"You cannot prove a negative" basically means that you cannot prove that some claim or statement is, was, and always will be false (without perfect knowledge of the past, present, and future).

That parenthetical is an important and almost always unstated axiom.

The general inability to prove a false statement does not mean you cannot prove that the answer to some equation is a number below zero. I am not really aware of the phrase being used in the context of math, but rather more often with examinations or experiments that are susceptible to evidence.

To be sure, "you cannot prove a negative" is itself unproven. It more a rule of thumb to remind you not to assume that though some statement is false now that it always was false and always will be false.

It's not perfect, but it's also not a law of logic or anything. It's just a guideline.

That phrase applies basically everywhere else except math. It's only possible to prove a negative in mathematics because it's a fully logical system.

On a side note Fermat's Last Theorem isn't a good counterexample because it hasn't been proven yet either.

Careful with that phrasing -- you need to prove that _only_ y are true.

I don't see a hard line between math, then physics, then the 'real' world. They are levels of formalization.

Both in math and in the real world we don't have 'perfect observation'. There is plenty of conjectures in math and the real world that lack a proof of something being true or false.

I think "can't prove a negative" is one of the least informative ways of trying to say something, I assume he meant to say "absence of evidence is not evidence of absence" or perhaps "absence and evidence don't commute"

Please elaborate