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Why are we so confident about black holes mathematically…

Why are we so confident about black holes mathematically, when we can't even solve problems in our own mathematics?

I have no physics background, this is just something that occurred to me. We can't even solve some problems in the mathematics we invented ourselves (like Goldbach's Conjecture or Riemann Hypothesis). So how can we speak so confidently about black holes using that same mathematics, when we've never actually observed one directly or been anywhere near one?
#science
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