When I was a maths teacher some curious students (Fez and Andrew) asked, “Does i, √−1, exist? Does infinity ∞ exist?” I told this story.
You explain to me what 4 is by pointing to four rocks on the ground, or dropping them in succession — Peano map, Peano map, Peano map, Peano map. Sure. But that’s an example of the number 4, not the number 4 itself.
So is it even possible to say what a number is? No, let’s ask something easier. What a counting number is. No rationals, reals, complexes, or other logically coherent corpuses of numbers.
Willard van Orman Quine had an interesting answer. He said that the number seventeen “is” the equivalence class of all sets of with 17 elements.
Accept that or not, it’s at least a good try. Whether or not numbers actually exist, we can use math to figure things out. The concepts of √−1 and ∞ serve a practical purpose just like the concept of ⅓ (you know, the obvious moral cap on income tax). For instance
- if power on the power line is traveling in the direction +1 then the wire is efficient; if it travels in the direction √−1 then the wire heats up but does no useful work. (Er, I guess alternating current alternates between −1 and −1.)
- ∞ allows for limits and therefore derivatives and calculus. Just one example apiece.
Do 6-dimensional spheres exist? Do matrices exist? Do power series exist? Do vector fields exist? Do eigenfunctions exist? Do 400-dimensional spaces exist? Do dynamical systems exist? Yes and no, in the same way.