FOM: "generality" for fom(?)

[Robert Tragesser]

Vladimir Kanovei wrote

">Date: Sat, 14 Feb 1998 07:52:15 -0500
>From: Robert Tragesser <RTragesser at compuserve.com>

>        Isn't "generality" a great virtue in fom investigations?  

0 = 0 is one of the most general mathematical facts, 
compatible with any theory (even with the topos theory, 
as it was indicated that the latter is essentially 
a restricted form of Zermelo). 

In view of this, would R.Tragesser consider 0 = 0 as 
even more important that the topos theory for f.o.m. ?"

        I don't think I was understood:

        Actually, I was playing (I guess misleadingly)
on what I think to be Steve Simpson's misplaced emphasis
on "generality" as a fundamental issue in the debate
over the potential role of topos theory in f.o.m. research.
It strikes me that first-order logic as a basic f.o.m.
tool and setting has value because of its great generality
in the sense that most naturally occurring mathematics can
be (more or less instructively) coded in it and in a way that
instructively (though in a special sense) typically "generalizes"
the mathematics coded (by virtue of sustaining unintended
models/interpretations/consistent extensions).
        I wanted to (ab)use this occasion to suggest another,
more scientific approach to the appreciation and evaluations
of foundational frames -- by treating such as "probes" and
insisting on determining what exactly one can,  and cannot hope to
learn from them.
        See my posting on shifting metaphor from foundations to
probings.

Robert Tragesser
Professor in the History and Philosophy
of Science and Mathematics at
Connecticut College