It is meaningful to talk about mathematical facts causing other mathematical facts. For example, if I knew the complete laws of physics but did not have enough computing power to determine all their consequences (which would be impossible anyways, as I’m living inside of them), my uncertainty about what is going to happen in the universe would be described by the exact same probability distribution as my uncertainty about the mathematical consequences of the laws of physics, and so both distributions would satisfy the causal Markov condition for the same causal graph* (modulo any uncertainty about whether the laws that I believe to be correct actually do describe the universe).
This works the same way with any other set of mathematical facts. I believe that if the abc conjecture is true, then Szpiro’s conjecture is also true and I believe that if the abc conjecture is false, then Shinichi Mochizuki’s proof of it is flawed. All of these facts can be put into one probability distribution which can then be factored over a Bayesian network. There is no need to separate the mathematical from the nonmathematical.
* Depending on how exactly you phrase the question, I would even say that these distributions are describing my uncertainty about the same thing, but that isn’t necessary here.
It is meaningful to talk about mathematical facts causing other mathematical facts. For example, if I knew the complete laws of physics but did not have enough computing power to determine all their consequences (which would be impossible anyways, as I’m living inside of them), my uncertainty about what is going to happen in the universe would be described by the exact same probability distribution as my uncertainty about the mathematical consequences of the laws of physics, and so both distributions would satisfy the causal Markov condition for the same causal graph* (modulo any uncertainty about whether the laws that I believe to be correct actually do describe the universe).
This works the same way with any other set of mathematical facts. I believe that if the abc conjecture is true, then Szpiro’s conjecture is also true and I believe that if the abc conjecture is false, then Shinichi Mochizuki’s proof of it is flawed. All of these facts can be put into one probability distribution which can then be factored over a Bayesian network. There is no need to separate the mathematical from the nonmathematical.
* Depending on how exactly you phrase the question, I would even say that these distributions are describing my uncertainty about the same thing, but that isn’t necessary here.
The thing is that mathematics seems to have an additional causal structure that seems to be (at least partially) independent from the proof structure.
I agree with this. I didn’t mean to give the impression that the causal structure is the same as the proof structure.