Correlation vs causation is not actually a complex mathematical problem. The issue is more philosophical.
Bayesian networks are a highly sophisticated and flexible framework for thinking about causation. And yet the essence of Bayesian networks can be captured in much simpler methods like Instrumental Variables or simply regressions with controls.
In all cases, the true distribution of observables (which we can estimate from samples) combined with some assumptions about the possible nature of causality (either very sophisticated in the case of Bayesian networks, or very simple in the case of instrumental variables) lead to the actual causal relationships.
Where do these assumptions come from? Consider a randomized trial. Even though physically, it is possible that some hidden cause influenced both the random number generator which selected patients into a trial, and also whether patients will get better. And yet people universally believe that there is no such mechanism, and so they believe that randomized trials prove the causal relationship between taking a drug and getting better.
No physicist ever wrote down an equation proving this. It is simply something we deduced from our human understanding of how nature works.
Put simply, causality is not a physical notion, neither is it a statistical notion. It is a part of our intuitive understanding of the physical world, in which a higher level notion of causality exists, beyond that described by special relativity.
To add to your point, we cannot prove or disprove the existence of causation (Try to conceive a falsifiable experiment about causation and I would argue you would end up with a metaphysical crisis). The 'is-ought' issue David Hume showed, where just because something 'is' a way does not indicate how it 'ought' to be or will continue to be, has highlighted to us how difficult it is to think this problem for hundreds of years. Another way to look at it is to ask "Who or what is to guarantee the law of physics will remain the same tomorrow? What is to stop the speed of light changing to 1 mile per hour?"
This isn't to say the article posted is of no use. Having a 'graph-like' mental model of how things work is incredibly useful, as most education simplifies real-world problems into a few key issues. Although most non-computer science issues can be reduced successfully using the 80/20 rule in real life, sometimes some problems require us to look at the 100s of contributing factors to allow us to solve the problem we're facing properly.
The more 'graph-like' problem solving becomes acceptable as way to solve issues the better of everyone will be.
Bayesian networks are a highly sophisticated and flexible framework for thinking about causation. And yet the essence of Bayesian networks can be captured in much simpler methods like Instrumental Variables or simply regressions with controls.
In all cases, the true distribution of observables (which we can estimate from samples) combined with some assumptions about the possible nature of causality (either very sophisticated in the case of Bayesian networks, or very simple in the case of instrumental variables) lead to the actual causal relationships.
Where do these assumptions come from? Consider a randomized trial. Even though physically, it is possible that some hidden cause influenced both the random number generator which selected patients into a trial, and also whether patients will get better. And yet people universally believe that there is no such mechanism, and so they believe that randomized trials prove the causal relationship between taking a drug and getting better.
No physicist ever wrote down an equation proving this. It is simply something we deduced from our human understanding of how nature works.
Put simply, causality is not a physical notion, neither is it a statistical notion. It is a part of our intuitive understanding of the physical world, in which a higher level notion of causality exists, beyond that described by special relativity.