I'm a lawyer, so I always ask "Why does the answer matter?" I cannot figure out why we care what label is hung on math or anything we might want to call math.

As a speaker of English, I would say that math is an invention in the way that maps are an invention. The map is invented, but the thing to which it maps is discovered. But so what? What changes if we reach the opposite conclusion? Who loses or gains? Who goes to jail or owes damages? Who gets promoted or demoted?

I'm reminded of Henry Kissinger's quip that academic politics is so brutal precisely because there is so little at stake.