Tuesday, February 7, 2012

Clarifying the Riemann Hypothesis

As stated in the last blog all mathematical symbols possess both quantitative and qualitative aspects.

Though in isolation it is indeed possible to seek to interpret numbers with respect merely to their quantitative properties, when using interdependent frames (implying a dynamic interactive approach), numbers are necessarily quantitative and qualitative (and qualitative and quantitative) with respect to each other.

So applying this again to prime numbers, one can indeed attempt to interpret the individual primes and their general distribution (in isolation) using a merely quantitative frame of reference.

However when we seek to understand the interdependence of specific primes and their general distribution, we must incorporate both quantitative and qualitative aspects of appreciation. And this poses insuperable difficulties for the conventional (Type 1) mathematical approach, which in formal terms is solely based on mere quantitative interpretation.

And the interdependence of primes, with respect to their individual identity and overall distribution, is clearly manifested in the relation of the trivial non-zeros on the one hand to the general prime number distribution (and the corresponding relationship as between the individual primes and the general distribution of the non-trivial zeros).

The non-trivial zeros themselves represent the unlimited possible solutions for s (where s represents a complex dimensional number of the form a + it). And as discussed in the last blog, the relationship of (base) numbers to their dimensional powers likewise is as quantitative to qualitative (and qualitative to quantitative).

Though this understanding is absolutely central to true appreciation of the nature of the Riemann Hypothesis, it completely eludes conventional (Type 1) analysis, which once again totally lacks, in formal terms, any distinctive qualitative aspect of interpretation.

As we have already seen the non-trivial zeros can be used in an ingenious manner (after a couple of other small adjustments) to gradually correct any remaining deviations arising from Riemann's general function for the prediction of prime number frequency. So in principle through using this approach we should be gradually able to zone in on the precise location of each individual prime, while ultimately correctly prediction the overall frequency of primes (up to a given number).

Now this is based on acceptance of the Riemann Hypothesis.

It is often stated in this manner that given the truth of the Riemann Hypothesis (i.e. that the real part of all these zeros of s lies on the line = 1/2, then in principle we can exactly predict the prime numbers (from their general frequency).

However much greater subtlety is required in this statement, which indeed is required to reveal the true nature of the Riemann Hypothesis.

As befits the proper distinction as between quantitative and qualitative, we need likewise to carefully distinguish as between (actual) finite and (potential) infinite meaning. By its very nature Conventional Mathematics inevitably reduces in any context (potential) infinite to (actual) finite notions of meaning!

So it is true as we progressively add in the corrections based on the non-trivial zeros that we move ever closer to the integer values of the primes.

However in actual terms this process can never be completed for no matter how many non-trivial zeros we seek to consider, an unlimited set of non-trivial zeros will remain. So there is an inherent uncertainty attached to this process whereby the successive approximation through determination of non-trivial zeros of the location of the primes is always based on an unlimited set of non-trivial zeros, which must remain indeterminate.

So in actual finite terms (which is the proper domain of quantitative interpretation)we can never exactly pinpoint the location of the primes, with an uncertainty thereby necessarily attached to their precise values.

However when we switch to a potential infinite context (which is the proper domain of qualitative meaning) we can indeed say that in potential terms, if the infinite set of non-zeros is included, that then we would indeed exactly obtain the discrete integer values of the primes. And in doing this our reconciliation of the primes with the trivial non-zeros would be complete.

However a purely potential state equally implies that no phenomenal identity can remain to the primes (in actual terms).

In other words the full reconciliation of the primes with the non-trivial zeros (both of which are mutually encoded in each other) points to an ineffable state with no phenomenal existence.

And as this process is based on the assumption that the Riemann Hypothesis is true, it thereby is pointing to this ineffable state.

So the Riemann Hypothesis is directly concerned with the ultimate reconciliation of quantitative and qualitative meaning (where finite and infinite can at last become identical).

So in this respect Hilbert was indeed correct. The implications of the zeros of the Zeta function in the context of Riemann's Hypothesis could not be more important.

The famous Buddhist heart sutra states this identity of finite and infinite in the following manner:

"Form is not other than Void;
Void is not other than Form"

The Riemann Hypothesis in fact is simply a restatement of this sutra related to the ultimate nature of mathematical meaning:

"The Quantitative is not other than the Qualitative;
The Qualitative is not other than the Quantitative"

The deeper implications of the true nature of the primes are awe inspiring.

There are two capacities that reveal themselves in nature, one for independence and the other for interdependence i.e quantitative and qualitative aspects (which ultimately relates to the nature of the prime numbers). In an original state - as mere potential for physical phenomenal existence - these two capacities are identical.

Then when operating through the veils of phenomenal reality, they become separated with full understanding of their identical nature again ultimately taking place in an ineffable spiritual manner.

So the task of understanding the mathematical (objective) nature of the primes cannot be ultimately divorced from the psychological nature of their (subjective) interpretation.

The mystery of the primes can therefore be validly seen as embracing the entire course of created evolution.

Once again it is all about the reconciliation of quantitative and qualitative notions of meaning.

And Mathematics would make enormous strides in simply grasping this key fact!

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