On 19 Nov 2012, at 19:46, meekerdb wrote:
On 11/19/2012 8:33 AM, Bruno Marchal wrote:
and the math shows that this will include some continuous/analog
observable.
How does it show that?
Intuitively: by dovetailing on each programs coupled with real
numbers. Each computations are done again with all possible streams of
real numbers, oracles, etc. Yes the UD is that dumb.
But this is probably needed for the measure question. We can even bet
that such a coupling has to be exploited by the "winner program" in
some special way, if it exists, because even the white rabbit
realities are multiplied into a continuum by the existence of that
coupling (which is unavoidable: you can't diagonalize against the UD
to build a UD avoiding those couplings).
Formally: the existence of such a semantics based on a continuum is
reflected in the possible topological semantics of the material
hypostases S4Grz1, Z1* and X1*, which gives rise to the arithmetical
quantum logics.
The formal reasons are different for S4Grz1, and the Z and X logics.
The topology is intrinsic to the S4Grz type of semantics, but it might
still be discrete at that level. For the X1* and Z1*, it comes in part
from the lack of necessitation, and the necessity to have infinite
sequences of neighborhood structures à la Scott-Montague.
If he quantum logic would have given only by S4Grz1, that would have
been an argument for loop gravity, and the continuum would have been
restricted to the frequency-statistical operator (like in Preskill and
Hartle or Graham).
If it would have appeared only in the Z and X logic, that would have
suggested that String theory might be the correct comp physics.
Amazingly the "arithmetical/comp" quantum logics seems to appear in
the three possible candidates (S4Grz1, Z1*, X1*).
Needless to say this is suggestive and it remains quite a lot of open
problems in logic to proceed. What I hope is that the arithmetical
quantum logics will give the quantum logic searched, but not found, by
von Neumann, which have the property that the probabilities can be
derived from the constrains given by the laws for the case of P(x) =
1. If this does not work, it means that we might use a stronger
definition of knowledge than the one given by Theaetetus.
Bruno
http://iridia.ulb.ac.be/~marchal/
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