By a curious and somewhat fanciful development of this conception the Pythagoreans drew up two parallel columns of antithetical principles in nature, ten in each, thus:-- Definite Indefinite Odd Even One Many Right Left Male Female Steadfast Moving Straight Bent Light Dark Good Evil Four Square Irregular Looking down these two lists we shall see that the first covers various aspects of what is conceived as the ordering, defining, formative principle in nature; and that the second in like manner comprises various {25} aspects of the unordered, neutral, passive, or disorganised element or principle; the first, to adopt a later method of expression, is _Form_, the second _Matter_. How this antithesis was worked out by Plato and Aristotle we shall see later on. [54] While, in a sense, then, even the indefinite has number, inasmuch as it is capable of having number or order imposed upon it (and only in so far as it has this imposed upon it, does it become knowable or intelligible), yet, as a positive factor, Number belongs only to the first class; as such it is the source of all knowledge and of all good. In reality the Pythagoreans had not got any further by this representation of nature than was reached, for example, by Anaximander, and still more definitely by Heraclitus, when they posited an Indefinite or Infinite principle in nature which by the clash of innate antagonisms developed into a knowable universe (see above, pp. 12, 16). But one can easily imagine that once the idea of Number became associated with that of the knowable in things, a wide field of detailed development and experiment, so to speak, in the arcana of nature, seemed to be opened. Every arithmetical or geometrical theorem became in this view another window giving light into the secret heart of things. Number became a kind of god, a revealer; and the philosophy of number a kind of religion or mystery. And this is why the {26} second grade of disciples were called Mathematicians; mathematics was the essential preparation for and initiation into philosophy. Whether that which truly exists was actually identical with Number or Numbers, or whether it was something different from Number, but had a certain relation to Number; whether if there were such a relation, this was merely a relation of analogy or of conformability, or whether Number were something actually embodied in that which truly exists--these were speculative questions which were variously answered by various teachers, and which probably interested the later more than the earlier leaders of the school. [56]