Thursday, July 18, 2013

What It's All About

I cannot stress strongly enough that a completely new paradigm is now required in Mathematics.

However an almost total blindness still exists in the mathematical community as to this fundamental issue.

Indeed, whenever I have suggested this - even to those with holistic sympathies - inevitably they have tried to filter my comments through the existing conventional approach. And because my suggestions clearly do not fit in with this existing paradigm, they have defended the status quo by convincing themselves that I must therefore know very little about Mathematics.

All I can say is that after more than 50 years of travelling on this lone journey, I am fully confident regarding my basic assertion. I have had the good fortune to glimpse many of the wonderful lands whose very existence my mathematical colleagues still resolutely deny.

The true nature of Mathematics is incomparably greater than what we presently imagine. Indeed what we have come to unquestioningly accept represents but a very limited special case (that admittedly is very important in its own context) but which pales in comparison to the glittering treasure that is truly Mathematics.

More importantly despite its great successes, Conventional Mathematics has led to mind-set that - if left unchallenged - is likely to have destructive consequences for our civilisation.

Such Mathematics creates the illusion of quantitative meaning (without reference to the qualitative). And as this Mathematics provides the essential basis for all the sciences, this means that science itself - which in many ways has become the new world religion - fundamentally entails, in every context, the reduction of qualitative to quantitative meaning.

So the overall scientific worldview (fundamentally rooted in Mathematics) is in fact hugely unbalanced and this is what so threatens global integration at every level whether social, economic, environmental etc.

We can truly say that the roots of this fragmented scientific worldview (so out of keeping with the genuine needs of the planet) are in the fact that our understanding of Mathematics itself is completely lacking a holistic integral perspective.

We have convinced ourselves, so wrongly, that Mathematics is exclusively concerned with quantitative meaning! And with the growing specialisation of this type of Mathematics through the millennia this has led to an incredibly defensive approach with respect to any criticism of its fundamental rationale.

However when one looks closely at the nature of the number system, it becomes quickly apparent - to anyone with the clarity to see - that quantitative interpretation is strictly meaningless in the absence of an equally important qualitative aspect. It also becomes apparent that the absolute notion of objective truth is a great fallacy (as all objective notions necessarily reflect the subjective mental lens of interpretation through which they are viewed).

Properly understood the Riemann Hypothesis provides the perfect vehicle for this necessary radical conversion with respect to mathematical understanding.

It has indeed struck me deeply in recent years how lack of a holistic vision is greatly preventing meaningful communication of fundamental results in both Science and Mathematics.

For example it seems to me that String Theory remains totally impoverished with respect to providing a coherent physical view of the Universe.

The reason for this is that physicists are still trying to filter insights through the accepted 1-dimensional paradigm (which is quite inadequate for the purpose). So it will need - literally - a radical new holistic vision of science before the insights of String Theory can be conveyed in a  manner that then can resonate intuitively with this newly acquired worldview.

It is somewhat similar regarding interpretation of the (non-trivial) zeta  zeros with respect to the Riemann Hypothesis.

Again because Mathematics is lacking any holistic framework, understanding of these zeros is necessarily being filtered likewise through a 1-dimensional paradigm which is hugely inadequate for their meaningful interpretation.

Because my own approach has always been strongly holistic, some years ago I set myself the task of  forming an understanding of these zeros in an intuitively meaningful fashion, that could express their crucial relevance for Mathematics. So I will summarise my findings now!

The relationship as between primes and natural numbers is indeed fundamental. However we must understand this relationship as possessing both quantitative and qualitative aspects.

Quantitative in this context is associated with notions of number independence; qualitative is associated with number interdependence.

So numbers are independent in a cardinal and interdependent (through their relationship with each other) in an ordinal sense. Both of these aspects quantitative and qualitative necessarily interact in a dynamic relative manner. Likewise external appreciation of number as objective and internal mental interpretation of number as subjective likewise interact in a dynamic relative manner.

Therefore the importance of the relationship between primes and natural numbers is that they provide the means through which both the quantitative and qualitative aspects of the number system are mediated (in a dynamic two-way fashion).

So from the (Type 1) cardinal perspective, the primes appear as the unique quantitative building blocks of the natural numbers; however from the complementary (Type 2) ordinal perspective, the natural numbers appear as the unique qualitative building blocks of each prime number. (For example, again from this perspective, 3 as prime, is necessarily comprised of its 1st, 2nd and 3rd members).

Thus in  a relatively independent manner, we therefore can identify - with respect to the primes and natural numbers - this two way relationship, by which quantitative and qualitative aspects are mediated (in an analytic fashion).

However the key issue of the corresponding (relative) interdependence of both quantitative and qualitative then arises. In other words how do we ensure the ultimate identity of both these aspects as fully consistent with each other?

This is where the (non-trivial) zeta zeros (Zeta 1 and Zeta 2) come into play! Just as we have two sets of analytic relationships connecting both quantitative and qualitative aspects of primes and natural numbers (in a relatively independent manner), we have likewise two sets of holistic relationships connecting quantitative and qualitative aspects of the primes and natural numbers (in a corresponding relatively interdependent manner).

Now the Zeta 2 provide a circular explanation whereas the  Zeta 1 provide a linear explanation of this relative number interdependence respectively.

Once again we will remind ourselves of what this means with reference to the first of the Zeta 2 zeros which relates to the 2 roots of 1! (Strictly the non-trivial solution relates to the root ≠ 1)!

So + 1 and –  1 have a relative quantitative independence as separate; however when combined qualitatively as interdependent, the (separate) quantitative aspects are eroded. 

Thus for the prime number 2 we are able to show a perfect harmony as between two (circular) numbers as - relatively - quantitatively independent with their overall qualitative relationship as interdependent. 

Therefore, the very point about the Zeta 2 zeros is that they provide this magical solution in a dynamic interactive manner as to how to preserve the relative independence of each natural ordinal number member of a prime number group (indirectly expressed in a quantitative manner) with the overall interdependence of the group in qualitative  terms.

In this way each prime number is given a unique holistic identity (through its ordinal natural number members).

So the very essence of the Zeta 2 zeros is to reconcile in a circular numerical manner the cardinal aspect of each prime with its natural number members (in ordinal terms).

By contrast the very essence of the Zeta 1 zeros is to reconcile in a linear manner the cardinal aspect of the (composite) natural numbers with the ordinal aspect of their constituent prime number factors.

We must remember the essence of multiplication is that it inherently is of a qualitative nature.

So we may say that 6 is uniquely expressed as the product of two cardinal primes i.e. 2 and 3.

However obtaining this product changes the dimensional (qualitative) identity of the primes involved. So the unique combination of prime factors (constituting each composite natural number) reflects a qualitative (ordinal) combination.

So the Zeta 1 zeros express for the number system as a whole this unique ordinal identity (resulting from prime number multiplication).

Thus once again a perfect harmony is preserved as between each individual zero and the overall collective interdependent identity of  the zeros as an entire group. So quantitative and qualitative aspects are reconciled here for the number system as a whole.

This in turn provides the deepest fundamental insight into the Riemann Hypothesis i.e. that all the (non-trivial) zeros lie on an imaginary line (through 1/2).

Again the significance of the imaginary line is that it provides - in holistic terms - the indirect means of expressing circular meaning (i.e. relating to holistic interdependence) in an analytic fashion.

1/2 basically relates to the harmony as between external (objective) and internal (subjective) mental interpretation.

Thus the requirement that all the zeros lie on a straight line simply points to the ultimate requirement for the consistent relating of both cardinal and ordinal notions of number in a linear (1-dimensional) context.

Thus the Zeta 1 zeros serve as the holistic requirement for the consistent interplay of the cardinal and ordinal notions of number in relation to the number system as a whole (mediated through the qualitative relationship of the prime number factors that thereby uniquely result in cardinal natural numbers of a composite nature).

The Zeta 2 zeros serve as the corresponding holistic requirement for the consistent interplay of cardinal and ordinal notions for each individual number (mediated through the qualitative nature of the ordinal natural numbers within each prime).

Clearly both simultaneously arise in an ultimately ineffable manner. Therefore once the number system becomes inherent in phenomena (as the most intrinsic encoding of their nature) this perfect simultaneity has been already broken with quantitative and qualitative aspects separated in space and time.

In fact moving towards this ultimate nature of the number system - which equally represents the ultimate nature of universal creation - requires:

(i) the ability to see the prime numbers and natural numbers as perfect mirrors of each other (through the dynamically interdependence of complementary cardinal and ordinal frames of reference) and

(ii) the ability the see the two sets of zeta zeros (Zeta 1 and Zeta 2) as perfect mirrors of each other again through the dynamic interdependence of complementary holistic frames of reference and

(iii) finally to see both the two sets of zeros and the prime and natural numbers as ultimately identical, which realisation approximates (insofar as is possible while still remaining in the phenomenal realm) to the pure spiritual experience of the original ineffable nature of the nature of the number system (and of the entire universe which is intrinsically encoded in number).

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