So I got to know many geniuses along the way.
My mind could only take me just so far, before it couldn't "grasp" higher concepts, like quaternions or octornions.
But some math people could.
One guy, a friend who was bound for a PhD used to say: "(boop), all we are doing is "discovering" things that already exist. We can noodle out mathematical concepts and pat ourselves on the back for "finding" them, but that's about it."
So the answer is YES, math exists outside of the human mind.
We can do our best to ATTEMPT an understanding of its existence.
Maybe take a swipe at trying to bring it down to our level.
But in the end, no matter how smart we think we are, mathematical concepts in 2014 are scratching the surface.
I asked my professor in functions of a complex variable: "What's really the POINT of this? Fine and dandy to know all of this, but why?"
The answer: "Because we are setting the foundation for the physics of 300 years from now."
Makes sense.
We use the math from hundreds of years ago for our physics today.
Mathematics does exist outside of the human mind. But not numbers. Numbers are a convention we use to do math.
Goedel showed that the idea of computation does not by necessity imply the axiomatic system. The axiomatic system we use (Peano postulates plus a few others) may have a basis in observed reality. But it is nonetheless convention and is not strictly formalizable.