The Arithmetic of Boethius

A tree diagram illustrates the hierarchy of "BEINGS" (ENTIA), connecting them to three levels: "Highest" (Summa), "Middle" (Media), and "Lowest" (Ima).
1. The "Highest" level relates to things separated from motion and matter, encompassing supramundane beings and metaphysics or wisdom.
2. The "Middle" level comprises the four mathematical disciplines (Mathesis): Arithmetic and Music, derived from "Number" (subdivided into those considered in themselves and those related to others), and Geometry and Astronomy, derived from "Magnitude" (subdivided into absolute and mobile).
3. The "Lowest" level pertains to beings joined to matter and motion, identifying with the study of nature (Natural Philosophy).

For when these things are considered apart from matter and in themselves, they avoid all change. Since, as is proven to Aristotle, apart from matter and a subject, there is no change at all—he concludes with certain and evident reason that a single and identical hyle prime matter is subject to contraries so that transformation may occur. And so, change happens to these things not when they are viewed in their own nature, essentially and abstractly, but only through their association with matter, to which they are contracted and within which they adhere as potentiality. Thus, in every transformation, matter must be presupposed; if you remove it, nothing remains at all. For how can it happen that a contrary approaches its contrary and, by its arrival, drives out and exterminates that which was weak, if there is not the same subject for both? Furthermore, it will soon be proven that transformation is an argument for imperfection. It has already been said that it depends on matter, in which one and the same indiscriminate subject is established for contraries. Therefore, those things that less adhere to or are associated with matter are more perfect. This applies to mathematical objects, which are above physical and sensible things. And indeed, by this reasoning, in individual things, the degrees of imperfection and perfection are weighed significantly. For that which has a greater mass of matter is exposed to more changes, and is therefore more imperfect. Thus, the earth is subject to more things than fire, and water than air; and in this way, earth is more imperfect than fire, and water than air. The same applies to mixtures: minerals and fossils undergo nearly countless accidental changes without destruction; plants, fewer; and of all things, man undergoes by far the fewest changes without detriment to his own nature, and thus he excels all mixtures, while fossils are numbered in the lowest grade. If you raise your mind higher, you will easily perceive that most pure act, namely God, who partakes of no coarseness and is subject to no change. Thus, there is no contrariety in Him, no opposition, but a singular coincidence. Those beings which are in the second place after Him do not adhere to sensible matter, nor do they admit of sensible motions, but they do not shun spiritual and divine influences, by which it is detected that they possess some matter, even if of a different nature and condition than that which is sensible. And since the rational soul is subject to contrary passions, from which warring habits of virtue and vice emerge, and since its powers are in large part discordant, that to which the body is associated introduces this; but because it is removed from that, it no longer experiences it. From these things, it is prompt to understand what Boethius implies, gathering quantities, qualities, forms, and accidents of that kind to those things which are not changed in their own nature. For these (as was premised) are not changed when they are apprehended in themselves, absolutely and removed from matter, but only when they are added to and associated with matter.

However, someone might perhaps encounter these things and say that increase and decrease are said once and again in physical texts to belong to quantity by their very nature, and intensity and remission to quality. It is so far from the truth that it does not seem far from the truth what the author asserts: that these are things which grow neither by intension nor decrease by retraction. This difficulty can be diluted significantly, namely that these are attributed to them not when they are considered as absolute and abstract in thought, but as annexed and joined to matter. And by this reasoning, the geometer, discussing magnitudes—whose consideration is abstract—does not attribute increase, decrease, rarity, and density to them, but the natural philosopher does, whose consideration is contracted, and whose consideration is, as Aristotle is the author, contrary to geometry, just as the curved is to the straight. And that he intends this, he shows from the following. These things (he says) are by nature incorporeal and flourish by the reason of an immutable substance. But by participation in the body, they are changed, and by the touch of a variable thing, they pass into a mutable inconsistency. But what is this other than that they are immutable by their own genius, but changed only by the companionship and touch of matter? For the fact that increase, rarity, and their opposites are attributed to quantity depends on matter. Separate quantity from matter after the manner of geometers, and you will find no rarity, and no increase that adheres to change. Thus, he infers that not even in reality are those things truly mutable, as they are not subject to any change by their own nature. Furthermore, what we call an accident is an imperfect being, not existing by itself...