Causality, if any, in mathematics itself

Question

Mathematicians often express comments like "X is true because Y and Z are true". One's sense of mathematical causation is also a major part of mathematical intuition.

But causality per se is not, as far as I know, a defined concept in mathematics.

Does there exist a rigorous concept of causality that allows one to state that a certain collection of mathematical truths "causes" another one to be true?

(Edit:) I should make clear that I am not just referring to when it is possible to prove that a statement is a logical consequence of a set of other statements.

Very often, after a subject in mathematics becomes mature, the eventual most widely accepted or repeated proofs of something may look very little like the original proof.

But then, many years later, that eventual proof has often remained the same.

It is then, when a subject is felt to be very well understood and perhaps even generalized, that it can be seen in context, and the sense of what causes what tends to become most striking.

I feel that as a field of mathematics becomes, asymptotically, more established [and here we stipulate that in truth no field is immune from potential upheaval, eventually], it becomes like the trunk of an old tree: quite firmly established and unlikely to change much in the near future, maybe not in over 100 years.

The reasons for this might include that the apparently "shortest" proof of a given theorem hasn't changed in a long time. So we're thinking metrically.

Or, it may turn out that — as with the Pythagorean Theorem — there are many paths to the same result and not so easy to choose one among them. Not to mention that there are different axiom systems that allow the same theorems.

These similes lead me to think that if there is a Book with the best proof of everything, the theorems somehow lie in a metric space, probably a tree, much as Conway's surreal numbers are born at a certain point and have a certain distance from each other.

And if this is all the case, then perhaps Mathematics in its (partially) finished state might resemble a riemannian manifold, with lots of paths to get from here to there, but (usually) only one shortest one. The Pythagorean Theorem might be the exceptional case that can arise, like with a pair of antipodal points on a n-sphere.

Answer 1

The short answer, as you surely suspected, is that there is no rigorous notion of the type you're asking for, that would allow us to say that (in a particular case) that $X$ definitely causes $Y$, or that $X$ definitely does not cause $Y$.

In an answer to another MO question, I gave some pointers to some mathematical approaches to studying causation. But I don't think that these really get at what you're interested in, which sounds a lot like an account of mathematical explanation, a notoriously open-ended topic.

Somewhat closer might be relevance logic. Relevance logic at least tries to address the issue that if you add irrelevant hypotheses to a theorem, everything looks fine as far as logical correctness goes, but ideally we'd like some way of formally identifying their irrelevance.

If you want to pursue this idea, then I'd recommend that you begin by collecting what you consider to be clear examples of the phenomenon you're interested in. They could be elementary (e.g., what about the proof that a chessboard with opposite corners removed cannot be tiled by dominos?) or more sophisticated (e.g., perhaps cohomological obstructions explain why certain things are impossible?). Negative examples could also be useful; e.g., what if something seems to be true by accident, or has many different proofs (Pythagorean theorem, quadratic reciprocity)? If nothing else, examples would help clarify your question.