One of my mentors in college was an expert on wireless communication systems, and he taught a class every few years about satellite communication. As I recall, the class project involved him giving students data about a signal from a satellite, and then they had to use the data along with everything they had learned about satellite orbits to calculate something about a particular point on the earth. Basically, they had to write their own GPS algorithm. So they had to learn all this math and physics about satellites orbiting the Earth, and it all worked - the math all made sense and matched what the satellites actually do in orbit as we see them from Earth.

Now, none of this proves that satellites are real. But what's interesting to me about it is that it does prove (to someone who knows the math well enough) that the story is self-consistent. The math and physics that tell us how gravity, planets, orbits, etc work is all essentially correct. So it would seem very very strange to me if all the math is right and the math tells us we love on a globe, but we nevertheless lived on a flat disc.