# [FOM] Bourbaki's general theory of isomorphism

Oran Magal oran.magal at gmail.com
Thu Jun 24 13:19:08 EDT 2010

```I confess I am unfamiliar with the Bourbaki book you mention, and I
wasn't on the list last year for that discussion, but I would like to
offer something in response.

What properties are such that two structures can be isomorphic, and
yet differ with regard to these properties? Informally speaking,
wouldn't those be: those properties that aren't structural properties?
Those properties would then have to be properties specific to an
_interpretation_ of the structure; something to do with their
'standard' interpretation, perhaps, such as the natural numbers for PA
or the 'intuitive' objects point, line, plane for the axioms of
Euclidean geometry.

But then, these are properties that are not given in any formalism,
but rather given in the usual concepts or associations or 'intuitions'
standardly connected with the structure in question. And given that
these are properties of a specific _interpretation_, how could any
other isomorphic interpretation of the same structure be identical
with respect to them without being the very same interpretation, and
therefore trivially identical to it?

What I have been trying to say, perhaps not too articulately, is that
I'm not sure what mathematical properties, which are not already
captured by isomorphism, you have in mind with respect to which the
question of identity/difference arises.

Best,

Oran Magal
McGill Univ.

On Wed, Jun 23, 2010 at 2:00 PM, Victor Makarov
<viktormakarov at hotmail.com> wrote:
>
> A year ago there was a discussion on FOM "When is it appropriate to treat isomorphism as identity?"    But the discussion went away without an answer.
>
>   It seems that an answer follows from the general theory of isomorphism suggested by Bourbaki in their book "Theory of sets".
>
>  If A is equal to B then, by Leibniz Law, every property of A is the same property of B.
>
>  If A is isomorphic to B, then every transportable property of A is the same property of B.
>
>  A property of A is transportable iff it preserved under all isomorphisms from A.
>
>  There is a paper by Victoria Marshall and Rolando Chuaqui "Sentences of Type Theory: the only sentences preserved under isomorphisms" - JSL, vol. 56, N3, Sep. 1991
>
> Victor Makarov
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```