How to show the truth of mathematical claims empirically?

In mathematics, we have many methods to prove the truth of mathematical theorems. However, many consider mathematical claims to be empirical claims; in other words, mathematical claims require evidence from the real world to validate.<br><br>Some examples of the empirical nature of mathematical claims: <br><br>* the Pythagorean theorem works in the real world<br>* geometry works in the real world (e.g., Apollo missions to the moon, architecture)<br>* the Fundamental Theorem of Calculus works in the real world (e.g., rocket trajectories)<br><br>When we want to show the truth of mathematical claims empirically, what methods do we use?