Commentary Aristotle Metaphysics
Lecture 8. The Pythagoreans’ doctrine of contraries
But the reason we have come [to examine these philosophers] is that we may also learn from them what they hold the principles of things to be, and how these principles fall under the causes already described. Now these men also seem to think that number is the principle of existing things both as their matter and as their attributes and states.
According to them, the elements of number are the even and odd; of these the latter is limited and the former is unlimited. The unit is composed of both of these, since it is both even and odd, and number is derived from the unit. And number, as has been stated, constitutes the whole heaven.
But other members of the same school say that the principles of things are ten in number, which they give as co-elements: the limited and unlimited, even and odd, one and many, right and left, masculine and feminine, rest and motion, straight and curved, light and darkness, good and evil, the quadrangular and rectangular.
Alcmaeon of Croton seems to have formed his opinion in the same way, and either he derived the theory from them or they from him; for Alcmaeon (who had reached maturity when Pythagoras was an old man) expressed views similar to those of the Pythagoreans.
For he says that many things in human affairs are in twos, calling them contrarieties, not distinguished as these men had distinguished them, but such as are taken at random, for example, white and black, sweet and bitter, good and evil, small and great. It is true that this philosopher threw out vague remarks about the other contrarieties, but the Pythagoreans have declared both what the contrarieties are and how many there are.
From both of these, then, we can gather this much, that contraries are the principles of existing things; but how many they are and that they are these we must gather from other thinkers. The way in which many principles can be brought together under the causes described is not clearly expressed by them, although they seem to allot their elements to the species of matter; for they say that substance is composed and molded out of these as something inherent.
From these remarks, then, it is possible to get an adequate understanding of the ancient philosophers’ meaning who said that the elements of things are many.
124. Here he states what the Pythagoreans had to say about the principles of things. In regard to this he does two things.
First, he expounds their opinions about the principles of things;
second, he indicates to what genus of cause the principles laid down by them are reduced, at from both of these.
In regard to the first he gives three opinions.
The second begins at the words, but other members;
and the third at Alcmaeon of Croton.
He says first, then, that the reason he came to examine the opinions of the Pythagoreans is that he might show from their opinions what the principles of things are and how the principles laid down by them fall under the causes given above. For the Pythagoreans seem to hold that number is the principle of existing things as matter, and that the attributes of number are the attributes and states of existing things. By “attributes” we mean transient accidents, and by “states,” permanent accidents. They also held that evenness of number is justice, because of the equality of division. Such a number is evenly divided into two parts right down to the unit: for example, the number eight is divided into two fours, the number four into two twos, and the number two into two units. And in a similar way they likened the other accidents of things to the accidents of numbers.
125. In fact, they said that the even and odd, which are the first differences of numbers, are the principles of numbers. And they said that even number is the principle of unlimitedness and odd number the principle of limitation, as is shown in the Physics 3; but in reality the unlimited seems to result chiefly from the division of the continuous. An even number is capable of division, but an odd number includes within itself an even number plus a unit, and this makes it indivisible.
He also proves this as follows. When odd numbers are added to each other successively, they always retain the figure of a square, whereas even numbers change their figure. For when the number three is added to the unit, which is the principle of numbers, the number four results, which is the first square, because two times two equals four. Again, when the number five, which is an odd number, is added to the number four, the number nine results, which is also a square number. And so on with the others.
But if the number two, which is the first even number, is added to the number one, a triangular number results—the number three. And if the number four, which is the second even number, is added to the number three, there results a septangular number—the number seven. And when even numbers are added to each other successively in this way, they do not retain the same figure. This is why they attributed the unlimited to the even and the limited to the odd. And since limitedness pertains to form, to which active power belongs, they therefore said that even numbers are feminine, and odd numbers masculine.
126. From these two (namely, the even and odd, the limited and unlimited), they produced not only number but also the unit itself—unity. For unity is virtually both even and odd, because all differences of number are virtually contained in the unit, all being reduced to the unit. Hence, in the list of odd numbers the unit is found to be the first. And the same is true in the list of even numbers, square numbers, and perfect numbers. This is also the case with the other differences of number, because even though the unit is not actually a number, it is still virtually all numbers. And just as the unit is said to be composed of the even and odd, in a similar way number is composed of units. Indeed, the heavens and all sensible things are composed of numbers. This was the sequence of principles which they gave.
127. But other members. Here he gives another opinion which the Pythagoreans held about the principles of things. He says that among these same Pythagoreans there were some who claimed that there is not just one contrariety in principles, as the foregoing did, but ten principles, which they give as co-elements, that is, by taking each of these principles with its co-principle, or contrary.
The reason for this position was that they took not only the first principles but also the proximate principles attributed to each genus of things. Hence, they posited first the limited and the unlimited (as did those who have just been mentioned), and subsequently the even and the odd (to which the limited and unlimited are attributed). And because the even and odd are the first principles of things, and numbers are first produced from them, they posited third a difference of numbers—namely, the one and the many, both of which are produced from the even and the odd. Again, because continuous quantities are composed of numbers, inasmuch as they understood numbers to have position (for, according to them, the point was merely the unit having position, and the line the number two having position), they therefore claimed next that the principles of positions are the right and left, for the right is found to be perfect and the left imperfect. Therefore, the right is determined from the aspect of oddness, and the left from the aspect of evenness. But because natural bodies have both active and passive powers in addition to mathematical extensions, they therefore next maintained that masculine and feminine are principles. For masculine pertains to active power, and feminine to passive power; of these, masculine pertains to odd number and feminine to even number, as has been stated.
128. Now it is from active and passive power that motion and rest originate in the world. Of these, motion is placed in the class of the unlimited and even, because it partakes of irregularity and otherness, and rest in the class of the unlimited and odd. Furthermore, the first differences of motions are the circular and straight, so that as a consequence of this the straight pertains to even number. Hence they said that the straight line is the number two, but that the curved or circular line, by reason of its uniformity, pertains to odd number, which retains its undividedness because of the form of unity.
129. And they posited principles not only to account for the natural operations and motions of things, but also to account for the operations of living things. In fact, they held that light and darkness are principles of knowing, but that good and evil are principles of appetite. For light is a principle of knowing, whereas darkness is ascribed to ignorance; good is that to which appetite tends, whereas evil is that from which it turns away.
130. Again, they said that the difference of perfection and imperfection is found not only in natural things and in voluntary powers and motions, but also in continuous quantities and figures. These figures are understood to be something over and above the substances of continuous quantities, just as the powers responsible for motions and operations are something over and above the substances of natural bodies. Therefore, with reference to this they held that what is quadrangular (that is, the square and the rectangle) is a principle. Now a square is said to be a figure of four equal sides whose four angles are right angles, and such a figure is produced by multiplying a line by itself. Therefore, since it is produced from the unit itself, it belongs to the odd numbers. But a rectangle is defined as a figure whose angles are all right angles and whose opposite sides alone, not all sides, are equal to each other. Hence it is clear that, just as a square is produced by multiplying one line by itself, in a similar way a rectangle is produced by multiplying one line by another. Hence it pertains to the even numbers, of which the first is the number two.
131. Alcmaeon of Croton. Here he gives the third opinion of the Pythagoreans, saying that Alcmaeon of Croton, so named from the city in which he was raised, seems to maintain somewhat the same view as that expressed by these Pythagoreans, namely, that many contraries are the principles of things. For either he derives the theory from the Pythagoreans, or they from him. That either of these might be true is clear from the fact that he was a contemporary of the Pythagoreans, granted that he began to philosophize when Pythagoras was an old man.
But whichever happens to be true, he expressed views similar to those of the Pythagoreans. For he said that many of the things in the realm of human affairs (that is, many of the attributes of sensible things) are arranged in pairs, understanding by pairs opposites which are contrary. Yet in this matter he differs from the foregoing philosophers, because the Pythagoreans said that determinate contraries are the principles of things. But he throws them in, as it were, without any order, holding that any of the contraries which he happened to think of are the principles of things, such as white and black, sweet and bitter, and so on.
132. From both of these. Here he gathers together from the above remarks what the Pythagoreans thought about the principles of things, and how the principles which they posited are reduced to some genus of cause. He says, then, that from both of those mentioned above (namely, Alcmaeon and the Pythagoreans), it is possible to draw one common opinion not expressed by earlier thinkers: the principles of existing things are contraries. This must be understood with reference to the material cause.
For Empedocles posited contrariety in the case of the efficient cause, and the ancient philosophers of nature posited contrary principles, such as rarity and density, although they attributed contrariety to form. But even though Empedocles held that the four elements are material principles, he still did not claim that they are the first material principles by reason of contrariety, but because of their natures and substance. These men, however, attributed contrariety to matter.
133. The nature of the contraries posited by these men is evident from the foregoing discussion. But how the aforesaid contrary principles posited by them can be brought together under (or reduced to) the types of causes described is not clearly expressed (or distinctly stated) by them. Yet it seems that such principles are allotted to the species of material cause, for they say that the substance of things is composed and molded out of these principles as something inherent, and this is the notion of a material cause. For matter is that from which a thing comes to be as something inherent. This is added to distinguish it from privation, from which something also comes to be but which is not inherent, as the musical is said to come from the non-musical.