LESSON 8

The Primary Sense of One. One in the Sense of Complete. One as the Principle of Number. The Ways in Which Things Are One. The Ways in Which Things Are Many

Chapter 6: 1016b 3-1017a 6

             430. For in general those things which do not admit of division are said to be one insofar as they do not admit of division. Thus, if two things do not admit of division insofar as they are man, they are one man; and if they do not admit of division insofar as they are animal, they are one animal; and if they do not admit of division insofar as they have continuous quantity, they are one continuous quantity. Hence many things are said to be one because they do or undergo or have or are related to some other thing which is one. But those things are said to be one in a primary sense whose substance is one; and they are one either by continuity or in species or in intelligible structure. For we count as many those things which are not continuous, or those whose form is not one, or those whose intelligible structure is not one.

             431. Again, in one sense we say that anything at all is one by continuity if it is quantitative and continuous; and in another sense we say that a thing is not one unless it is a whole, i.e., unless it has one form. Thus in looking at the parts of a shoe which are put together in any way at all, we would not say that they are one, except by reason of their continuity; but if they are put together in such a way as to be a shoe and to have a certain form, there would then be one thing. And for this reason among lines the circular line is one in the highest degree because it is whole and complete.

             432. But the essence of oneness is to be a principle of some number; for the first measure is a principle, because that by which we first come to know each class of things is its first measure. Unity, then, is the first principle of what is knowable about each class. But this unity or unit is not the same in all classes; for in one it is the lesser half tone, and in another it is the vowel or consonant; and in the case of weight the unit is different; and in that of motion different still. But in all cases what is one is indivisible either in quantity or in species. Thus a unit is indivisible in quantity as quantity in every way and has no position; and a point is indivisible in every way and has position. A line is divisible in one dimension; a surface, in two; and a body, in three. And conversely, that which is divisible in two dimensions is a surface; in one, a line; and quantitatively indivisible in every way, a point and a unit. If it has no position, it is a unit; and if it has position, it is a point.

             433. Further, some things are one in number, some in species, some in genus, and some analogically or proportionally. Those things are one in number which have one matter; in species, which have one intelligible structure; in genus, which have the same figure of predication; and proportionally, which are related to each other as some third thing is to a fourth. And the latter types of unity always follow the former. Thus things which are one in number are one in species, but not all which are one in species are one in number; and all which are one in species are one in genus, but not all which are one in genus are one in species, although they are all one proportionally. And not all which are one proportionally are one in genus.

             434. Moreover, it is evident that things are said to be many in a way opposite to that in which they are one. For some things are many because they are not continuous; others, because their matter, either the first or ultimate, is divisible in species; and others because they have many conceptions expressing their essence.

COMMENTARY

             866. Here the Philosopher reduces all senses in which things are said to be one to one primary sense, and in regard to this he does two things. First (430:C 866), he makes this reduction; and second (431:C 870), to those senses in which things are said to be one, which have already been given, he adds another ("Again, in one sense").

             He accordingly says, first (430), that it is evident from what precedes that things which are indivisible in every way are said to be one in the highest degree. For all the other senses in which things are said to be one are reducible to this sense, because it is universally true that those things which do not admit of division are said to be one insofar as they do not admit of division. For example, those things which are undivided insofar as they are man are said to be one in humanity, as Socrates and Plato; those which are undivided in the notion of animality are said to be one in animality; and those which are undivided from the viewpoint of extension or measure are said to be one in quantity, as continuous things.

             867. And from this we can also derive number and the types of unity given above, because what is one is indivisible either in an absolute sense or in a qualified one. If it is indivisible in an absolute sense, it is the last type of unity, which is a principle; but if it is indivisible in a qualified sense, it is so either in quantity alone or in nature. If it is indivisible in quantity, then it is the first type. If it is indivisible in nature, it is so either in reference to its subject or to the division which depends upon the form. If it is divisible in reference to its subject, it is so either in reference to a real subject, and then it is the second type, or to a logical subject, and then it is the third type. And indivisibility of form, which is indivisibility of intelligible structure, or definition, constitutes the fourth type.

             868. Now from these senses of the term one certain others are again derived. Thus there are many things which are said to be one because they are doing one thing. For example, many men are said to be one insofar as they are rowing a boat. And some things are said to be one because they are subject to one thing; for example, many men constitute one people because they are ruled by one king. And some are said to be one because they possess one thing; for example, many owners of a field are said to be one in their ownership of it. And some things are also said to be one because they are something which is one; for example, many men are said to be one because each of them is white.

             869. But considering all of these secondary senses in which things are said to be one, which have already been stated in the five ways given above, we can say that those things are one in the primary sense which are one in their substance. For a thing is one in substance either by reason of its continuity, as in the first way; or because of the species of the subject, as in the second way; and again in the third way because the unity of the genus is somewhat similar to the unity of the species; or also because of the intelligible structure, as in the fourth and fifth ways. That some things are said to be one in these ways is clear from the opposite of one. For things are many in number, i.e., they are counted as many, either because they are continuous, or because they do not have one species, or because they do not have one common intelligible structure.

             870. Again, in one sense (431).

             Then he gives an additional sense in which the term one is used, which differs from the preceding ones. This sense is not derived from the notion of indivision, as the foregoing are, but rather from the notion of division. He says that sometimes some things are said to be one because of continuity alone, and sometimes they are said to be one only if they constitute a whole and something complete. Now this happens when the thing has one form, not in the sense that a homogeneous subject is said to have one form, which pertains to the second type given above, but in the sense that the form consists in a kind of totality requiring a definite order of parts. Thus it is clear that we do not say that a thing is one, for example, some artifact such as a shoe, when we see the parts put together in any way at all (unless perhaps it is taken to be one insofar as it is continuous); but we say that all parts of a shoe are one when they are united in such a way that the thing is a shoe and has one form--that of a shoe.

             871. And from this it is clear that a circular line is one in the highest degree. For a circular line is not only continuous like a straight line, but also has a totality and completeness which a straight line does not have; for that is complete and whole which lacks nothing. Now this characteristic belongs to a circular line; for nothing can be added to a circular line, but something can be added to a straight one.

             872. But the essence (432).

             Then he indicates a property which flows from oneness or unity. He says that the essence of one consists in being the principle of some number. This is clear from the fact that the unit is the primary numerical measure by which every number is measured. Now a measure has the character of a principle, because measured things are known by their measure, and things are known by their proper principles. And it is clear from this that unity is the first principle of what is known or knowable about each thing, and that it is the principle of knowing in all classes.

             873. But this unity which is the principle of knowing is not the same in all classes of things. For in the class of musical sounds it is the lesser half tone, which is the smallest thing in this class; for a lesser half tone is less than a half tone since a tone is divided into two unequal half tones one of which is called a lesser half tone. And in the class of words the first and smallest unity is the vowel or consonant; and the vowel to a greater degree than the consonant, as will be stated in Book X (831:C 1971). And in the class of heavy things or weights there is some smallest thing which is their measure, i.e., the ounce or something of this kind. And in the class of motions there is one first measure which measures the other motions, namely, the simplest and swiftest motion, which is the diurnal motion.

             874. Yet all of these have this feature in common that the first measure is indivisible in quantity or in species. Hence, in order that something be one and first in the genus of quantity it must be indivisible, and indivisible in quantity. It is called a unit if it is indivisible in every way and has no position, and a point if it is altogether indivisible in quantity but has position. A line is something divisible in one dimension only; a surface, in two; and a body, in all, i.e., in three dimensions. And these descriptions are reversible; for everything that is divisible in two dimensions is a surface, and so on with the others.

             875. Again, it must be noted that being a measure is the distinctive characteristic of unity insofar as it is the principle of number. But this unity or one is not the same as that which is interchangeable with being, as has been stated in Book IV (303:C 557). For the concept of the latter kind of unity involves only being undivided, but that of the former kind involves being a measure. But even though this character of a measure belongs to the unity which is the principle of number, still by a kind of likeness it is transferred to the unity found in other classes of things, as the Philosopher will show in Book X of this work (814:C 1921). And according to this the character of a measure is found in any class of things. But this character of a measure is a natural consequence of the note of undividedness, as has been explained (432:C 872). Hence the term one is not predicated in a totally equivocal sense of the unity which is interchangeable with being and of that which is the principle of number, but it is predicated of one primarily and of the other secondarily.

             876. Further, some things (433).

             Then he gives another way of dividing unity, and this division is rather from the viewpoint of logic. He says that some things are one in number, some in species, some in genus, and some analogically. Those things are one in number whose matter is one; for insofar as matter has certain designated dimensions it is the principle by which a form is individuated. And for this reason a singular thing is numerically one and divided from other things as a result of matter.

             877. Those things are said to be one in species which have one "intelligible structure," or definition; for the only thing that is defined in a proper sense is the species, since every definition is composed of a genus and a difference. And if any genus is defined, this happens insofar as it is a species.

             878. Those things are one in genus which have in common one of the "figures of predication," i.e., which have one way of being predicated. For the way in which substance is predicated and that in which quality or action is predicated are different; but all substances have one way of being predicated inasmuch as they are not predicated as something which is present in a subject.

             879. And those things are proportionally or analogically one which agree in this respect that one is related to another as some third thing is to a fourth. Now this can be taken in two ways: either in the sense that any two things are related in different ways to one third thing (for example, the term healthy is predicated of urine because it signifies the relationship of a sign of health [to health itself]; and of medicine because it signifies the relationship of a cause to the same health); or it may be taken in the sense that the proportion of two things to two other things is the same (for example, tranquillity to the sea and serenity to the air; for tranquillity is a state of rest in the sea, and serenity is a state of rest in the air).

             880. Now with regard to the ways in which things are one, the latter types of unity always follow the former, and not the reverse; for those things which are one in number are one in species, but not the other way about. The same thing is clear in the other cases.

             881. Moreover, it is evident (434).

             From the ways in which things are said to be one he now derives the ways in which things are said to be many. He says that things are said to be many in just as many ways as they are said to be one, because in the case of opposite terms one is used in as many ways as the other. Hence some things are said to be many because they are not continuous, which is the opposite of the first way in which things are one.

             882. Other things are said to be many because their matter is divisible in species, whether we understand by matter "the first," i.e., their proximate matter, or the final or ultimate matter into which they are ultimately dissolved. Indeed, it is by the division of their proximate matter that wine and oil are said to be many, and by the division of their remote matter that wine and a stone are said to be many. And if matter be taken both for real matter and for conceptual matter, i.e., for a genus, which resembles matter, many in this sense is taken as the opposite of the second and third ways in which things are said to be one.

             883. And still other things are said to be many when the conceptions which express their essence are many. And many in this sense is taken as the opposite of the fourth way in which things are said to be one.

             884. But the opposite of the fifth way in which things are one does not have the notion of many except in a qualified sense and potentially; for the fact that a thing is divisible does not make it many except potentially.