ezietsman wrote
Mathematics is not a human invention. The whole of our current mathematical knowledge can be derived from simple numbers using logic only. Think of it as a very complex system describing how numbers (and there are many kinds of numbers) work together. Mathematical notation however, is a human invention. I'm pretty sure an advanced alien civilization will also know about calculus and algebra and how to find the slope of a parabola but their notation may (probably) look very different to ours.
Nature however, doesn't exactly follow mathematics. We have very nice mathematical formulae for many things we observe in nature e.g. the relationship between frequency, length and linear mass of a guitar string, but these are all based on simplified physics making assumptions (in the case of the string the big assumption is that the displacements are very small).
I don't think we'll solve this debate here, as it's a debate that goes on all over the place. Do "numbers" actually exist in the real world, or are they just ideas we attach to stuff we perceive? Can. Worms. Big. ? I know which side of the fence I'm on...
It's interesting how this relates to music though. In fact, I see music itself as very much an invention, an abstraction. If a random bunch of water droplets in a cave make up a pleasing little "melody", it's only the human being listening that perceives the music. The music doesn't exist on its own (or does it?). Music theory, is then a further abstraction which gives us a framework and a system for understanding and creating music of our own.
It's amazing that all these bits of noise lumped together are the focus of so much human time, effort and emotion. ?