Thursday, October 25, 2012

Incredible Nature of the Zeta Zeros (17)

As we know from Einstein’s famous equation E = MC2, matter and energy are equivalent.

When we look at numbers in an appropriate dynamically interactive manner, an equal equivalence applies, whereby from the analytic perspective, number can be seen in abstract material terms as quantitative; however from the equally valid holistic perspective number can be seen as ultimately representing pure energy.

In the actual dynamics of experience, the analytic aspect (as matter) is provided directly through reason; however the holistic aspect (as energy) is directly provided through intuition.

However because the conventional mathematical paradigm formally reduces interpretation in a mere rational manner, the very notion of number as representing energy, thereby can have little meaning from such a perspective. Thus in the context of the accepted appreciation of number, the idea that number directly embodies an energy state has - literally - no intuitive resonance.

In other words it does not resonate with the standard appreciation of number (as mere form).

However in the dynamics of experience – when appropriately interpreted – a continual transformation takes place due to the inevitable interaction of (analytic) reason with (holistic) intuition.

So number itself - through the interactions of its analytic and holistic aspects - is itself subject to continual transformation.

Thus one effective way of interpreting the conventional (Type 1) approach is as that which represents the lowest possible energy state of number!

Therefore, though it may be informally accepted that a certain degree of psychological energy (through intuition) is necessary to fuel mathematical activity, in explicit terms such activity is interpreted in a merely rational fashion - literally - as formal material.

However this poses enormous problems for interpretation of the nature of the non-trivial zeros, for in truth these lie at the other extreme of understanding, as representing - as close as is consistent while still mainatining phenomenal form - the notion of number as representing pure energy states.

Therefore from the psychological perspective, due appreciation of these zeros requires a very refined interaction of both intuitive and rational understanding, where both can be seamlessly integrated with each other.

From the complementary physical perspective, the zeros likewise reflect pure transition states of energy (where any distinction as between matter and energy dissolves).

At a deeper level this entails that the very notion of number cannot be ultimately separated from either the physical or psychological domains, but rather represents the most fundamental interpretation (in phenomenal terms) of their intrinsic nature.

Expressed in an equivalent fashion, the standard (Type 1) notion is based on interpretation of numbers as independent entities (in quantitative terms).

However there exists an equally important – though still unrecognised (Type 2) treatment - based on the holistic interpretation of numbers as interdependent with each other in qualitative terms.

The non-trivial zeros – in this enlarged context of number appreciation – represent the marriage of both the quantitative (Type 1) and qualitative (Type 2) interpretations respectively of number, where both approach mutual identity with each other (and which can be referred to as the Type 3 treatment).

However once again, appropriate appreciation of their nature cannot be attempted through standard type analysis, which is as futile as trying to understand the chemical composition of water with mere reference to the existence of oxygen atoms!

So the message which needs to be continually reiterated is that the most radical revolution in mathematical history is now required to take place, where both its quantitative (analytic) and qualitative (holistic) aspects are recognised as equal partners.

And such a revolution can in no way take place through an attempted extension of the present paradigm to deal with the neglected qualitative aspect (as it is based on an utterly distinctive type of understanding).

This is the very reason why the extreme specialisation that has taken place with respect to the quantitative (Type 1) aspect is so damaging, as it has greatly eroded the distinctive intuitive recognition that is required for the Type 2 aspect.

Indeed so greatly has this ability been undermined that the very recognition of its possible existence has largely disappeared (especially within the recognised Mathematics profession).

This is why this necessary mathematical revolution will be initiated by genuine seekers of truth outside the profession, who can appreciate the proper nature of Mathematics within a much more comprehensive integrated perspective and thereby not remain bound by its current artificial restrictions.

The spread between primes can be seen as an inverse measurement of the independence of the number system which increases as we ascend the natural number scale.

A simple measurement for this spread is given by log t.

So for example in the region of 100 (i.e. t = 100), log t = 4.60517.. so that the average spread (or gap) as between primes is about 5. Thus, we would expect around 1 in 5 numbers to be prime.

However in the region of 1,000,000, log t = 13.81551. So the average spread is now about 14 with 1 in 14 numbers prime!

So, because between 100 and 1,000,000, log t has approximately trebled in value, we could therefore express this inversely by saying that the independence of the natural number system (in the regularity of the occurrence of the primes) has fallen to a third of its initial measurement.

Expressed in an equivalent manner, the probability of finding a prime in the region of 1,000,000 is about 1/3 of its probability in the region of 100.

However there is equally another side to the coin as it were in that the interdependence of the number system thereby steadily increases as we ascend the number scale. Therefore whereas in the region of 100 we would expect a run of approximately 4 (i.e. 5 – 1) composite numbers for every prime in the region of 1,000,000 we would expect about 13 (i.e. 14 – 1).

So therefore as the frequency of the primes (as the independent numbers without factors) decreases, the frequency of the composite natural numbers (as representing the interdependent numbers with factors) increases in an inverse manner. And for large t, this inverse relationship would steadily improve in precision!

However, whereas the linear number scale is the proper home of number notions as independent, the circular number scale is the appropriate home of number notions as interdependent.

So the interdependence of 4 numbers – as I have illustrated elsewhere - is expressed through the four roots of 1, which geometrically lie as equal points on the circle of unit radius in the complex plane.

Once again the Type 2 explanation is of a very subtle nature combining both quantitative and qualitative type interpretation of the same number symbols.
So for example in a 4-dimensional qualitative interpretation, the quantitative nature of the 4 roots 1, – 1, i and – i, as – relatively - separate, is matched by the holistic qualitative interpretation of these same roots as interdependent.

And the geometrical representation of these roots combines the notions of line and circle, with each root represented as a line drawn from the centre of the circle to its circumference.

So just as in quantitative terms we can give expression to the notions of line and circle, equally in qualitative terms we have complementary notions with respect to linear and circular type logical understanding respectively.

However if we wish to convert this Type 2 appreciation of the dynamic nature of independence and interdependence in a Type 1 fashion, then we can simply concern ourselves with the distance as between the equidistant points on the circle of unit radius.

Now as interdependence is inversely related to prime independence (as log t) with the circumference of the unit circle = 2π, therefore the gap between each point (on the circle as a measurement of interdependence) is given as 2π/log t (which as with the prime number theorem will become ever more accurate for large n).

Now the formula for the spread or gap as between the non-trivial zeros is given as 2π/log (t/2π).

However for large t, log (t/2π) approximates ever more closely to log t.
For example when t = 10^9000, log (t/2π) = 20721.4 and log t = 20723.3 (correct to 1 decimal place). So the two results are already very similar, with the relative deviations continually falling for larger t!

This means in effect that just as the pattern of primes represents the independent aspect of the number system (i.e. as numbers with no factors) the non-trivial zeros represent the opposite extreme of the interdependent aspect of the same system (where natural numbers increasingly tend to be composed of the product of prime constituents).

Put more precisely, whereas the conventional understanding of the primes represents interpretation, where the independent aspect (in their individual nature) and interdependent aspect (in their collective behaviour) are separate from each other, the non-trivial zeros – when appropriately understood – represent a completely new interpretation of number where both the independent and interdependent aspects of number are – relatively – identical.

However as the presentation of the non-trivial zeros is in linear format, this requires the ability to convert the true ordinal interpretation of interdependence (which relates directly to Type 2 understanding) in Type 1 terms (i.e. on an imaginary scale). Now the process of analytic continuation on the complex plane enables such a quantitative transformation. However ultimately any meaningful interpretation of the results arising, requires the holistic type appreciation associated with Type 2 understanding.

So this combined use of Type 1 and Type 2, which the non-trivial zeros encapsulate, represents what I refer to as Type 3 understanding.

So the non-trivial zeros of the Riemann Zeta Function, provide once again on an imaginary linear scale, the points where the natural and prime numbers coincide.

However as these largely relate to holistic rather than (linear) analytical type understanding, their true meaning cannot be grasped in the standard manner of mathematical interpretation.

Not surprisingly, a much simpler equivalent way therefore exists for expressing this coincidence of the primes and natural numbers through the Type 2 aspect of the number system.

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