Sunday, December 18, 2011

Riemann's Zero

As is well known the Riemann Hypothesis amounts to the statement all the non-trivial zeros of the Riemann Zeta Function lie on the real line whose value = 1/2.

Now in conventional terms, mathematicians have been trying to understand this problem from a merely quantitative perspective.

My persistent point however is that Mathematics equally contains an important (largely unrecognised) qualitative aspect.
Furthermore as the Riemann Hypothesis properly relates to the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical understanding, its significance cannot be appreciated in a merely (reduced) quantitative manner.

Indeed ultimately the Riemann Hypothesis points to the relationship as between dual and nondual meaning.

Imagine a circle that is drawn with unit radius. Now in conventional terms the length of the line diameter of this circle = 2 units. So the midpoint of this line at the centre of the circle from linear perspective divides the line into two equal parts. This point lies exactly halfway (1/2) on the total line.

However if we look at this midpoint from a circular perspective we would represent it as 0. In other words if the radius to the right is + 1, then the radius in the opposite direction is - 1.

Now this behaviour has a direct qualitative significance. Linear understanding is inherently dualistic (which represents the holistic meaning of 2). So the midpoint of the circle at 1 unit represents thereby half of the total line.

However to understand the measurement of the line from the centre of the circle in complementary positive and negative directions we require - literally - nondual insight (so that dual notions are rendered paradoxical). So the midpoint from this perspective is 0.

Properly understanding the Riemann Hypothesis requires both dual and nondual understanding. Indeed it points to the ultimate state where both dual and nondual notions (or alternatively quantitative and qualitative notions) of number are identical. And this state is utterly mysterious and thereby cannot be grasped in a phenomenal manner.

Thus the mysterious order governing the nature of the prime numbers is already inherent in all number behaviour representing a non-phenomenal reality (where qualitative cannot be distinguished from quantitative notions).

The attempt to prove the Riemann Hypothesis in a merely (reduced) quantitative manner is thereby utterly futile.

The significance of the Riemann Hypothesis is ultimately of a truly breath taking order in that - properly understood - phenomenal reality as we know it would not be even possible if it did not hold.

In other words underlying all of visible reality is reality is a secret code ensuring a perfect harmony of the prime numbers with respect to their specific (quantitative) and holistic (qualitative) interaction.

It is this harmony that enables all subsequent phenomenal events (representing varying interactions of this original prime number number code) to unfold.

This once again suggests that ultimately the underlying nature of reality - insofar as it can be phenomenally investigated - is purely mathematical!

In other words at the deepest level, phenomena represent the interaction of a fundamental mathematical code that governs the subsequent behaviour of all natural events. However though we can only come to knowledge of this code through phenomena, its nature - and indeed in reverse fashion what we know as nature - is ultimately ineffable.

So at some stage, Physics will have to abandon the quest for the ultimate particles before arriving at a purely mathematical appreciation of reality that underlies all manifestations of such particles.

Indeed properly understood it has already arrived at this point. As I would see it, string theory represents an elaborate fiction that the ultimate physical particles are strings. Strictly speaking however these have no manifest physical reality but really operate as the vehicle of ever purer mathematical notions.

However my key point all along is that the very scope of Mathematics needs to be radically extended to explicitly include both its qualitative as well as quantitative aspect. In this light all particles ultimately emerge as the dynamic interaction of a prime number mathematical code that is designed to preserve the perfect harmony of both its quantitative and qualitative aspects.

Though this point is largely lost because of the reductionist nature of current scientific understanding, the truly great wonder of reality is how both the finite and infinite - though utterly distinct - yet successfully coexist with each other at all levels of understanding.
Thus before we can even for example engage in conventional mathematical activity, we must already presume this meaningful correspondence of finite and infinite. Indeed as I have frequently pointed out it underlines the very notion of mathematical proof!

Looked at another way, the Riemann Hypothesis is the necessary condition for such correspondence to exist. So it represents in fact a massive act of faith in the subsequent meaning of the whole mathematical enterprise.

As the Riemann Hypothesis thereby already underlies conventional mathematical proof (as the starting condition for its meaningful interpretation), The Riemann Hypothesis cannot therefore be proven (in conventional terms).

However far from this representing a defeat, true realisation of this fact has the capacity to open up appreciation of a greatly enlarged scope for Mathematics where every number, symbol, relationship has a unique qualitative - as well as quantitative - significance. And with this will come an enormous enrichment of the true nature of both Mathematics and Science.