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I'd like to know, for instance, why we write '$(G, *)$' as opposed 'the set $G$ regarded as a group with $*$ as the operator'. There is an obvious answer, namely, that '$(G, *)$' is more concise. Are there other reasons?

When speaking of a group $G$ with operator $*$ we're not speaking of the tuple $(G, *)$, for $(G, *)$ is merely a representation of the group $G$ with operator $*$. To illustrate this, we could've just as well defined the group as $(*, G)$; either way, we're talking about the same thing.

Are there known instances in non-foundational and non-computational mathematics in which it's clearly convenient to regard a representation of an object as the object itself and that representation isn't a tuple?

  • The answer to the last question is no for the obvious reason: everything else in the representation is the rest of the tuple $\ulcorner X\urcorner=(X,(\text{whatever else you throw in}))$. – user10354138 Feb 21 '21 at 06:11
  • Well, $G$ alone is not the group and $$ alone is not the group. At some point you have to define what a group is, and that has to be some kind of $G$-together-with-$$. Using $(G,)$ and $(,G)$ one can define the concept of group in two equivalent ways. When we speak of $G$ (alone) as group when hopefully it is clear what $$ is, that is an abuse of language. -- [Okay, strictly speaking, one might be able to extract $G$ from $$ as its co-domain] – Hagen von Eitzen Feb 21 '21 at 06:12
  • @HagenvonEitzen The fact that you can define a group in two equivalent ways elucidates that these definitions aren't actually what a group is. The ways are distinct, i.e., they're not the same thing. A group can't be distinct from itself. –  Feb 21 '21 at 06:15
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    The formal definition of a group is as a tuple, because that's the way we bundle information together (the set and the operation). Saying that the set "is" a group is technically incorrect but very conventional. – Karl Feb 21 '21 at 07:14
  • @Karl "The formal definition of a group is as a tuple, because that's the way we bundle information together (the set and the operation)." I'm aware, but why? "Saying that the set "is" a group is technically incorrect but very conventional." Tomato, tomato. –  Feb 21 '21 at 08:24
  • I'd say it's because we want to be precise. Tuples are defined in the underlying set theory (or type theory or whatever), so defining something as a class of tuples feels well-founded and gives clear answers to questions of equality, etc. (E.g. if we can regard one set as a group in two different ways, are the "groups" equal?) – Karl Feb 21 '21 at 17:16

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