FOM: sharp boundaries/tameness

[Harvey Friedman]

>From: "Harvey Friedman" <friedman at math.ohio-state.edu>
>> 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".
>
>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.

I am obviously not talking about temporary uses of predicates. In addition,
there is no issue about sharp boundaries for this, and so it is not related
to the discussion at hand.

>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.

This is why the overwhelming majority of mathematicians stopped making a
study of well orderings of the reals. This is part of the reason why they
spend their time on such things as algebraic curves and surfaces, algebraic
number fields, Diophantine equations, finite groups, differentiable
manifolds, etcetera.

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

No. This property is commonplace and ordinary. Can you publish the
following in a Math Journal?

THEOREM. Let x be an infinitesmal. Then x < 17.
>
>> 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".
>
>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?

You haven't given any examples or significant facts that pertain to
"arbitrary objects". There might be an interesting theory of arbitrary
objects that does have examples with significant properties, but I don't
know about it.

GENERAL COMMENT: It is a lot of work to polish philosophical remarks so as
to be immune to criticism of the sort in the recent postings of Frank and
Pruss, etcetera. Generally, one is pushed into weakening the remarks so
much as to be unproductive and not useful for progress in foundations of
mathematics. So I invariably avoid trying to do this kind of work unless I
think that such polishing would lead to something productive for
foundations of mathematics. And then I will only do such polishing as long
as it is or is likely to become productive. Obviously, I would find it an
easy exercise to write such responses to my own postings (as Frank and
Pruss have done)! I even thought of all these points while I was writing my
postings!! However, I don't mind if FOM people wish to respond in this way
- especially if they try to direct this into productive channels.