My old friend Allen Downey has a thoroughly sensible post about correlation and causation.
It is true that correlation doesn’t imply causation in the mathematical sense of “imply;” that is, finding a correlation between A and B does not prove that A causes B. However, it does provide evidence that A causes B. It also provides evidence that B causes A, and if there is a hypothetical C that might cause A and B, the correlation is evidence for that hypothesis, too.
If you don’t understand what he means by this, or if you don’t believe it, read the whole thing.
I do think he’s guilty of a bit of rhetorical excess when he says that “the usual mantra, ‘Correlation does not imply causation,’ is true only in a trivial sense.” I think that the mantra means something quite specific and valid, namely something like “correlation does not provide evidence for causation that’s as strong as you seem to think.” One often sees descriptions of measured correlations that imply that the correlation supports the hypothesis of causation to the exclusion of all others, and when that’s wrong it’s convenient to have a compact way of saying so.
But that’s a small quibble, which gets even smaller if I include the next phrase in that quote from Allen,
The point I was trying to make (and will elaborate here) is that the usual mantra, “Correlation does not imply causation,” is true only in a trivial sense, so we need to think about it more carefully [emphasis added].
I couldn’t agree more, and everything Allen goes on to say after this is 100% correct.