Arithmetic of Boethius

A diagram illustrates a logic tree classifying "Beings" into three levels. At the top are the "Highest Beings," which are supramundane and studied through Metaphysics or Wisdom: these are abstract and separate from motion and matter. In the "Middle" are mathematical entities, divided into Number (which includes Arithmetic and Music) and Magnitude (which includes Geometry and Astronomy). At the "Lowest" level are the "Beings of Nature" studied through Physics, which are tied to the matter and motion of the natural world. Lines connect these sub-categories to show the division of the sciences according to the principles of Boethius.

All? matter is changeable but? not? all change is matter?, which is proven? by Aristotle as not? subject to it?.

Indeed, when these things are taken apart from matter and perceived by thought, they flee from all change. When they are outside of matter and a subject, there is no change at all. This is proven by Aristotle, who concludes with certain and evident reason that the same and single matter original: hylēn is placed under opposites so that transformation can occur. And so change happens to these things not when their nature is viewed in itself as abstract, but only through their partnership with matter. In every transformation, matter must be assumed, for if you take it away, nothing remains at all. How can one opposite come to another and drive out the weaker one if they do not share the same subject? Furthermore, it will soon be proven that transformation is an argument for imperfection. Change depends on matter, which remains the same underlying subject while opposites are established within it. Therefore, things that are less attached to matter are more perfect. This perfection belongs to mathematics, which sits above physics and the things we perceive with our senses. By this reasoning, the degrees of perfection and imperfection in individual things are weighed. Whatever is covered by a greater mass of matter is open to more changes, and is therefore more imperfect. Thus, the earth is subject to more change than fire, and water more than air. Earth is more imperfect than fire, and water more than air. The same is true in mixed things: minerals and fossils undergo almost countless accidental changes without dying, while plants undergo fewer. Of all these, man undergoes the fewest changes without the destruction of his nature. Therefore, man excels all mixed things, while fossils are counted at the lowest level. If you lift your mind higher, you will easily weigh that purest act original: actum purissimum and God, who shares in no thickness and is subject to no change. There is no contrariety in him, no opposition, but a singular coincidence. Those beings that are in the second place from him do not cling to sensible matter and admit no sensible motions. But because the soul does not flee from spiritual and divine influences, it is understood to possess some matter, even if it is of a different nature than that of the senses. Because the rational soul is subject to opposing passions from which virtue and vice arise as battling habits, it often experiences discord. This discord is brought in by the body to which it is joined. Once exempt from the body, the soul no longer experiences it. From these things, it is easy to understand what Boethius suggests: he adopts quantities, qualities, forms, and that kind of accident as things that do not change in their own nature. For these are not changed when they are grasped by themselves, absolute and removed from matter, but only when they are joined to matter. But someone might object that growth and decrease belong to the nature of quantity, and intensity and remission belong to quality, as is said often in physics. The author claims these things neither grow by stretching nor diminish by pulling back. This difficulty can be resolved because these traits are attributed to things not when they are viewed as absolute and abstract in thought, but as they are woven and joined to matter. For this reason, the geometer discussing magnitudes does not attribute growth, decrease, or density to them because his consideration is abstract. It is the natural philosopher who considers them in a combined state. Aristotle is the authority for this distinction between the geometric and the physical. He shows what he wants in the following words: "These things," he says, "are by nature incorporeal and thrive by the reason of immutable substance. But by participating in a body, they are changed, and by the touch of a variable thing, they pass into changeable inconstancy." What is this but to say they are immutable by their own nature and only changed by the partnership and touch of matter? That growth and its opposites are attributed to quantity depends on matter. Separate quantity from matter like the geometers do, and you find no rarity and no growth that clings to change. Therefore, he concludes that these are truly the things that are subject to no change by their nature. Furthermore, we call an accident an imperfect being, not subsisting by itself...

By whatever greater abstraction? things are taken, the more? immutable? they are taken?. Then knowledge? is more? harmful to? the perfection of wisdom?.
In every? way? to things? and? composites?.
Since it is as? an accident? at an angle? and? will be changed?, but? when? it opposes? it remains? immutable?.

Book I

...and not resting on its own support. This is how it is seen in physics and the subsistence of nature. But when the author asserts that things in the category of accidents truly exist, he refers that existence to an abstract and mathematical state. In that state, they exist truly, and far more truly than in matter. Thus, that plane is true, that line is true, and that point is most true which the geometers discuss. The Stagirite Aristotle does not keep silent about this in the thirteenth book of his Metaphysics, where he proves the excellence of mathematics. It should be noted here that Boethius gives the name of "wisdom" to the mathematical sciences. One might suspect he does this more for their praise than for strict truth. For it should not be granted that mathematical things excel supramundane things, since those divine things exist above all motion and time in a most happy age. Yet I will freely grant him that mathematics is a certain part of wisdom, because it deals with immutable things, which are the subject of wisdom. If by the name of mathematics one hears not only the knowledge of numbers and magnitudes, but also the divine contemplations original: theōriai included in that same scope, then I will gladly acknowledge the name of wisdom given to them. This is not far from the Pythagoreans, who embrace the mystical significance of numbers. I believe Boethius took them in this way. Now we must try to show that those things that are not subject to motion are more perfect, and that all motion is a proof of imperfection. Every change is led toward either a perfect or an opposite form, by which it is completed. Whatever is led toward achieving perfection is not yet standing in that perfection, and therefore shows itself to be minimally perfect. Nothing moves toward what it already possesses. Thus, when a motion tending toward perfection is present, the thing is not yet perfect. That motion is an argument for a perfection not yet held. It is even more so for motion toward an opposite, which is an imperfection of the moving thing. This sounds like a certain weakness in the thing undergoing it. Nature seeks to avoid such things. To be driven toward something by motion is a sign of the weak and powerless. Thus it is clear that motion is a proof of imperfection. We have taught that motion depends on matter and is joined to it. Therefore, things that approach matter also approach motion and are more imperfect. Because of this, philosophers say matter is almost nothing and a "non-being," as it is the receptacle of all motion. On the other hand, things that move away from matter also move away from motion and change. Therefore, they move away from "not-being," especially since matter is closest to non-being. But what is it to move away from matter if not to approach the reason of pure act? To move away from motion is to approach stability. To move away from non-being is to approach being. To move away from one opposite is to approach the other. If things rise by this retreat, those that move away most must sustain being the most. God moves away from matter by an infinite amount; therefore, he is infinite act. But how can there be infinite act if he has any potentiality? It is as impossible as for an immense light to have some darkness. He is also immense stability and immutability. What are these other than eternity? Therefore, God is eternal. Furthermore, God is the greatest and most immense being. How is he the greatest being if something else were more perfect than him? How is he immense if something else were equal to him? If something were equal, it would be measured by an equal measure. Therefore, God is one, eternal, and immense. In the same way, it is not difficult to see that beings separated from matter are more perfect than those bound to matter. Plato asserted that supramundane things are superior to mathematical things, and these are superior to sensible things. He also stated that mathematical beings hold a middle place between the "Ideas" (which he called separate) and sensible things. Secondly, mathematics distinguishes the beings within its scope, making the quadrivium four-fold path itself. Furthermore, when he calls mathematical beings the "essences of things," it seems this should be taken symbolically original: symbo-likōs, as will be clearer in the next chapter. The subsistence of a thing does not exist without a limit and a distinction. That which is indeterminate and infinite is said to be "not being" rather than "being." Thus, the being of a thing is taken from its "act," not from its matter. Magnitude expresses the limit of each thing, and number expresses its distinction. To that extent, number and magnitude are called the essences of things. I am not unaware that some Pythagoreans and almost all Platonists say mathematical beings are the essences of sensible things, as if they are the things from which they consist. Aristotle refutes them in the 13th and 14th books of his Metaphysics. However, I would prefer that Nicomachus, whom Boethius interprets here, took them symbolically, as did Plato.

Physical essence through natural substance is the same as one science formally or by sight.
Because that which is changed is less perfect, it is changed toward the better.
So that which is more perfect by motion is the very likeness to perfect things, while motion and nature are more imperfect.
By as much as it moves away from matter, by so much more it approaches the reason of act.
By which he understands his wisdom above mundane things; these excel sensible things. Plato asserted mathematics exists in a middle place between the ideas and the sensible things which he called separate.
In the reasoning of numbers and magnitude, they are called the essence of things.