The thing about map id = id etc. probably has more to do with equational reasoning (can use equals to substitute terms, since there are no side effects, at least in Haskell), but I don't see the connection to lambdas/closures.
Nitpick: Lambdas and closures are different things. A closure is a semantic notion of a function captures its local environment. A lambda is a mostly-syntactic notion of defining a function without giving a name. Whether a lambda is a closure depends on the language's scoping rules.
Well yeah, that's what I meant by higher order functions.
> Lambdas are just a notation for function values.
But regular (named) functions can still be used as function values. So this doesn't explain why you need things like lambdas in order to implement function composition.
> Typed lambda calculus is the internal language of cartesian closed categories and function values are then called internal morphisms. The composition above is then the internal composition of internal morphisms.
Ok bud.
> But if you want to be able to define function composition within the language, you need to have something like lambda.
Well I could implement function composition without the syntactic construct lambda:
(.) g f x = g (f x)
I am not using any lambdas, in the sense of anonymous functions or closures. To implement function composition with a lambda is more of a stylistic choice, in this case. Granted, maybe functions-used-as-values are also lambdas, for all I know.
Interesting. I might say that partial application is a kind of closure. Certainly, it winds up the same - "function carrying some data that it uses internally". I think you are correct that compose and apply does not require closures of any sort.
g . f = \x -> g(f(x))
(Here \ denotes lambda)
Why would it be useful to have function composition in your language? Well it gives you similar power as "method chains" in an object oriented language, without being tied to specific classes, especially if the language also supports polymorphic functions. It also interacts nicely with other abstractions usually found in functional languages: For example consider map, of Map-Reduce fame
map : (Functor f) => (a -> b) -> (f a -> f b)
then one has
map (g . f) = map g . map f
Now imagine that map would cause the function to be send to thousands of nodes in a cluster, then the above identity tells you that instead of doing that twice, once for f and once for g, you might aswell take g . f and send it out once. Also say you would for some reason know that f . g = id, the identity function, then
map id = id,
so you would not need to do anything. This might appear trivial, but if you can teach the compiler about those cases, you can do interesting stuff with it. In the case of GHC (the Glasgow Haskell Compiler), it is able to use such rules in its optimization phase, which allows people to write apparently inefficient but declarative code and let the compiler eliminate intermediate values. See for example https://hackage.haskell.org/package/repa.