[Edited to provide better context.]
In a comment on meta, JɛﬀE suggested that this would be a good topic for a question to ask here.
why is a Turing machine defined as a 5-tuple?
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Short meta preamble — Despite my misgivings (in particular, despite the comment on meta, I don't really think that this question is parallel to the question of why one should define topologies in terms of open sets), I think it may be important for the sake of the exercise and in particular as a point of empiricism to see if we can squeeze out a meaningful answer to this particular question.
We have to recall that the theory of computation began a few decades after mathematicians and logicians became quite keen on formalization, and in particular on simple, universal foundations for mathematics. In particular, when Turing defined his machines, people had recently been fretting about the best way to define ordered pairs in terms of sets. This sort of pre-occupation with minutiae probably strikes some of us as a little bit funny today, but it was the context in which definitions were set out.
As I hint in my comment above about groups — that although they are sometimes described as an ordered pair $(G,\ast)$ consisting of a set $G$ and some binary operator $\ast$, most people talk about $G$ itself as though it were the group — the notion that a Turing machine is a tuple is really beside the point; the role served by the tuple description is just to say that the machine is characterized by a collection of mathematical objects which have some relationships to one another. Furthermore, whether it is a 5-tuple, or a 7-tuple, depends on how you want to present the particular mathematical pieces. It makes a lot of sense to lump some of them together, as I argue below, so that (if you care about tuples) you can have some sort of argument as to the "best" arity of the tuple.
The set of states $Q$ is simple enough and could stand on its own, but doesn't really have any meaning on its own either, without e.g. the transition function alongside it to give it the semantics of the state of a "machine" which depends on "time".
The initial state, and the halting states, of the machine are just special elements of $Q$. These can be supplied separately (with the proviso that they are elements of $Q$), or we might say that $Q$ is not merely a set but some more exotic object in which the distinguished elements are supposed to be given. For instance, perhaps $Q$ is a pointed set in which the initial state is an intrinsically special element, so that we could easily describe the unions of languages by joins of nondeterministic Turing machines which accept those languages. Perhaps some other clever special roles could be given to accept and reject states; or perhaps those could just be states designated specially by the transition function, rather than along with $Q$ or separately on their own.
The alphabet on which the machine acts can again stand on its own, and it makes an enormous amount of sense to consider this again as a pointed set, where the blank symbol is the special element, rather than supplying that blank symbol separately.
The tape is some function from a set such as $\mathbb N$ to the alphabet in which all but finitely many symbols are blank; so the tape immediately has some relation to the alphabet as a pointed set.
Finally, the transition function is impossible to define without the set of states, the alphabet, and the tape — but it is what gives them meaning as a model of computation by giving some sort of sense of evolution via repeated application.
Recognizing that this is what we care about in the definition of a Turing machine, the only thing that a tuple (of any arity) serves to do is to present them all together, in definite positions so that (if your foundations of mathematics are so thoroughly set-theoretic that you would be constitutionally unable to distinguish the number zero from the empty set) you can tell which is supposed to be the set of states $Q$ and which is supposed to be the transition function.
Indeed, if you want to describe Universal Turing Machines, in which a machine can read an encoding of another machine, some encoding must be used; and so describing a Turing machine as a tuple of some sort is a reasonable conceit. But does this mean that a Turing machine is really "a tuple", as opposed to being presentable (really, encodable) as a tuple? Not meaningfully so. They might be presented just as well or better in terms of categories, or some class of mathematical structures of which Turing machines are the first interesting exemplar.
Mathematical objects, Turing machines included, derive their meaning from what they do. What tuples do is present several things at once; they are a data structure. What Turing machines do is allow us to express a robust notion of computability through the relationships of several related mathematical structures, and through our understanding of how to apply those relationships to things that we care about.