*When I was a math 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.

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Tags: ℝ, √−1, ∞, complex numbers, education, equivalence class, equivalence classes, i, imaginary numbers, math, mathematics, maths, philosophy, science, why, Willard van Orman Quine

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