FOM: AFA again

[Sean C Stidd]

Apologies in advance if I mischaracterize anyone's position.

In an earlier debate on this list, Professor Hazen suggested that
anti-foundational set theories (hereafter AFA, likewise with apologies to
Finsler and Scott) served to 'knock the iterative conception of set off
its metaphysical pedestal'. The reason he gave might be paraphrased thus:
models of AFA and ZFC are canonically recoverable from one another; AFA's
conception of set is not an iterative conception; hence we could have
gotten the mathematical 'meat' of set theory without recourse to the
iterative conception; hence the iterative conception is not essential to
our understanding of what sets are.

To Hazen and some others Professor Simpson seemed to reply as follows: 
models of AFA and ZFC are canonically recoverable from one another; AFA's
conception of set is not an iterative conception; but there is no
mathematical 'meat' AFA offers us that ZFC did not already; hence there
is no motivation to adopt AFA's conception of set; hence the iterative
conception stands secure as a foundation for our understanding of the
domain set theory is about.

In other words, it seems as though there is an agreement on the
mathematical facts and a disagreement about their significance.

A position like Hazen's would seem to be derivable from philosophical
structuralism, in the sense that two more-or-less-interdefinable theories
are being treated as 'about' the same kinds of things in spite of the
fact that both their axioms and their intended models are clearly
distinct. (I don't say this about Hazen's position as he expressed it
because towards the end of his long second post on the subject he seemed
to be trying to articulate a more general philosophical concept of set
across these different theories.)

On the other hand, a position like Simpson's would seem to be more
traditionally Platonistic, in the sense that the axioms and intended
model seem to be taken to characterize a particular mathematical domain -
the domain of the set theorist's concern - irrespective of the fact that
it has strange models that don't correspond to the normal picture of that
domain.

If I've characterized the difference right it's pretty clearly a
philosophical rather than a mathematical one. I'd be interested in
comments on or defenses of either position in spite of that, however,
being a philosopher.