[FOM] Re: Sharp mathematical distinction between potential and actual infinity?

[Timothy Y. Chow]

On Mon, 29 Sep 2003, Harvey Friedman wrote:
>And I respond by saying that "I am baffled by stories such as this".

Think of it as a reductio ad absurdum argument.  It's supposed to show
that if skepticism about the determinate character of N is justified,
then skepticism about rules is justified.  If your reaction is that the
Kripkenstein argument is absurd, then hopefully your conclusion is that
skepticism about N is equally absurd.  (Of course if you're independently
convinced that skepticism about N is absurd then you may still not see
the point of trying to tie it to skepticism about rules.)

If you object to the inference from skepticism-about-N to skepticism-
about-rules, then what is it that allows you to draw a sharp line between
the two?  This question drives at something close to your suggestion: "It
seems rather subtle to set up the foundations of simple-rule-following, so
that it satisfies certain criteria of clarity and self containment."

Additionally, I find that the Kripkenstein argument helps me see exactly
how pervasive my presuppositions about infinity are.  For example, when I
naively think of physical spacetime, I think about R^4, not rho^4 for some
set rho of nonstandard reals, even though maybe there's no experiment to
distinguish between the two.  Maybe this is obvious, but I've seen some
people, who otherwise insist on some kind of "indistinguishability"
between standard and nonstandard numbers, turn around and try to point
to physical experiments as being able to nail down this distinction.

>Wittgenstein never did, to my knowledge.  I have never seen such criteria
>laid out properly by anyone.

I agree.  For example, the original conclusion of Wittgenstein, and also
Kripke perhaps, about private and public languages seems to me to miss
the point of the skeptical argument.  But I don't want to derail this
discussion into a textual study of Wittgenstein.

Tim