FOM: sharp boundaries/tameness

[Harvey Friedman]

>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.

It is realatively easy to object my statement as you have done, and much
harder to fix it so that this sort of objection is hard to do.

The difference between the infinitesimal situation and the situations you
talk about is so enormous that I can retract the statement that I made and
replace it with the following.

In their "completely rigorous" mode - the normal polished professional
mode - mathematicians insist on using only concepts whose boundaries are
sufficiently sharp or precise
or absolute that they can easily give a variety of examples and establish
significant features of them. Infinitesmals fail on this account because of
such challenges as "give me an example of an infinitesmal and establish
some significant facts about it". In contrast, "give me an example of a
transcendental number and establish some significant facts about it" can be
answered with e or pi.

This paragraph may also be applicable to "arbitrary objects".