[Nodrog]

The axiomisation of either a particular branch of mathematics (eg geometry/group theory), or of mathematics as a whole (eg ZFC) normally occurs long after the major results have been discovered through largely non-logical means such as geometric arguments, heuristics, 'intuition', etc The axioms are then explicitly deliberately designed to be capable of reproducing the already known results. It's not like someone sat down one day and created logic and an axiom system and then mathematicians went off and deduced results from it - that isnt how mathematical discoveries are generally made. The formalisation normally occurs after the results, not before; mathematical logic wasnt even invented till the 19th century, yet mathematians had largely managed fine without it (with a few notable problems).

Its backwards to say that mathematics is true because it follows from axioms; the axioms (and formal system) were invented after the fact in order to ground results which were largely already known.