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

[Arnon Avron]

This is a somewhat late addition to the discussion
when is it appropriate to take isomorphism as identity,
caused by coming across another example where 
(in my opinion) such an identification is counter-intuitive:

It is not difficult (see Fitting's new book "incompleteness
in the land of sets") to define a p.r. relation E on N
(the natural numbers) so that <N,E> is isomorphic
to the structure <HF,\in> (where HF is the set of hereditarily
finite sets, and \in is the "epsilon" relation). So
presumably <HF,\in> and <N,E> are "identical". On
the other hand <N,E> and the usual standard structure
<N,0,S,+,*> are identical in the same sense that 
<N,0,S,+,*> and <N,0,<,S,+,*,> are identical: E is definable
in terms of 0,S,+,*, and vice versa.
So the conclusion should be that <HF,\in> and <N,0,S,+,*> 
are "identical". But are they really? I do not think so
(to start with: it is not a trivial mathematical fact
that these two structures are "identical", but
how can one even *formulates* this nontrivial mathematical fact
without distinguishing first between the two structures?).

Another famous, very common identification which 
seems to me problematic is that of the Euclidean plane
with R^2 (where R is the "set" of "real" numbers).
The concept of a "real number" was historically
problematic, and in my opinion it still is, since it 
depends on the very questionable powerset axiom. 
On the other hand (And I know that I am in a minority
here, in the present cultural climate), I agree with Kant 
and Frege that we have a direct intuition concerning
the meaning and TRUTH of the theorems  of the Euclidean
Geometry. I was never able to persuade myself otherwise
(and the many beautiful pictures I have seen of the 
"Mandelbrot set" tell me that so do many mathematicians
and other scientists, at least in some days of the week).

Arnon Avron