Perhaps one of the earliest examples would be with the [Pythagoreans](http://en.wikipedia.org/wiki/Pythagoreanism), who held that any two magnitudes were <a href="http://en.wikipedia.org/wiki/Commensurability_(mathematics)">commensurable</a>, 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](http://en.wikipedia.org/wiki/Irrational_number#Ancient_Greece) numbers, such as $\sqrt{2}$. But it may be anachronistic to refer to the fundamental Pythagorean beliefs or principles as "axioms".