Through all his criticisms on the popular sources of information—sense, language and public opinion—Plato refers to an ideal of perfect knowledge which he assumes without being able to define it. It must satisfy the negative condition of being free from self-contradiction, but further than this we cannot go. Yet, in the hands of a metaphysician, no more than this was required to reconstruct the world. The demand for consistency explains the practical philosophy of Socrates. It also explains, under another form, the philosophy, both practical and speculative, of his disciple. Identity and the correlative of identity, difference, gradually came to cover with their manifold combinations all knowledge, all life, and all existence. It was from mathematical science that the light of certainty first broke. Socrates had not encouraged the study of mathematics, either pure or applied; nor, if we may judge from some disparaging allusions to Hippias and his lectures in the _Protagoras_, did Plato at first regard it with any particular favour. He may have acquired some notions of arithmetic and geometry at school; but the intimate acquaintance with, and deep interest in them, manifested throughout his later works, probably dates from his visits to Italy, Sicily, Cyrênê, and Egypt. In each of these places the exact sciences were cultivated with more assiduity than at Athens; in southern Italy they had been brought into close connexion with philosophy by a system of mystical interpretation. The glory of discovering their true speculative significance was reserved for Plato. Just as he had detected a profound analogy between the Socratic scepticism and the Heracleitean flux, so also, by another vivid intuition, he saw in the definitions and demonstrations of geometry a type of true reasoning, a particular application of the Socratic logic. Thus the two studies were brought into fruitful reaction, the one gaining a wider applicability, and the other an exacter method of proof. The mathematical spirit ultimately proved too strong for Plato, and petrified his philosophy into a lifeless formalism; but no extraneous influence helped so much to bring about the complete maturity of his constructive powers, in no direction has he more profoundly influenced the thought of later ages.