Depends. By the way, just because 2 + 2 = 4 does not follow that a2 + b2 = c2, as your example next to it suggests (which is true of a hypotenuse of a right angle triangle). If you square an equation such as a + b = c then (a + b = c)2 is not a2+ b2 = c2 but a2 + 2ab + b2 = c2. In other words 22 + 22 ? 42 just because 2 + 2 = 4. Rather 22 + 22 = (2.8284271...)2 or just plain 8.
Ptolemy's navigational system "works" even though his model is flawed (there are no epicycles), because it accounts for all the motions and positions consistently, but does not necessarily represent reality and therefore is not true. The fact that it "works" cannot be used as "proof" that it represents the way the world is, that it is a true picture of the world. Neither can quantum mechanics.
We know that Ptolemy's method is not reality because we can prove it through reality. Quantum mechanics by definition cannot be proven by reality, because the reality "breaks down" in that model. Who knows, maybe another Leibniz and/or Newton will emerge one day serpenditiously and proclaim a new math that will provide answers to our indeterminate equations.
But math itself must be verified by the perceived world. Math itself does not determine what the real world is. Math is still a human method that attempts to describe the perceived world, and is real only to the extent that it can be verified by the real world. Math is no different than words; it is a descriptive language restrained by rules.
So, then, math can be sued to describe real events but it can also be used as fiction, to describe imaginary world, just as we can write a "realistic" story of people who don;t exist, or even a fairytale, where physical laws "break" down and all sorts of "magic" takes place.
Well, duh. It was two different examples. You mentioned real numbers and "theoretical integers, i.e. x,,y,z, etc." (whatever that is.) So I gave examples of numbers and variables .
Quantum mechanics by definition cannot be proven by reality, because the reality "breaks down" in that model.
No, gross mistaken over-simplification. Theory is tested and proved by "reality." In experiments. Bell's theorem for example, the wave/particle duality of light, photons, sub-atomic particle cloud chambers, the Large Hadron Collider, etc. (For the record, these are different examples.)
So, then, math can be used to describe real events but it can also be used as fiction, to describe imaginary world…
Certainly, there is abstract math - though it follows the real world rules of all math - but the examples I gave are not abstract, they are set theory, polynomials, and basic math. I hope you're not next going to exclude calculus as not "real world."
Your initial reply requested specificity in "facts, truth, proof." Math is the most specific, most precise language. Unfortunately for the argument that "provable" and "true" are identical sets, math proves it wrong.