The psychology of the Pythagoreans was greatly modified by their physical, and still more, by their moral tenets. The soul was ἀριθμὸς ἑαυτὸν κινῶν--a self-moving number or Monad, the copy (as we have seen) of that Infinite Monad which unfolds from its own incomprehensible essence all the relations of the universe. This soul has three elements, Reason (νοῦς), Intelligence (φρήν), and Passion (θυμός). The two last, man has in common with brutes, the first is his grand and peculiar characteristic. It has, hence, been argued that Pythagoras could not have held the doctrine of "transmigration." This clear separation of man from the brute, by this signal endowment of reason, which is sempiternal, seems a refutation of those who charge him with the doctrine. In the department of morals, the legislator of Crotona found his appropriate sphere. In his use of numerical notation, moral good was essential unity--evil, essential plurality and division. In the fixed truths of mathematical abstractions he found the exemplars of social and personal virtue. The rule or law of all morality is resemblance to God; that is, the return of number to its root, to unity,[443] and virtue is thus a harmony. [Footnote 442: That is, 1+2+3+4=10. There are intimations that the Pythagoreans regarded the Monad as God, the Duad as matter, the Triad as the complex phenomena of the world, the Tetrad as the completeness of all its relations, the Decad as the cosmos, or harmonious whole.] [Footnote 443: Aristotle, "Nichomachian Ethics," bk. i. ch. vi.] Thus have we, in Pythagoras, the dawn of an _Idealist_ school; for mathematics are founded upon abstractions, and there is consequently an intimate connection between mathematics and idealism. The relations of space, and number, and determinate form, are, like the relations of cause and effect, of phenomena and substance, perceptible _only in thought_; and the mind which has been disciplined to abstract thought by the study of mathematics, is prepared and disposed for purely metaphysical studies. "The looking into mathematical learning is a kind of prelude to the contemplation of real being."[444] Therefore Plato inscribed over the door of his academy, "Let none but Geometricians enter here." To the mind thus disciplined in abstract thinking, the conceptions and ideas of reason have equal authority, sometimes even superior authority, to the perceptions of sense. [Footnote 444: Alcinous, "Introduction to the Doctrines of Plato," ch. vii.]