The right way to disprove "Nothing other than correlation implies causation." would be to find one causation whose existance we know from something other than causation.
But you won't find it. Think about any causation. Now ask yourself why we now that it exists. The answer will always be correlation.
Causal calculus, mutual information, random forest importance scores, various hypothesis tests and other methods can all imply causation as well or better (especially in the case of non-linear or multivariate association) then correlation. All these methods and more are widely used in literature.
Those methods still rely on correlation. To be clear: We are not only talking about linear correlation here, which is only one of several kinds of correlations.
No they don't. They work directly with underlying estimates of probability distributions, entropy or impurity decrease in machine learning models.
Another example: mendelian inheritance patterns in a pedigree study.
If you know of a good measure of non linear correlation please let me know. And publish a paper in science or nature like the MIC/MINE people did (a measure that has issues in practice).
To estimate probability distributions, you need data that is non-random. Non-random means there's a pattern. That's another word for correlation.
In using those methods you may never calculate correlation as a number, but when those methods find something you still rely on the fact that there is a correlation.
How about this? Playing soccer causes broken hips, but soccer playing is in fact anti-correlated with broken hip incidence. Why? Because most broken hips occur among extremely old people, who generally don't play soccer.
[I'm not sure whether this is actually statistically true, but you get the idea, and I'm sure there's a similar example which actually is true.]
Just because old people tend to break their hips more often than younger people wouldn't necessarily make soccer anti-correlated with hip injuries. In your example, if you compared two groups that are identical, the only difference being one plays soccer and the other doesn't, you'd find a correlation between soccer and broken hips.
