Tuesday, February 21, 2012

Riemann Zeta Function (2)

Though I initially started to use the analogy of crossroads as a means of highlighting the "missing" qualitative aspect of Mathematics, I have since come to realise that in fact it serves a more direct purpose in its ability to properly identify key features with respect to the Riemann Hypothesis.

The very depiction of the Riemann Hypothesis is in the form of a crossroads.

So in horizontal terms we have the real axis (corresponding in turn to the real dimensional values (s) of the Riemann Zeta Function. Then we have a vertical line drawn through the real axis where s = .5 which represents the imaginary part of the complex number values for s that generate the non-trivial zeros.

And of course if the Riemann Hypothesis is true, the imaginary part of all non-trivial zeros will lie on this vertical line through .5!

Now, as we have seen at a crossroads we are faced with opposite polarities that can shift location (depending on convention). Thus from one valid perspective, travelling "up" a road when I unambiguously label one turn as left, the other turn at the crossroads (in relation to this turn) is necessarily right; then travelling "down" the road again from above the crossroads, when I unambiguously label one turn as left the other turn is then - relatively right.

However when combining both terms of reference "up" and "down" (and "down" and "up") we create a purely circular logical interpretation, where what is left is also right and what is right is also left. And such circular logic is the very essence of the interpretation of interdependence (which cannot be approached with isolated reference frames).

And of course this poses a central problem for conventional mathematical interpretation (based on independent frames in accordance using linear rational understanding).

In particular it poses intractable problems with respect to interpretation of prime numbers (whose central feature relates to such interdependence).

Now putting it more mathematically i.e. in holistic (Type 2 terms), left and right represent a specific example of positive and negative polarities (which in dynamic interactive terms change location depending on context).

And in the present context positive and negative apply to the dynamic interaction of both quantitative and qualitative type meaning.

Now a full understanding requires that we combine both 1-dimensional (using independent frames of reference) with 2-dimensional interpretation (where reference frames are interdependent). This requires in turn that we employ linear logic (in the case of the fixed frames) and circular logic (in the terms of the simultaneous combination of these frames as interdependent).

So we will cut right to the chase now (before elaborating later in considerable more detail).

Riemann came up with a fascinating transformation formula (Functional Equation) that associates the values for ζ(s) to the RHS of .5 on the real axis, with corresponding values on the LHS of .5 for ζ(1 - s) with the one exception where s = 1.

Now remember .5 in this context represent the intersection of our crossroads with - literally - right hand and left hand designations.

The crucial fact to grasp is that right hand and left hand correspond here - relatively - with both quantitative and qualitative interpretation respectively.

In other words if we fix values on the real axis to the right of s with quantitative interpretation of values of the Zeta Function, then values to the left of s thereby must - in this context - be given a qualitative interpretation.

Likewise if we alternatively fix values to the left of .5 with quantitative interpretation of the value of the Zeta Function, then values to the right must - in this alternative context - be given a qualitative interpretation.

So the Zeta Function therefore can be given both quantitative and qualitative interpretations on both the LHS and RHS of .5 respectively.

Now the midpoint (.5) of this horizontal crossroads represents the pure intersection of both quantitative and qualitative type interpretations (as simultaneously identical).

And this again is what the Riemann Hypothesis is all about i.e. in establishing the key condition for the identity of both quantitative and qualitative aspects (which are inherent in the primes).

So the additional requirement that the value of the Zeta Function be then = 0, requires that in the case of all the imaginary parts of s for which this is true, that the real part = .5. In other words the imaginary part values of s, for which both ζ(s) and ζ(1 - s) = 0, must necessarily lie on the imaginary axis through .5.

Now, this in no way constitutes a proof, as the Hypothesis is prior to any proof and already inherent in any axioms that might be used for its establishment. Rather it is a demonstration of the nature of what already necessarily is the case!

As I say, I will tease out in more detail the exact nature of this relationship as between quantitative and qualitative in future blog entries!

I will just leave here with one revealing observation.

The only point on the real axis where ζ(s) is not defined occurs for s = 1.

1 in this context directly refers to a dimensional value and as I have stated so many times before, Conventional (Type 1) Mathematics is defined - literally - by its 1-dimensional approach, where the qualitative aspect is thereby reduced to the quantitative.

Foe all other dimensional values, both quantitative and qualitative preserve distinctive meanings (that are not directly confused with each other).

So once we accept that the Riemann Zeta Function properly relates to the complementary matching of both quantitative and qualitative aspects of interpretation, then this cannot apply in the case of 1 (as by definition it can be given no matching qualitative interpretation!)

Putting it more starkly, it highlights how unsuited conventional (Type 1) interpretation in fact is as a means of appreciating the nature of the Riemann Hypothesis.

So once again I will return to the nature of this complementary quantitative/qualitative matching in future blog entries with a view to much greater clarification of what is involved.

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