Wednesday, October 13, 2010

The Invisible Gap

In the most fundamental sense, the Riemann Hypothesis relates to the invisible gap, as it were, that divides quantitative (linear) and qualitative (circular) type interpretation of reality. In psychological terms this relates to the inevitable interaction that necessarily takes place with respect to all understanding (including of course mathematical) between rational (analytic) and intuitive (holistic) type processes of understanding.

Unfortunately the problem for Conventional Mathematics is that it in formal terms it recognises solely the role of rational interpretation. Therefore it can only deal with this interaction in a reduced manner i.e. by attempting to explain - what properly relates to - the rational and intuitive, in merely rational terms.

Alternatively, it can only attempt to deal with the relationship of both quantitative and qualitative type mathematical understanding in a reduced quantitative manner.

And this is in a nutshell is the very reason why a satisfactory "proof" of the Riemann Hypothesis has proven so elusive.

Quite simply - when the nature of the problem is properly appreciated - the Riemann Hypothesis can have no solution in conventional mathematical terms.

This same problem can be expressed in other ways that directly impinge on the understanding of the true nature of the Hypothesis.

For example the Riemann Hypothesis lies in that invisible gap where the discrete and continuous interpretation of number is united. And this is central to appreciation of the true nature of prime numbers in the attempt to successfully unite their discete individual identities with the continuous nature of their overall frequency among the natural numbers.

There is a key problem here which again is not properly recognised. The study of individual primes and their overall general frequency have both quantitative and qualitative aspects that are of a (conscious) analytic and (unconscious) holistic nature with respect to each other. Just we can choose in isolation to investigate an atomic particle with respect to either its particle or wave aspect in quantitative terms, likewise we can attempt to study in isolation both individual primes and their general distribution with respect to their quantitative characteristics.

However this approach will inevitably break down in the simultaneous integration of both aspects (which are now complementary). So here we must incorporate both quantitative and qualitative type appreciation. Conventional Mathematics by its very nature is not geared for this task. Not alone can it not resolve the Hypothesis, it is not even capable of providing a coherent explanation of its true nature.

Another way of expressing the same problem is that all classical systems have counterpart systems that are quantum mechanical in nature. Once again we can indeed attempt to study both systems in a separate manner with respect to their - mere - quantitative aspects. However if we wish to properly relate both types once again we have to broaden appreciation to include both quantitative and qualitative modes of interpretation.

Finally, as is now evident from particle physics, the Riemann zeros can be given a coherent explanation in terms of the energy states of certain chaotic quantum mechanical systems. This again clearly points in my mind to an inevitable interaction as between two distinct modes of behaviour with properties that are analytic and holistic with respect to each other.

However through all this the elephant in the room is ignored.

There is an entirely distinctive holistic interpretation that can be given to every mathematical symbol (in what I term Holistic Mathematics) that constitutes the missing qualitative aspect of mathematical understanding.

It is in the very relationship of the two aspects of Mathematics (quantitative and qualitative) that the simple resolution of the Riemann Hypothesis is found not as a proof but rather as a fundamental axiom that underlies all the lesser axioms on which conventional mathematical interpretation is based.

When asked once what was the most important problem in Mathematics, the great mathematician Hilbert - as claimed in Constance Reid's book "Hilbert" - replied.

"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important"

And in fact Hilbert was right!

There is a famous sutra in Buddhism which ultimately is equivalent to the Riemann Hypothesis

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

One could equally say with respect to Mathematics.

"The quantitative aspect is not other than the qualitative; the qualitative is not other than the quantitative."

So it is in this mysterious intersection of quantitative and qualitative aspects that the Riemann Hypothesis resides.

So the end the resolution of the Hypothesis is attained not through reason but in that ineffable spiritual experience where quantitative and qualitative distinctions are no longer necessary (nor even can be made).

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