What can science prove?

[Ryft]

Not only are a priori facts defacto conclusive, but so are a posteriori facts.

(1) All fish now living on earth eventually die
(2) Tuna is a type of fish now living on earth
(3) Therefore, all tunafish now living on earth will eventually die

The statement that "all fish now living on earth eventually die" is neither scientific nor conclusive, if science has anything to do with observable results (and I'm pretty sure it does). The only way that statement could be considered scientific and conclusive is for all fish now living on earth to be under controlled observation until they die. Although that's possible in theory, it's not feasible in practice because of the sheer number of (i) fish that would have to be found and (ii) tracking devices built—even if we restricted it to just tuna fish.

I think there are good reasons to accept the statement as true, even with a high degree of certainty, but those reasons are philosophical, not scientific, because it's a conclusion drawn from inductive inference, not empirical observation. That is to say, the premises may be proved scientifically, but the conclusion is reasoned philosophically. (Incidentally, such reasoning is inductive, not deductive, because the move is from specific facts to a general conclusion: from "all fish observed die" to "all fish die.")

Science says the things I can see exist.

Reality check #1.

Science does not say that. Science assumes it (as surely it must). It might be helpful to consider the word "says" (in this context) as idiomatic, that is, 'what science says' is an idiom whose literal meaning is 'what science concludes'. The things you can see exist, yes. But that is not a conclusion reached through scientific methodology; in fact, the issue is not even available for empirical review in the first place. What exists? What does it mean for something to exist? When does something cease to exist? This is philosophy, not science. The branch of philosophy known as metaphysics is where conclusions are drawn about ontology (existence). Science does not, and cannot, draw those conclusions. It assumes them.

Science relies on the assumption of materialism.

Reality check #2.

Science does not rely on that assumption. Insofar as science is occupied with the study of natural causes and events, it assumes that material things exist (as noted above) but it does not assume that ONLY material things exist. You are conflating an epistemological assumption (which science relies on) and an ontological assumption (which science does not rely on), the latter being the presupposition that "nature is all there is, and all basic truths are truths of nature" (The Encyclopedia of Philosophy. Macmillan, 1996). Even if metaphysical naturalism were shown to be complete bollocks, we would still have the natural sciences because material things do exist.

Is it possible to KNOW what would happen, say, by taking an action for which you know of the direct reaction? Knowing what would happen based on the knowledge and understanding of physics? Is this knowing something?

In a colloquial sense, yes. But strictly speaking that's not knowledge so much as belief. Given what I know about the relevant factors involved, I believe (on good reason) that the ball will do such and so. Since a proposition cannot be an item of knowledge without the 'true' element (where knowledge is "justified true belief"), I would say it is not more than justified belief.