Le 18-juin-06, à 06:35, Tom Caylor a écrit :

> I don't know much about Church Thesis, but I want to learn more.  Even
> though it seems easy to recite, as in the existence of a Universal
> Language, it seems very deep and mysterious.

I think so.



>  Almost like stating a
> Unified Field Thesis.


I agree.



> You just state it, and then see where it leads,


Yes.



> at least as far as you are able to follow it.

? I mean we follow it. Comma. Of course in case we get a contradiction 
we will abandon it. If we get just weirdness, well, personnaly I follow 
Deutsch dicto: let us take our theory seriously, it is the only chance 
to make them wrong ...



>
> But on the surface, the very prospect of someone trying to disprove
> Church Thesis is funny to me.



I think so too. To be honest, one paper on continuous (quantum) 
teleportation made me doubt on Church thesis during several hours ... 
but no more.





> Perhaps I am missing something.  To
> think of a counterexample seems like a contradiction.  Didn't Turing
> describe his Turing machine as equivalent to what a mathematician can
> do with a pencil and paper?  But the counterexample would have to be
> something that someone (probably a mathematician ;) could think up!  So
> a counterexample would be a function F that
>
> 1) a mathematician could think up
> 2) without using a pencil and paper!  Oooo.



Not really, because Church thesis asks for the "pencil and paper". So a 
counterexample of Church thesis would be a FINITE description of how to 
compute a function, description which can be send by radio wave or by 
pedestrian mailing and understandable by human but not by computer.
The main problem is that we cannot define really what we mean by 
"FINITE". It is as hard to define the term "finite" (without using its 
meaning implicitly) than the term "consciousness".
(and in my opinion the term "matter" is still more complex, and that 
almost follows from comp in fact, cf UDA).




>
> Actually, I am aware that people like Penrose actually say that
> something is going on in the brain (quantum-mechanically) that could
> never happen on a piece of paper.


After having read his cute "road to reality" book I think he has 
changed his mind.



>  But putting it in the above way
> makes it sound funny.  And I think actually Penrose might claim that
> the function G is just such a function.


Penrose knows perfectly well that no known human can compute g. (G can 
be compute partially and is in general undefined in a predictible way 
(for example on its own code)).
So Penrose has concentrated his non-comp argument on the formal version 
of that argument, that is on Godel second incompleteness theorem. He 
needs to say that humans know their are consistent and thus escape the 
godelian fate, but Judson Web already refutes such attempts. I propose 
we come back to this when we will go the fact 1 and to to the facts I 
and II (cf Smullyan). We need the notion of theory for doing that.



> He doesn't say much in his
> Shadows of the Mind about Church Thesis.



Interesting. I will try to reread what he says about Church thesis.



> But, Bruno, your posts on
> this seem to be assuming Church Thesis and then seeing what the
> conclusion is about G, which is perhaps the opposite of Penrose.



Sure. remember I put explicitly Church thesis in the definition of 
computationalism.  Penrose postulates non-comp at the start (of its two 
best-sellers book).
penrose uses Godel as an argument for non-comp. I take it as the most 
lucky possible event for the mechanist, because it saves mechanism from 
reductionism.
With a good understanding of Church thesis you can realize that comp is 
the most anti-reductionist philosophical standpoint possible. Church 
thesis and comp are the rationalist road to the mystical notion of 
unconceivable freedom. Unfortunately, although we have practice 
rational mysticism for a millenium (from Pythagoras to Proclus), we 
have (irrationally, after mixing religion and state) separate them 
since about 1500 years.



>
> Do you think that there is a possibility that Church Thesis has the
> same status as the Continuum Hypothesis in this sense:  the Continuum
> Hypothesis has been shown to be independent of the axioms of
> arithmetic, i.e. both the truth and the falsity of the Continuum
> Hypothesis is consistent with the axioms of arithmetic.



You mean with the axioms of Zermelo Fraenkel set theory, I guess. (The 
continuum hypothesis is not even expressible in first order axiomatized 
arithmetic, like Peano).




> Could the
> Church Thesis be independent of... what?... The problem is: what body
> of knowledge is there that is in the pursuit of truth, and is also
> intimately affected by the Church Thesis?  The mind-body problem, I
> guess.  Could the Church Thesis be independent of the mind-body
> problem?



Quite interesting question. In 1922, Emil Post, the real first 
discoverer of "Church thesis" (except perhaps for Babbage and Ada) 
considered it as a law of mind, and speculated that progress in 
psychology could perhaps settle it.
Together with arithmetical realism I make Church Thesis part of comp. 
Just for making precise the digital, numerical aspect of it.

Again, we talk about the classical Church Thesis (CT). There are 
intuitionistic and epistemological variants of CT which can be 
formalized, and then prove to be independent of some axiomatization of 
intensional or intuitionistic mathematics, but this should not concern 
us before we met the arithmetical notion of (first, first plural, 
third) notion of person (the "hypostases").

Bruno


http://iridia.ulb.ac.be/~marchal/


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