Plato moved from the Socratic search of ethical definitions to the theory of Ideas. He contrasted the world of existence with the world of appearance, which is only a manifestation of ideas.

Plato divided the cosmos into two separate regions: the world of being , and the world of becoming . The former he regarded as perfect, unchanging, the essence of what he called Ideas . All those concepts sought by Socrates - justice, virtue, etc - existed in perfection in the world of Ideas. To Plato, the world of Ideas was the real world; the material world, though seeming real to our senses, was only an illusion. Plato used his famous 'Allegory of the Cave' to make the distinction between illusion and reality. (See www link below).

For example, to tune a muscical instrument, a player adjusts his strings according to arithmetical ratios that define particular notes. These arithmetical relations express harmonic intervals, and never get out of tune. The harmonic relations cannot be heard, but are understood by the mind. In the material world, the strings of the instruments go constantly out of tune. Therefore Plato said that only the universal ratios or ideas of harmony have true being. The senses are unreliable Plato argued and he had faith only in the intellect, which can be trained to lead us to the truth.

The best preparation, Plato thought, was training in mathematics: the geometer considers not the ill-defined drawing, but the perfect, ideal triangle which truly exists in the world of Ideas.