In short, the ultimate consequence of the appearance of a FASI would be that every intelligent being in the universe would become omnipotent.

Even if we kept all your hypotheticals , you still need to consider that we may be forever lost to these civilizations because of distance. Nick Bostrom said the following:

However, cosmological theory implies that, due to the expansion of the universe, any life outside the observable universe is and will forever remain causally disconnected from us: it can never visit us,communicate with us, or be seen by us or our descendants

I just noticed this argument, I hope I'm not too late in expressing my view.

Premise: I want to live in the universe with the least amount of pain.

And now for some calculations. For the sake of quantification, let's assume that that the single tortured person will receive 1 whiplash per second, continuously, for 50 years. Let's also assume that the pain of 1 whiplash is equivalent to 1 "pain unit". Thus, if I chose to torture that person, I would add 3600 "pain units" per hour to the total amount of pain in the universe. In 1 day, the amount of pain in the universe would increase of 360024 = 110400 pain units. In 1 year, approximately 110400365+36006 = 40317600 pain units. In 50 years, approximately 4031760050 = 2015880000 pain units. And now, let's examine the specks. They were described as "barely enough to make you notice before you blink and wipe away the dust speck". In other words, while they can be felt, the sensation is insufficient to trigger the nociceptors. This means that each speck increases the level of pain in the universe of 0 pain units. So, if 3^^^3 people received each a dust speck in one of their eyes, the amount of pain in the universe would increase of exactly 0*3^^^3 = 0 pain units! This is why I would definitely choose SPECKS.

"I wish that wishes would be granted as the wisher would interpret them".

Hi DevilMaster, welcome to LessWrong!

Generally, the answer to your question is Bayes' Theorem. This theorem is essentially the mathematical formulation of how evidence ought to be weighed when testing ideas. If the wikipedia article doesn't help you much, Eliezer has written an in-depth explanation of what it is and why it works.

The specific answer to your question can be revealed by plugging into this equation, and defining "proof". We say that nothing is ever "proven" to 100% certainty, because if it were (again, according to Bayes' Theorem), no amount of new evidence against it could ever refute it. So "proof" should be interpreted as "really, really likely". You can pick a number like "99.9% certain" if you like. But your best bet is to scrap the notion of absolute "proof" and start thinking in likelihoods.

You'll notice that an integral part of Bayes' Theorem is the idea of how strongly we would expect to see a certain piece of evidence. If the Hypothesis A is true, how likely is it that we'll see Evidence B? And additionally, how likely would it be to see Evidence B regardless of Hypothesis A?

For a piece of evidence to be strong, it has to be something that we would expect to see with much greater probability if a hypothesis is true than if it is false. Otherwise there's a good chance it's a fluke. Furthermore, if that evidence is something that we wouldn't expect to see much either way, than it's not very informative when we don't see it.

So you see how this bears on your examples. I'm not especially familiar with astronomy, so I don't know whether it's true that we haven't seen other galaxies with planets, or how powerful our telescopes are. But let's assume that what you've said is all true.

If we know our telescopes aren't powerful enough to see other planets, then the fact that they don't see any is virtually zero evidence. The probability of us seeing other planets is basically the same whether they're out there or not (because we won't see them either way), so our inability to see them doesn't count as evidence at all. This test doesn't actually tell us anything because we already know that it will tell us the same thing either way. It's like counting how many fingers you have to determine if the stock market will go up or down. You're gonna get "ten" no matter what, and this tells you nothing about the market.

The same reasoning applies to the bacteria example. If we're not more likely to see them given that they're real than we are given that they're not real, then our inability to see them is not evidence in either direction. The test is a bad one because it fails to distinguish one possibility from the other.

But all this isn't to say that it would be valid to reject these notions based on the absence of these evidences alone. There may be other tests we can run that would be more likely to come out one way or the other based on whether the hypothesis is true. So no, it wouldn't make sense to reject the existence of planets or bacteria, because in both of your examples people are using tests that are known to be useless.

If absence of proof is not proof of absence, but absence of evidence is evidence of absence, what makes proof different from evidence?

Example: we currently have no evidence supporting the existence of planets orbiting stars in other galaxies, because our telescopes are not powerful enough to observe them. Should we take this as evidence that no galaxy except ours has planets around its stars?

Another example: before the invention of the microscope, there was no evidence supporting the existence of bacteria because there were no means to observe them. Should've this fact alone been interpreted as evidence of absence of bacteria (even though bacteria did exist before microscopes were invented)?