An individual exists at point A. Point A exists. That same individual becomes aware of point B, but is not aware of a way to reach point B. In finding a way to reach point B, he has discovered point B, but not created it. Neither has he created the way he took to reach point B; that way, and a theoretically infinite number of other ways (despite the fact that some of them are impossible to access) exist regardless of his knowledge of them.

Mathematics exists similarly. Consider a proof for Pythagoras' theorem. That proof is certain, and it provides us a way to understand the "point" that the theorem represents. Through this proof, we have obtained the personal and shared knowledge that Pythagoras' theorem is true. Next, imagine that the proof is lost. The personal knowledge remains the same, and we understand that despite our inability to remember the proof, the theorem remains true. That point is inaccessible, but it remains visible. In that way, the concept is emancipated from the proof. The concept is discovered, lost, rediscovered, but it exists regardless of our observation or understanding. 

Andrew Wiles, in creating a proof for Fermat's last theorem, discovered a truth. All mathematical truths that exist, in my view, are linked by the fact that they can be reached, or validated, by sound reasoning. Mathematical fallacies are more difficult to prove, because we have to know that there is no way they can exist on that plane of legitimate mathematical concepts. Because there are so many different potential paths, we will arguably never know conclusively that there is not one that works, meaning that we can never have a truly deductive way of knowing that there is no proof to an argument. Additionally, how does our language in mathematics tend to function? Do we say we've created an answer? It is far more common to say "found" an answer, which is accepted to already exist.

One of the biggest problems here is semantics. If mathematics is discovered, to what extent can anything be created? Consider music. A composer is working on a simple melody, containing only a single line. While no human has ever composed that line before, it could be said to exist, in the sense that the notes of which it is composed of exist, and a program that randomly arranges notes and rests would eventually arrive at that melody, no matter how long it might take. In the same way, the complete works of Shakespeare could be written verbatim by a randomising program given enough time. As for works of art, even the most abstract images are, on some level, random arrangements of pixels.

What makes math fundamentally different from this is that math, while it does to a certain extent reflect our culture, is far more objective, and far more relevant to the universe itself, rather than our specific world. While an alien culture might never paint a human, because the idea would never occur to them, they would almost certainly be familiar with the formular for exponential decay, as they would have witnessed the decay of radioactive substances. The same is true for an observation of Pythagoras' theorem. While the measurements might never be exact, they would observe the phenomenon that occurs in every case of a right triangle. There is no place in the universe that we know where the theorem would not hold true.

Whether or not these ideas exist in nature is the next question. We are certain that they are practical, but are they inherent to the world, or to the way our mind functions? What it really seems to come down to is a question of superposition. The real world exists physically, and conceptually, mathematics can exist as a layer above it. It is the nature of the inquisitive, advanced intelligence to attempt to quantify the world, to simplify it and make it understandable. It doesn't really matter how we perceive, because mathematics is a simplification of our perception. It doesn't matter how detailed a mountain might be, because we can look at it and imagine a line running from its peak to the ground below. Mathematics doesn't really have to exist within the real world. Maybe humans just got lucky, in that what we naturally do (for instance, building four sided houses with corners that are right angles) happens to be very mathematically significant. What if we preferred to divide circles into six sectors, or ten? It would be more difficult for us to see the correlations that we do. However, those correlations still exist, regardless of whether they are easily perceived. The mathematical world is objective, in that a mathematical statement can be true. A piece of art cannot be true, nor can a piece of furniture. But math rises above that, and true statements have a whole new significance across cultures. While the way they are expressed may vary drastically, I hold the belief (and yes, I did say belief) that any sufficiently advanced mind could, with enough time, grasp their meaning and significance to the real world.