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 .
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,