Can statistics prove correlation, causality, both, or neither?
Strongly support: both.
Statistics can provide evidence for correlation, and if, in an attempt to find and eliminate lurking variables, repeated experimentation yields consistent correlation results, then this can provide evidence for causation.
Note, however, that statistics can not (mathematically) prove correlation or causation; it can only provide strong evidence in favour of it (or show that there is no strong evidence against it).
"Smoking causes cancer." We've all heard this before. But what does it really mean? Surely, there must be someone out there who has smoked at least once in their life and did not get cancer. Conversely, there must be someone out there who does have cancer and has never smoked even once.
Clearly, the phrase "smoking causes cancer" can not mean "it is guaranteed that if you smoke, you will get cancer," or even "it is impossible to get cancer if you do not smoke." So, what does it really mean?
It means that, after trying to remove/account for other possible factors/explanations, repeated experimentation shows there is a high correlation between smoking at time A and having cancer at a later time B. With high enough correlation and sufficient attempt to eliminate other possible explanations, the null hypothesis of causation can not be rejected.
Once again: this does not prove causation. It merely states there is no strong evidence for an alternative hypothesis.