  All of this is rather abstract at this stage. It is now time to turn to a concrete mathematical case where such structural connections have been found. The question, of course, will be whether the mathematics makes for a convincing extrinsic case for new axioms. And the question of pluralism at this level—where we are presupposing ZFC and trying to solve the problem of selection for second-order arithmetic—will rest on the nature of the case. 