Didascalicon, Book II: the sciences, their parts and their discipline
Chapter IV. On mathematics
Mathematics is called the doctrinal science. For mathesis, when it is written without the aspirate, is interpreted vanity, and signifies the superstition of those who place the fates of men in the constellations; hence such persons were called mathematicians. When, however, it has the aspirate, it means doctrine. This science considers abstract quantity. “Abstract” quantity, moreover, “is said to be that which, separating it in the understanding from matter or from other accidents, such as even, odd, and the like, we treat in reasoning alone, which doctrine does, not nature.” Boethius calls this the intelligible, which comprehends in thought and understanding the first part, the intellectible: namely the things of all the heavenly works of the higher divinity, and whatever beneath the lunar globe is of a more blessed mind and purer substance, and lastly of human souls, all of which, when they had been of that prior intellectible substance, degenerated by the pull of bodies from the intellectibles to the intelligibles, so that they are rather understood than understand, and are then the more blessed in the purity of understanding as often as they apply themselves to the intellectibles. For the nature of spirits and souls, because it is incorporeal and simple, is a sharer in intellectible substance. But because it descends by the instruments of the senses, not uniformly, to grasp sensible things, and draws their likeness to itself through imagination, it in a certain way forsakes its simplicity, whereby it loses the character of simplicity. For what is like a composite cannot in every way be called simple. The same thing, then, in different respects is at once intellectible and intelligible. Intellectible, because its nature is incorporeal and cannot be grasped by any sense. Intelligible, because it is indeed a likeness of sensible things, yet is not sensible. The intellectible is that which is neither sensible nor a likeness of the sensible. The intelligible is that which is itself perceived by the intellect alone, but does not perceive by the intellect alone, because it has imagination or sense, by which it comprehends what lies under the senses. By touching bodies, therefore, it degenerates; for while it runs over the visible forms of bodies through the impressions of the senses and draws them, attracted, into itself through imagination, it is divided from its simplicity as often as it is informed by any qualities of contrary impression. But when, rising from this distraction to pure understanding, it gathers itself into one, it becomes more blessed by participation in the intellectible substance.