[FOM] Counterfactuals in relative computability theory

[Richard Heck]

On 08/14/2016 05:37 PM, Timothy Y. Chow wrote:
>
> The Church-Turing thesis, by its very nature, asserts an equivalence
> between a mathematically precise abstract object (e.g., recursive
> function or Turing machine) and an abstract object that is not
> mathematically precise (algorithm).  In particular, it is not the kind
> of assertion that is susceptible to mathematical proof or disproof. 
> (At least, this is the conventional reading of the thesis.  There is a
> minority view that the Church-Turing thesis might be something that
> could be proved.  But even this minority view acknowledges that the
> first step would consist of further clarifying the "algorithm" side of
> the equation to the point where it is mathematically precise.)

This is not at all clear, though it does represent a sort of orthodoxy.
The problem is that it ignores the possibility of a "squeezing argument"
of the type pioneered by Kreisel. Not only is such an argument possible
in principle, but such an argument has been given by Peter Smith [1,2].
I'll not try to evaluate the argument here. The point I am making
requires only the possibility that such an argument can be given.
Whatever one thinks of the specific argument Kreisel gave, or Smith
gives, or another such argument that I've given [3], it's long since
past time for people to stop saying, "It's impossible in principle to
prove the equivalence of an intutive notion with a rigorous one". At
least, people should stop saying it for the kind of reason Tim has given.

Richard Heck

[1] http://www.logicmatters.net/resources/pdfs/KreiselSqueezing.pdf
[2] http://www.logicmatters.net/resources/pdfs/SqueezingChr.pdf
[3]
http://rgheck.blogspot.com/2015/01/the-squeezing-argument-in-is-freges.html

-- 
-----------------------
Richard G Heck Jr
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