philosophy. And the great aim of education is the cultivation of the habit of abstraction. This is to be acquired through the study of the mathematical sciences. They alone are capable of giving ideas of relation, and of arousing the dormant energies of thought.
Mathematics in the age of Plato comprehended a very small part of that which is now included in them; but they bore a much larger proportion to the sum of human knowledge. They were the only organon of thought which the human mind at that time possessed, and the only measure by which the chaos of particulars could be reduced to rule and order. The faculty which they trained was naturally at war with the poetical or imaginative; and hence to Plato, who is everywhere seeking for abstractions and trying to get rid of the illusions of sense, nearly the whole of education is contained in them. They seemed to have an inexhaustible application, partly because their true limits were not yet understood. These Plato himself is beginning to investigate; though not aware that number and figure are mere abstractions of sense, he recognizes that the forms used by geometry are borrowed from the sensible world (vi. 510, 511). He seeks to find the ultimate ground of mathematical ideas in the idea of good, though he does not satisfactorily explain the connection between them; and in his conception of the relation of ideas to numbers, he falls very far short of the definiteness attributed to him by Aristotle (“Met.” i. 8, § 24; ix. 17). But if he fails to recognize the true limits of mathematics, he also reaches a point beyond them; in his view, ideas of number become secondary to a higher conception of knowledge. The dialectician is as much above the mathematician as the mathematician is above the ordinary man (cp. vii. 526 D, 531 E). The one, the self-proving, the good which is the higher sphere of dialectic, is the perfect truth to which all things ascend, and in which they finally repose.
This self-proving unity or idea of good is a mere vision of which no distinct explanation can be given, relative only to a particular stage in Greek philosophy. It is an abstraction under which no individuals are comprehended, a whole which has no parts (cf. Arist. “Nic. Eth.” i. 4). The vacancy of such a form was perceived by Aristotle, but not by Plato. Nor did he recognize that in the dialectical process are included two or
