The Arithmetic of Boethius

An even number accepts a division into two unequal parts, but in such a way that in neither division is parity mixed with imparity, nor imparity with parity. This is true for all except the number two binarius the dyad, which is the leader of parity; it does not accept an unequal section because it consists of two units and arises from the first parity of two.

What I mean is this: If an even number is set down, it can be divided into two equal parts, as ten is divided into fives. Furthermore, it can be divided into unequal parts, as the same ten is divided into three and seven. But it works in this way: when one part of the division is even, the other is also found to be even; and if one is odd, the remaining part does not differ from that imparity. Consider the same number ten. When it is divided into fives, or when it is divided into three and seven, both parts in each portion were odd. If, however, this or another even number is divided into equals, such as eight into four and four, and likewise by unequals, as the same eight into five and three: in the first division both parts were made even, and in the second both were odd. It can never happen that when one part of the division is even, another can be found to be odd, or when one is odd, another can be understood as even.

An odd number, however, is that which, in any division, is always divided into unequal parts, so that it always shows both species of number. One species is never without the other: one part is assigned to parity, and the other to imparity. For example, if you divide seven into three and four, one portion is even and the other is odd. This same pattern is found in all odd numbers. Never, in the division of an odd number, can these twin species be apart from one another, for they naturally compose the power and substance of number.

If these must also be defined through their alternate species, an odd number is said to be that which differs by a unit from an even number, either by increase or decrease. Likewise, an even number is that which differs by a unit from an odd number, either by increase or decrease. For if you take one away from an even number, or add one to it, it becomes odd; or if you do the same to an odd number, an even one is immediately brought forth.

COMMENTARY ON CHAPTER THREE

Number is only? in the soul? according to the mind? of philosophers? but Boethius? was otherwise?

A decorative woodcut initial letter Q depicts a classical figure or child within an ornate rectangular frame decorated with floral patterns.

Since the parts of number were explained a little while ago, the author now proceeds to explain its substance further by definition. And since quantity is twofold—continuous and discrete—number is associated with discrete quantities, not continuous ones. There is, however, a certain number belonging to continuous things, which serves as a measure, such as the parts of time and magnitude. Hence time, like a continuum, is called a number, and it borrows many things from numbers. It is, however, a numbered number, not a numbering one. Indeed, the numbering number—which we touched upon slightly—exists subjectively in the soul. It has such a close proximity to the soul that some of the philosophers said the soul is number. But properly speaking, the "numbering" aspect is primary, not the "numbered." For number is the unique instrument of distinction.

Perhaps the shared name for both "numbering" and "numbered" gave the modern thinkers an excuse for error. For Aristotle expresses in several places that number exists in things and is also distinct from them. Boethius moved to make number identical with the things themselves, whereas the Nominales Nominalists, casting aside the things themselves more than is fair and favoring the logic of names, clearly indicate how they define number. This identity with things is true of the "numbered" number, which is all they notice. The Reales Realists, however, distinguish the "numbering" number from things, which is correct. But Aristotle does not approve of their claim that it is immersed in things, and Boethius even less so. This is not difficult to grasp from Aristotle’s First Philosophy Metaphysics, especially books 13 and 14, in which he refutes various opinions of the ancients regarding numbers. And this fits well with the measures of continuous things:

Book I

which are considered twofold: the one that measures and the one that is measured, to which the measuring act is applied. A wooden yardstick and a yard of cloth clearly demonstrate this. This distinction helps significantly in understanding what follows. For how could you not divide a unit if you make number a sensible, physical thing? And in what way would you recognize triangles, squares, or other figured numbers as they are described, or their mutual relationships? I omit the fact that in sensible things, the proportions of numbers are found not primarily, but only symbolically—unless you were to say, babbling into your cups, that two horses make a double interval to one horse, or three oxen to two flies make a sesquialterum a ratio of 3 to 2. Such a claim is surely ridiculous in arithmetic.

Number is thus defined by Boethius as a collection of units, or a heap of quantity poured out from units. Furthermore, since every definition ought to express a cause, the first definition surely expresses the formal cause, though without neglecting what is ascribed to the material cause. For the name "units"

The name Unit expresses? the matter;? Union truly? expresses the form from the connection.?

expresses the matter itself. But the "union" and the "connection" express the form, or at least that which is analogous and corresponding to the form. That collection and connection of units happens by the craftsmanship of our mind. To that extent, number is recognized as the first composite of our mind. For what the Divine Mind is to creatures, the human mind is to its numbers. And just as creatures proceed from God by divine art, so numbers proceed by the craft of the human mind. Likewise, just as every creature which is made one and held as one has that from the Divine Mind, so too the number of our mind, which is made one and held as one, has that from our mind. If you take away the mind, there is no number, much less a single one.

THE SECOND definition of number seems to express, beyond those things, the efficient cause, since it declares that heap and that discrete quantity and multitude to be "from units." This implies that the unit is the principle of numbers. Truly, however, the mind itself is the primary cause of numbers. The unit is in the second place, acting as the instrument of our mind in forming the number. Nor is anything else implied by such a definition than that number proceeds from the unit by a kind of flow, much as a line is formed by the flow of a point, a surface by the flow of a line, and a body by the flow of a surface. Nonetheless, it consists of units. In this, it takes its distinction from a point: although a point is a boundary of a line, it is not a part of it. This is because a point has position in a continuum, and no continuum is composed of indivisible parts; thus, by the addition of one indivisible to another, nothing is said to be made or increased. On the contrary, this happens in discrete quantity. For by the additions of units, which are indivisible, numbers are increased according to their distinction.

Therefore, the second definition expresses the principle of numbers: from what they proceed and from what they are constituted. Indeed, insofar as number is from the unit, the unit carries a certain logic of an efficient cause. But insofar as number is constituted from it, the unit seems to be linked to the material cause. From this, it becomes known in some part: just as all things are from the Divine Mind, so in a way all things are from our mind. For what God is in the creation of things, our mind is in the production of numbers. The Divine Mind discerns all things; our mind also discerns all things. But God’s discernment is the production of things in their own subsistence. Our discernment, however, is only of numbers, which are likenesses of the divine discernment.

Truly, we are granted a way to rise through the unity of our mind to that divine and incomprehensible Unity. For the fact that the unit is the beginning of all numbers—as it is that from which every number flows—and also the end of all—as it is that into which every number is resolved—is a trace of the divine unity. The unit does not draw its origin from anything else, nor is it cut into any number. It can exist without numbers, but numbers are so far from being able to exist without it that it is most intimate to them. God is the beginning and end of all things, so that He is not without reason called the Alpha and Omega the first and last letters of the Greek alphabet, opening and closing all things. Nothing exists before or after Him. He is so far from taking His origin from creatures that He preceded them by an interval, existing without them. Creatures, conversely, cannot exist without Him. Their being, living, feeling, reasoning, understanding, and whatever else is found in creatures belongs to that supreme Unity.

That supreme Unity gives being to things far more than a creature gives being to its own image. Thus the Crater of Hermes The Mixing Bowl of Mercurius Trismegistus warns us to think from the monad unit of our mind toward the true Monad. He says: "The Monad, that is, Unity, is the principle, root, and origin of all things; without a principle, however, there is nothing. A beginning is not of a principle, but of something else. Therefore the Monad is the principle, containing every number, but contained by none. It begets every number, but is begotten by no number. Whatever is begotten is imperfect, divisible, increasing and decreasing. But to that which is perfect, none of these things happen. Indeed, that which increases is increased by the power of the Monad, but it vanishes by its own weakness when it can no longer contain the Monad." These are the words of the Crater. Nor is it without a shadow of divine light that the unit is the limit and measure of all numbers, measuring all, communicating its name to all numbers, and yet nameable by none.