Here’s what I think:
Yes, the things that math seeks to explain are already there. If humans did not create math, this language, to explain those things, however, math would not exist. Math is a human creation, a human invention we use to explain natural phenomena that we discover.
Contrast this with the work of scientists. Like that radio show that @Rarebear posted pointed out, a discovery can be made by many people at the same time or by no one at all, but it exists regardless independent of humanity. Indeed, Darwin rushed to publish his Theory of Evolution because he was afraid someone else would beat him to it. *
What math seeks to explain exists regardless of humanity, but math is a tool created by man that would not have existed without man.
Then you get to the level of math, beyond what I am capable of fully comprehending, where you start to discover things that can exist within the man-made framework of the language itself that do not yet exist – to the knowledge of mankind – in reality.
Math is an iterative process, leading mankind beyond the cutting edge and yet owing its birth to man. It would seem math is both an invention and a discovery…kind of like Hal.
* As a side note, this logic is why I think genes should not be allowed to be patented as they are discoveries, not inventions. We cannot know with any certainty that the genes created in lab do not already exist in the world somewhere – where we have not yet thought to look. What they have “invented” may have already existed in the world but has yet to be discovered. After all, they have discovered that it is possible for that gene to exist in the world, which is kind of like the pathway to proof that their discovery is not an invention.