Suppose someone played idly manipulating small objects, like bottlecaps, in the real world. Suppose they formed (inductively) some hypotheses, picked some axioms and a formal system, derived the observed truths as a consequence of the axioms, and went on to derive some predictions regarding particular long or unusual manipulations of bottlecaps.

If the proofs are correct, the conclusions are true regardless of the outcome of experiments. If you believe that mathematics is self-hosting; interesting and relevant and valuable in itself, that may be sufficient for you. However, you might alternatively take the position that contradictions with experiment would render the previous axioms, theorems and proofs less interesting because they are less relevant.

Generic real numbers, because of their infinite information content, are not good models of physical things (positions, distances, velocities, energies) that a casual consumer of mathematics might think they're natural models of. If you built the real numbers from first-order ZFC axioms, then they do have (via Lowenheim-Skolem) some finite-information-content correspondences - however, those objects look like abstract syntax trees, ramified with details that act as obstacles to finding an analogous structure in the real world.