[FOM] Identity of isomorphic structures

Martin Davis martin at eipye.com
Mon Apr 7 15:48:28 EDT 2014

Steve Awodey's provocative and interesting article "Structuralism,
Invariance, and Univalence" in the February 2014 issue of Philosophica
Mathematica discusses the proposal to regard isomorphic structures as
identical. (This proposition is a consequence of an even more widely
embracing proposed axiom.)  Examples are given of situations in which
mathematicians routinely "identify" two isomorphic structures with one
another, such as identifying the Cauchy-sequence-of-rationals real numbers
with the Dedkind-cuts-in-rationals real numbers.

I would be interested to learn what proponents of this view would say about
the well-known isomorphism between the group of positive real numbers under
multiplication and the group of all real numbers under addition. Note that
in this case it's the isomorphic mapping itself that is important: it
underlies the use of logarithms in computation that  had such a significant
role before modern computers.

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