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Philosophical questions concerning the difference between equality, isomorphism, equality upto (unique) isomorphism, undistinguishability, and the like are not very popular among practicing mathematicians: there's a difference between equality upto isomorphism and equality upto unique isomorphism, and that's it (not to forget about isomorphism and natural isomorphism).

But personally, I'm not totally satisfied with this stance when looking at truly abstract structures like unlabelled graphs (finite or infinite, countable or uncountable), conceived as nothing-but-dots-and-arrows.

Are two abstract structures not to be considered equal in the strongest sense - being one and the same - as soon as there is an isomorphism between them, regardless of being unique, natural, and/or not?

The existence of isomorphisms in turn tells us something about the symmetries of the abstract structure, but of one and only one.

Maybe there are no such abstract structures per se, but only concrete structures (models) and/or concrete presentations of them (like adjacency matrices). Then the question misses a subject. But if so?

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    @Mariano: When one defines "abstract structure" set-theoretically to be an isomorphism class of structures, and isomorphism between abstract structures to be isomorphism between representatives, then the question becomes trivial, and the answer is "yes", isn't it? My problem seems to be that I try to see "abstract structures" on an equal footing with "concrete structures": among all members of an isomorphism class, I imagine a distinguished one: the abstract one, the base set of which consists solely of undistinguishable dots (atoms). (I should remove the "category-theory" tag!)2011-02-18

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In practice, God does not hand you abstract structures: you construct them from other structures, and the point of caring about isomorphisms is that you want to keep track of this construction process. For example, you are almost never handed a vector space $V$ and its dual $V^{\ast}$. Usually you are performing some construction (e.g. on the tangent space $T_p(M)$ of a manifold $M$ at a point $p$) which naturally involves elements of $T_p(M)$ as well as elements of the cotangent space $T_p(M)^{\ast}$, and if you are foolish enough to think that they are the same space then you will literally not be capable of doing calculations in this setting (e.g. changing coordinates).

One way to say this is that abstract structures often arise functorially, and even if $F(c), G(c)$ are isomorphic where $F, G : C \to D$ are functors and $c \in C$ is an object, the functors $F, G$ need not be naturally isomorphic, and usually we actually care about the functors, not the objects $F(c), G(c)$ in isolation. In the above example taking duals is a contravariant functor $\text{Vect} \to \text{Vect}$, and since it is contravariant it does not behave at all like the identity functor.

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    @Hans: perhaps the following analogy might help you see what people have been trying to tell you. You are going up to a group of carpenters and asking them "why do you use so many different kinds of wood? It all comes from trees, right?" All they are trying to tell you in response is "maybe you should make some chairs and then get back to us." Certainly this is a much better way to learn something about carpentry than thinking about carpenting all the time.2011-02-17
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Let me draw your attention to the newborn founding endeavor of Voevodsky, which attempts, among other things, to capture exactly that isomorphic structures are indeed identical; this is supposed to be the content of his so-called "univalence axiom" (as explained by Awodey in relevant lectures).

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    Sorry for the late response! I'm afraid I can't explain Voevodsky's work---I've hardly studied it myself. I just happen to know a couple of slogans like the above. For info of varying degree of depth, you might want to look at the "HoTT" site---ah, this western bent on clever acronyms... :-) --- http://homotopytypetheory.org/.2012-10-01