"When one compares the pre-Greek and Greek understanding of the concepts of mathematics... [one] notes the sharp transition from the concrete to the abstract... One advantage of treating abstraction is the gain in generality. ...Another advantage ...abstracting ..frees the mind from burdensome and irrelevant details ...The emphasis ...was part and parcel of their outlook on the entire universe. ...Pythagoreans and Platonists maintained that truths could be established only about abstractions. ...impressions received by the senses are inexact, transitory, and constantly changing ...truth, by its very meaning, must consist of permanent, unchanging, definite entities and relationships. ...man may rise to the contemplation of ideas. These are eternal realities and the true goal of thought, whereas mere "things are the shadows..." Thus Plato would say that... reality is in the universal type or idea... Beauty, Justice, Intelligence, Goodness, Perfection, and the State, are independent of the superficial appearances... of the flux of life... of... biases and warped desires... they are ...constant and invariable, and knowledge concerning them is firm and indestructible. ...physical or sensible objects suggest the ideas just as diagrams of geometry suggest abstract geometrical concepts... but one must not lose himself in trivial and confusing minutiae. The abstractions of mathematics possessed a special importance for the Greeks. ...to pass from a knowledge of the world of matter to the world of ideas, man must train ...These highest realities blind the person ...The study of mathematics helps make the transition from darkness to light. ...man learns to pass from concrete figures to abstract forms ...this study purifies the mind by drawing it away from the contemplation of the sensible and perishable and leading it to the eternal ideas. ...to lift the mind above mundane considerations and enable it to apprehend the final aim of philosophy, the idea of the Good."