Thursday, October 18, 2012

Incredible Nature of the Zeta Zeros (14)

As we have seen, from the enlarged dynamic perspective, every mathematical symbol can be given both a coherent analytic and holistic interpretation.

The very nature of the non-trivial zeros relates to the ultimate identity of both aspects.

So these zeros can be viewed as point singularities on the imaginary scale, drawn through ½ (on the real axis), where both analytic and holistic meanings with respect to each number involved, directly coincide.

Much has been written (to a tremendous level of detail) on the analytic nature of the zeta zeros.

However as yet, the holistic nature of these zeros has not been recognised.
And quite simply we cannot appreciate their true significance without deep interpretation of this crucially important aspect.

As we know the Riemann Hypothesis postulates that all the non-trivial zeros lie on the imaginary line drawn through ½.

The holistic significance of ½ derives from the Riemann Zeta Function. This – in the enlarged dynamic interpretation – establishes complementary relationships on the real axis as between number results on the RHS which can be given the recognised analytic interpretation (i.e. which are intuitively concur with linear interpretation) and corresponding results on the LHS which requires a distinct holistic interpretation (i.e. based on circular type reason).

Now these results are connected through the Riemann Functional Equation, which establishes a relationship as between the analytic interpretation of ζ(s) on the RHS and the corresponding holistic interpretation of ζ(1 – s) on the LHS.

So for example when s = 2, the Functional Equation thereby establishes a relationship as between the analytic interpretation of ζ(2) on the RHS = (π^2)/6 and the corresponding holistic interpretation of ζ(– 1) = – 1/12 on the LHS.

Thus the first i.e. ζ(2) = 1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + … = (π^2)/6 is intuitively in accordance with linear notions of magnitude; however the latter ζ(– 1) = 1 + 2 + 3 + 4 +… = – 1/12 corresponds rather to a more complicated circular type interpretation.

The pure analytic type interpretation for values of s > 1, thereby establishes a complementary relationship with corresponding holistic type interpretation for values of s < 0.

What is known as the critical region where 0 < s < 1, therefore establishes a relationship as between both analytic and holistic aspects with respect to the Functional Equation on both sides.

Now the clear condition for establishing the identity of both aspects is that s = ½, for it is at this value that ζ(s) = ζ(1 – s). So from this new enlarged perspective, the Riemann Hypothesis can be seen as the simple condition required to ensure the mutual identity of both the analytic (quantitative) and holistic (qualitative) interpretation of number. However no solutions for the equation ζ(s) = ζ(1 – s)= 0 exist, where s is defined solely in real terms.

The solutions of this equation relate to complex numbers, which must therefore of necessity must have a real part = ½. The imaginary part then varies along a line drawn through ½ on the real axis. So every solution of the form, ½ + it is identified with a corresponding solution of the form ½ - it.

The next step is to provide a holistic interpretation of the nature of the imaginary part. We have seen that the real part relates to the conscious aspect of interpretation in a linear rational manner.

This effectively means that number is viewed independently as quantitative! The corresponding imaginary part then relates to the unconscious holistic aspect that is directly intuitive and interpreted in a real circular manner (using both/and logic based on the complementarity of opposites).

This circular formulation is then indirectly conveyed as linear in an imaginary manner. What this means in effect is that even though the non-trivial zeros all appear as linear numbers (that literally fall on the same vertical line drawn through ½ on the real axis) their interpretation now relates to the opposite notion of the interdependence of number.

So this is the key issue that remarkably has yet to be properly addressed by the professional mathematical community in that it is strictly meaningless to attempt to view numbers in a merely absolute quantitative fashion as independent entities!

Properly understood, all numbers must necessarily possess both independent aspects (whereby they can be viewed as relatively separate from other numbers) and interdependent aspects (whereby they are seen to have a relationship with other numbers).

Therefore Conventional Mathematics in formally recognising merely the quantitative aspect of number operates in a grossly reduced manner leading ultimately to total misrepresentation of its corresponding qualitative nature.

Now this reductionism is especially exposed in attempting to understand the relationship of the primes to the natural numbers (and the natural numbers to the primes).

This relationship cannot be interpreted in a merely quantitative manner but rather as a relationship of quantitative to qualitative and qualitative to quantitative meaning respectively. This then requires establishing the ultimate condition for the mutual identity of both quantitative (analytic) and the qualitative (holistic) aspects of number.

So if we are to properly appreciate the true nature of the non-trivial zeros, we require a radically distinctive interpretation of number. Thus in the conventional interpretation – based on abstract notions of independence – the quantitative is clearly separated from the qualitative aspect, with numbers appearing in static terms as absolute entities. However we are now faced with the opposite extreme where numbers represent the ultimate state possible – consistent with maintaining a phenomenal existence – of the full interdependence of both their quantitative and qualitative aspects.

So the correct interpretation here of number is of such highly dynamic nature as to be bordering on what is completely ineffable. Indeed correctly understood this most fundamental notion of number – as the non-trivial zeros - represents the finest partition possible bridging the phenomenal world from what is utterly ineffable.

This dynamic nature can be further understood with respect to the transcendental nature of the imaginary parts of these zeros. When one looks carefully at the nature of a transcendental number, one realises that it necessarily combines both finite and infinite aspects (relating to quantitative and qualitative interpretation respectively).

This is clearly demonstrated through the nature of π (which is perhaps the best known transcendental number). Now π can most simply be demonstrated as the relationship of its circular circumference to its line diameter in quantitative terms. In a corresponding holistic qualitative manner, π represents the relationship between both circular and linear type understanding.

Now in quantitative terms the one point that is in common (as both the midpoint of the circle and its line diameter) coincides at the centre. In corresponding holistic terms the midpoint where both linear and circular type understanding are identified is at the centre. However this point - though having a location – is literally pointing to an ineffable reality. It is quite similar with the non-trivial zeros.

The transcendental nature of these zeros relates to the fact that they represent a relationship as between both the quantitative and qualitative notion of number (where both are identical). Therefore though it appears as non-intuitive in terms of conventional notions of number, the non-trivial zeros represent points on an imaginary number scale where both the analytical (quantitative) and holistic (qualitative) interpretation of number are identical.

Not surprisingly therefore, these zeros serve as the means of reconciling the prime numbers with the natural numbers (and the natural numbers with the primes). Therefore from one perspective we can use the non-trivial zeros to move from the general frequency of prime distribution (among the natural numbers) to their precise location; equally from the other perspective, we can use the knowledge of the prime numbers (distributed throughout the natural number system) to precisely locate the non-trivial zeros.

There therefore exists a two-way interdependence as between the non-trivial zeros and the prime numbers (in their relationship with the natural numbers). And once again it has to be emphatically stated that it is strictly meaningless to attempt to understand such a key relationship in the absence of genuine holistic notions!

Thus instead of the existence of just one number system, properly understood in dynamic interactive terms, we now have three! So firstly we have the Type 1 system relating to the analytic understanding of number in a relative quantitative manner.

Thus though we can cover here all the same ground as in Conventional Mathematics a more refined interpretation operates (whereby implicit recognition is given to the “shadow” qualitative aspect of number).

Secondly we have the Type 2 system relating to the holistic understanding of number in a dynamic qualitative manner.

More correctly this system relates to the dynamic understanding of number entailing the interaction of both quantitative and qualitative aspects (where both still enjoy a relative degree of separation).

Whereas the Type 1 system is of a linear (1-dimensional) nature, the Type 2 system relates potentially to all other finite dimensions (which are unlimited in scope).

The Type 3 system then establishes the mutual identity of both quantitative and qualitative aspects.

The importance of this system is that it establishes the means through which both quantitative and qualitative aspects can be consistently related to each other in all phenomenal contexts.

This is something that is totally taken for granted in Conventional Mathematics (which gives no formal recognition to the qualitative aspect). However properly understood, underlying all mathematical axioms is a prior assumption that quantitative (relating to finite) and qualitative (relating to infinite) notions can be consistently related.

For example this assumption underlies mathematical proof whereby what is proven for the general (infinite) result is assumed to apply to all specific (finite) cases within its class!

In terms of the Type 2 system it represents the extreme case where dimensional interpretation is infinite.

So in this necessary reformulation of the number system Type 1 and Type 3 exist as two extreme cases.

Once again in the Type 1, quantitative is clearly separated from qualitative meaning in the independent (analytic) interpretation of number. In the Type 3, quantitative is directly united with quantitative meaning in the holistic interpretation of both the quantitative and qualitative aspects of number as fully interdependent.

Then between these two extremes in the Type 2 system quantitative and qualitative aspects are understood in dynamic relative terms (combining both elements of independence and interdependence).

The Riemann Zeta Function – when appropriately interpreted – provides the required framework through which all 3 aspects of the number system can be understood.

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