FOM: Re: PenMaddy's set-reductionism,Re: Friedman,fom,& Sheep's.Shop.

[penelope maddy]

>[1] I'd like to see an exact formulation,  and solid justification of,
>the claim that "all mathematical objects are ultimately (modelled on)
>sets".  ... Are
>the reasons empirical [i.e."we can in fact...for all known..."]?  Or 
>something much stronger [e.g., something  a priori -- and definitely not
>understanding "a priori" in the Kantian sense,  as for example 
>Bill Tait would hope it was not understood in the Kantian sense]?

I would say that the reasons are empirical, in a loose sense.  As set
theoretic methods developed in the work of Dedekind and Cantor, Zermelo,
etc., it only gradually emerged that all classical mathematics could be
modelled in these ways and all classical mathematical theorems proved from
these assumptions.  It seems to me that this should rank as one of the
fundamental foundational discoveries of our time.  For now, set theory
provides a unified ontology for mathematics, giving final answers to
questions of existence and proof.  In the fullness of time, mathematics
could develop in ways that set theory cannot handle, and it would lose this
role.  (Colin might say that this has already happened.)  But it seems to me
that providing a foundation of this sort is one of the central goals of
contemporary set theory, and (I suspect) this is one of the reasons it is so
keen on maximizing principles of various sorts.

PM