## Thursday, June 23, 2016

### Riemann Hypothesis in Context

I would like to convey here the true significance of the Riemann Hypothesis.

However it is vital to appreciate from the outset that this significance cannot be properly understood within the accepted confines of Conventional Mathematics, which solely recognises, in formal terms, the quantitative (analytic) interpretation of its symbols.

Indeed from one important perspective, the Riemann Hypothesis - when adequately interpreted - points directly to a critical limitation in the overall accepted framework of Mathematics.

The number system is fundamental to the reality we call Mathematics. And what in turn is central to the number system is the relationship between the primes and the natural numbers.

Now a prime has no factors (other than itself and 1). So 7 for example is an early example of a prime. All other natural numbers (i.e. composites) can then be uniquely expressed as the product of primes. So for example 30 ( = 2 * 3 * 5) is therefore composed of 3 prime factors and does not allow for any alternative factor combination.

Therefore the primes have been long accepted in conventional wisdom as supremely important, which - rather like atoms in physics - serve as the basic “building blocks” of the natural number system.

However it has also been keenly recognised that the individual behaviour of the primes is highly unpredictable with no regular pattern occurring.

Then at the beginning of the 19th century it was discovered that their overall frequency did indeed correspond to a simple log pattern.

Thus an overall dichotomy became apparent as between the - apparent - random behaviour of each individual prime and an increasingly regular pattern that characterised their overall frequency.

It was then Bernhard Riemann who on 1859, made immense strides in reconciling this discrepancy as between whole and part (i.e. the  regular collective order of the primes with their random individual behaviour).

However it is my strong contention that the overall philosophical significance of his seminal paper on the primes, has never been properly grasped.

For to put it simply, Riemann's findings point to the fact that underlying the conventionally accepted natural number system of a discrete “particle” (i.e. part) is a highly intricate continuous “wave” like system of equal importance, that in truth is dynamically inseparable from the manifest nature of natural numbers.

This discovery took place long before the quantum revolution that shook the physical world in the 1920’s. Here paradoxical features of behaviour were revealed as the inherent nature of sub-atomic matter. So for example at this level it was now understood that physical behaviour manifested itself in a complementary dual fashion with both particle and wave like features.

One might ask why parallels were not then drawn as between this behaviour at a physical level and the inherent nature of number revealed by Riemann more than half a century earlier!

However quite simply, the accepted  framework of Mathematics is so rigid that no proper accommodation of these findings could be made within its accepted confines.

Then starting from the 1970’s evidence began to emerge suggesting striking parallels as between the “Riemann zeros” (to which the wave like features number relate) and findings regarding energy levels in atomic physics.

However though mathematicians can no longer ignore these findings, they still lack the means of properly explaining their nature.

In fact the real position is so revolutionary that it undermines the very paradigm that Mathematics has been built on now stretching back over several millennia!

So what has wrongly emerged in our culture is the view that numbers - and indeed all mathematical relationships - can be interpreted abstractly in an objective manner with respect to their mere quantitative (analytic) characteristics.

However in truth numbers - and by extension all mathematical relationships – are inherently experiential in nature, comprising the dynamic interaction of both quantitative (analytic) and qualitative (holistic) characteristics.

And it is only in this relative interactive context (comprising both analytic and holistic aspects) that the true relationships as between the particle and wave features of number can be properly interpreted.

Putting it simply, one overriding confusion lies at the heart of all accepted mathematical interpretation. This relates to the attempted reduction - in every context - of qualitative notions of interdependence in a merely quantitative manner (i.e. as independent).

So for example we can indeed attempt to look at the individual primes in a quantitative analytic manner (as independent).

However the overall collective relationship of the primes (to the natural numbers) properly relates to their corresponding qualitative nature (as interdependent). However we cannot then also attempt - without resorting to gross reductionism - to express this qualitative nature solely in an analytic manner! Rather we must now switch - in complementary relative terms - to a corresponding holistic interpretation of symbols.

In psychological terms this clearly entails incorporation of both conscious (rational) and unconscious (intuitive) modes of understanding, with the unconscious aspect indirectly conveyed in a subtle circular (paradoxical) rational manner.

However no recognition whatsoever exists - at a formal level - in present Mathematics of the qualitative holistic interpretation of mathematical symbols (which in truth is of equal importance to the analytic).
This is why I stress once again without a hint of hyperbole that the greatest revolution yet in our mathematical history - indeed in our intellectual history - is now required where both quantitative (analytic) and qualitative (holistic) aspects of interpretation are accepted as equal interacting partners, both in relation to the fundamental nature of the number system and by extension to all mathematical and scientific relationships.

So in this context the present accepted analytic approach represents but a limited special case of an altogether much more comprehensive understanding.

The great German mathematician Hilbert in responding to what he considered the greatest problem in Mathematics once replied,

“The problem of the zeta zeros, not only in Mathematics but absolutely (the most important)”.

Now once again the “zeta zeros” relate to the variety of solutions to a certain key equation - the Riemann zeta function - from which the wave pattern of number is constructed.

Over the past 10 years or so, I have come to agree with Hilbert, though for reasons that he would have been loath to consider!

The famed Riemann Hypothesis is also based on a certain assumption regarding these zeros i.e. that they all lie on an imaginary line drawn through .5 on the real axis.

In fact, when properly interpreted this assumption serves as the central condition for the ultimate identity of both the quantitative (analytic) and qualitative (holistic) aspects of number.

In other words the Riemann Hypothesis serves as the key requirement for the mutual consistency of mathematical symbols in both quantitative and qualitative terms (which is fundamental for all subsequent mathematical interpretation).
However once again this clearly cannot be properly appreciated through the present mathematical paradigm (which recognises solely the quantitative aspect).