Re: "But you also can't prove a negative"
From a logical point of view you CAN prove the negative of a general statement. It is enough to observe ONE instance that contradicts the general statement. On the contrary, in order to prove a general statement true, you should observe all possible instances of that statement (in natural science all occurrences of a phenomenon): in principle infinitely many.
In natural science this is relevant given the additional assumption that we can, in principle, make the measurement error, on an specific quantity, as small as we like.