The discovery of irrational numbers triggered the first mathematical crisis. First of all, this is a fatal blow to Pythagoras' philosophy, which relies entirely on integers. Secondly, irrational numbers seem to contradict common sense. Geometric correspondence is also surprising, because contrary to intuition, there are incommensurable line segments, that is, line segments without common units of measurement. Because the Pythagorean school's definition of proportion assumes that any two similar quantities are reducible, all the propositions in Pythagorean proportional theory are limited to reducible quantities, so their general theory of similar shapes is also untenable.

This also reflects that intuition and experience are not necessarily reliable, but reasoning is reliable. From then on, the Greeks set out from the axiom of "self-evident" and established the geometric system through deductive reasoning, which was a great revolution in mathematical thought and a natural product of the first mathematical crisis.