The mathematician Dirichlet (1805—1859), around 1830, proposed a definition for “function” that is essentially the modern set-theoretic definition of function, namely:

- If and
- If ,
- Then .

See a more detailed explanation here.

The cool thing about this definition is that we don’t have to think about functions as “formulas” or “black boxes” anymore. It’s an ontological change: it opens up a lot of possibilities for what a function can BE and what it can DO.

Dirichlet used the following function as an example of one that would be hard to make sense of without this newer definition.

*Where is the Rational numbers.*

The domain of this function is ALL of , the Real numbers. But the range is only the tiny set of two elements. This is a pain in the ass to graph, and it would be rather tough to come up with a “formula” for it in the traditional sense. But with the set-theoretic definition it is trivial.

*Now go lift something heavy,*

Nick Horton