## Plato’s Philosophies of Mathematics

The philosophy of mathematics begins with Pythagoras, who believed that mathematics gave us the key to understanding reality, but it is Plato who first gave it articulate form. In the Meno he proves that mathematics is known a priori-that is, without appeal to sense experience.1 He starts talking to a slave boy, and by a series of questions elicits from him a method of constructing a line 2 as long as a given one, using a special case of Pythagoras’ theorem. The general proof of Pythagoras’ theorem is difficult: for two thousand years it was the Pons Asinorum for school boys. In the Meno, however, Plato considers the special case of an isosceles right-angled triangle, where even someone who has never done geometry in his life can be brought to see how to construct a square with area twice that of a given square.