In functional programs, you very explicitly _do not_ need to understand an entire program. You just need to know that a function does a thing. When you're implementing a function-- sure, you need to know what it does. But you're defining it in such a way that the user should not know _how_ it works, only _what_ it does. This is a major distinction between programs written with mutable state and those written without. The latter is _much_ easier to think about.
I often hear from programmers that "oh, functional programming must be hard." It's actually the opposite. Imperative programming is hard. I choose to be a functional programmer because I am dumb, and the language gives me superpowers.
I think you missed the point. I understand that if you writing a simple function with an expected interface/behaviour then that's all you need to understand. Note this isn't something unique to a functional approach.
However, somebody needs to know how the entire program works - so my question was where does that application state live in a purely functional world of immumutables?
Does it disappear into the call stack?
It didn't disappear; there's just less of it. Only the stateful things need to remain stateful. Everything else becomes single-use.
Declaring something as a constant gives you license to only need to understand it once. You don't have to trace through the rest of the code finding out new ways it was reassigned. This frees up your mind to move on to the next thing.
> Only the stateful things need to remain stateful.
And I think it is worth noting that there is effectively no difference between “stateful” and “not stateful” in a purely functional programming environment. You are mostly talking about what a thing is and how you would like to transform it. Eg, this variable stores a set of A and I would like to compute a set of B and then C is their set difference. And so on.
Unless you have hybrid applications with mutable state (which is admittedly not uncommon, especially when using high performance libraries) you really don’t have to think about state, even at a global application level. A functional program is simply a sequence of transformations of data, often a recursive sequence of transformations. But even when working with mutable state, you can find ways to abstract away some of the mutable statefulness. Eg, a good, high performance dynamic programming solution or graph algorithm often needs to be stateful; but at some point you can “package it up” as a function and then the caller does not need to think about that part at all.