessant inner bombardment takes place exposing one’s hidden faults and failings!

One sobering point however to consider is this!

Just as energies outside the natural light band are not visible to the naked eye, likewise all these energies (and the understanding associated with them) cannot be appropriated through linear reason (informed by common intuition).

This therefore creates insuperable difficulties for Conventional Mathematics in trying to appropriate such understanding within its accepted axioms and definitions!

Now again in the spiritual esoteric traditions, these higher bands of understanding are often referred to as higher “dimensions”.

Now the key here is to recognise that number is used to represent dimensions in Mathematics.

So therefore associated with each number – now understood in a qualitative holistic sense – is a unique dynamic form of understanding where a more refined quality of intuition interacts with the ever more circular (paradoxical) use of reason!

I have often mentioned in this context the fact that Conventional Mathematics is based on linear reason i.e. literally in a 1-dimensional manner.

Therefore though the quantitative expression of other dimensional numbers (as powers or exponents) is indeed accommodated, this all takes place within a default qualitative interpretation (= 1).

And the very nature of such interpretation is that distinctive holistic meaning (in any relevant context) is necessarily reduced in a mere quantitative manner.

Using the language of string theory, we could say therefore that Conventional Mathematics represents the lowest possible energy state (with respect to its understanding) where entities such as numbers thereby assume a (misleading) absolute identity.

More accurately we could say that Conventional Mathematics is exclusively defined in 1-dimensional terms (where 1 is defined in a qualitative holistic manner).

However the remarkable truth that needs to be embraced is that mathematical relationships can be validly defined in an unlimited number of possible ways (where each number as dimension represents a unique overall manner of interpretation).

Now in all cases (except 1) these dimensional interpretations imply unique configurations with respect to the dynamic interaction of both the analytic (particle) and holistic (wave) aspects of mathematical understanding.

So in making this very point, I am illustrating how the well-known fact that the Zeta Function remains uniquely undefined on the complex plane for s = 1, can be given both a quantitative (analytic) and qualitative (holistic) interpretation.

If you can appreciate this then you can understand immediately why the Riemann Zeta Function (and its associated Riemann Hypothesis) cannot be properly interpreted from within the standpoint of Conventional Mathematics.

As – when correctly interpreted - it fundamental relates to the dynamic interaction of the quantitative (analytic) and qualitative (holistic) aspects of the number system, this remains uniquely undefined in linear (1-dimensional) terms where the qualitative aspect formally is not recognised.

And of course from this enlarged perspective, as the Riemann Hypothesis relates to the fundamental condition for the ultimate identity of both quantitative (analytic) and qualitative (holistic) aspects, it is futile trying to prove it through the axioms of Conventional Mathematics (which give no formal recognition to the qualitative aspect).

From what we have said it is but a short journey to appreciating that the number system itself represents the ultimate spectrum (in both physical and psychological terms).

So the vibrations of all dynamic systems (physical and psychological) can thereby be ultimately understood as the original vibrations of this number system with respect to both its quantitative and qualitative aspects.

Therefore all natural phenomena thereby represent but a secondary expression of an original system where number - when dynamically interpreted with respect to both analytic and holistic aspects – inherently exists as their deepest possible means of encoding.

Once again the recognised non-trivial zeros (of the Zeta 1 Function) are - when properly understood - fully complementary with the Zeta 2 zeros (which I have been explaining).

Whereas the Zeta 2 can be identified initially with each natural number except 1 (that constitutes each prime) in ordinal terms, The Zeta 1 inversely can be identified initially overall with the entire group of prime numbers (that comprise the natural number system except 1) in cardinal terms.

The solutions from both perspectives establish the identity of the quantitative and qualitative aspects of the number system. Ultimately through simultaneous appreciation of Zeta 1 and Zeta 2, we can perfectly see (in Zeta 3 terms) the primes and natural numbers (and indeed both sets of non-trivial zeros) as perfect mirrors of each other with an ineffable identity.

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