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I'm told that an isomorphism is a kind of underdetermined term, unlike say, group isomorphism or ring isomorphism. Why couldn't we just say $\phi$ is an isomorphism on object $A$ if for all operations (or primitive relations) $R$ on $A$, and all $a, b \in A$, $\phi(a) R \phi(b) = \phi(a R b)$ and it is a bijection. I realize it requires some second-order quantification of relations, but that's not exactly unheard of.

If I've been careful, then this definition of isomorphism when applied to groups gives us group isomorphisms, and when applied to rings gives us ring isomorphisms. If we wanted to discuss things that preserved some but not all relations on an object, say for example we wanted to talk about preserving the abelian group operation on a ring, we could just refer to that as a group isomorphism on the ring or something like that.

William Bell
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    There are mathematical objects whose structure is not captured by "relations" or "operations". The wikipedia page https://en.wikipedia.org/wiki/Isomorphism points to some examples. I think category theory is the abstraction you are looking for. – Ethan Bolker Sep 07 '19 at 02:46
  • My understanding of isomorphism is that it's just a relabeling of the elements of a set. – Wuestenfux Sep 07 '19 at 13:21

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This (appropriately further generalized to higher arities) is indeed the definition of isomorphism in the broad context of first-order structures, which vastly generalizes groups and rings; indeed, operations and relations of infinite arity are not problematic in this context either. However, the point is that some things are not (infinite-arity) first-order structures, or fruitfully interpreted as such: we don't want to a priori limit ourselves to any one particular notion of "mathematical object," and this prevents us from whipping up a precise definition of "same-ness" that we're totally certain will be applicable in all contexts.

This sort of concern crops up in a serious way in category theory, but that's a bit down the road; for now, I would simply take it as a given that there will always be more kinds of mathematical structure yet to see, and that a notion of isomorphism which is satisfactory in all contexts so far may not be applicable in all situations in the future.

Noah Schweber
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