Why is it the case that $\rm\langle Socrates\rangle = Socrates$, while $\rm\{Socrates\}\neq Socrates$? In Nolt's book called Logics (part III, chapter 7, § 7.1 Sets and $n$-tuples) one can read:

Ordered one-tuples, unlike unit sets, are the same things as the object they contain. Thus , for example, the one-tuple $\rm (Socrates) = Socrates$; and more generally, for any object $x$, $(x)  = x$.

Could you please explain Nolt's (too) short explanation (at least for me)? 
Is this explanation a way of saying that a "$1$-tuple" is not a well defined object? 
Is there a way to derive the formula $(x) = x$ from the set theoretic definition of an $n$-tuple?
 A: The explanation identifies a $1$-tuple with its object. So while $x$ is generally not the same as $\{x\}$, the $1$-tuple $(x)$ is identified with $x$ itself.
This is perfectly reasonable to do, and so people do it. But does that clash with the set theoretic definition? Well, that depends on your definition of a $1$-tuple to begin with. Since the language of set theory only has $\in$, any definition of $1$-tuples is just a way to interpret this concept of a sequence of length $1$.
Generally, we think about $n$-tuples as functions from $\{0,\ldots,n-1\}$, so a $1$-tuple would be a function from $\{0\}$ into something. But this is not something which is hardcoded into set theory, it's just one way to interpret $n$-tuples. You can do it just as well in a different way, or even explicitly decide this is fine for $n>1$, and for $n=1$, $(x)$ is just $x$ itself (which then raises the question what is a $1$-tuple whose object is a $3$-tuple?).
This is also why we can't require $\{x\}=x$ in general. Since we understand $\{x\}$ as a set whose unique member is $x$. And then $\varnothing\neq\{\varnothing\}$, since exactly one of these has an element. But tuples are looser, since as we see, they are not hardcoded into set theory like singletons.
A: In the world of logics and foundations various authors have different views about "self understood identifications". E.g., if ${\tt socrates}$ is the identifier used for the human being Socrates then some people think that ${\tt socrates}$ is the same as the singleton set $\{{\tt socrates}\}$, whereas I would write $${\rm the}\bigl(\{{\tt socrates}\}\bigr)={\tt socrates}\ ,$$
whereby the "$\,{\rm the}\,$" extracts the single element from a singleton set. "Datawise" the objects $\{{\tt socrates}\}$ and ${\tt socrates}$ are of course the same.
Similarly, an $n$-tuple is a function $f:\>[n]\to X$ with a predefined target set $X$, and a $1$-tuple is a function $f:\>\{1\}\to X$ resulting in a single value $a=f(1)\in X$. "Datawise" the function $f:\>[n]\to X$ is encoded as $(a_1,a_2,\ldots, a_n)$, since the domain $[n]$ is considered as "self understood". It is then easy to identify the $1$-tuple $(a)$ with the element $a\in X$.
What I have written here is a lot of "abstract nonsense". It should tell you that in each longwinded argument it should be clearly indicated what the tacit simplification assumptions are made, but that there are various such simplifications possible.
A: The most important thing to understand is a set or a list or a collection of an object is not the same thing as the object that is in the set or list or collection.  That is $\{Socrates\}$ is a set containing Socrates whereas Socrates is a guy of flesh and blood who drank hemlock millenia ago.  
Now consider an ordered pair.  If I write:  $(3,2)$ that represents a point in 2-dimensional space where the 1st coordinate is $3$ and the 2nd coordinate is $2$.  Now when I write it out as "$(3,2)$" it looks like I am writing a list containing $3$ and $2$.  But it isn't a list.  It's a single point.  The thing is the only way to represent the point is to list out its components.  But it is understood that by so listing we are representing the point and not the list itself.
Same thing if we do a $1$-tuple, a point in $1$ dimensional space.  But a $1$-tuple with only one coordinate is just that... it's coordinate.  The one-tuple with a coordinate of $3$ is... the number $3$.
So whereas $\{3,2\}$ is a set of $3$ and $2$; and "$(3,2)$" is a listing of $3$ and $2$ inside parenthesis.  What the list "$(3,2)$" represents is the $2$-tuple $(3,2)$.
Likewise $\{3\}$ is a set of $3$; and "$(3)$" is a listing of $3$ inside parenthesis.  What the list "$(3)$" represents is the $1$-tuple $3$ or $(3)$ which is simply the value $3$-- a one-dimensional value with a coordinate value of $3$.
A: Your book's author was being careless and accidentally contradicted set theory. There are two ways tuples are defined, and neither of them allow for $\{x\} = x$.


*

*A function $T : \mathbb{N} \to S$ together with a collection of projection functions $\pi_i : \{i\} \to \mathbb{N}$ such that $\pi_i \circ T$ is the ith element of $T$. Here obviously $(x) \neq x$.

*Some set theoretic definition involving the Kuratowski pair. And in these definitions, $(a) = \{a\}$ which according to the axiom of regularity cannot be equal to $a$.
So no, your book has made a mistake.
