[FOM] When is it appropriate to treat isomorphism as identity?

[Monroe Eskew]

Dear Andrej,

On Tue, May 12, 2009 at 11:51 PM, Andrej Bauer <andrej.bauer at andrej.com> wrote:
> The kind of reductionism that is present in today's foundations of
> mathematics is dangerous. All mathematical objects and ideas tend to
> be expressed in classical set theory. I believe this kills many useful
> mathematical intuitions, in particular those related to geometry and
> computation. We're like logicians after Aristotle who were unable to
> imagine anything beyond his logic. We're unable to imagine anything
> beyond classical set theory. (Well, there are people who can, but they
> a tiny minority.)

I disagree with this characterization.  I know a prominent set
theorist who made the following analogy:  Most mathematics is "the
software" and set theory is "the machine language."  Set theorists
don't ask that people change their methods.  What they do is study a
system that unifies almost all ordinary mathematics.  Every classical
result can be translated into and proven within set theory, but we
also want to justify the ordinary methods in terms of set theory.
This kind of thing shows that (a) in the eyes of set theorists,
ordinary methods are not in need of revision, and (b) models of set
theory are models of classical mathematics, properly interpreted.  (b)
can be useful.

Best,
Monroe