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> Technically X is not Y, but only isomorphic to it Not sure this is a useful argument. If two structures are isomorphic, there is no way to tell them apart. I
by tobinfricke 5y ago
> Technically X is not Y, but only isomorphic to it
Not sure this is a useful argument. If two structures are isomorphic, there is no way to tell them apart. If you can't tell them apart - maybe they are the same thing.
- nick__m 5y agoIsomorphism implies that there is a complete reversible mapping (a bijection) between structure A and B . Real number under addition are isomorphic¹ to positive real number under multiplication. Yet they are not the same objects. 1- The mapping is f(x)=e^x. See https://math.stackexchange.com/questions/573794/prove-that-mathbbr-and-the-interval-0-infty-have-the-same-cardinality https://math.stackexchange.com/questions/573794/prove-that-m... for the proof
- Tainnor 5y agoWhen it comes to group theory, they are the same object. When we talk about things like rotation groups, we're not usually concerned with the way they are represented (much as we don't usually how the real or the complex numbers are constructed, for both of which there exist multiple different constructions).