A couple of quick thoughts "out loud".
My previous thought on the possible connection between the OR/AND dual
(along with addition/multiplication) and the Riemann Hypothesis might
be extended by looking at the Riemann zeta function. Notice that the
infinite series form of the zeta function's uses addition and the
infinite product uses multiplication. Could it be that the zeta
function somehow gives information on observer moments? (Or instead of
the zeta function, maybe it would end up being something more general
function like the Dirichlet-L series which appears in the generalized
Riemann Hypothesis.)
Suppose an observer moment is specified by s, or maybe the imaginary
part of s. One possibility is something like this, heuristically: The
zeta series could describe the 1st person pov, and the zeta product the
3rd person pov.
For the 1st person pov, each term in the series, 1/n^s, could somehow
correspond to the probability of having a particular "next observer
moment". Then the whole series describes the probability of (observer
moment #1) OR (observer moment #2) OR... (ad infinitum?, with exclusive
ORs).
For the 3rd person pov, each term in the product, 1/(1-p(^-s)), could
somehow correspond to the probabilities of NOT having a particular
"next observer moment". Then the whole product describes the
probability of (NOT observer moment #1) AND (NOT observer moment #2)
AND... (ad infinitum?, with somehow finally having a selection of ONE
observer moment out of the infinite(?) possible next observer moments).
The equating of the series and product forms is analogous to a
deMorgan's law (however with an exclusive OR and the ANDing of one
affirmative selected observer moment).
A possible variation on this starts with the observation that the zeros
of the Riemann zeta function lie in 0 < Re(s) < 1 and are symmetrical
about the critical line, Re(s) = 1/2. Could it be that the real part
of s corresponds to the probability of a next observer moment given a
current observer moment specified by the imaginary part of s? The
zeros of the Riemann zeta function could somehow describe the 1st
person indeterminacy, with the Riemann Hypothesis corresponding to a
50/50 chance.
Along those line, I notice that Chaitin (referencing du Sautoy) says
that if it could be proved that the Riemann Hypothesis is undecidable
then it is true, since if it were false then it would be decidable by
finding a zero off of the critical line.
(
http://maa.org/features/chaitin.html). But could it be that the
Riemann Hypothesis follows quantum indeterminacy in something like the
following way? Just role-playing: "The Riemann zeta function does
indeed have zeros which are off of the critical line (or even, it has
zeros having real parts taking on every real value between 0 and 1.)
This is the non-computable truth. However, whenever a zero of the
Riemann zeta function is actually computed (observed), it falls on the
critical line."
Just having fun,
Tom
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Received on Mon Apr 17 2006 - 13:54:54 PDT