When you define something in mathematics, can you do no appealing to what it does?
For example, I can define the a function by its outcome. f(x) = 2x is something that doubles x. But, I wouldn't be able to tell you what it is.
In contrast to real life doodads where you can define, for example, a knife by its shape + material or by what it does (cut).
Are mathematical doodads defined just by that, by what they do?