Calude shows that incompleteness implies uncertainty
Eray Ozkural exa
to Reality?]
First some weblinks for Calude's work on uncertainty, which argues
(strongly!) that Heisenberg's uncertainty principle implies
incompleteness.
http://www.cs.auckland.ac.nz/seminars/index.php?year=2004&pid=125
http://www.cs.auckland.ac.nz/CDMTCS/researchreports/235cris.pdf
[Actually I found the above report after writing the post, so 90% of
what I say below is redundant (even inaccurate and wrong!), Calude has
made everything quite clear in his paper.]
In response to some of Stephen Harris's comments, I agreed on the
relation of incompleteness to the uncertainty principle. In my
opinion, it is conceivable that for a discrete system s of complexity
H(s), some (many?) _observational_ statements about s will turn out to
be non-trivial (in the Post sense), and hence undecidable in s. That
is still very far from uncertainty (since I seem to say nothing about
Planck constant), but the idea is that this is a matter of degree, I
don't think I know the physics well enough to construct the right
sentence. It's still elusive for a lousy computer guy.
At any rate, I would be interested in reading about other works which
try to relate algorithmic incompleteness to uncertainty principle.
Ideally, we would like to derive uncertainty from incompleteness
directly (analytically!), but this does not seem to be possible
without first admitting a discrete physical model (a RUCA?) first. The
numbers would depend on the model. Otherwise, we are making
metaphysics, e.g. since algorithmic incompleteness deals with limits
of knowledge in possible (discrete!) worlds. [The converse relation
shown in Calude's paper seems to relate the formal Heisenberg
uncertainty relation to a formal uncertainty relation in AIT, which
looks OK, but I cannot tell if that is decisive, since I'm no
physicist]
I don't know if Calude's above paper can be seen to be the complete
answer, but it is surely very interesting in that it gives the
conjecture the substance it deserves.
"Stephen Harris" <cyberguard...@yahoo.com> wrote in message news:<eyUkd.20482$6q2....@newssvr14.news.prodigy.com>...
> To tie this back into Penrose and randomness/non-computability.
> Chris Calude an associate of Chaitin has published a recent paper.
> His thesis is that Godelian Incompleteness (GI)is an implication of the
> Heisenberg Uncertainty Principle (HUP). He is trying to build a bridge
> between mathematical uncertainty and the uncertainty of quantum mechanics.
Chaitin's proof of straightforward Godelian incompleteness may be
thought of as a logical abstraction of the uncertainty principle at
work. I think the bridge will be eventually built. Too bad I don't
have the knowledge and capacity to carry it out.
It's pretty easy to see that uncomputability in the universe is a
MATTER OF DEGREE and not a 0/1 thing as Penrose or Godel may have
thought it is.
That is because in a computable universe things are computable, you
can't just loop out of computability with a snap, it comes in degrees,
in bits, in discrete increments of complexity.
In fact, that is one of the reasons why I think Turing computability
and Chaitin incompleteness has more explanatory value than predicate
logic and Godelian incompleteness.
It is obvious that Chaitin's proof of his Theorem LB in AIT is a
physical one, because it explicitly talks about finite quantities. You
can't put 2 liters of water in a 1 liter cup, and that is how
Chaitin's proof proceeds, instead of talking about real numbers that
do not, and cannot nomologically, exist.
Regards,
--
Eray
PS: If anybody cares to discuss on a formalization of these ideas for
RUCAs and other models of digital physics, I would be a happy
listener. The other direction may be worked out too, we should be able
to show that incompleteness implies uncertainty for a discrete model.
PS2: Thinking of RUCA regions with very small complexity, one might
conceive the possibility of making quite exact measurements for some
properties of the region, but all of these may turn out to be
"unlikely", for instance. Or, it may turn out that uncertainty
"emerges" beyond a certain complexity.
Tim Tyler
> [forwarded from comp.ai.philosophy, from thread |anyone read the Road
> to Reality?]
>
> First some weblinks for Calude's work on uncertainty, which argues
> (strongly!) that Heisenberg's uncertainty principle implies
> incompleteness.
> http://www.cs.auckland.ac.nz/seminars/index.php?year=2004&pid=125
> http://www.cs.auckland.ac.nz/CDMTCS/researchreports/235cris.pdf
I don't rate this very highly :-|
You can get Godel's incompleteness straight from math - without the
need to assume any physics. The physics-related assumptions
presented (i.e. the U.P.) thus seem unnecessary - and irrelevant.
> At any rate, I would be interested in reading about other works which
> try to relate algorithmic incompleteness to uncertainty principle.
> Ideally, we would like to derive uncertainty from incompleteness
> directly (analytically!), but this does not seem to be possible
> without first admitting a discrete physical model (a RUCA?) first.
...and I'm certain that this is impossible. There's nothing about
embedded observers in a universe that means that they can have no
way to access the exact details of state of some subset of the system
at some previous point in time.
I.e. one can easily imagine universes without an uncertainty principle.
--
__________
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