Plato argued that the things we see are imperfect copies of perfect, unchanging 'Forms'. Any circle you draw is slightly flawed, yet you can think of a perfect circle; for Plato, that perfect Form was more real than anything in the physical world.
Plato set out the theory in dialogues such as the Phaedo and the Republic, written in Athens in the fourth century BC, usually through the voice of his teacher Socrates. His Greek words for the Forms, idea and eidos, both come from seeing and mean 'form' or 'look'; English 'idea' descends from them. The Forms, such as beauty, justice or the perfect circle, can be grasped only by reason, not by the senses. In the Republic's Allegory of the Cave, prisoners who have only ever seen shadows on a wall take them for reality, much as we do with the physical world. Plato's own student Aristotle rejected separate Forms, arguing that form exists only in real things. The debate lives on: many mathematicians, Kurt Gödel among them, have held that numbers exist independently of us.





