Question

Why can we never prove that a hypothesis is true?

Answer 1

A theorem can be proven (from axioms or prior theorems), using logic.

A hypothesis can be supported by evidence. The more evidence in support of the hypothesis, the more likely the hypothesis is to be correct. However, you’re always at the mercy of contrary evidence appearing in the future, to reduce the likelihood or even invalidate a hypothesis.

A (mathematical) proof suffers no such vulnerability to future evidence, as long as you hold the axioms of the theory to be true, and as long as there was no flaw in the construction of the proof.