[FOM] Counterfactuals in relative computability theory

[Matthias Jenny]

On Thu, Aug 18, 2016 at 5:51 PM Richard Heck <richard_heck at brown.edu> 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.
>

Richard, I think I agree with your view of squeezing arguments the way
you've described it. But I'm curious whether you agree with the following
additional claim: Whether the theory of C that we use in the squeezing
argument is the correct theory of C cannot be proved in the same (or at
least a similar) sense in which we prove things in mathematics. Rather, in
Kreisel's, Smith's, and your squeezing arguments, we either know that the
theory of C is correct because the axioms of the theory are conceptual
truths or else we have some sort of overwhelming abductive evidence for the
theory (the latter would be the view of someone like Williamson). I take it
that which of these two conjuncts is right depends on one's more general
views on issues such as conceptual truth and a priori reasoning, so I don't
want to decide that here. But it does seem to me that the epistemology of
the theory of C is of a different sort from the epistemology of proofs; it
strikes me that the epistemology of the theory of C is of the same kind as
the epistemology of, say, the axioms of ZFC.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20160818/6428edda/attachment-0001.html>