You are making the assumption we are "discovering" math, not inventing it. This is a position that does need to be supported before assumed.
And how exactly do you propose we differentiate between the two? Discovery implies that something exists before it was observed or known about; in this case, I would argue that mathematics *does* exist prior to our knowledge about it in the sense that it represents the physical and logical laws which govern the universe. It is conceptual, to be sure, but no more so than an expression of Newton's laws that does not rely upon mathematics. "Matter tends to maintain inertia" is no less conceptual than "2+2 = 4" and, as far as mankind is aware, both statements reflect reality.
I'd wager that it will not match exactly with predictions made using an abstract proof. It is ridiculous to claim it will. Perhaps you meant it will "match close enough for engineering and stuff."
How is it ridiculous? If you measure out a right triangle using the most accurate methods available, its dimensions will exactly match the predictions made by the Pythagorean theorem and other relevant mathematical proofs. The only limit on the degree to which these things can be verified against reality is the limit of our ability to measure things.
The Pythagorean theorem, and other similar maths, are not empirically-determined approximations. They are logical constructions based on the way we define the world around us. Your argument implies that even simple things like distance measurements along a straight line are only inaccurately expressed by mathematics. If I draw a line along the x-axis beginning at the origin and stop at x=6, then the line will be exactly 6 "units" long. If I put a long piece of lumber on the ground and measure exactly 6 feet from one end then cut the lumber, the lumber will be exactly 6 feet long. Further, I can accurately predict this result beforehand using the mathematical technique of drawing a line on a coordinate axis.
Simply put, mathematical results are accurate depictions of reality because the simplest axioms and definitions are derived from real concepts, and because deductive logic guarantees that a valid conclusion drawn from a set of true premises is also a true conclusion. If you have beef with the veracity of mathematical representation of reality, then you have beef with deduction itself.
Now, if you're merely pointing out that sometimes mathematics is used as an approximation to reality, then that's a different story. In this case specifically, it is clear that the haversine formula for calculating distances is perfectly accurate only for perfectly spherical surfaces. If RET is correct and Earth is a globe, however, then it is also true that the Earth is only approximately spherical and is really a bit deformed. Haversine calculations, then, are not going to be exactly precise -- but because the deformation is small relative to the size of the Earth, they will still be more accurate than straight-line calculations based on flat maps.
When one looks into hyperreality, similculcrum and Jean Baudrillard. I imagine some of them are very pertinent to this discussion as well, however, like I said, I'll have to leave that to others to discuss.
I've never put much stock in Baudrillard's arguments. I think he over-defines his notion of what constitutes reality, and ends up excluding a lot of things that I believe should be included as "real." His position is only tenable if you accept the argument that the meaning of something (or our interpretation of this meaning) can never change without rendering it inaccurate or somehow less meaningful.
Kantian dualism, I think, is a little closer to the mark -- the actual physical manifestation of the universe is just as real as the conceptual framework that we use to describe it, in the sense that we interact with and are influenced by both. The map and the territory may be separate entities, but both are real and each can be used to influence the nature and existence of the other.
That said, however, I do agree with Baudrillard that it is possible for representation to entirely overtake reality. But I wonder, what would be the negative effects? In Baudrillard's ultimate conclusion of a situation in which our representations of reality no longer are actually representative of reality, we would have interactions only with those representations and no longer would interact directly with reality. In this case, I would argue that the representations themselves constitute reality and that it would no longer matter what the independent reality actually was.
If Neo had taken the blue pill and stayed in the Matrix, would it have mattered from his perspective? In all honesty, I'm with Cypher on this one: the Matrix (simulacra) can be just as real as the real world, and if there's nothing telling you the difference or no way at all to tell the difference, then from the point of view of someone inside there is in fact no difference at all. The Matrix *is* the real world, and whatever Baudrillardan simulacra we create are, or eventually become, the truth of reality.