The underlying question:
What does lambda calculus do for us that we can't do with the basic function properties and notation generally learned in middle school algebra?
First of all, what does abstract mean in the context of lambda calculus? My understanding of the word abstract is something that is divorced from the machinery, the conceptual summary of a concept.
However, lambda functions, by doing away with function names, prevents a certain level of abstraction. For example:
f(x) = x + 2 h(x, y) = x + 5 y
But even without defining the machinery of these functions, we can easily talk about their composition. For example:
1. h(x, y) . f(x) . f(x) . h(x, y) or 2. h . f . f . h
We can include the arguments if we want, or we can abstract away completely to give an overview of what's happening. And we can quickly reduce them to a single function. Let's look at composition 2. I can have student layers of detail I can write with depending on my emphasis:
g = h . f . f . h g(x, y) = h(x, y) . f(x) . f(x) . h(x, y) g(x, y) = h . f . f . h = x + 10 y + 4
Let's perform the above with lambda calculus, or at least define the functions. I'm not sure this is right, but I believe the first and second expressions increment by 2.
And to multiply by 5y.
Rather than be abstract, this seems to get into the very machinery of what it means to add, multiply, etc. Abstraction, in my mind, means higher level rather than lower level.
Furthermore, I am struggling to see why lambda calculus is even a thing. What is the advantage of
h(x) = x + 5 y
or a combined notation
or even Haskell's notation
h x y = x + 5 * y
Again, what does lambda calculus do for us that we can't do with the f(x)-style function properties and notation many are familiar with.