Perhaps one of the earliest examples would be with the Pythagoreans, who held that any two magnitudes were commensurable, measured as integer multiples of a smaller common unit, a belief that was connected with their mystical religious views and also with their mathematical theory of musical harmony. The Pythagoreans were shocked by the discovery of incommensurable numbers, such as $\sqrt{2}$.

But it may be anachronistic to refer to the fundamental Pythagorean beliefs or principles as "axioms".