[FOM] Counterfactuals in relative computability theory

[Timothy Y. Chow]

Matthias Jenny wrote:
> I'm not sure I have a clear grasp of the notion of a mathematically 
> precise object. Take a sorites series of tiles where the first tile is 
> red, the last tile is orange, and there is no distinguishable difference 
> between any two adjacent tiles. It's unclear in this case whether the 
> definite description 'the number of red tiles in the series' succeeds in 
> picking out a particular natural number. According to some theories of 
> vagueness, it does, and according to others, it doesn't. Suppose it 
> does, and suppose the number it picks out is n. Surely, n is a 
> mathematically precise object.
>
> It's true that I was supposing that 'algorithm' determinately picks out 
> exactly one property. And I realize that that's potentially 
> controversial, so I should have made that explicit. Steward Shapiro 
> thinks that at least in the 1930's, 'algorithm' didn't pick out a 
> determinate property. But I'm not sure he thinks that it doesn't pick 
> out a determinate property today.

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.

> 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.

Tim