How? How can you prove that gravity works the way we think it does?
You get the short version here: have you read OPAR chapter 5? Having ruled out arbitrary claims as unworthy of our attention, we focus on evaluating evidence relevant to the claim at hand. Sometimes you have some supporting evidence, but not enough to rule out alternative claims and perhaps evidence that is equivocal. As far as the latter is concerned, there may be an instrumental reading that appears to contradict the claim, but you also know that under those conditions, the instrument is not fully reliable – i.e. you are not sure what the facts are in this case. That gives you a reason to doubt, and it tells you where you should direct your attention. You can reach that state where all of the evidence points towards the claim, and no evidence indicated that a competing claim is possible. That is when you are certain, and the claim is proven.
I think we can be fairly certain, but I won’t rule out the possibility that we’re wrong.
You really should read OPAR 5. Human knowledge is limited and it is contextual. The possibility of error does not preclude the notion of proof.
As for Popper and his nihilism, again I suggest Stove for a free read. The essence of Popper’s claim is that a “for all X” statement cannot be proven true except by inspecting an infinity of entities, which cannot logically be done; but you can refute such a statement by pointing to an X where the claim is false. The “it’s not logically possible” claim rests on the false assumption of actual infinities, and without actual infinities there is no argument from principle. There is of course the argument from practicality, so it’s not at all practical to inspect all men in order to validate a claim about “all men”, because we can’t determine things about some men decades dead or ones who live in the remote wilds of Patagonia. There are also many scientifically untestable claims, ones which we don’t yet have the machinery for. The latter is apparently not a problem for Popperian scientology, since we don’t have to be limited to the “merely practical”, and we can allow claims that cannot be practically tested. The foundation of the “cannot test all cases” claim is also imaginary – there are no actual infinities. So it’s imaginarily possible to do an exhaustive test, and thus no logical problem for Popper to overcome. Imagination is a tricky thing.
Virtually every scientific observation is founded on a very deep hierarchy of universal truths. Suppose you have a physical claim X which predicts that if you mix A with B, the temperature should rise 20c in 2 minutes. The story goes, if you replicate that result a million times, you can’t be sure whether you’ll get the same results the next time you do the experiment. On the other hand, if the temperature rises only 4c, then the claim is supposedly falsified and you now know that you didn’t know X – you have no knowledge, other than that X is not so. Without the falsifier, you can strongly suspect but not know that X is true; with the falsifier, you can know that X is false, or so they claim,
Of course if it is not true that the temperature rose only 4c, then rejection of X is wrong. You may feel firmly convinced that you saw a certain number on the meter at one point and another number on the meter two minutes later, but relating that to the claim that there was a temperature rise depends crucially on accepting as valid a particular universal physical theory Y about temperature and meters. If theory Y has been disproven, then of course it isn’t a valid basis for refuting another theory. So we have to assume, then, that theory Y is plausible but still unknown. The consequence is that because you don’t know that the theory of thermometers is true, you don’t know that the supposed temperature rise recorded at that time is actual, and therefore you don’t know if X has been refuted. So more correctly, in Popperian nihilism, you can neither know for certain that a claim is true, or that a supposedly refuting observation is true (and this the claim is false). If you cannot know that something is true or that it is false, you know nothing, which is nihilism.