Warning: The following is very long. Feel free to read only parts of it. For those who care, my qualifications for speaking on the math that follows are a Master's Degree in Mathematics which will be conferred in August as part of a pursuit of a Ph.D in the same subject. Unless you actually come to the graduation ceremony I obviously can't prove it to you though =P
Nothing is provable.
As long as we base our knowledge on epistemological evidence and allow society's prejudices to cloud our judgement, truth is nonexistant. However, we have no other realistic, objective way of seeing the world. Therefore, this argument is meaningless. In fact, all arguments are meaningless. Nothing can ever be proven for sure.
Hi thanks for playing. You are now no longer part of any rational discussion so long as you hold this axiom because if I assume you are right then I have absolutely no reason to believe any of your arguments. The proof that there are no proofs is self-contradictory on its face.
As a few have said it is in fact quite possible, and for mathematicians is a somewhat mundane affair, to prove a negative. Impossibility theorems are rife throughout mathematics and since all science is based on math at the end of the day, the only reason they don't like that is because it can't be done experimentally(as pointed out so humorously in the posted video).
A prime example is one of the so called "problems of antiquity" and it deals with the problem of trisecting a given angle. As commonly formulated the idea is that given ONLY the tools of an unmarked straight edge and a collapsible compass, it is impossible to construct an algorithm to trisect a given arbitrary angle. This is not to say that some angles can't be so trisected, but rather the impossibility of doing it in general. And the impossibility of doing so is in fact provable. With such a proof in hand I know ahead of time that if anyone claims to be able to trisect an arbitrary angle with just an unmarked straight edge and collapsible compass, that they are either mistaken or lying. I don't even have to look at their algorithm. I know it isn't possible, and unlike scientific "impossibilities" which historically have often proven to be wrong, this one is iron clad. It simply is not possible.
In general this is how the Reducio Ad Absurdum (i.e. proof by contradiction but with fancy foreign words) works. As others have mentioned you assume that it is in fact possible and derive a contradiction. Like I said, mathematicians do it all the time.
What you fail to realize (or refuse to) is that the claim your imaginary person is refuting has already been proven. Look at a high school physics book and you will see how the scientific community has laid down explanations for all the concepts they hold. The reason why the scientific community wouldn't need to refute your hypothetical person is because a convincing case for their theories already exists, and is readily available for anyone interested in learning about such concepts, so it falls on hims to prove his theory.
Please stop being ridiculous, you are trying to claim that because religion has existed for so long it does not need explanation. To this day there is no convincing evidence for the existence of god, nothing that can be easily analyzed or reproduced. The burden of proof always falls to those who make an exceptional claim, saying there is a supernatural being who created everything and sees everything all the time is far more outlandish them saying "but there is no evidence of such".
Actually what you fail to realize is that his point is that at one point the existence of God was the established belief. The general falling away from that belief was not one of scientific inquiry at all and now you have two camps with firmly established beliefs and the fact is that whichever one you march into, if you make the claim that they are wrong (i.e. walking into a church and claiming God doesn't exist or walking into a skeptics conference and claiming that He does) then the onus IS in fact on you to support your claim.
Again, while the posted video was wrong about being able to prove a negative, though he was quite right about the inability to do it experimentally, he was absolutely right that the burden wasn't on him because he wasn't a debunker. If he WAS a debunker than the burden of proof would be on him. Since he knows he can't prove a negative experimentally, and since he is loathe to dabble into any other methods of proof, that is a profession which he has declined, but that doesn't change the fact that he knows full well that if he was a debunker, e.g. if he was going into a church and making the claim that God didn't exist then the burden of proof would be his, not the church's.
Personally if it doesn't apply to my senses and isn't provably by science then it doesn't exist
In a lot of ways this is pure and utter nonsense, but its an attitude that I see on this thread a lot. From the probability argument all of human history, having only happened once, is immensely improbable. Probability itself cannot be upheld by any type of scientific experimentation, and the entire idea of a Universe which conforms to some sort of uniformity, under which alone probability has any meaning, is itself accepted on pure faith by the entire scientific community.
To wit: The Battle of Waterloo, like all historical truths, is one which is impossible to test by experimentation. And yet we feel reasonably certain that it happened. Believe it or not, the many Christians, and other religious folk around the world, who DO believe in a personal God do not do so because we believe He was just dreamed up. We believe it on what we feel, rightly or wrongly, to be good historical evidence, combined with our own transcendental experiences whose consistency with the reports of fellow believers makes us confident that they are not mere illusions.
I feel I have sound logical proof that this interlocking system of nature cannot be all there is (read Miracles by C.S. Lewis for a good treatment on the subject if you want. I don't feel like getting into it here), and that something outside nature (and so supernatural by definition) exists. But I can no more scientifically, or rigorously prove that God exists than I can prove any other historical fact, or fact about something which is literally outside the observable universe, and the thing is I accept already that such a proof will never be possible. That is even if science were ever able to attain all knowledge which lies within its ability to obtain, assuming such a thing is possible, I know science will never be able to prove positively or negatively that God exists. It simply lies outside science's ability.
All that to say that you do your fellow man a great disservice by assuming that because he believes something on faith that he has no reasons whatever to support it. The Judeo-Christian faiths of the west (I confess to being too ignorant of eastern traditions to comment accurately about them) do not claim to have simply thought up God but claim that He revealed Himself to them. If you believe a priori that there is no God then all such claims are automatically ludicrous on their face, but in such a case you need to have proof that no such God exists, just as I need a proof that it is impossible to trisect an angle with collapsible compass and unmarked straightedge before I dismiss all claims to the contrary out of hand. Otherwise you must be willing to evaluate the historical evidence on its face, or, much more likely, come to a conclusion about the possibility of such claims, and if possible, the probability of such claims, before they can be so evaluated. Looked at with either a mind open enough to accept other historical facts, or a mind which has settled positively on both the possibility and probability of such a fact, I do believe the historical evidence holds up, but both the possibility and probability are things which must be handled completely logically and outside the realm of experimentation because I tell you now that no such experimentation is possible.
Ok at this point I know I'm getting way too long and wordy and lets face it these are pretty weighty matters to be discussing on what, at the end of the day, is a video game forum, and I won't blame anyone who goes "TL;DR". But for those who do read I appreciate any thoughts on the matter.