I believe this is largely the reason Galileo is often considered the first scientist. When Copernicus argued for a heliocentric world-view, he was largely doing so on the basis of aesthetic value: it simplified the mathematics. But it was always theoretically possible for another mathematical model to make the geocentric perspective more appealing, so anybody who was attached to it could simply cast his model off as a mathematical trick.

But Galileo built a telescope that anybody with a pair of eyes could use, no matter your aesthetic preferences, and he was compelled to transform his world-view on the basis of the facts he observed in it. He didn't really have to make any mathematical argument at all; he just had to explain what he observed, and it directly contradicted the Aristotelian model of reality.