You can not Prove a Negative
A popular phrase used by skeptics is that "you can not prove a negative". The idea behind this phrase is that in the eyes of a believer in pseudo scientific concepts, the paranormal, cryptobiology or the like, it is often impossible to provide enough evidence of the absence of something to convince the believer beyond all doubt that "the something" does not exist. The phrase infers:-
"you can not prove a negative proposition beyond all doubt to a true believer".
A negative proposition is saying that something either does not exist or that something has no effect. Examples would be "extraterrestrial aliens are not visiting the Earth", "the Loch Ness Monster does not exist", "homeopathy does not work", "the Earth is not flat".
To prove that extraterrestrials are visiting the Earth, all we need is a single piece of hard evidence. To prove that they are not visiting the Earth, you can never provide a single piece of evidence to prove your negative position. You can come to a conclusion that something is very, very, very unlikely when no solid evidence turns up despite so many people looking for it. Evidence is showing up, but being found to be manufactured, or from a known cause, or of such bad quality that it could be of just about anything. The more people turn their cameras to the sky only to catch no more concrete evidence, the more likely it becomes that we are not being visited by extraterrestrials in spaceships. Sure, the absence of evidence becomes increasingly evidence of absence, but it never quite becomes proof of the negative position. There is always a slim chance that concrete evidence may surface tomorrow.
"The Earth is not flat" is a statement supporting the overwhelming evidence that the Earth is an oblate spheroid. This evidence comes in the form of photographs from space, observations from Earth and a myriad of other evidence that the Earth is not flat. The evidence is overwhelming to the point that almost everyone on Earth accepts it, but the Flat Earth Society will give you a myriad of reasons why things appear the way they do to most people. To the people of the Flat Earth Society, you can not prove that the Earth is not flat.
In the context of the statement "you can not prove a negative" it is not being inferred that you can't prove a negative equation. Nor is it concerned with observations that if you throw in a double negative, you can make any positive statement appear to be a negative. For example, "the Earth is a spheroid" versus "the Earth is not not a spheroid". Simply expressing something the second way does not automatically make it a negative proposition. It is still a positive proposition because two negatives always cancel each other out.
We Can Come Very Close
Having established what we are actually referring to when we say "you can not prove a negative", I will now refer you to an article by Steven D. Hales in which he discussed this very issue. However I would caution that Professor Hales takes a little time to get to the real meaning of the phrase. You can virtually skip to the second page to get to the part addressing the nature of the claim.
Professor Hales says "you can prove a negative beyond all reasonable doubt", which is exactly how reasonable people think. He makes a strong argument for inductive reasoning as evidence of absence, or proving a negative proposition, which is fine for the vast majority of reasonable people. Whilst Professor Hales may not have understood the intent of the "can not prove a negative" phrase, his final paragraph, in trying to explain why reasonable people can't dismiss inductive reasoning, actually supports the intended nature of the original claim. The claim "you can not prove a negative" is aimed at unreasonable people, the true believers of any claim that reasonable people would consider to be extremely fanciful. So in the intended nature of the claim it is correct, you can not prove a negative. Evidence of absence is not absolute proof, and absolute proof is often the only thing that will convince a true believer. In other words, you can not convince a true believer. You can not prove a negative.