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.
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.