I often tell students when introducing new (to them) concepts that when mathematicians hit a brick wall in pursuing some investigation, they invent something to knock down the wall. Mathematics is purely an invention of the human mind.
Another way of looking at the same thing is that the tool to knock down the wall was there all along just waiting for someone with the need to discover it. All of mathematics exists as a fundamental part of the universe independent of us.
In thinking about this, I realized that I use the terms “invent” and “discover” interchangeably when talking about mathematics, and yet they are not the same.
Theoretical astrophysicist Mario Livio has an article in the August Scientific American titled “Why Math Works” where he examines both sides of this philosophical argument … and comes down squarely in the middle.
The article is at www.sciam.com; however, I think you need to be a subscriber to read the whole article (unless the college has a site license).