[FOM] Counterfactuals in relative computability theory

[Timothy Y. Chow]

Matthias Jenny wrote:
> 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.
[...]
> 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.

Given what you say here, it strikes me that the most novel thing in your 
account isn't anything about relative computability theory or the vacuity 
thesis.  The most novel thing is your account of "algorithm."

As I understand it, you have in mind a concept of "fully precise" 
according to which "algorithm" is "fully precise," but you balk at 
"mathematically precise" because the latter concept is unclear to you. 
(I think that for most people, "mathematically precise" is the clearer 
concept since all mathematicians recognize that mathematical precision is 
the quality that a definition needs for one to prove theorems about it, 
whereas there's no obvious way to understand what "fully precise" means if 
it's different from "mathematically precise.")  Also, "algorithm" is 
supposed to "pick out a set of objects in a determinate way"; your slashes 
between "is a name" and "picks out" and "expresses" suggests that any of 
these is equally fine, despite Kripke's sharp distinction between names 
and definite descriptions, the latter of which some people might think 
"picks out" objects without being a name.

If you can give a coherent account of all this then I think that that 
should be the main focus of your work, since you'd be frying a much bigger 
fish than the vacuity thesis.

Tim