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

Timothy Y. Chow tchow at alum.mit.edu
Thu Oct 2 10:16:39 EDT 2003

One reason I have delayed subscribing to FOM until now has been the fear
that it would take up too much of my time---that people would demand that
I reply to their messages on pain of an accusation of "ignoring" them.
Nowadays I care less than I used to about such accusations, but I mention
this just to assure Vladimir Sazonov and others that if I do not reply to
their message, it is not because I am ignoring them, but because I don't
have time to reply, and/or because I don't have anything valuable to add.

On Mon, 29 Sep 2003 Vladimir Sazonov wrote:
>The point was that a rule or a formalism or a deduction is not
>only finite, but feasible (physically presentable).

Kripkenstein, or my version of him, argues that rules are not physically
presentable.  All you can physically present is a finite approximation to
a rule, sort of like how a finite automaton can only multiply numbers up
to a certain length whereas a Turing machine requires a mysterious
unbounded tape in order to truly implement the multiplication rule.

On Tue, 30 Sep 2003 Vladimir Sazonov wrote:
> "Additional essential point is that WE DO NOT NEED any theory of
> formal systems to use their rules: we only need to be well trained
> for that. (Likewise, we do not need any theory of bicycles to ride.)

This much I agree with.

> These notes allow us to avoid any serious infinite regress in
> understanding the nature of mathematical rigorous reasoning."

But now you're talking about not just following rules, but *understanding
the nature* of following rules.  We then need to step back and look with
a critical eye at all these machines and people around us that are
allegedly "following rules."  Even if there really are such things as
rules and these machines are really following them, how can we possibly
come to know these facts?  The totality of our physical observations is
finite and is consistent with the claim that "there are no such things
as rules."

I do not want to get into a long debate over these points.  As my subject
line indicates, I am more interested in understanding if there are ways of
drawing a sharp mathematical distinction between potential and actual
infinity, not in rehearsing yet again the P-F debate (or worse, rehearsing
a debate over the best way to describe the P-F debate in terms that are
maximally fair to all parties concerned and that exhibit superb
philosophical sophistication beyond what the average mathematician is
capable of).  I brought up the P-F debate and Kripkenstein because I
thought they might help clarify my question and maybe provide useful
fodder for formulating, or beginning to formulate, an answer.

So please excuse me if I fail to follow up to every objection to the
points I have made above.


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