In another thread on Science 2.0 we got onto the nature of existence and what we can say for certain about the universe (if anything).  Given that physics and math seem so intimately linked in practice, I often wonder if this link is "real".  In other words:

Do numbers and other mathematical entities actually exist, or are they made up by the human mind?

This is a question I've asked in a number of forums before and have only gotten one interesting reply.  Derek Abbott said he doesn't know, but thought it was a paradox.  To paraphrase Derek: "Nowhere in the universe does a perfect circle exist; only approximations.  Yet we can imagine a perfect circle in our minds, and unless you believe cognition is epiphenomenal, our minds exist in the universe, and thus so do perfect circles...."

I struggled on whether to post this in Physics or Math, but I'm more curious as to the physicist's answer.  Anyone?