Does/has type theory influenced how to treat functions as values? Does/has type theory influenced how to treat functions as values?
Particularly, I still believe that it's general to think of functions with some input, e.g. $f(x)=x$ to be the value of the function, given the input. So the R.H.S., while being a function, is also a value. While the L.H.S. is a value.
However, from type theory perspective, one could perceive that $f(x) \space != \space x$, since they differ in types.
This is particularly confusing in programming languages such as Haskell, where e.g. id:: a -> a is not the same as x::a, even if id could map to the same value as a.
Then again, in Church encoding, there seems to be a separate "value container" meant to apply the function to its value.
 A: I do hope type theory starts influencing other folks in mathematics, because it generally handles this thing precisely in a way it's often not. For instance, your question seems to have a bit of confusion still in it.
For instance, you write $f(x) = x$, which as someone who's studied a lot of type theory, I generally take to mean a specification for what $f$ is. Note that I wrote $f$ on its own there, because that is how I'd refer to the function itself. $f(x)$ would be more like an expression with the function applied to a (free) variable. And when you go on to say $f(x) \neq x$, it doesn't make much sense in that context, because the specification of $f$ says that $f(x)$ probably reduces to $x$ (or at least is somehow equivalent to it).
What I would say is that $f \neq x$. Or, if you want to refer to the definiton of the function instead of the name, $(\lambda y. y) \neq x$. Note also that I changed the variable used, because the actual name used is irrelevant for bound variables.
However, it is definitely true that a lot of informal mathematics conflates a lot of these things in a confusing way, and mathematicians in general aren't as careful about all of this as type theorists and logicians tend to be (maybe even some type theorists aren't careful enough, especially if you read older presentations). I think this article (and its comments) explain this and more a lot more thoroughly, if you're interested.
