FOM: sharp boundaries/tameness

[Matthew Frank]

On Wed, 13 Feb 2002, Harvey Friedman wrote:

> 2. In their "completely rigorous" mode - the normal polished professional
> mode - mathematicians insist on using only concepts with sharp or precise
> or absolute boundaries. Infinitesmals fail on this account because of such
> questions as "which reals are infinitesmal and which are not infinitesmal?"

I disagree.  Many people have said (especially in connection with the
continuum hypothesis) that the concept of subset is vague; nonetheless
mathematicians use it in their most rigorous modes.  The concept of
infinity is vague, but we use it all the time.  In each case we have
formal frameworks to use with the concept, whose syntax and allowed rules
of inference are clear enough.  We have such frameworks for infinitesimals
too.

--Matt