On 28 May 2015, at 02:18, Russell Standish wrote:
On Wed, May 27, 2015 at 06:28:58PM +0200, Bruno Marchal wrote:
On 27 May 2015, at 17:48, John Clark wrote:
On Wed, May 27, 2015 Bruno Marchal <marc...@ulb.ac.be> wrote:
The Church-Turing thesis says something about intelligence but
not consciousness, it says that any real world computation, like a
intelligent action, can be translated into a equivalent program on
a Turing machine.
Church thesis does not invoke ideas of real-world computation
Wrong yet again. From Wolfram Mathworld, the makers of Mathematica:
"The Church-Turing thesis (formerly commonly known simply as
Church's thesis) says that any real-world computation can be
translated into an equivalent computation involving a Turing
machine. In Church's original formulation (Church 1935, 1936),
the thesis says that real-world calculation can be done using the
lambda calculus, which is equivalent to using general recursive
functions."
http://mathworld.wolfram.com/Church-TuringThesis.html
This is either a confusion between [...]
Obviously somebody around here is very confused indeed!
Citing Wolfram is a per-authority argument. That is not valid.
The expression "real-world" is ambiguous, what does it mean?
A function computable by a physical device,
A function intuitively computable, or mechanically computable, that
is having some algorithm describable in some language.
Better read the original papers. Buy the cheap Davis books in the
Dover edition. His book "computability and unsolvability" is quite
good, and the Dover edition contains his paper on the Hilbert 10th
problem, leading to (Turing) Universal Diophantine Polynomial.
I would probably have to side with John Clark here - in Platonia,
hypercomputers such as David Deutsch's example of the Hilbert Hotel
are possible,
The gods --- or highly non computable sets--- exists in the
arithmetical reality, too.
They are not in the sigma_1 (computable) reality.
so the CT thesis really is saying something about what
is physically possible, not what is mathematically possible.
This has just nothing to do with Church's thesis.
And does not follow from what you say above (which is anyway non
correct).
By definition, COMP assumes the CT thesis, so already the ontology is
constrained from what is potentially possible in abstract mathematics.
Church's thesis has nothing to do with ontology, above the fact that
we need some amount of arithmetical realism, which is just the
acceptance of the excluded middle principle on the arithmetical (even
only sigma_1) formula.
Church's thesis is not related to physics at all, nor to general
classical mathematics.
It has to do with algorithmic, and the existence or inexistence of
procedure to compute functions or to generate sets.
John Clark is worst on Church thesis than on step 3, I'm afraid.
We have the set N^N of functions from N to N. A computable function
from N to N is a function that is computable by following a a non
ambiguous procedure. Church thesis allows us to define that class of
function mathematically. For example we can say that a function f =
(lambda x f(x)) is computable if there is a SK combinator which
computes it, in the math sense of computing with combinators (there is
a sequence of combinators which get the answer, by the application of
the contractions:
- Kxy = x (1) and
- Sxyz = xz(yz) (2).
Exemple SKKS = KS(KS), by 2. KS(KS) = S (by 1). The sequence SKKS-
KS(KS)-S.
All computations (including a simulation of a quantum computer by a
Fortran program) can be simulate through combinator computation.
There is no physics evoked, as you can see.
Bruno
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Visiting Professor of Mathematics hpco...@hpcoders.com.au
University of New South Wales http://www.hpcoders.com.au
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