Le 15-juin-06, à 13:53, Tom Caylor a écrit :

> OK.  I think I understand what you are saying, on a surface level.  Of
> course a surface level will never be able to expose any contradictions.
>  I'm just riding this wave as long as I can before deciding to get off.
> It seems that there are very deep concepts here.  We are standing on
> the shoulders of computability giants.  I think it would take a Godel
> or a Church or Turing to find any problem with your whole argument, if
> there is any.  My feeling is that any "problem" is actually just lack
> of deep enough insight, either on the part of the attempting-refuter of
> the argument, or on your part, or both.
> By the way, I am also cognizant that what you are covering here
> actually is pretty standard stuff and actually has been pored over by
> the giants of computability.  So like I said, I'm riding the wave.
> On a certain level, it bothers me that Church's Thesis is said to not
> have any proof.  But maybe it is sort of like Newton's gravity.  It is
> just a descriptive statement about what can be observed.  And yet... we
> still don't really understand gravity.  Here we are at the level where
> all there is is falsifiability.

OK. Note that Church thesis has a unique status in math. I will come 
back on this.

> And, by the same diagonalization
> argument, you'd have to be God to falsify this "stuff".

Ah... but here you are wrong. Church thesis, although not entirely 
mathematical, still less physical, is completely refutable in the sense 
of Popper (and thus "scientific" in the "modern" acceptation of the 
word). To refute Church thesis it is enough to find a function F such 
that you can show:

1) F is computable, and
2) F is not Turing emulable.
Some people, like Kalmar, has thought they got such a function, but 
this has been rebuked.

Eliot Mendelson has argued that CT is provable, and sometimes I share a 
little bit that feeling. I did believe that CT is provable in "second 
order arithmetic" for a time because it did seem to me that CT relies 
above all on the intuition of the finite/infinite distinction. But then 
it could still be subtler than that, and I am not sure at all CT could 
really be proved.

Note that I talk only on the classical Church thesis, not about its 
intuitonistic variants, which I do believe are false for the first 
person associated to the machine (and this has been partially confirmed 
by a result due to Artemov which shows that some "computabiliy version" 
of constructive mathematics (like the so called Markov principle) is 
false in S4Grz (with quantifiers). But intuitionist CTs really asserts 
a different thing. The only roles intuitionistic CTs have in my work 
are for the explanation of why (first person) machine find so hard to 
say yes to the doctor and also to clarify the non-constuctivity feature 
of the "OR" in "Washington OR Moscow" self-duplication experiments.

Godel did miss Church thesis, and he takes some years for him to 
eventually assess it and then to describe it as a "sort of 
epistemological miracle". At the same time I would say Godel never got 
it completely because he will search for an equivalent miracle for the 
provability notion, but this can be shown being highly not plausible 
... from Church thesis.

I have evidence that Charles Babbage (and perhaps its friend Ada 
Lovelace) got Church thesis, once century before the others. The 
evidence comes from a book by Jacques Lafitte(*) who said in 1911 that 
Babbage discovered that his notation system for describing its 
analytical machine was somehow cleverer than his machine. Now, the 
first who really discovered explictly "Church thesis" and its relation 
with both computability and provability, is Emil Post in 1922, 
according to my knowledge.

Must go now. I intend to comment Tom and Stathis' post  later, perhaps 
Saturday because I have exams all the day tomorrow.


(*) LAFITTE J., 1911, 1932, Réflexions sur la science des machines, 
Vrin 1972 (New Ed.), Paris.


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