My opinion on Randi went down, of course you can prove negative statements. (Edit: In fact, here's a PDF: http://departments.bloomu.edu/philosophy/pages/content/hales... But the most obvious instance is that anything you prove true you can also prove is not false.) Anyway, it is important to note that "prove" in this English, not mathematical, context really means "suggests", if you want to be technical and pedantic. It suggests that other users who share similar traits to the person in the video will also experience difficulties.
> However, it would be a grievous mistake to insist that someone prove all
> the premises of any argument they might give.
[...]
> So why is it that people insist that you can’t prove a negative? I think it
> is the result of two things. (1) an acknowledgement that induction is not
> bulletproof, airtight, and infallible, and (2) a desperate desire to keep
> believing whatever one believes, even if all the evidence is against it.
You can prove negative statements within a set of assumptions. But, as even this
author acknowledges, it is impossible to absolutely prove anything in all cases, and
I think that is what Randi basis his argument on: that nothing can be absolutely proved
false in all cases.
Disclaimer: I'm not an epistemologist, nor a philosopher, nor an experienced logician.
I may have misread this PDF and/or Randi's speech.
I actually believe it's primarily a third reason the author didn't mention: (3) not understanding what a "proof" is and what it means to "prove" something. Of course it ties into the 2nd reason in that many people aren't motivated to learn what a proof actually is.
The "impossible to absolutely prove" phrase has a misuse of "prove". A proof is absolute. A proof is either valid or invalid judged solely on whether it follows the rules of the predicate calculus (or the rules of some other proof-framework you're using which is likely using the predicate calculus or an extension behind the scenes anyway). A proof's conclusions are "true" if and only if they are true. ( http://yudkowsky.net/rational/the-simple-truth )
It is true (so far as we know, we might be wrong in the end) that we can't be absolutely certain that something is false, but it's the same case for being absolutely certain that something is true. (Here "absolutely certain" means "no admittance to even the possibility of being incorrect".) This is more generally identified as the problem of induction, but it's more of a law than a problem. One can guess from the name it has to do with inductive arguments that rely on guessing+evidence rather than deductive proofs that rely on accepting premises (which may be true or false) and a proof framework like the predicate calculus.
Edit: Have a Feynman video on UFOs. :) http://www.youtube.com/watch?v=wLaRXYai19A When he says "I can't prove it's impossible", the interpretation you should take is "I can't produce a set of premises we can all agree with that leads to a deductive proof that UFOs are impossible."
James Randi's lecture [1] on proving a negative (in this case, "windows 8 doesn't work for users") is highly relevant and poignant.
[1] http://www.youtube.com/watch?v=qWJTUAezxAI
EDIT: fixed link formatting