Correlation vs. causality - how can you prove the latter?
I've read a couple of documents at work over the last few months that used correlation to suggest causality. For instance, since there's a correlation between a country's investment in research and its economic success, these documents imply that investing in research will benefit a country's economy. I always come away from such arguments thinking that it may just as well be the other way around: economically successful countries are more likely to have more money that they can invest in research. There needs to be more evidence to show that A causes B.
Mathematically, how would one go about trying to prove a degree of causality in such a case? Is it even possible? I'm not asking for an extremely detailed description so much as being pointed in the general direction of the principles applied when trying to show that A isn't just correlated with B, it's actually one of the causes of B. Any explanations would be much appreciated.
"Nothing is gonna save us forever but a lot of things can save us today." - Night in the Woods