The math bothers me

I don't have a problem with math as an attempt to model systems, but when it attempts to be its own meta reality it just seems stupid. I can say:
"What is the square root of the color of cats?"
and make as much sense as some concepts that have been adopted. What is bugging me right now is the product of two negative numbers is a positive number. Now, who thought up that silliness? What exactly does it have in common with anything in the known universe. It is like having a math co processor that adds a little endian int and a float and calls it a long, then stores it as a big endian string. It just leads to strange situations and the most notable is sqrt(-1).

The La^Tex for the equation at the head of this post.

\varphi = \frac{q}{\hbar} \int_P \mathbf{A} \cdot d\mathbf{x},