You're adhering to certain misconceptions for no good reason. An exhaustive check of all the cases in the model that supports the four-color conjecture is the "

You're adhering to certain misconceptions for no good reason. An exhaustive check of all the cases in the model that supports the four-color conjecture is the "empirical" (here, meaning experimental) data required to falsify conjectures. And even physicists don't deny the value of falsification in narrowing down models to truth. But I think they are much more loose in what they are willing to stipulate as true if it means advancing other conclusions. Furthermore mathematicians do accept certain kinds of probabilistic arguments, but they need to take a form of proving that an object will be guaranteed to have a certain property as a parameter on the object approaches a statistical limit. So partial evidence can sometimes point to the possible truth of a mathematical conjecture, but you are still burdened with the requirement of deductive demonstration if you desire logical certainty. The tautological notion you take of math is a half-truth. On the one hand, models in math seek to be sound over the objects they specify. This is opposed to physics where deliberate simplifications must be made to make your idealizations of a system manageable or relevant. At the same time, no mathematical system is complete. So there will be truths that require other axiomatizations to access via proof. This is one source of mathematical creativity requiring judgment beyond following tautologies.