I am trying to figure something out. I am looking for relations between
mathematical limits on computations and physical limits on computations. My
hypothesis implies that there should be an isomorphism between the class of
mathematical computations and the class of physical systems that could
implement those computations. There has to be a reason why we only observe
computations that either terminate, or do not terminate in less than the
age of the universe, 14e9 years. We have no hope of observing a computation
that requires an infinite number of discrete steps. (This seems to make the
falsification of Bruno's result difficult.)
From astronomical observations, our universe appears to be finite and is
calculated to have a finite quantity of matter in it. If this is all the
result of computations, an infinite quantity of computations (as each
observer in the universe is defined as the intersection of an infinite
quantity of computations -is that Bruno's exact definition?-) then how is
it that the universe we observe is finite?
A related question: If the matter in the universe where the "hardware"
for the computation of the universe, would the universe that any observer
in that universe be required to be finite? Observers seem to involve,
individually, an infinite number of computations so my question above
follows again: Why do we observe a finite universe?
I can see two reasons:
1) Mathematical: Infinite sets are such that there is a bijection between
the infinity and its proper subsets, thus if an observer is defined by some
infinity it is also defined by any proper subset of that infinity. It seems
to me that finite sets that are subsets of an infinite set, will not
observe the infinite proper subsets of the infinity as they cannot be
distinguished from each other or the whole! There is no difference that
makes a difference between them. Computations that do not terminate are
those that run forever or get stuck in a loop.
2) Physical: Assuming the universe is finite, there will only be a finite
quantity of computation associated with any subset of that universe. We can
identify the resources required to "run" a computation that terminates in
some finite time. A finite universe with a certain set of properties could
be simulated in finite time, assuming maximal efficiency, etc. There will
be computations that cannot be run is a finite universe because they either
do not terminate 'in time' or they do not terminate at all.
I am assuming that any realistic use of clever configurations of matter
allowed by the laws of physics, which seem to disallow for hypercomputing
ala Tipler <http://en.wikipedia.org/wiki/Frank_J._Tipler> and others. The
expansion of the distance between cosmic objects would not improve
prospects of hypercomputing
<http://en.wikipedia.org/wiki/Hypercomputation>unless the total
quantity of matter and "fields" in the universe changes;
the computational resources available do not change (or they go down as
they go outside of each others event/causal horizons).
A conjecture of mine is that the reason why we observe a finite universe
is that only finite processes and products of finite processes, like our
physical bodies, can only observe finite things because we can only
communicate about finite things. A reality requires the ability to
coherently communicate between its observers.
On Fri, Jan 17, 2014 at 6:02 PM, LizR <lizj...@gmail.com> wrote:
> On 18 January 2014 11:39, Stephen Paul King <stephe...@provensecure.com>wrote:
>> Dear John,
>> I invite your comment on a statement and question: *There is not
>> observable difference between "X is non-computable" and "there does not
>> exist sufficient resources to complete the computation of Y".*
>> Are X and Y effectively the same thing, everything else being equal? If
>> there is a difference that makes a difference, what might it be?
> In other words, is anything non-computable because of some theoretical
> reason, rather than "merely local geographical" ones (which might cease to
> be restrictions if, say, our local even horizon expands, or we construct
> wormholes to other universe) ?
> Surely the halting problem?
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