[FOM] Counterfactuals in relative computability theory

[Richard Heck]

On 08/15/2016 11:33 AM, Timothy Y. Chow wrote:
> On Sun, 14 Aug 2016, Richard Heck wrote:
>> The problem is that it ignores the possibility of a "squeezing
>> argument" of the type pioneered by Kreisel.
>
> I don't agree.  All the squeezing argument says is that if you have a
> concept C, and you are willing to affirm that any set C' that
> faithfully represents C must contain a certain precisely defined set S
> and must also be contained in some other precisely defined set T, then
> you can prove some things about C' that then imply some things about
> C.  But it's only the part of argument that involves C' alone that is
> *mathematical*.  To arrive at an assertion about C, you have to affirm
> some kind of relationship between the informal concept C and the
> mathematically precise concept of "any set between S and T."  

As I understand such arguments, you have to be able to affirm:
(1)    Cx → x ∈ T
(2)    x ∈ S → Cx
where S and T are precisely defined notions. There's no need to bring C'
into the picture.

> This part of the "proof" is not mathematical.

Unless "mathematical" is supposed to be the same as "formal", then I
don't see why that should be so. Of course, one has to assume that we
have a clear enough grip on the informal notion C to be able to argue
for (1) and (2). But there's no reason that shouldn't be so. (This may
involve some "cleaning up" of the intuitive notion, though that may fall
far short of giving us anything mathematically precise.) And there's no
reason that the arguments given for (1) and (2) can't be "mathematical".
After all, mathematicians were giving proofs of various facts about
continuity long before a rigorous definition of the notion was given.

Richard Heck