Real and imagined mathematics

Do we discover mathematics, or do we create it? Though this question is an old one, it has fascinated me lately. Firstly, an operational definition of mathematics appears necessary to discuss the subject. I would define mathematics (and I believe reasonably so) as the investigation of the properties of logical systems. These logical systems have several attributes: definitions (the meanings of various terms and operations in mathematics e.g. axis, addition), axioms (statements we take as givens e.g. the shortest distance between 2 points is a straight line), and theorems (implications of the axioms). The job of the mathematician is to deduce conclusions (on significant occasions labelled 'theorems') from the definitions and axioms. Seeing as such conclusions follow directly from the axioms and definitions via a logical progression (e.g. axiom: shortest path between two points is a straight line; conclusion: a curve is not the shortest path between 2 points) are we discovering mathematics? It would seem reasonable to say we are discovering the implications of the definitions and axioms, as those inevitably ensue within any system with given definitions and axioms. However, seeing as the axioms and definitions were decided upon by ourselves, one might also argue that we create the definitions and axioms and therefore create mathematics.

The issue with this line of reasoning, it seems to me, is that while we may agree on certain axioms and definitions, I don't believe this implies we've created them. We have simply arrived upon certain ideas and agreed upon them. Before man defined what a line was, did the concept not exist? I think it's important to specify that I am not talking about real world approximations to the concept of an ideal line, which undoubtedly existed well beforehand, but rather the concept of an ideal line itself. Is it fair to say that we arrived upon this concept and accepted it, or that we invented it? Indeed, to say we invented it would seem to imply that the definition of a line depends entirely upon the mental capacity of human beings. This might imply by extension, that if our mental processes functioned differently, the definition of a line would change, a notion that seems utterly at odds to with me. We might call it differently, and refer to it in different ways, but the notion of a line remains the same in my opinion irrespective of human existence.

If we examine the other line of reasoning (which I am more inclined to), that we discover mathematics, this would imply that mathematical concepts have an independent exist of their own thereby allowing us to discover them. The immediate question here, is if mathematical concepts have an independent existence, where do they exist? Here we find ourselves returning to platonic ideas: the theory of forms. The theory of forms states that abstract concepts such as mathematical ones have an independent and immaterial existence outside any parameters we associate with the physical world, such as space and time. This is a hard notion to ever accept, and one for which by definition we can never produce empirical evidence. Nonetheless, it works for me.