It may be different in mathematics, but—

On the original question:—    No. Demanding that one prove a negative is on the formal list of logical fallacies. The matter has been argued since Aristotle's time, and it always comes out the same. It's called the Fallacy of the Appeal to Ignorance (Proving a Negative). The rule is "an argument that asserts a claim is true because no one can prove it is untrue attempts to shift the burden of proof from the claimant to the opponent."

The way this works is I say, "What do you mean, I'm fishing through the bowl of mixed nuts and picking out all the cashews? I'm not the one who's doing it. There's an invisible gremlin (or tiger, if you insist) in this room, and he's doing it." You say, "That's crazy! There's no such thing as an invisible gremlin." I say, "Oh, yeah? Prove there isn't!"

Unfair. And illogical. Now, I could try to claim that your statement "There's no such thing as an invisible gremlin" is an assertion and that you have to prove it, but since I'm the one who brought up the gremlin in the first place, it falls to me to supply the proof. Basic freshman logic, Philosophy 115.

Our system of jurisprudence is based on this. You are innocent until proven guilty. If you are accused of a crime, it is the responsibility of the accuser to supply the proof. You are not required to prove your innocence.

Anyone who wants to engage in spirited discussion and not wind up throwing a lot of logical gutterballs ought to spend some time with this page:   http://chuma.cas.usf.edu/~pinsky/logicguide.htm. [I tried to put the URL in a link, but for some reason, Mudcat's "Preview" told me it was a "forbidden HTML tag" and woundn't display it, so you'll have to crank it by hand.] Good stuff, and fairly concise. Print it out. Keep it handy.

Don Firth