In statistics, it is a common practice to say that "correlation does not mean causation", and mostly the proof for this is given by examples. While that is good for the intuition, it's not rigorous. Ideas such as a third variable which may be causing both is often cited, but again, that's an example.
Can someone give me a Mathematical argument to why this is true? And possibly, give the mathematical condition for when it does hold true? Any leads or answers are much appreciated :) Thanks in advance!