FOM: Relevance and mathematical modeling: a case in point

[Jon Barwise]

Lou,

1.  I didn't say that there is not a principled response.  But certainly
there are mathematicians who do not know one.  Frege didn't, as you point
out.  I like Martin's answer better.

2. But perhaps your point is deeper, that we have faith that mathematics is
a consistent unity and that everything is relevant to everything else. An
attractive picture. You admit that ZF MIGHT be inconsistent. If so, it is
unlikely that the inconsistency  would shed any light whatsoever on RH. It
would be irrelevant just because ZF would be irrelevant to mathematics.

3. Next, and somewhat irrelevantly (joke), your response does not really
address the heart of Russell's paradox.  It is not patently absurd to
assume there is a set of all sets.  There are set theories that have such
things and even have some principled motivation, though they necessarily
involve a very different conception of set than axiomatized by ZF.  But
there is no set of all sets that are not members of themselves.  So such
set theories have to give up full separation.  But why?  What is really
going on that makes the one sensible and the other not.

4.  I know most of you will not think the idea of a set of all sets is
sensible.  I did not think so either.  But if you think of sets not as
built up as stages but as collections up to bisumulation (observational
equivalence), then it does make sense.  Call them something else if you
want: collections up to bisumulation, say.  (See Chapter 20 of my book
<Vicious Circles> with Larry Moss for a discussion of this conception.) And
on that conception you can see what goes wrong with Russell's "set".  The
collections up to bisumulation are basically the closed sets in a large
topological space and, like closed sets in general, are not closed under
compliments.  Any attempt to form it lets in other things as well, through
the closure operation.

5.  BUT: The point of my message really had little to do with Russell's
paradox.  It was more pointed than that.  It is that logic hacking, without
thinking about how formal logics model informal proof methods, is an
ill-motivated exercise.  I was simply using relevance logician's objection
to the rule
	the absurd
	----------
            B
as an excuse to make that impolite point.  If you get rid of that rule then
you had been modify v-Elimination to have a natural system, one that can
model the reasoning mathematicians (and others) DO use all the time.

6. And that point was only made in order to stress the importance of
remembering what I claim we are doing with these formal languages and
logics, modeling real mathematical activity which does not take place in a
formal language with fixed formal rules of inference, but in a meaningful
human language.

Jon