FOM: Importance of (topos) "generality" for fom(?)
Robert Tragesser
RTragesser at compuserve.com
Sat Feb 14 07:52:15 EST 1998
Steve Simpson wrote:
"Here it is again: the argument that topos theory is wonderful because
it is so very very general and has so many many different
interpretations that differ so very very wildly from each other. Why
do topos people find this charming? My colleagues in the Penn State
Math Department tend to regard excessive generality as a strategic
error."
Isn't "generality" a great virtue in fom investigations? One
is perhaps justified in thinking that the value of to fom investigations
of first-order logic, despite its not being the natural logic of
"mathematical practice", is its "generality"-- as embodied in the logical
phenomena associated with it --
countable compactness, lowenheim-skolem...effects. I think (wrongly?)
of first order logic as a more sensitive probe than higher order logic,
indeed good for discerning in the latter what is otherwise lost to
sight there (an apt metaphor?).
At the same time, that sort of "generality" is balanced against
built in constraints (mechanical effectiveness).
Someone (such as I am) interested in the relation between
physico-geometric conceptuality and set-theoretic (re-)construction might
find
(as I did) topos theory very instructive in its distinctions between
the well-pointed and non-well-pointed and the "pointless". . .for the
light it sheds on what I am beginning to see (rightly?) as the
traumatic "primal scene" separating set topology from algebraic
topology -- the "approximating" of topological spaces by complexes.
(Working within approximations amounts to a virtual loss of
pointedness...(?))
Of course a good probe with have a wide range (good generality)
but it should not be cluttered with distinctions that make no real
differences. Maybe that's what is wrong with topos theory from
Steve's pov.
So should one ask, if topos theory might however be too general
to be a genuinely mathematically instructive probe? But might it not be
good (in the way the geometric dynamics is good) as a source of discerning
metaphors (as geometric dynamics is for investigating bionatural systems),
but next to useless for actually nailing anything down (in the way
that geometric dynamics is useless for nailing down any sufficiently
nontrivial biosystem). This would contrast with first order logic as
a probe with some good generality but which is also strict enough
to nail things down -- e.g. into exact hierarchies of consistency
discerning powers (?).
Again I raise the probl;em (approved by Feferman) of asking
just what fom-question first-order probes in contrast to topos theoretic
probes are good for exploring? (Why must there be such an either/or?)
robert tragesser
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