On 05 Jun 2011, at 18:45, Rex Allen wrote:
On Sat, Jun 4, 2011 at 4:09 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 04 Jun 2011, at 19:06, Rex Allen wrote:
On Sat, Jun 4, 2011 at 12:21 PM, Jason Resch
<jasonre...@gmail.com> wrote:
One thing I thought of recently which is a good way of showing how
computation occurs due to the objective truth or falsehood of
mathematical
propositions is as follows:
Most would agree that a statement such as "8 is composite" has an
eternal
objective truth.
Assuming certain of axioms and rules of inference, sure.
But everyone agree on the axioms of arithmetic.
I’m not sure what you mean here. “Agree” in what sense?
In the usual sense. Everyone agree that the axiom of Robinson or Peano
arithmetic are sound for elementary arithmetic.
Agreeing means that they say, in front of the axioms, that they are
OK, and proceed.
Everyone agrees that the axioms of arithmetic are...what?
Interesting? Useful?
Indispensable. We use them consciously or unconsciously since a very
long time.
Who is “everyone”?
The Löbian universal numbers.
Does everyone also agree that there are other axiomatic systems?
Yes. Everyone agree with this. Since Gödel we know there is an
infinity of them, and that none are both mechanical and complete.
And we could take any
universal (in the Turing sense) system instead. The physical laws
cannot
depend on the choice of the "universal base".
What exactly are “physical laws”?
The laws we are using in our everyday life to predict our observation
(notably).
You’re really saying “the regularities in our experience cannot depend
on the choice of the universal base”?
Yes. I can prove it from the mechanist assumption.
Other recursive formulae may result in the evolution of structures
such as our universe or the computation of your mind.
Is extraordinary complexity required for the manifestation of
"mind"?
If so, why?
Is it that these recursive relations cause our experience, or are
just
a way of thinking about our experience?
Is it:
Recursive relations cause thought.
OR:
Recursion is just a label that we apply to some of our implicational
beliefs.
I think you are confusing computability, which is absolute
(assuming Church
thesis), and provability, which is always relative to theories,
machines,
entities, etc.
What are your justifications for assuming the Church thesis?
The conceptual justification is the closure of the set of partial
computable functions for diagonalization.
The empirical is the diversity of equivalent systems and the failure
to get anything stronger than a universal Turing or Church universal
system (with respect to computability, not provability!).
Do oracles exist in Platonia? In HyperPlatonia perhaps? If not, what
precludes their existence?
It is almost a matter of convenience to add them in Platonia. With
arithmetical Platonia, we can explain why machines believe correctly
in random oracle, even if they don't belong in the ontological
reality. It is absolutely undecidable for machine if infinite objects
exist or not 'ontologically'. It is provable, with comp, that they
exists epistemologically.
Jason is right, computation occurs in "arithmetical platonia", even
in a
tiny part of it actually, independently of us.
Ya, I have my doubts about that.
If comp is correct, all Löbian machine doubt that, as they have to
doubt on comp itself.
This tiny part is assumed in the rest of science, and comp makes
it necessarily enough (by taking seriously the first and third person
distinction).
What is science in a deterministic universe? What is science in a
probabilistic universe? What other kinds of universes could there be?
Science is the attempt made by people to figure out what is.
Comp makes the physical universe, or multiverse, or multi-multiverse,
or ..., unique, except for possible clusters of comp inaccessible
realities, which can neither interact, nor interfere with our physical
reality (and so are not interesting, or Löbian-interesting).
Bruno
http://iridia.ulb.ac.be/~marchal/
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