[voxfrege]

Well, probably the word "discovered" would fit better. :)

Here is a paragraph from "Funktion und Begriff" (1891):

> Wie nun Funktionen von Gegenständen grundverschieden sind, so sind auch Funktionen, deren Argumente Funktionen sind und sein müssen, grundverschieden von Funktionen, deren Argumente Gegenstände sind und nichts anderes sein können. Diese nenne ich Funktionen erster, jene Funktionen zweiter Stufe.

For non-german speakers: Frege makes a distinction between functions that take things as arguments and functions whose argument are and must be functions. He calls the former ones "first order functions" and the latter ones "second order functions".

Today we call functions whose order is greater one "higher order".

[mafribe]

Thanks, that's a nice quote. I had read that text as a student, but overlooked this morsel.

I wonder if Frege's older Begriffsschrift (1879) doesn't already discuss, or at least mention, higher-order functions. After all, in this text Frege explains his new conception of function.

I also wonder if Cantor would have been aware that this is possible.

[voxfrege]

I haven't read the "Begriffsschrift", but you are right. It is probable that he had developed his formal apparatus already then.