I always struggled with math, so I just don’t get how an astrophysicist could look at some numbers and symbols and say, “Aha, it’s proof of black holes!”
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Math and Physics are developed by observing what you see, trying to make a rule that describes what you see, and then testing that rule in other situations to make sure it holds up (basically the “scientific method.”).
Math showed the existence of black holes from two directions. One direction was that we observed planets and stars breaking existing rules, by having strange orbits or by stretching and visually warping in strange ways. These anomalies _would_ match existing rules, only if there was an extremely dark, and extremely dense object there that we couldn’t see.
The other direction is that we had existing rules and formulas that allowed the existence of such objects, but in order for the equations to work in a way that matched our observations, the object would have to be infinitely dense and it’s gravitational pull would be so strong not even light could escape.
We started with rules developed using the scientific method, but we also have also been using the scientific method in our steps to figure out the black hole: we observed something weird, we came up with rules to match it, and now all that was left was to test these rules. This is where things like observing gravetational waves and attempts to photograph a black hole come in, as they allow us to see if our predictions were correct.
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