to be entirely fair, it's not how we teach math and logic to students and readers, but all our axioms (which we still can't verify) are the result of abductive

to be entirely fair, it's not how we teach math and logic to students and readers, but all our axioms (which we still can't verify) are the result of abductive reasoning. Sometimes we discovered that a specific abductive reasoning was superfluous and could be derived from other axioms, turning the ex-axiom into a theorem. So the current axiomatic systems and their applications seem entirely deductive usages, but the axioms are in fact still of an abductive nature... this is much more visible in domains like physics, where the teeth of time has not yet gnawed as thoroughly as in the foundations of mathematics... Edit: just adding: Some day physics might appear like an entirely deductive system (apart from a very small set of axioms or postulates), and hard to discern from the field of mathematics