Are mathematics discovered or invented?

The more you study physics and the real world, the more you marvel at the pure mathematical nature of existence. So you ask a very interesting question, one that could be, and has been, debated for years.

My take is this, what if aliens show up and they're using the identical math as us? Then we discover math.

Math doesn't exist in nature.  Or in other words science is an attempt to describe the physical world but math isn't.  Math is a closed system unto itself.  It's a human construct that is used as a tool to help us do science.  Or it can be done simply for it's own sake.  Many mathematical solutions precede their usage in the real world, if indeed there are any.  Sometimes Math is even an art.  One can do a Math degree in the Faculty of Arts if one wants (discrete math for example).

Mathematics developed as humans needed the discipline.