I think one can get far enough by applying known mathematical facts. (Proof: elementary school math is useful.)

The problem is, usually you need to combine known mathematical facts to solve a problem. Now, how do you know that you combined them properly?

Yup. You need to know what a proof is.

This.

Applying mathematical facts isn't math, it's calculation at best.

That's what most programmers already do. They throw things together in the hope they work without proof they got all edge cases covered.

And it works pretty good, because often the edge cases aren't that bad.