"Plato is very fond of appealing to mathematics to show that exact reasoning is possible, not about things which are seen and heard, but about ideal objects which exist in thought only. ... Plato is apparently convinced that the mathematicians will agree with him. And indeed, when we deal with line segments which one sees and which one measures empirically, the question as to the existence of a common measure has no sense; a hair's breadth will measure integrally every line that is drawn. The question of commensurability makes sense only for line segments as objects of thought. It is therefore clear that Theodorus could not appeal to intuition to prove the incommensurability of the sides of his squares."