Keywords to find documentation for this syntax

Question

Mathematica allows this syntax, but I cannot find any documentation (or books) that discuss the various uses/advantages of it

g[1] := Plus;
g[2] := Times;
g[1][3, 5]
(* 8 *)

g[2][3, 5]
(* 15 *)

What keywords can I use to find this in the documentation?

Answer 1

Your g[1] and g[2] are simply acting as Head:

g[1] := Plus;

So there is no mystery in this syntax:

 {g[1][a, b], Plus[a, b]}

{a + b, a + b}

 Head /@ {g[1][a, b], Plus[a, b]}

{Plus, Plus}

So you need to read:

• Head
• Everything Is an Expression

But maybe there is a bit more to it than meets the eye. You actually almost wondered into programming concept called Currying which according to Wikipedia "is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument (partial application)."

So you can do things like:

f[x_][y_] := Sin[x y]

f[x] /@ Range[5]

{Sin[x], Sin[2 x], Sin[3 x], Sin[4 x], Sin[5 x]}

Documentation mentions it here. For deeper insight see discussion in @SalMangano "Mathematica Cookbook".

BTW, @acl nice addition (to see how Mathematica thinks) can be visualized as

TreeForm[Trace[g[1][a, b]]]

So the thinking goes from top to bottom and from left to right.

Answer 2

Just to add to Vitaliy's answer: You can see what happens with

g[1] := Plus;
FullForm /@ (g[1][3, 4] // Trace)

So, on evaluating g[1][3,4], Mathematica looks up g[1] and sees it evaluates to Plus; it's then left with Plus[3,4] which evaluates to 7.