Is it possible in principle to ever establish or prove a causal relationship exists between two variables or events?
3 Answers
We 'Prove' things by performing experiments.
So I would say we can prove something to within the degree of accuracy/uncertainty of the relevant experiments.

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To a philosopher who does not "do" science, the answer is probably no, in the sense that no philosophical viewpoint is privileged over any other.
To a physicist who does not "do" philosophy, the answer is yes, in the sense that equations have solutions.
To a recovering ex-engineer who spent 30 years debugging hi-tech production lines, the answer is yes unless some manager in the plant knows how to invoke the underdetermination of theory by evidence, in which case you are screwed.

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Physics (and other sciences) proceed by drawing conclusions from empirical observations - proof belongs to mathematics. So the key question is how exactly would you observe a causal relationship ? What would it look like ?
Even if we see that whenever A happens then B also happens and vice versa, this does not establish a causal relationship between A and B - they could both be caused by some as yet unobserved event C. Or it could just be coincidence - maybe tomorrow we will observe A without B and B without A.
And the outcomes of most science experiments are far less clear cut than this. Usually they are more along the lines of “usually when we observe A we also observe B and vice versa but occasionally we observe one without the other - but this is not too surprising since we know our instruments are not infallible”.

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See https://philosophy.stackexchange.com/questions/8216/is-it-possible-to-know-anything-with-certainty
– Jagerber48 May 17 '22 at 00:11