Mathematicians stopped asking whether axioms are true and started asking what follows
For Euclid, a starting assumption in geometry was a plain fact about the world, like the ability to draw a straight line between any two points. Modern mathematicians treat axioms more like rules of a game. Change one, as happened with Euclid's parallel postulate, and a whole new geometry opens up rather than collapsing.
The Greek axioma meant something thought worthy or fit, from a root about being in balance or having equal value. Postulate, by contrast, comes from a word for demanding: Euclid asks his reader to grant that certain constructions can be done. Ancient writers kept the two apart. Axioms were self-evident truths shared by many sciences, such as the idea that taking equal amounts from equal amounts leaves equal remainders, while postulates were particular to one field and rested on experience. Proclus reports that Geminus thought Euclid's fourth postulate, that all right angles are equal, really belonged among the axioms.
Aristotle and Euclid saw deduction from such starting points as a way to avoid error. Nothing beyond empty tautologies can be proved from nothing, so every body of theorems needs some unproved base.
Over roughly the last 150 years, mathematics learned to strip meaning out of its assumptions. Giuseppe Peano and fellow Italians championed undefined primitive terms, and structural subjects such as group theory and topology were built with no particular application in view. Dropping Euclid's fifth postulate yielded hyperbolic geometry, teaching that words like line and parallel can flex. The axioms defining a field are best read as a checklist: any system of addition and multiplication meeting them instantly inherits a mass of known results.
The formalist hope was to derive all mathematics from one consistent set of axioms. David Hilbert showed Euclidean geometry could be axiomatised consistently, but Russell's paradox exposed cracks in naive set theory, and Kurt Gödel proved that any sufficiently rich system, including Peano's arithmetic, contains statements it can neither prove nor refute and cannot establish its own consistency. Paul Cohen's forcing later showed that Cantor's continuum hypothesis is independent of the standard Zermelo–Fraenkel axioms, whose own consistency nobody knows how to demonstrate.
Source: Axiom
