Yes. Functions are arbitrary.
In modern mathematics, and in particular in set theory, a function is just a set of ordered pairs which have some properties.
One can ask, what is a natural number? The answer, intuitively, would be "you know... like 1,2,3,4and so on.", but that's not a mathematical answer. The mathematical answer, circular as it may be, is probably along the lines of "an element of the standard model of the Peano axioms".
And so a function is not something which necessarily coheres with our intuition, like f(x)=x+5or so. Functions are just sets which satisfy a certain property which makes them functions.
manpreet![Tuteehub forum best answer]()
Best Answer
2 years ago
I'm learning about functions between sets. I get the concept. But I'm having a hard time drinking the kool-aid, so to speak. There's some part of my mind that thinks (loudly),
But I get that -really- all a function between two sets is doing is saying is
But still, intuitively, it strikes me as potentially problematic. For instance, we used to be able to assign any element we wanted to a set - that turned out to be a problem. I don't necessarily wonder whether functions would lead to the same problem, but perhaps to some other problem arising from arbitrarily relating things?
Do functions have the potential to lead to that kind of a problem, or any other problem? Is there a better way to think of them than I've described?