Arithmetic of Boethius | Book I

But these points were proposed in the earlier sections where he speaks of mathematics.

He solves an objection which could have arisen from the preceding points in this manner: Number increases toward the infinite, such that no "last" number is given. Likewise, magnitude decreases so that the one dividing it is never allowed to stop. However, the infinite, being restrained by no limits, cannot be known. Indeed, all our knowledge is comparative, using a mean of proportion, and exists in the proportion of the known to the unknown; yet no proportion is accommodated to the infinite.

Indeed, this dissolution provided by Boethius is ready at hand. Philosophy spontaneously rejects this infinity, since nothing of such a kind can be comprehended. Instead, it only takes from the infinite heap of numbers those which are finite. Nor does it consider the sectioning of parts which they call proportions without end, a thing more suitable for those who pervert every calculation of reason into a circle, but rather those things which consist of a determined and defined quantity. From this, it is easy to recognize that those who ask for a line to be divided in all parts of the same proportion are no longer acting mathematically, since the principle of infinity does not permit it.

Notice that saying number is infinite in its increase, while magnitude is infinite in its decrease, seems in no way dissonant with Plato. To that extent, he posited a double infinite: the Great and the Small. He attributed the Great to increase, and the Small to decrease; to that extent, he made the Great and the Small the principles of things.

So that it may be clear through those things which were said concerning the Great and the Small, and how multitude and magnitude arise from one or the other.

Furthermore, whoever would attribute the principle of the infinite small to matter, by which it is seen to be continuously cut and distributed into portions according to the demands of forms, but would attribute the principle of the infinite great to forms on account of their continuous increase and plurality, would see that all beings of nature consist of the great and the small, and that one efficient cause presides over them. But these things are said according to the mind of Plato. For as much as I can gather by the conjecture of my own mind, I would contend that Plato constituted the universe from numbers and magnitudes, namely by the Great and the Small.

If he took this symbolically, as his noble expositors seem to divine, that assertion will not seem very unreasonable. That which he finally joins is this: it is that quadrivium the fourfold path of mathematical sciences by which those must travel who have a more excellent mind, being led from the senses born with us to the certainties of intelligence. This expresses the good principle of mathematics, that it consists in divine theories.

From Boethius, he proceeds to the love of mathematics.

By these things born with us, which are subjected to our senses, we are led to the certainties of intelligence and those divine things which the intellect alone grasps. Nor does any more apt trace of discussing divine things in a human way occur, as has already been stated.

In the fourth place, he shows the order to be observed in the progress of mathematics. For there are set orders and certain progressions in the disciplines, by which the disciplines are more easily acquired. But this is especially true in these which illuminate the eye of the internal mind. In this genre are those things which the Greeks call mathēmata the mathematical sciences and the Latins call disciplinae disciplines. Furthermore, what Nicomachus asserts—namely that the eye of our mind, submerged and blinded by bodily senses, is again illuminated by these disciplines—seems to allude to Plato. Plato held that the intellect had sciences created along with it, but that an oblivion of all things occurred through its submersion into the body. Truly, the works of mathematics are again illuminated with a full torch, and these things are anamnēskesthai to be remembered through recollection. Indeed, it is consonant with reason that a certain native judgment is implanted in our intellect, and that reason is not deprived of its own light. This is why it draws out the principles of things with almost no labor, from which it proceeds to the recognition of conclusions through the discourse of reason. However, the Peripatetics followers of Aristotle do not approve of the idea that sciences are created with the soul, but were lost due to submersion in the body. It is believed to have been done by the Lord so that the soul, created without virtues and sciences, would be infused into the body, so that it might not persist in idleness, but rather, using the body as an organ, might acquire these things for itself.

However, the order of mathematics is this: arithmetic is established as the first of all. And he proves this not only symbolikōs symbolically:

The exemplary number, which is nothing other than a 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), and it is from the previous definition.

insofar as the exemplary number which is nothing other than a 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 also rationally, by assuming the definition of the "prior." For the "prior" is that from which the consequence of existing is not reversed. When it is taken away, that which is posterior is removed; but when it is posited, the posterior is not necessarily posited. To that extent, "animal" is prior to "man," and every genus is prior to its species. For if there is no animal, there is no man; but if animal is posited, it is not therefore necessary that man be posited. Many animals exist that have pulses but are not men. By a similar reason, arithmetic is found to be first of all. Indeed, if numbers are taken away, geometric figures are removed. For if "three" does not exist, how can there be a triangle? Likewise, if the number which is "per se" existing in itself is taken away, the number "in relation to something" is also removed. Therefore, arithmetic is prior both to geometry and music. And since astronomy is posterior to these same ones, as it takes much from them, it is clear that arithmetic is prior to them as well, which can be gathered without much effort from the text. Therefore, we must pass on to the rest.

God established all things in number, measure, and weight. Wisdom of Solomon 11. And he always holds them so.

ON THE SUBSTANCE OF NUMBER. CHAPTER II.

The world is like a most beautiful bond of things.

An ornate drop cap 'O' contains a small landscape scene with a seated figure, likely representing the Creator or a philosopher contemplating nature. ALL THINGS, whatever have been constructed from the primeval nature of things, seem formed by the ratio of numbers. For this was the principal exemplar in the mind of the Creator. From here the multitude of the four elements was borrowed; from here the changes of the seasons, the motion of the stars, and the revolution of the heavens. Since these things are so, and since the status of all things performs its function through the binding together of numbers:

Nothing seems to be composed of similar things.

it is also necessary that this number remains in its own substance, always possessing itself equally, and that it is not composed of diverse things. For what would join the substance of number, when its own example had joined all things? Rather, it seems to be composed of itself. Furthermore, nothing seems to be composed of similar things, nor of those which are joined by no proportion of reason, and which are discrete from one another in all substance and nature. It is certain, therefore, since number is joined, that it is neither joined from similar things, nor from those which cling to one another by no ratio of proportion. Therefore, there will be primary numbers which are joined into a substance, which consist and remain always for nothing can be made from non-existing things, and they are themselves dissimilar and possess the power of being composed.

Nor to cling to one another from innumerable joined things by no ratio of proportion. It is impossible because something can be made from non-existing things.

These things of which number consists are: the even and the odd. These, by a certain divine power, although they are disparate and contrary, nevertheless flow from one birth and are joined into one composition and modulation.

Number consists of evenness and oddness which, by a certain divine power, are contraries, yet they are united and flow from one birth.

COMMENTARY ON THE SECOND CHAPTER.

Nicomachus in this part follows the Pythagoreans, who compose all things from even and odd, just as from the one and the other, the equal and the unequal, the great and the small. This is because they give the name of oddness to unity and equality on account of their indivisible and discrete nature, while they give the name of evenness to the other, seeing their ratio by a divisible difference. Furthermore, because number and otherness cling together inseparably, that which precedes number also precedes otherness. If you take this away, change can in no way remain. Therefore, the highest unity precedes all plurality and is prior to any number. Therefore, the highest unity is prior... for the highest is the highest connection... And if division is taken away, how can change or otherness remain? Therefore, the highest connection is eternal.

An ornate drop cap 'A' features a seated scholar figure working at a desk with a book, symbolizing the commentator or Boethius himself. NOTICE that Boethius in this second chapter, as in many others, follows Nicomachus. In this part, he agrees with those Pythagoreans who compose every number from the even and the odd, just as from the "one" and the "other," the equal and the unequal, the great and the small. They give the name of oddness to unity and equality for the reason that these have an indivisible and indiscrete nature. Conversely, they give the name of evenness to "otherness" and inequality, seeing that their ratio involves a divisible difference. Furthermore, "otherness" and inequality do not so much aim at number and cling to it, as much as the ratio of number is compatible with nothing except those things in which these are present. For things which are in no part different cannot even constitute a plurality. For instance, you would not say that Marcus Tullius Cicero is "more than one," because one thing is the subject of these three names. From this place, a location clearly offers itself to rise toward the eternity of the highest unity and connection. Namely, because number and otherness cling together inseparably, it is certain that whatever precedes number also precedes otherness. If you take that away, change can in no way remain, especially since there is no change at all without an exceeding part and an exceeded part, which are different and diverse from one another. Yet the highest unity precedes all plurality and is prior to any number. Therefore, it is also prior to otherness and even to change. Accordingly, it escapes being subject to any change at all, and to that extent is eternal. By a similar reason, the highest equality which precedes all inequality, to which change is joined is proven eternal. Thus no change exists without inequality, since it is necessary for the "greater" and the "lesser," which constitute inequality, to be present. Furthermore, since unity is the cause of connection, while the "binary" the number two is the cause of division, and unity is prior to the binary, the highest connection is recognized without effort. Whatever is prior to division must be considered the cause of the prior effect. But if division is taken away, how can change or otherness remain? Therefore, the highest connection is eternal.

And it is impossible for there to be many eternal things: for since every plurality is bounded by unity, if some "many" were established as eternal, there would be something prior to the eternal.

Furthermore, it is impossible for there to be many eternal things. For because unity bounds all plurality behind itself, if multiple eternals were established, there would be something prior to the eternal. It is certain, therefore, that those things are eternal, and nonetheless they are one. And it is a wonder,