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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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    No. A major lesson of category theory is that in practical situations it is not enough to know that two things are isomorphic; one needs to keep track of _how_ they are isomorphic because in practice one has to work with multiple such isomorphisms at the same time. (Hans, more generally might I suggest that you spend less time thinking about mathematics and more time _doing_ mathematics? Then you would get to learn these lessons the hard way.)2011-02-17
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    I also disagree with the premise of the question; higher category theorists care a lot about different notions of isomorphism.2011-02-17
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    But having to do with two abstract structures is not a practical situation: in practical situations you have to do with two *concrete* structures or with two *concrete* presentations, don't you? Only then you can ask *how* they are isomorphic, or is this flawed thinking? (You might ask, why bother about non-practical situations? But they occur, e.g. when trying to understand conceptual foundations. And that's what I try to do. Sorry for bothering you by asking the wrong questions ;-)2011-02-17
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    Is it, that we cannot even *think* and *talk* about truly abstract structures? (In the finite we seem to be able: unlabelled graphs).2011-02-17
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    You say, there are two isomorphic abstract structures ("of dots-and-arrows"), that cannot be seen as one and the same structure. So there has to be a difference between them. Can you give me a glue (an example).2011-02-17
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    Can you exhibit an *abstract structure* for us to see?2011-02-17
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    Only in the finite case: I consider an unlabelled directed graph to be an abstract structure ("just dots and arrows"). If finite, I could draw it for you, or give you an adjacency matrix as a representative. In the same vein I can exhibit finite groups, vector spaces and topologies to you.2011-02-17
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    My premise is: It makes sense to say, that two equivalent adjacency matrices or drawn graphs present *one and the same abstract graph/structure*.2011-02-17
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    @Hans: One situation where the existence of an isomorphism is not considered "strong enough" is a finite dimensional vector space and its dual. Although they are "abstractly isomorphic", the **real** isomorphism is the one between a vector space and its double dual. The reason is that although $V$ and $V^*$ are isomorphic, the isomorphisms depend on a choice of basis, whereas there is a "coordinate-free" isomorphism between $V$ and $V^{**}$. So, here, while we think about $V$ and $V^{**}$ as being *very strongly* "the same", this is not the case with $V$ and $V^*$.2011-02-17
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    @Arturo: Thanks for the hint. I know this argument, but nevertheless I cannot figure out, what makes V and V* - stripped off of all their peculiarities, reduced to dots and arrows - "more different" than V and V**.2011-02-17
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    @Hans, so the only way you have to give me an abstract structure is by giving me a concrete one?2011-02-17
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    Honestly, I can but agree with Qiaochu... *Doing math* will make it evident to you why $V$ and $V^*$ are felt to be different.2011-02-17
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    @Mariano: ... or at least, not "as equal" as $V$ and $V^{\ast\ast}$. To paraphrase George Orwell, all $n$-dimensional vector spaces are isomorphic, but some are more isomorphic than others. Hans: It wasn't a hint, it's a fact that becomes apparent with use. Trying to work with $V^\ast$ as if it were "exactly the same" as $V$ leads to a lot of problems (especially when dealing with inner products and representability of functionals), whereas this is simply not the case with $V$ and $V^{\ast\ast}$; the coordinate-free nature of the latter isomorphism *is* key to many important properties.2011-02-17
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    Does no one manage to stand inside my shoes and understand what I'm asking for? Is my question so hopelessly senseless? ("Do the hard work" cannot be all that is to be said.)2011-02-17
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    @Mariano: I wanted to give you concrete (= specific) abstract structures. Why do you say, I gave you concrete structures, when I obviously tried to make sense of "abstract structures"?2011-02-17
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    @Hans: the problem is, I guess, that you are asking a non-mathematical question.2011-02-17
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    @Hans: I think perhaps you are not communicating entirely clearly what you are going for, but to me it seems like you are saying "I do not see why specific kinds of isomorphisms might be more 'important' than just arbitrary isomorphisms". We are saying that there are situations where they turn out to be different, such as the case of the isomorphism between $V$ and $V^*$. But unless one tries to actually *use* the equivalences, it is indeed completely unfathomable why we would not consider *any* isomorphism as "just the same" as any other. (cont...)2011-02-17
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    I'm somewhat confused by the terms 'abstract' and 'concrete' structure... Are these things well defined or just made up for the sake of asking this question?2011-02-17
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    @Hans (...cont) So it's not "just do the hard work", but rather "I know it's hard to see why we distinguish between these, but that's because you are standing a little too far off and you can't see the finer points. Come, stand a bit closer so you can take a closer look and see what we're pointing to"2011-02-17
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    @Mariano: My question concerns abstract structures, and you will find many characterizations of mathematics as the study of abstract structures. So my question cannot be totally non-mathematical.2011-02-17
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    @Hans, Can you provide a definition of "abstract structure"? I'm sure you'll agree mathematics deals with objects with definition... The claim that "mathematics is the study of abstract structures" is sufficiently vague that it is pretty meaningless, so I simply cannot use it as a reference for anything.2011-02-17
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    In any case: Thanks to all of you for your valuable comments! (No irony!!) Let's stop now.2011-02-17
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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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    Thanks, Qiaochu. The phrase "the point of caring about isomorphisms is that you want to keep track of this construction" is really enlightening. That gives me a lot of thinking.2011-02-17
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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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    May I ask you to explain a bit more what Voevodsky's work entails, and how it relates to the question Hans posed?2012-07-25
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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