FOM: sharp boundaries/tameness

Alexander R. Pruss ap85 at
Thu Feb 14 19:34:54 EST 2002

From: "Harvey Friedman" <friedman at>
> 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
> such challenges as "give me an example of an infinitesmal and establish
> some significant facts about it".

Surely the ability of giving examples is not necessary.  First of all, the
concept of a finite properly skew field seems to be a concept a
mathematician can use in professional mode, but there are no
examples--indeed, the main use of the concept is to show that nothing
satisfies it.  And if one doesn't like the idea of concepts that have no
extension, take the concept of "a well-ordering of the reals", something one
can't give an examples of.

Significant facts about an infinitesimal x?  Sure: x<17.  Isn't that

> In contrast, "give me an example of a
> transcendental number and establish some significant facts about it" can
> answered with e or pi.
> This paragraph may also be applicable to "arbitrary objects".

I missed most of the discussion, so this may be off-base.  If "arbitrary
objects" are "objects of any sort whatsoever", then certainly examples can
be given: I, you, and my father-in-law's dog.  Significant facts?  Well, it
depends on what is meant by "establish" and "significant."  Each object
exists and is one.  It has whatever features the correct ontological system
of the world shows objects to have--and this may include various very
significant facts (e.g., if some people are right, each object is either a
substance or an attribute of a substance), though whether one counts them as
"established" depends on how strict one's criteria for that are.


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