On 16 Oct 2014, at 21:43, John Clark wrote:
On Thu, Oct 16, 2014 at 12:01 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
>> there is no logical reason or empirical evidence to think that
the halting oracle exists in the physical world or even in Plato's
abstract Platonia.
> Time implement the halting oracle. There is a result by
Schoenfield to that effect, which shows that the fuincion computable
with the halting oracle are the function computable in the limit (in
the computer scientist sense of limit).
I assume you mean the halting oracle would work by making use of
closed timelike curves, in other words a time machine. There is not
one scrap of evidence that closed timelike curves exist in the
physical universe and if they involve logical self contradictions as
most think they do then they don't even exist in abstract Platonia.
Gödel shows that there are solution of Einstein's equation of
gravitation with closed timelike curves, making them consistent. But I
was not alluding to closed timelike curves.
I was alluding to the usual time. It tells you which machines stop and
which does not stop if you wait a long time enough (which can be great
and is not computable, but is always finite (indeed bounded by the
busy beaver function on the number of K describing the machine). I can
prove this if you want, it is an easy exercise).
>> If you work in the fundamental It's also a big assumption that
Platonia exists.
> No, because it is the most least Platonia ever. You need only to
believe that 2+2=4 is true independently of you.
If the physical world didn't exist
Which is already impossible if 2+2=4 is independent of you and
computationalism is true (but to grasp this you need to unstuck your
mind in step 3). But I will accept momentarily your assumption to see
the point.
and there wasn't 4 of anything and never has been, would 2+2=4 have
any meaning?
Yes, because you don't need things for 2+2=4. You need things only for
applying it.
And even if it did would it matter, who would be around to
understand that meaning?
In this case nobody, but that would hardly change the truth of that
fact. The big-bang did not need an observer to proceed.
You have always just assumed that mathematics is more fundamental
than physics and maybe it is,
Only arithmetic. I don't not assume set theory, infinities, etc.
but with the recent discovery that information is physical
I agree with Liz on this. Shannon, or Chaitin-Kolmogorov-Löf-Solovay
type of information are mathematical. Physics makes information
important, and input some requirement on its physical implementation.
Landauer's result is very important, but it does not make information
physical. This is only a sort of metaphor.
we can at least question that assumption. It's also odd that
computers need to be made of matter not abstract stuff from Platonia
before they can actually do anything intelligent and presumably
before they can become conscious; but I don't think anyone knows yet
which is more fundamental, the real numbers or superstrings.
Natural numbers (or Turing equivalent), provably so if you assume
computationalism.
Nobody knows, but we can derive it from the assumption that we are
discussing. If 2+2=4 is not independent of you and me, then Church
thesis and computations, and thus classical computationalism, are
meaningless. This does not mean that some form of non-classical
computationalism, with some non standard intuitionist version of
computations can make sense. For some intuitionist ALL functions are
computable and continuous. It is interesting, but the theoretical
computer behind it is less rich, and not appropriate for the
fundamental inquiry. It is good for some type of engineering assuring
control of the program behavior. There is no computer or universal
machine in such theory, in fact there is no "other people" possible
there, and the form of comp studied here does allow universality, with
the price that we are not able to control all machines, but then we
have other people and get out of the solipsism of the intuitionists.
Bruno
John K Clark
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