The Arithmetic of Boethius

Arith. Boetij

He resolves the objection, which might have arisen due to the aforementioned points, in this way: Number increases infinitely, such that no last number is given. Likewise, magnitude decreases, such that there is nowhere for a divider to stop. But the infinite, confined by no boundaries, cannot be known. Since all our knowledge is comparative, using the medium of proportion, it exists in the proportion of the known to the unknown. Yet, no proportion is accommodated to the infinite.

However, this dilution by Boethius is prompt. Philosophy repudiates this infinity of its own accord, because nothing of such a nature can be comprehended. He only assumes the infinite heap of numbers which are finite; nor does he consider the section of parts (which they call proportional) without an end, which would rather accommodate those who pervert all calculation of reason, but rather those which exist of determined and defined quantity. Whence it is prompt to recognize that those who seek to divide a line into all parts of the same proportion are no longer acting mathematically, since, otherwise, the method of infinity does not permit it.

Observe that saying number is infinite by increase, and magnitude, on the contrary, is infinite by decrease, seems in no way dissonant to Plato. To that extent, he posited a twofold infinite: the great and the small. He attributed the great to increase, and the small to decrease. He made the great and the small the principles of things. Furthermore, he who would attribute to matter the nature of the "infinitely small"—by which it seems to be cut continuously and distributed into portions according to the requirement of forms—and to forms, on account of their continuous increase and plurality, the nature of the "infinitely great," would see that all entities of nature consist of the great and the small, and that one efficient cause presides over them. But these things are said outside the mind of Plato. For as far as I can conjecture, I would contend that Plato constitutes all things by numbers and magnitudes, namely by the great and the small. And if he took this symbolically, as his noblest expositors seem to augur, that assertion will not seem very unreasonable.

What he finally adds—that this is the quadrivium The fourfold path of mathematical studies: arithmetic, geometry, music, and astronomy. by which those must travel for whom the mind, more excellent than the senses born with us, is led to the more certain things of the intellect—expresses that a good portion of mathematics consists in divine theories. For by these—the senses born with us, and those things which are subjected to our senses—we are led to the more certain things of the intellect, and those divine things which the intellect alone apprehends. Nor does any trace of reasoning humanly about divine things occur to us more aptly, as already stated.

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In the fourth place, he shows the order to be observed in the progress of mathematics. For in disciplines there are established orders and certain progressions by which disciplines are more easily acquired, but especially in those which illuminate the internal eye; in which genus are those things which the Greeks call mathemata learnings/disciplines, and the Latins call disciplines. Furthermore, what Nicomachus asserts—namely that the eye of the mind, submerged and deprived of bodily senses, is illuminated again by these disciplines—seems to allude to Plato, who believed that sciences were concreated with the intellect, but that through submersion in the body, a forgetting of all things had occurred. But through the work of mathematics, it is again illuminated by a full torch, and thus anamnesis recollection occurs.

For it is consonant with reason that a certain innate judgment is bestowed upon our intellect, and that reason is not deprived of its own light; and that is not a small thing. And that is by which it extracts the principles of things with almost no labor, from which, through the discourse of reason, it proceeds to the recognition of conclusions. But the Peripatetics do not approve that the sciences were concreated with the soul, but that it suffered loss due to immersion in the body. For it is believed to have been done by the Lord so that the soul, created without virtues and sciences, might be infused into the body so that it would not remain idle, but rather, using the body as its instrument, would acquire those things for itself.

But this is the order of mathematics: that arithmetic be established as the first of all. And he proves this not only symbolikos symbolically—insofar as the exemplary number (which is nothing other than the divine mode of knowing) was the first exemplar of the divine mind in the creation of things (which is that God created all things discrete and ordered)—but rationally, and with the assumption of the prior definition. For "prior" is that from which the consequence of existing does not convert, and with which removed, that which is posterior is taken away; but with it posited, it is not necessary that the posterior be posited. To that extent, "animal" is prior to "human," and each genus to its species. For if there is no animal, there is no human. But with "animal" posited, it is not necessary that "human" be posited. For there exist many animals that are not humans. By a not dissimilar reason, arithmetic is found to be the first of all. For with numbers removed, figures are removed. For if there is no "three," how can it be that there is a triangle? Likewise, with the number which is per se removed, it is worth the effort to also remove number relative to something else. Wherefore arithmetic is prior to both geometry and music. And since astronomy is posterior to them, as it takes many things from them, it is evident that it, too, is posterior to arithmetic. These things cannot be gathered from the letter without effort. Wherefore, we must pass on to the remaining points.