[FOM] Counterfactuals in relative computability theory

[Timothy Y. Chow]

On Thu, 18 Aug 2016, Richard Heck wrote:
> 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.

I agree that the words "mathematical" and "formal" aren't always 
synonymous in ordinary discourse.  But we are currently discussing the 
Church-Turing thesis, where the distinction between "informal" and 
"formal" takes center stage, so I took it for granted that when I was 
contrasting "informal" with "mathematically precise," the latter should be 
understood to mean "formally precise."  In other words, I was just making 
the standard claim that an assertion that an informal concept coincides 
with a formal concept is not capable of formal proof.  Squeezing arguments 
don't disprove this claim.  The formal part of the argument must deal only 
with a formal object C' that we informally ascertain to be a sufficiently 
faithful representation of the informal concept C for our purposes.

Tim