Arithmetic of Boethius | Book I

b.j.

Abstract things are better known to us through their own abstract instances.

Some assert that Pythagoras made these claims in his own assertions, but we have not thought it necessary to discuss that in the present work. Therefore, those things which express the essences of things to us are quantities, and they are twofold: one is continuous, the other is discrete. For the parts of some quantities are gathered and joined by a common boundary. I call it a common boundary when it serves as the beginning of one part and the end of another, just as a point connects a line, a line connects a surface, and a surface connects a body. For the parts in the middle of a line are soldered and joined together by a common atom indivisible point. Likewise, the parts of a surface are joined by a common line. Finally, the middle parts of a body are joined by a common plane. These are counted among the continuous numbers. Furthermore, the parts of other quantities are not connected by any common point at all, but remain discrete in themselves, which is easily recognized in numbers. For the parts of the number two, namely the units, have no boundary that acts as a glue or bond between them. In that case, there is nothing that acts as the beginning of one and the end of the other. These are the discrete quantities. Therefore, one quantity is continuous, and the other is discrete. Magnitude is continuous, while number is discrete.

Every quantity is known through itself, even the thing from which the quantity is made.

Note that Boethius uses concrete and contracted terms here instead of abstracts, which Aristotle is also seen to observe in many places. For concrete things are better known to us than the abstracts themselves. For who would doubt or wonder that those who possess a quality are better known than the quality itself by which they are named? Thus, a grammarian is better known than grammar. Similarly, things that have size are more known than the quantity by which they become sized. Thus, a large thing, or a discrete thing, is better known than magnitude or discretion itself. For this reason, it is not irrelevant that when he wishes to reveal continuous quantity, he assumes a "large thing," and for discretion and number, he assumes a "thing discrete in number." In this way, the human intellect is stirred by concrete notions so that it may form abstract ones, just as it is persuaded of universal things by means of singular ones. I would not have thought this annotation necessary if I had not discovered that among the Neoterici modern scholars, quantities are not distinguished from the things that are quantified. I feared that they might persuade others that Boethius supports their error, using this passage as a poorly understood excuse. They should eliminate this view everywhere, as it is far from his actual opinion, which is not difficult to grasp from his commentaries. For what else does he mean in the fourth chapter of his first book On the Trinity when he says: "For 'one' is a single thing, but 'unity' is that by which it is called one. Again, 'two' exists in things, such as stones, but 'duality' is that by which two men or two stones are so named." And he speaks the same way of other things. Therefore, in the number by which we count, the repetition of units makes a plurality. But in the counting of things, the repetition of units does not make a plurality. Thus far Boethius. From this, everyone understands that counting number is distinguished from concrete number. Boethius thinks that number is a discretion of our mind, but he does not think that things themselves are so discrete. Magnitudes are even less so in his view, for whoever does not distinguish these from their subjects destroys the entire abstraction of mathematics. It is a sin to attribute this to Boethius, who is the foremost among the Latins in mathematical skill. This is the one point that some defend with their assertions, which seem less true than they appear. It is not clear in their view how mathematics is more abstract than physics, since they do not even allow for abstracts. They go so far as to stain and defame the abstractions of mathematics with lies. This is true even if Aristotle, easily the prince of philosophers, should reclaim and resist this. If these things are the same as corporeal substance, what was the point of asserting abstracts in various places without lying, and ascribing them to things that strike the senses, especially since this is diametrically opposed to substance? Those who drag the words of the philosopher in the physical and supernatural disciplines toward the naming of words seem to me to have little wisdom. When they should investigate and discuss the properties of things, they flee to the phantasias original: phantasias; mental appearances of words, not recognizing that logical and rational things must be taken logically, and physical things physically. I would not deny that things are considered in the rational disciplines, but that is for the sake of reasons and is not the primary focus. Likewise, in physics there are reasons, but they are for the sake of the things themselves and are secondary. But to discuss these things more fully is a matter for another task. I pass over the rest in silence, not because it is trivial, for it is of the greatest importance; those who do not do these things close off the path for themselves to bring forth the theorems of mathematics.

Quantities in things where they are not perceived? but known?.

How a thing like a tree or stone might be called [quantity] by Boethius, the Pythagoreans, and mathematicians is proven?.

Therefore, when he calls a tree, stones, and corporeal substances "magnitudes," it must be understood as meaning that these things are large and possess magnitude. I will add what does not displease our friend Faber referring to Jacques Lefèvre d'Étaples: Boethius in this art, as in music, follows Nicomachus and certain Pythagoreans at times, rather than his own opinion. Aristotle says in his works on supernatural things the Metaphysics that some of the Pythagoreans made magnitudes into substances, just as the Moderns do. I believe this must be resolved in one of two ways: either Boethius is not introducing this as his own opinion,

A diagram shows a horizontal line segment with three points labeled "a", "c", and "b". Point "c" is in the center. To the right is a small vertical line. This represents the continuous nature of a line compared to the discrete nature of numbers.

a c b
The distance of continuous and discrete quantity.

or it is to be understood according to the previous notes.

The parts of a continuum are said to be distributed with no boundaries because they do not have separate actual boundaries. Instead, that which belongs to the previous part as an end is the same as the beginning of the following part, and this is the joining point of both. This is easily seen in the line $a c b$, where the middle point is $c$. It has its halves, $a c$ and $c b$, joined rather than divided. That which is the end of the first middle portion is the same as the beginning of the remaining one. Conversely, the parts of a number are discrete. The end of one is not the same as the beginning of the other. Instead, each enjoys its own beginning and its own end. This is no different in number than what happens with syllables in a word. For the two halves of the number six do not have the same beginning, because there is no unit there that finishes the first three and starts the rest. Rather, each is completed by its own unit. Thus, the parts of discretion are enclosed by their own ends.

Absolute number does not look toward material things.

Boethius’s number properly looks toward things.

Again, he subdivides these categories to adapt the proper subject genus to each part of mathematics. Discrete quantity, if considered absolutely and not in relation to something else, is assigned to the theorems of arithmetic. However, it should not be denied that some things in arithmetic are viewed relatively, especially as arithmetic prepares the way for music. In this way, it even embraces some figures, but only insofar as they lead one by the hand to geometry. But relative number that pertains to sound is assigned to music. For no tone or consonance is recognized unless these things are compared to one another. In the same way, geometry contemplates magnitude separated from motion. That which is attached and joined to motion is the subject of astronomy. Thus the quadrivium four-fold path of study is divided into four ways of contemplating quantities.

Furthermore, he adds that without these, the truth can never be found, nor can anyone think rightly. This strongly tastes of the opinion of Plato and the Pythagoreans. How this is so must be explained in a few words. Multitude and number refer to discretion. Magnitude expresses the whole of a thing. Number discerns and is the discretion of things. Magnitude terminates and is the boundary of things. He who understands the truth of a thing recognizes it by discerning it from all other things and does not go beyond the whole of each thing. Therefore, the comprehension of all things falls under number and magnitude. Since these are brought together into one mathematics as the path through which they become accessible, the hidden truth of the essence of things is clearly revealed. You will recognize that he divided the essences of things into magnitude and multitude so that it might be understood more fully through symbols. All things fall under magnitude and multitude because the demonstration of all things happens according to the power of one or the other. Thus it is established that a definition has the power of magnitude, as it contains the whole substance of the thing defined. Division has the power of multitude, as it discerns the essence of the genus by specific differences. A syllogism original: φ, likely representing "philosophical argument" possesses both. For a syllogism with three terms and three enunciations belongs to multitude and discretion. A syllogism with universals and particulars belongs to magnitude. This moved the Pythagoreans to speak of individual things through numbers and the Platonists to do so through magnitudes. This is the foundation of their approaches. Thus, the descent from the highest being to the creatures belongs to multitude, while the ascent belongs to magnitude. This will be weighed more fully in what follows. For this reason, it is not possible to philosophize correctly without the quadrivium. The power of numbers is constrained in arithmetic and music, from which discretion arises in things. In geometry and astronomy, the knowledge of magnitude is contained, from which the entire comprehension of the integrity of things flows. Furthermore, while the skill of mathematics is useful for all things, it especially brings one prepared to a high level for divine things. Since invisible things are not grasped by us except through visible ones, we busy ourselves with grasping truth through images. The more certain the images, the more suitable they are for weighing the truth. But the beings of nature have no stability because of constant motion and change. Therefore, we are less able to look upon truth itself through them than through abstracts. According to the previous method of ascending, the further they recede from matter, the closer they come to greater stability. Furthermore, certainty is annexed to stability. Therefore, they also approach greater certainty. Thus, the beings of mathematics are more certain than the beings of nature because they are more abstract. So much so that the greatest philosophers said that mathematics holds the first grade of certainty, both because of the certainty and immutability of the beings it deals with, and because of the efficacy of the demonstrations, which is greatest in mathematics. If you continue to ascend, you will find the supramundane beings to be both the most certain and the most true, but they are grasped especially through the way of reason, by the medium of numbers and magnitudes.

With the Pythagoreans and Plato as [a path] to divine things...

All mathematical things are more certain than natural beings because they are more abstract.

Whatever pertains to magnitude... number... to the divine and mystic... certainty... just as perfection is not taken away from creatures but every creature has its own perfection from the same. Likewise for Father Boethius.

In the third place, he declares the power of unity in numbers and of magnitude in the sectioning of parts. For the power of unity is recognized as not being exhausted in numbers, no matter what number is assigned. Likewise, the power of magnitude is not exhausted, no matter what sectioning is placed upon it. But what else do these things express in symbols than that the divine power is not limited in creatures? Likewise, magnitude and perfection are not taken away from the creatures, even though each creature has its own perfection from that same source. Followed by b.j.