[FOM] Counterfactuals in relative computability theory

[Matthias Jenny]

On Mon, Aug 15, 2016 at 4:59 PM Timothy Y. Chow <tchow at alum.mit.edu> wrote:

>
>
> I'm not sure if I correctly follow your reasoning here, but it seems to me
> that what you need for your argument is that the word "algorithm" is a
> *name* for a certain mathematically precise object.  Then, at least if we
> buy Kripke's argument about the rigidity of names, the name "algorithm"
> would correspond to the same object in all possible worlds.
>
> But if the word "algorithm" only "picks out" (whatever that means) a
> mathematically precise object, then that's not enough to imply rigidity.
> Surely definite descriptions also "pick out" things, but they aren't
> rigid.
>
> And claiming that the word "algorithm" is a name is definitely going to be
> controversial.
>

That's not quite what I had in mind. What I need is an argument that the
word 'algorithm' is a name/picks out/expresses a property in a determinate
way. We can think of properties just as sets of objects for our purposes.
So if 'algorithm' picks out a set of objects in a determinate way, then I
don't see how an argument of the kind you had in mind that appeals to the
supposed imprecision inherent in algorithms would work. But maybe I don't
have a clear grasp of that argument yet.


>
> > But just because some axioms can't be proven mathematically doesn't mean
> > that the concept that they axiomatize aren't precise.
>
> True, but given the overwhelming consensus that the word "algorithm" in
> the Church-Turing thesis refers to something *not* mathematically
> precise---otherwise it would be the Church-Turing theorem and not the
> Church-Turing thesis---the burden of proof is on you to demonstrate that
> the word "algorithm" is a name for a mathematically precise object.
>

I think I'm still not sure what you mean by 'mathematically precise.' I
think that 'algorithm' is fully precise, but I agree that it *isn't* obviously
mathematical in the sense that we can't define it with the usual
mathematical tools. I agree that if we could do that, then it would be the
Church-Turing theorem. But it seems to me that the difference between a
theorem and a thesis is epistemological: we can't be as certain of theses
as we are of theorems (modulo squeezing arguments, on which I am tempted to
agree with you *pace *Richard Heck). But just because we can't be
absolutely certain of some proposition doesn't mean that it isn't true in
all possible worlds. According to Kripke, the proposition that Mark Twain
is Samuel Clemens is true in all possible worlds. But we can't be
absolutely certain of it. Perhaps Mark Twain had some elaborate scheme to
fool us into thinking that he's Samuel Clemens. So to be more precise, we'd
have to say that the proposition that Mark Twain is Samuel Clemens is true
in all possible worlds, *if true*. (That's a material 'if.') Similarly, I
would say that the Church-Turing thesis is true in all possible worlds, if
true (again understanding the 'if' as material).
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