On 22 Jan 2014, at 22:42, Stephen Paul King wrote:
Dear Bruno,
On Wed, Jan 22, 2014 at 2:30 PM, Bruno Marchal <[email protected]>
wrote:
On 22 Jan 2014, at 20:05, Stephen Paul King wrote:
Dear Bruno,
On Tuesday, January 21, 2014 8:51:14 AM UTC-5, Bruno Marchal wrote:
On 20 Jan 2014, at 21:17, Alberto G. Corona wrote:
> Computation is understood as whatever made by a digital computer or
> something that can be emulated (or aproximated) by a digital
computer.
OK. That's a good definition, and it is correct if ... we assume
Church's thesis.
> So everything is a computation.
Goddam! Why. Even just about what is true in arithmetic cannot be
emulated by any computer.
I am afraid you might not really grasp what a computer is,
conceptually. See my answer to stephen yesterday, which shows wahy
Church thesis entails that most attribute of *machines* cannot be
computed by a machine.
Or think about Cantor theorem. The set of functions from N top N is
not enumerable, yet the set of *computable* functions is enumerable.
That is a theorem that takes certain axioms as true... We can build
theories with other axioms...
Always. But that would made sense only if you provide the other
axioms.
I wrote this.
Axioms like the anti- foundation axiom, finite versions of the axiom
of choice, axioms that imply alternatives to the Cantor continuum
hypothesis, etc.
I didn't write this.
We can design our theories toward some goal.
To solve some problem. yes.
This could be said to be cheating and assuming what one wishes to
proof, but I submit that canonical logical has done this all along.
For example the use of the foundation axiom to prevent self-
containing sets - which prevent self-reference...
The Gödel self-reference can be done with or without foundation or
anti-foundation axiom.
Anyway, set theory is not assumed at all in the comp TOE.
I wish to escape the prison of the Tennenbaum Theorem!
This looks non sensical to me. But even if there were some sense
here, I remind you that I gave you two days ago, a constructive
proof of the existence of non computable functions, based on a
constructive diagonalization procedure (unlike the one by Cantor),
(and Church's thesis).
Just that with Cantor's result, it is more easy.
It is becoming clear that going with what is "easy" is a problem.
So reread my less easy proof, in that case. We can do it in elementary
arithmetic.
Nature does not obey our wishes of convenience.
Nature is not assumed, nor its absence.
It is she who we must obey and modify our assumptions so that our
models and theories match empirical data.
Maybe I am falling victim to a wish, maybe not, but the Tennenbaum's
theorem's prohibition of no countable nonstandard model of Peano
arithmetic (PA), and thus no recursive functions for computation
makes some assumptions.
For example:
http://en.wikipedia.org/wiki/Tennenbaum's_theorem
"A structure in the language of PA is recursive if there are
recursive functions + and × from to , a recursive two-place
relation < on , and distinguished constants such that
where indicates isomorphism and is the set of (standard) natural
numbers. Because the isomorphism must be a bijection, every
recursive model is countable. There are many nonisomorphic countable
nonstandard models of PA."
Why must this isomorphism always a bijection?
http://en.wikipedia.org/wiki/Isomorphism#Isomorphism_vs._bijective_morphism
"...there are concrete categories in which bijective morphisms are
not necessarily isomorphisms (such as the category of topological
spaces), and there are categories in which each object admits an
underlying set but in which isomorphisms need not be bijective (such
as the homotopy category of CW-complexes)."
The Stone duality that I am considering for a solution to the mind-
body problem is a subset of the greater Physical things-
Representations duality. You start with AR which, I claim, is
equivalent to an axiom that only representations (in the form of
Arithmetic) exist.
This is nonsense. the belief that 2+2=4 does not entail only
representations in the form of Arithmetic exist.
The hypostases contradicts this immediately. The 1p is not
representable in arithmetic, for example.
You then use the fact that representations can be of themselves, via
the Godel numbering or equivalent schema, to work out a brilliant
result that shows that the physical world can not be an ontological
primitive. But it has an open problem: What is an Arithmetic Body?
Yes, what is it? That notion does not make sense at all. What we call
a "physical body" does not exist. There are only "bodies perception"
which comes from the FPI, and admits substitution level, and thus can
be locally approximated (comp). Only my relative state of mind is
encoded.
If an Arithmetic body is a topological space that is the Stone
dual of the logical algebra of the computations and there are many
mutually irreducible (via the non-isomorphism of countable
nonstandard models of PA) "bodies". These "bodies" can share a set
of functions (Hamiltonians?) that have a morphism into the countable
recursive functions. ISTM that will allow us to obtain the Church
Thesis as a special case. We can also get much more and possibly
address questions of interaction and concurrency that cannot even be
stated in the definition of a Turing Machine.
Assuming that the Integers and Arithmetic are all that exist is a
gilded prison for our minds.
I can't make sense of this. Sorry.
Bruno
Free your mind!
Bruno
> That is a useless definition. because
> it embrace everything.
For a mathematician, the computable is only a very tiny part of the
truth.
>
> Everything is legoland because everything can be emulated using
lego
> pieces? No, my dear legologist.
Not veything can be emulated by a computer. few things actually in
usual math. Some constructivist reduces math so that everything
becomes computable, but even there, few agree.
In Brouwer intuitionist analysis he uses the axiom "all function are
continuous" or "all functions are computable", but this is very
special approach, and not well suited to study computationalism
(which
becomes trivial somehow there).
>
> What about this definition? Computation is whatever that reduces
> entropy.
It will not work, because all computation can be done in a way which
does not change the entropy at all. See Landauer, Zurek, etc.
Only erasing information change entropy, and you don't need to erase
information to compute.
> In information terms, in the human context, computation is
> whatever that reduces uncertainty producing useful information and
> thus, in the environment of human society, a computer program is
used
> ultimately to get that information and reduce entropy, that is to
> increase order in society, or at least for the human that uses it.
The UD generates uncertainty (from inside).
>
> A simulation is an special case of the latter.
>
> So there are things that are computations: what the living beings
do
> at the chemical, physiological or nervous levels (and rational,
social
> and technological level in case of humans) . But there are things
that
> are not computations: almost everything else.
That is the case with the definition you started above, and which is
the one used by theoretical computer scientist.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Stephen Paul King
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