Pythagoras’ System of Numbers
Pythagoras, then, declared the originating principle of the universe to be the unbegotten monad, and the generated duad, and the rest of the numbers. And he says that the monad is the father of the duad, and the duad the mother of all things that are being begotten—the begotten one (being mother) of the things that are begotten. And from the duad, again, as Pythagoras states, (are generated) the triad and the succeeding numbers up to ten. For Pythagoras is aware that this is the only perfect number—I mean the decade—for that eleven and twelve are an addition and repetition of the decade; not, however, that what is added constitutes the generation of another number. And all solid bodies he generates from incorporeal (essences). For he asserts that an element and principle of both corporeal and incorporeal entities is the point which is indivisible. And from a point, he says, is generated a line, and from a line a surface; and a surface flowing out into a height becomes, he says, a solid body.
— The Refutation of All Heresies VI.XVIII
The Categories of Pythagoras
There are, then, according to Pythagoras, two worlds: one intelligible, which has the monad for an originating principle; and the other sensible. But of this (latter) is the quaternion having the iota, the one tittle, a perfect number. And there likewise is, according to the Pythagoreans, the i , the one tittle, which is chief and most dominant, and enables us to apprehend the substance of those intelligible entities which are capable of being understood through the medium of intellect and of sense. (And in this substance inhere) the nine incorporeal accidents which cannot exist without substance, viz., “quality,” and “quantity,” and “relation,” and “where,” and “when,” and “position,” and “possession,” and “action,” and “passion.” These, then, are the nine accidents (inhering in) substance, and when reckoned with these (substances), contains the perfect number, the i .
Wherefore, the universe being divided into the intelligible and sensible world, we have also reason from the intelligible (world), in order that by reason we may behold the substance of things that are cognised by intellect, and are incorporeal and divine. But we have, he says, five senses—smelling, seeing, hearing, taste, and touch. Now, by these we arrive at a knowledge of things that are discerned by sense. Nothing, he says, of intelligibles can be known to us from sense. Neither, again, by reason is it possible to arrive at a knowledge of any of the things discernible by sense.
Things being thus constituted, the arrangement of things that have been made and are being made is observed to happen in conformity with numerical (combinations). For in the same manner as, commencing from monad, by an addition of monads or triads, and a collection of the succeeding numbers, we make some one very large complex whole of number; (and) then, again, from an amassed number thus formed by addition, we accomplish, by means of a certain subtraction and re-calculation, a solution of the totality of the aggregate numbers; so likewise he asserts that the world, bound by a certain arithmetical and musical chain, was, by its tension and relaxation, and by addition and subtraction, always and for ever preserved incorrupt.
— The Refutation of All Heresies VI.XIX
The Pythagorean Cosmology
In their system, Love forms the world incorruptible (and) eternal, as they suppose. For substance and the world are one. Discord, however, separates and puts asunder, and evinces numerous attempts by subdividing to form the world. It is just as if one severs into small parts, and divides arithmetically, the myriad into thousands, and hundreds, and tens; and drachmæ into oboli and small farthings. In this manner, he says, Discord severs the substance of the world into animals, plants, metals and things similar to these. And the fabricator of the generation of all things produced is, according to them, Discord; whereas Love, on the other hand, manages and provides for the universe in such a manner that it enjoys permanence.
— The Refutation of All Heresies VI.XX