from a certain perspective, it's hard to talk about currying without talking about sets. "function" itself is defined as a special type of relation between sets. imo, the first paragraph does a good job introducing both the formal and practical meaning of currying to people who have enough background to find the topic useful.

simple wikipedia does take your approach though:

> Currying is a technique used in mathematics and computer science that consists of changing a function that takes several arguments into a number of functions that each take one argument. Mathematicians Moses Schönfinkel and Gottlob Frege laid the groundwork for this technique, which is named after Haskell Brooks Curry. Currying is used in Lambda calculus. Some programming languages, such as ML and Haskell say that functions can only have one argument. [0]

that's actually the entire article. not sure how useful it is, but at least they tried.

[0] https://simple.wikipedia.org/wiki/Currying