Suppose you test Fermat's Last Theorem for n up to 10^10, and don't find a counterexample. How much evidence does that give you for FLT being true? In other words, how do you compute P(a counterexample exists with n<=10^10 | FLT is false), since that's what's needed to do a Bayesian update with this inductive evidence? (Assume this is before the proof was found.)

I don't dispute that mathematicians do seem to reason in ways that are similar to using probabilities, but I'd like to know where these "probabilities" are coming from and whether the reasoning process really is isomorphic to probability theory. What you call "heuristic" and "intuition" are the results of computations being done by the brains of mathematicians, and it would be nice to know what the algorithms are (or should be), but we don't have them even in an idealized form.