The Arithmetic of Boethius

...not subsisting, and not resting upon its own support; this happens when considered in physics and the subsistence of nature. But when the author asserts that things in the category of accidents truly and properly exist, he refers that very thing to the abstract and mathematical existence. By which indeed they truly are, and far more truly than in matter. Thus, that plain is true, that line is true, and that point is most true, concerning which geometers differ—a thing which the Stagirite Aristotle does not undermine in the thirteenth book of the Metaphysics Supernatural/Metaphysical books, thereby confirming the excellence of mathematics.

It is to be noted here in passing that Boethius attributes the name of wisdom to the mathematical arts, which it may be suspected he does more for their greater commendation than for the sake of truth. For it should not be attributed to what are called mathematical that they excel the supramundane, since these, excluding all sensible transformation, endure above all motion, change, and time in a most happy age. Nevertheless, I will ingenuously grant that mathematics is a certain particle of wisdom, and this because it deals with immutable things, of which wisdom consists. But if by the name of mathematics is meant not only the knowledge of numbers and magnitudes, but also the divine theoria contemplation concluded within that same scope, and this not far from the Pythagoreans, who embrace the mystical significance of numbers, then I will gladly acknowledge that the name of wisdom is attributed to them. And I would believe that Boethius has taken it in this way.

Otherwise, that those things are more perfect which are not subject to motion, and that all motion is an argument for imperfection, must now be attempted to be shown. Every change is brought toward a perfect or opposite form, namely by which it is completed and terminated. But that which is led toward the attainment of its own perfection—which is mobile—proclaims that it has not yet undergone this, and is therefore in no way perfect; since nothing is moved at all toward that which it already possesses. And this happens: so that when motion directed toward perfection is present, that thing is not yet perfect. Thus, that motion is an argument of a perfection not yet possessed. But it is much more so with that which is moved into the opposite, and into the imperfection of the mobile thing, as this sounds of a certain impotence in the thing undergoing it, especially since there is nothing in the appetite of nature that embraces such a thing, but rather it recoils; why it is driven to this through motion is the sign of an impotent and weak thing. And so it is certain that motion is entirely an argument of imperfection. We have taught that motion depends on matter and is joined to it. Therefore, those things that approach it more are likewise seen to approach motion more; consequently, they are more imperfect and exist less. Thus, it is not without reason that matter is called by philosophers almost nothing and non-being, as it is the receptacle of every motion whatsoever. On the other part, those that recede from it are also convinced to recede from motion and change. Therefore, they also recede from non-being, especially since it is the same as being next to non-being. But what is receding from matter except approaching the reason of act? And receding from motion, except approaching stability? Receding from non-being, except approaching being? For to recede from one of the opposites is to approach the other. But if those things arise from recession, those which recede more, the same also obtain more. If to the greatest degree, then to the greatest degree. And if by the immense, it is worth the effort that they obtain those very things by the immense. But God recedes from matter by the immense; therefore, He is an immense act. But how can an immense act exist if it has any potentiality? For that fights as much as immense light having some shadows. There is also immense stability and immutability. But what are these other than eternity? Therefore, God is eternal. Moreover, God is the maximum and immense being. But in what way is it the maximum being if anything possesses more perfection? How is it immense if some other thing is equal to it? Since, if it is equal, it becomes known to be an equal measure. God is, therefore, from the mode of ascending: one, eternal, and immense. By the same, it is not difficult to detect that beings separated from matter are more perfect than those that are addicted to matter. And the more separated, the more so. And urged by these reasons, Plato asserted that the supramundane excel the mathematical, and these excel the sensible. He also asserted that mathematical beings obtain a place in the middle between the ideas (which he called separated) and sensible things.

2 Secondly, mathesis mathematics distinguishes between the beings it includes in its scope, wishing to make the quadrivium passable to that extent. Furthermore, that he calls mathematical beings the essences of things, it seems this should be taken symbolikos symbolically, which will be seen more in the following chapter. For the subsistence of a thing is by no means without limit and discretion. For that which is interminate, infinite, and which obtains no discretion, is said not so much to be as not to be. Thus, the being of a thing is taken from act, not from matter. But magnitude expresses the limit of each thing; number, however, expresses discretion. Thus, to that extent, number and magnitude are called the essences of things. Nor is it hidden from me that some Pythagoreans and almost all Platonists say that mathematical beings are the essences of sensible things, as if they are things from which they consist. Aristotle refutes them in the 13th and 14th books of the Metaphysics supernatural books. I would prefer, however, that Nicomachus, whom Boethius interprets here, takes them symbolically. The same for Plato and...