6-2
BOOK X
UNITYLesson 1
Kinds of one
1920. Above in Book IV of this work the Philosopher showed (548) that this science has for its subject being and the kind of unity which is interchangeable with being. Therefore, having drawn his conclusions about accidental being (1172) and about the kind of being which signifies the truth of a proposition, which he does in Book VI (1223), and about essential being as divided into the ten categories, which he does in Books VII (1245) and VIII (1681), and as divided into potency and actuality, which he does in Book IX (1768), his aim in this tenth book is to settle the issue about unity or oneness and the attributes which naturally accompany it. This is divided into two parts. In the first (1920) he establishes what is true of unity in itself; and in the second (1983) he considers unity in relation to plurality.
The first part is divided into two members. In the first he explains the different senses in which the term one is used. In the second (1937) he establishes a property of unity or oneness.
The first part is divided into three members. In the first he establishes the different senses in which the term one is used. In the second (1932) he reduces all these to one common meaning. In the third (1933) he explains the different ways in which the term one is used of the things of which it is predicated.
In regard to the first he does three things. First, he gives two senses in which the term one is used. Second (1927), he exposes the notion of unity contained in these two senses. Third (1929), he gives two other senses of the term one.
1921. In treating the first member of this division he gives, first, the primary senses in which the term one is used. He says that he has explained in Book V (749) the different meanings of the terms which pertain to the study of this science; for it was pointed out there (842) that the term one is used in many senses. And while this is true, there are four principal senses in which it is employed. But let us speak of those senses in which the term one is used primarily and essentially and not accidentally; for what is accidentally one has different modes of its own.
1922. (1) Now one of the senses in which things are said to be essentially one is that in which the continuous is said to be one; and this can be taken in two ways: either (a) the continuous in general (i.e., anything continuous in any way at all) is called one; or only the continuous (b) by nature is called one by continuity. And this latter is what is continuous in the fullest sense of the term, and not that which is continuous by force or by art or by any kind of contact (as is evident in the case of pieces of wood), or by any kind of continuity (as is evident in the case of things which are continuous or held together by a nail or by any other bond).
1923. And the phrase continuous by nature designates two things: what is a (+) uniform whole, as a straight line or even a circular one, and what is not a (~) uniform whole, as two lines which constitute the angle in which they are connected.
And of these, lines which are said to be straight and those which are said to be circular are one to a greater degree than those which form an angle, and they are one anteriorly. For a straight line must have one motion, since one part cannot be moved and another at rest, or one be moved in this way and another in that; but the whole must be moved simultaneously and by one motion. The same holds true of a circular line.
1924. But this does not apply to two continuous quantities which form an angle; for we can imagine either that one line is at rest and the other is moved closer to it so as to form a smaller angle, or that it is moved away from it so as to form a larger angle, or even that both lines are moved in opposite directions. Hence he says that a continuous quantity whose motion is more indivisible and simpler is one to a greater degree.
1925. (2) Then he gives a second sense in which things are said to be essentially one; and here we must consider that what "is such," i.e., continuous, is not only said to be one but also has something more; i.e., it is a whole having some form or specifying principle, just as an animal is one, and a triangular surface is one. Hence this sense of one adds to the oneness of continuity the kind of unity which comes from the form by which a thing is a whole and has a species.
1926. And since one thing is a whole by nature and another by art, he added that "a thing is one to the greatest degree" if it is such by nature and not by force. For example, all those things which are united by glue or by some such bond so as to become a whole are joined by force. But whatever is joined by nature is one to the greatest degree, because it is clearly the cause of its own continuity; for it is such by its very nature.
1927. Then be clarifies the meaning of unity contained in these two senses of the term one. He says that a thing is such, i.e., continuous and one, because its motion is one and indivisible both as to place and to time; as to place, because whithersoever one part of a continuous thing is moved another part is also moved; and as to time, because when one part is moved another is also moved.
1928. Hence, if a thing that is continuous and whole by nature is said to be one because its motion is one, then it is evident that, if anything continuous and whole has within itself a principle of the primary kind of motion, this will be the primary kind of one in the realm of continuous quantity; for example, of all motions the primary kind is local motion, and of local motions the primary kind is circular motion, as is proved in Book VIII of the Physics. And of bodies which are moved by circular motion there is one which contains the principle of such motion, i.e., the body which is moved circularly and causes the circular motion of other bodies by a daily motion. It is evident, then, that this is, the one primary continuous quantity which contains the first principle of the primary kind of motion.
Hence two senses of the term one are evident, namely, that in which the continuous is called one, and that in which a whole is called one.
1929. Then he gives the other ways in which things are said to be one. He says that certain other things are said to be one, not because their motion is one, but because their intelligible structure is one. And things of this kind whose concept is one are those which are apprehended by a single intellectual act. And such things as are said to be apprehended by a single intellectual act are those of which there is a single apprehension of an undivided object.
1930. This can be so for two reasons: either (3) because the undivided, object apprehended is specifically one, or (4) because it is numerically one.
Now what is numerically undivided is the singular thing itself, which cannot be predicated of many things; and what is specifically one is undivided because it is a single object of knowledge and acquaintance.
For in distinct singular things there is no nature numerically one which can be called a species, but the intellect apprehends as one that attribute in which all singulars agree. Hence the species, which is distinct in distinct individuals in reality, becomes undivided when apprehended by the intellect.
1931. And since substance is prior in intelligibility to all the other genera, and the term one is used in these senses because it has one meaning, then it follows that the primary sort of one in these senses is what is one in substance, i.e., what causes substance to be one, just as in the first two senses the primary sort of one was the continuous quantity which is moved circularly.
1932. Here he reduces the senses of one given above to a single meaning by summarizing what he had said above. He says that the term one is used of four things: first, (1) of what is continuous by nature; (2) second, of a whole; (3) third, of a singular thing; and (4) fourth, of the universal, for example, a species.
And all of these are said to be one because of one common aspect, namely, being indivisible; for properly speaking, a one is an undivided being.
But the term one is used in the first two senses because a motion is undivided, and in the latter two senses because an intelligible structure or concept is undivided, inasmuch as the apprehension of a particular thing is also included under this.
1933. Here he shows how the term one is predicated of things which are said to be one. He says that it must be borne in mind that the term one should not be taken to mean the same thing when a thing is said to be one and when someone expresses the essence of oneness, which is its intelligible structure; just as wood too is not said to be white in the sense that whiteness is the essence of wood, but in the sense that it is an accident of it.
1934. Then he gives the following explanation of a statement which he had made, saying that, since the term one is used in many senses (as has been stated), a thing is said to be one because some one of these senses applies to it, i.e., continuous, whole, species, or singular thing. But the essence of oneness sometimes applies to something that is one in some one of the foregoing senses, as when I say that what is one in continuity is one (and the same holds true of the others); and sometimes it is attributed to something which is nearer to the nature of one, for example, what is undivided but contains within itself potentially the senses of one given above; because what is undivided as regards motion is continuous and whole, and what is undivided in meaning is singular or universal.
1935. He adds to this the example of elements and causes, viewed in the problem of identifying them in things, as when we say that such and such a thing is an element or cause by defining the term; for example, we say that that is a cause which has the essence of a cause. And in this way we say that fire is an element or "the indeterminate itself," i.e., what is unlimited in itself (which the Pythagoreans posited as a separate entity and the element of all things), or anything else of this sort for whatever reason it can be called an element. But in a sense fire is not an element, and neither is the indeterminate; for fire does not constitute the essence of an element, because the notion of fire is not the same as that of an element. It is an element, however, as existing in reality or in the natural world. But when the term element is predicated of fire, it signifies that something "has become accidental to fire," i.e., that fire is that of which something is composed as a primary constituent, and this is the formal note of an element. He says "constituent" in order to exclude privations.
1936. What has been said about an element also applies to cause and to one and to all such terms; because the things of which they are predicated are not the very things which the terms signify; for example, white man is not the very thing which the term white signifies, for white signifies a quality.
Hence the essence of oneness consists in being undivided, i.e., in being an individual thing; and this is proper to a thing which is inseparable as to place or to form or in whatever other way it is inseparable.
Lesson 2
One as a measure
1937. Having explained the various senses in which unity is predicated of things, and having stated what its essential note is, to which all its usages are reduced, i.e., being indivisible, here the Philosopher infers a property of unity from its essential note, namely, that it is a measure. This is divided into two parts. In the first he shows how the notion of a measure belongs to unity and to the various classes of accidents. In the second (1961) he shows how unity in the sense of a measure is found in substances ("It is necessary").
In regard to the first part of this division he does two things. First, he indicates the class of things in which unity in the sense of a measure is primarily found, and how it is transferred from this class to the others with the proper notion of a measure. Second (1956), he explains how it is transferred figuratively to the other classes ("And for the same reason").
In treating the first part he does two things. First, he indicates the class of things in which unity in the sense of a measure is first found, and how it is transferred from this class to the others. Second (1950), he makes a study of measures ("However, a measure").
In regard to the first he does three things. First, he shows how unity as a measure is found in quantity, and how it is transferred from this category to the others. Second (1939), he indicates the species of quantity in which it is first found ("And that by which"). Third (1940), he shows how it is transferred to other species of quantity ("And the measure").
1938. He accordingly says, first, that, since the essential note of unity consists in being indivisible, and what is indivisible in each genus is somehow the measure of that genus, unity must be said to be in the highest degree the first measure of each genus. This is said to apply most properly to quantity, and it is from this class that the notion of a measure is transferred to other classes of things. Now a measure is nothing else than that by which a thing's quantity is known, and this is known by the unit or by a number: by a unit, as when we say one furlong or one foot; and by a number, as when we say three furlongs or three feet. Again, every number is known by the unit because the unit taken a certain number of times gives a number. It follows, then, that every quantity is known by unity. To "quantity" he adds "as quantity," intending that this be referred to the measure of quantity; for the properties and other accidents of quantity are known in a different way.
1939. Then he indicates in what species of quantity unity or measure is primarily found. First, he makes it clear that the notion of a measure is primarily found in discrete quantity, which is number. He says that that by which quantity is first known is "unity itself," i.e., the unit which is the principle of number. For in other species of quantity the unit is not unity itself but something of which unity is an attribute, as when we speak of one hand or of one continuous quantity. Hence it follows that unity itself, which is the first measure, is the principle of number as number.
1940. Second, he shows how unity is transferred to other species of quantity; and in regard to this he does two things. First, he indicates the species of quantity to which it is transferred. He says that it is from this class, i.e., from number and from the unit, which is the principle of number, that the notion of a measure is transferred to other quantities as that by which each of them is first known. And whatever is the measure in each class of things is the unit in that class.
1941. He gives examples of this in three classes of things, i.e., in dimensions-length, breadth and width; in weight, or in what he calls heaviness; and in speed, or in what he calls rapidity, which refers to the measure of time.
In the case of dimensions no one doubted that they were quantities and that they were properly susceptible to measurement, but in the case of weight and of speed there could be a difficulty because these seem to be qualities rather than quantities.
1942. He therefore explains how these pertain to the genus of quantity, and how they are susceptible to measurement. He says that heaviness and rapidity have something in common with their contraries because one contrary is found in the other; for what is heavy is in some sense light, and the reverse; and what is rapid is in some sense slow. For each of these terms is used in two senses. (1) In one sense the term heavy is used without qualification of anything that has an inclination to be borne towards the center of the earth, without taking into consideration how great its inclination is; and in this sense heavy does not refer to the category of quantity, and it is not susceptible to measurement. (2) In the other sense it is used of one thing in comparison with something else, namely, of what exceeds something else in terms of the abovementioned inclination; for example, we say that earth is heavy in comparison with water, and that lead is heavy in comparison with wood. Therefore it is by reason of this excess that some notion of quantity and measure is found.
The term rapid is similarly used in two senses. In one sense it is used without qualification of anything that has any motion; and in a second sense it is used of anything that has an excessive motion. And in one respect the notions of quantity and measure properly apply to it, and in another respect they do not.
1943. With a view to clarifying his statement about the condition of heaviness and rapidity in reference to contraries he adds that rapidity is found in something that is slow inasmuch as what is simply and unqualifiedly slow is more rapid in comparison with something that is slower than itself. And in a similar way heaviness is found in light things; for example, air is light in comparison with earth, and heavy in comparison with fire.
1944. Then he shows how the notion of a measure is transferred from number to other kinds of quantity. He immediately makes this clear, first, in the case of dimensions and in that of weights; and second (1947), in that of the rapidity of motions ("And they also measure").
He accordingly says, first, that the notion of a measure is transferred from number to the other kinds of quantity in this way that, just as the unit which is the measure of number is indivisible, so too all the other kinds of quantity have something that is one and indivisible as their measure and principle. For example, in measuring lines men use "the foot measure," i.e., the measure of one foot, as something indivisible; for wherever something indivisible is sought as. a measure, there is something simple either in quality or in quantity; in quality, as whiteness in the case of colors, which is in a sense the measure of colors, as will be mentioned below (1968); and in quantity, as the unit in the case of numbers, and the foot measure in the case of lines.
1945. Further, he points out why a measure must be something indivisible. The reason is that an exact measure must be something which can be neither added to nor subtracted from. Thus the unit is the most exact or certain measure, because the unit which is the principle of number is altogether indivisible, and whatever unity is not susceptible either to addition or to subtraction remains one. The measures of the other classes of quantity resemble this unit which is indivisible inasmuch as men take some smallest thing as a measure to the extent that this is possible. For if anything large were taken, as the furlong among distances and the talent among weights, it would escape our notice if some small portion were subtracted from or added to it. And this would always be more true of a larger measure than of a smaller one.
1946. Hence all men take this as a measure both in the case of liquids, such as oil and wine, and in that of solids, such as grain and barley; and also in that of weights and dimensions, which are designated as heaviness and continuous quantity. And this is first found to be such that nothing perceptible can be subtracted from it or added to it that might escape our notice. And men think they know the quantity of a thing exactly when they know it by the smallest measure of this kind.
1947. Then he makes the same thing clear with regard to the rapidity of motions. He says that men also measure motion "by that motion which is simple," i.e., the motion which is uniform and quickest, because it takes the least time. Hence in astronomy they take such motion as the basis of measurement; for they take the motion of "the first heaven," i.e., the daily motion, which is regular and quickest, and they judge and measure all other motions by this.
1948. And because the low and high pitch of sounds results from the quickness and slowness of motions, as is established in the science of music, he adds as an example the measurement of sounds. He says that in music the first measure is the "diesis," i.e., the difference between two half tones; for a tone is divided into two unequal half tones, as is proved in the science of music. And similarly in speech the measure is the letter, because the shortness or length of a word is a natural consequence of the quickness or slowness of a motion.
1949. Now all these something one, not in measures are the sense that some measure is common to all, but in the sense that any measure in itself is something one, as has been pointed out.
1950. After having shown in what class of things unity as a measure is primarily found, here the Philosopher clears up certain points that have to be investigated about measures.
The first of these is that, although a measure is understood to be one thing inasmuch as it comes close to being indivisible, it is not necessary that a measure be something numerically one; but sometimes many things are measures; for example, in the case of musical sounds "there are two dieses," i.e., two half tones. However, because of their smallness they are not distinguished by the sense of hearing, for the senses do not perceive the difference between two things that are very small; but their difference is perceived "in their ratios," i.e., in the different ratios which comprise their proportions, because they are caused by different numerical proportions.
1951. Similarly the things by which we measure words are also many; for the quantity of one meter or of one foot is measured by different syllables, some of which are short and some long.
The same thing is true of the diameter of a circle and of the diagonal of a square, and also of the side of a square.
And any continuous quantity is measured by two things, for an unknown quantity is found only by means of two known quantities.
1952. Having said this he brings this part of his discussion to a close by summarizing what has been said above, namely, that unity constitutes the measure of all things. The reason for this is that unity is the term of division. And those principles which constitute the substance of each thing are known by the division or dissolution of the whole into its component parts, whether they are quantitative parts or specific parts such as matter and form and the elements of compounds. Therefore what is one in itself must be indivisible since it is the measure by which a thing is known, because in the case of singular things whatever is first in the process of composition and last in the process of dissolution is indivisible, and it is by means of this that the thing is known, as has been explained.
1953. Yet indivisibility is not found in all things in the same way. (1) Some things are altogether indivisible, such as the unit which is the basis of number, whereas (2) others are not altogether indivisible but only to the senses, according as the authority of those who instituted such a measure wished to consider something as a measure; for example, the foot measure, which is indivisible in proportion [to the things measured] but not by nature. "For perhaps everything continuous is divisible"; and he says "perhaps" because of the difficulty facing those men who claimed that continuous quantity is composed of indivisible elements, or that natural continuous quantities are not infinitely divisible, but only mathematical quantities. For it is possible to find the smallest amount of flesh, as is mentioned in Book I of the Physics.
1954. Then he gives the second point that has to be investigated about a measure. He says that "the meter," i.e., the measure, should always be of the same kind as the thing measured, i.e., of the same nature or measure as the thing measured; for example, a continuous quantity should be the measure of continuous quantities; and it is not enough that they have a common nature, as all continuous quantities do, but there must be some agreement between the measure and the thing measured in the line of their special nature. Thus a length is the measure of lengths, a width of widths, a vocal sound of vocal sounds, a weight of weights, and a unit of units.
1955. "For this is the view which must be taken" in order that we may speak without being criticized, "but not that number is the measure of numbers." Now number does not have the notion of a first measure but unity does; and if unity is a measure, then in order to signify the agreement between the measure and the thing measured it will be necessary to say that unity is the measure of units and not of numbers. Yet if the truth of the matter be taken into consideration, it will be necessary to admit also that number is the measure of numbers or even that the unit may be taken in a similar way as the measure of numbers. But it does not seem equally fitting to say that the unit is the measure of units and number of number or unity of number, because of the difference which appears to exist between the unit and number. But to observe this difference is the same as if someone were to say that it is fitting for units to be the measure of units but not the unit, because the unit differs from units as things expressed in the singular differ from those expressed in the plural. And the same argument applies to number in relation to the unit, because a number is nothing else than a plurality of units. Hence to say that the unit is the measure of number is merely to say that the unit is the measure of units.
1956. Then he shows how the term measure is transferred in a figurative way to another class of things. He says that, since it has been stated that a measure is that by which the quantity of a thing is known, we may say that intellectual knowledge is the measure of that which is knowable intellectually, and that sensory perception is the measure of that which is perceptible; because we know something by means of them, namely, sensible objects by means of perception and intelligible objects by means of intellectual knowledge; but we do not know them in the same was as we do by a measure. For something is known by a measure as a principle of knowledge, whereas in sensation and knowledge we are measured by things that are outside ourselves.
Lesson 3
1969. And because in colors we look for something that is first and one, namely white, it is clear that if all beings were colors, they would have some number, not in the sense, however, that number would constitute subsisting things themselves, but in the sense that there would be a number of subsisting things of a particular sort, i.e., colors. And then there would be something that is the subject of unity, namely, that which is white.
1970. The same thing would be true if all things were tunes; because beings would be of a certain number, that is, a number of minor half tones or tones. Yet number is not the very substance of beings, and consequently it would be necessary to look for something which is one, namely, the minor half tone; but not in such a way that unity itself would be a substance.
1971, In a similar way too if all beings were sounds, they would be a number of beings, because there are a number of particular subjects of number, namely, "of elements," or letters. Hence the vowel, which is the primary letter (since consonants cannot be pronounced without vowels) would constitute their unity.
And in a similar way if all figures were rectilinear figures, there would be a number of subjects, namely, figures; and the triangle, which is the primary rectilinear figure, would constitute their unity; for all such figures are reducible to the triangle. The same reasoning applies to every category.
1972. If it is in this way, then, that number and unity are found in all other categories: in affections, qualities, and quantities, and in motion; and if number and unity are not the substance of the things of which they are predicated, but number is predicated of certain substances, and if unity similarly requires some subject which is said to be one, the same thing must be true of substances, because being and unity are predicated in the same way of all things. It is evident, then, that in any category of things there is some nature of which the term one is predicated, not because unity itself is the nature of a thing, but because it is predicated of it.
1973. And just as when we speak of unity in the case of colors we are looking for some color which is said to be one, so too when we speak of unity in the case of substances we are looking for some substance of which unity may be predicated. And this is predicated primarily and chiefly of what is first among substances (which he investigates below, 2553-66), and subsequently of the other classes of things.
1974. Since he had given the same argument for being and for unity, he now shows that unity and being somehow signify the same thing. He says "somehow" because unity and being are the same in their subject and differ only in meaning. For unity adds to being the note of undividedness, because what is one is said to be an indivisible or undivided being. He gives three reasons why unity signifies the same thing as being.
1975. (1) The first is that unity naturally belongs to all of the different categories and not just to one of them; that is, it does not pertain just to substance or to quantity or to any other category. The same thing is also true of being.
1976. (2) The second reason is that, when a man is said to be one, the term one does not express a different nature from man, just as being does not express a different nature from the ten categories; for, if it did express a different nature, an infinite regress would necessarily result, since that nature too would be said to be one and a being. And if being were to express a nature different from these things, an infinite regress would also follow; but if not, then the conclusion of this argument must be the same as that of the first one.
1977. (3) The third reason is that everything is said to be one inasmuch as it is a being. Hence when a thing is dissolved it is reduced to non-being.
1978. [Objection] Now in this solution of the question the Philosopher seems to contradict himself; for he first said that unity and being are not the substance of the things of which they are predicated, but here he says that unity and being do not express a nature different from the things of which they are predicated.
1979. Hence it must be noted that the term substance is used in two senses. (1) In one sense it means a supposit in the genus of substance, which is called first substance and hypostasis, to which it properly belongs to subsist. (2) In a second sense it means a thing's quiddity, which is also referred to as a thing's nature. Therefore, since universals are subsistent things according to the opinion of Plato, they signify substance not only in the second sense but also in the first. But Aristotle proves in Book VII (1572) that universals are not subsistent things, and therefore it follows that universals are not substances in the first sense but only in the second. And for this reason it is said in the Categories that second substances, which are genera and species, do not signify particular things, which are subsisting substances, but "they signify the quiddity of a thing," i.e., a nature in the genus of substance.
1980. The Philosopher accordingly proved above that unity and being do not signify substance in the sense of this particular thing, but it is necessary to look for something that is one and a being, just as we look for something that is a man or an animal, as Socrates or Plato.
Later he shows that these terms signify the natures of the things of which they are predicated and not something added, like accidents. For common attributes differ from accidents in this respect (although they agree in not being particular things), that common attributes signify the very nature of supposits, whereas accidents do not, but they signify some added nature.
1981. And Avicenna, who did not take this into account, claimed that unity and being are accidental predicates, and that they signify a nature added to the things of which they are predicated. For he was deceived by the equivocal use of the term one, because the unity which is the principle of number and has the role of a measure in the genus of quantity signifies a nature added to the things of which it is predicated, since it belongs to a class of accident. But the unity which is interchangeable with being extends to everything that is, and therefore it does not signify a nature which is limited to one category.
1982. He was also deceived by the equivocal use of the term being; for being as signifying the composition of a proposition is predicated accidentally, since composition is made by the intellect with regard to a definite time. Now to exist at this or at that particular time is to be an accidental predicate. But being as divided by the ten categories signifies the very nature of the ten categories insofar as they are actual or potential.
Lesson 4
Ways one and many are opposed
1983. After having treated of one considered in itself, here the Philosopher deals with one in comparison with many; and this is divided into two parts. In the first (1983) he treats one and many and their concomitant attributes. In the second (2023) he establishes what is true about the contrary character of one and many; for the investigation of this involves a special difficulty.
The first member of this division is divided into two parts. In the first part he shows how one and many are opposed. In the second (1999) he considers their concomitant attributes.
In regard to the first he does three things. First, he indicates how we should understand the opposition between one and many. He says that, although one and many are opposed in many ways, as will be made clear below, none the less one of these ways, and the most important one, concerns one and many insofar as they are opposed as something indivisible is opposed to something divisible, because this mode of opposition pertains to the proper notion of each.
1984. For the essential note of plurality consists in things being divided from each other or in being divisible. He says "divided" because of the things which are actually separated from each other and which are for this reason said to be many. He says "divisible" because of the things which are not actually separated from each other but come close to being separated, for example, moist things such as air and water and the like, of which we use the term much because they are easily divided; thus we speak of much water and much air.
1985. But the formal constituent of unity or oneness consists in being indivisible or in being undivided; for the continuous is said to be one because it is not actually divided, although it is divisible.
1986. Second, he makes clear to what kind of opposition the aforesaid manner of being opposed is ultimately reduced. He says that, since there are four kinds of opposition, one of which is based on privation, it is evident that one and many are not opposed as contradictories or as relative terms, which are two kinds of opposition, but as contraries.
1987. That they are not opposed as (~) contradictories is evident because neither of them applies to non-being, for non-being is neither one nor many. But the second member of the contradiction would have to apply to being as well as to non-being. That they are not opposed as relative terms is likewise evident, for the terms one and many are used in an absolute sense.
1988. And although he had said that one and many are opposed as what is indivisible and what is divisible, and these appear to be opposed as privation and possession, none the less he concludes that one and many are opposed as contraries; for the opposition between privation and possession is the basis of the opposition between contraries, as will be made clear below (2036). For one of the two contraries is always a privation, but not a pure privation; otherwise it would not share in the nature of the genus, since contraries belong to the same genus. Each of the two contraries, then, must be a positive reality, even though one of them shares in the nature of the genus with a certain deficiency, as black in relation to white, as has been stated above (1967). Therefore, since unity does not signify a pure privation, for it does not designate the mere lack of division but the very being which is undivided, it is evident that one and many are opposed not as pure privation and possession but as contraries.
1989. [Objection] Third, he answers an implied question. Because he had said that one is related to many as what is indivisible to what is divisible, and what is indivisible seems to be the privation of what is divisible since privation is subsequent to possession or form, it seems to follow that one is subsequent to many, although he had said above (1939) that one is the principle of many, from which it becomes known.
1990. In order to see the solution of this difficulty, then, it must be borne in mind that things which are prior and better known by nature are subsequent and less well known to us, because we derive our knowledge of things from the senses. Now the first things to be perceived by us are composite and confused things, as is said in Book I of the Physics; and this is why the first things to be known by us are composite things. But simpler things, which are prior and more intelligible by nature, are known by us only derivatively; and this is why we define the first principles of things only by the negations of subsequent things; for example, we say that the point is what has no parts; and we know God by way of negations inasmuch as we say that God is incorporeal, unchangeable and infinite.
1991. Accordingly, even though what is one is prior by nature to what is many, yet in our knowledge it is defined and gets its name from the privation of division. This is why the Philosopher says that "what is one is described," i.e., named, "and made known," i.e., understood, "in reference to its contrary," just as the indivisible is known from the divisible. And for this reason many things are able to be perceived more easily than one thing; and what is divisible is able to be perceived more easily than what is indivisible, not in the order of nature but because of sensory perception, which is the foundation of our knowledge.
1992. [Objection] But a twofold difficulty arises with regard to those things which the Philosopher is expounding. The first concerns his statement that one and many are opposed as contraries. For this appears to be impossible, because unity is the basis of plurality, whereas one of two contraries does not ground the other but rather destroys it.
1993. Hence it must be noted that, since contraries differ formally, as is said below (2120), when we say that things are contraries, each of them is to be taken (+) insofar as it has a form, but not (~) insofar as it is a part of something having a form.
(+) For insofar as body is taken without the soul, as something having a form, it is opposed to animal as the non-living is opposed to the living. (~) But insofar as it is not taken as something complete and informed, it is not opposed to animal but is a material part of it.
We see that this is likewise true of numbers; for insofar as the number two is a kind of whole having a determinate species and form, it differs specifically from the number three; but if it is taken insofar as it is not made complete by a form, it is a part of the number three.
1994. Therefore insofar as unity itself is considered to be complete in itself and to have a certain species, it is opposed to plurality; because what is one is not many, nor is the reverse true. But insofar as it is considered to be incomplete as regards form and species, it is not opposed to plurality but is a part of it.
1995. [Objection] The second difficulty has to do with the statement that plurality is prior in intelligibility to unity; for, since the concept of plurality or multitude involves unity, because a plurality is nothing else than an aggregate of units, if unity is subsequent in intelligibility to plurality, it follows that the notions of unity and plurality involve circularity, i.e., in the sense that unity is intelligible in terms of plurality and vice versa. But circularity of definition is not admissible in designating the intelligible structures of things, because the same thing would then be known both to a greater and to a lesser degree. This is impossible.
1996. The answer to this difficulty, then, must be that nothing prevents one and the same thing from being prior and subsequent in intelligibility according to different traits which are considered in it. For in multitude it is possible to consider both multitude as such and division itself.
Thus from the viewpoint of division multitude is prior in intelligibility to unity; for that is one which is undivided. But multitude as multitude is subsequent in intelligibility to unity, since a multitude means an aggregate of units or ones.
1997. Now the division which is implied in the notion of that kind of unity which is interchangeable with being is not (~) the division of continuous quantity, which is understood prior to that kind of unity which is the basis of number, but is (+) the division which is caused by contradiction, inasmuch as two particular beings are said to be divided by reason of the fact that this being is not that being.
1998. Therefore what we first understand is being, and then division, and next unity, which is the privation of division, and lastly multitude, which is a composite of units.
For even though things which are divided are many, they do not have the formal note of a many until the fact of being one is attributed to each of the particular things concerned. Yet nothing prevents us from also saying that the notion of multitude depends on that of unity insofar as multitude is measured by one; and this already involves the notion of number.
1999. Here he indicates the attributes which stem from unity and plurality; and in regard to this he does two things. First, he gives the attributes which naturally stem from unity and plurality. He says that sameness, likeness and equality flow from unity, as has been pointed out above in Book V (911), where he divided or distinguished the various senses in which things are said to be contrary; for those things are the same which are one in substance; those are like which are one in quality; and those are equal which are one in quantity.
2000. And the contraries of these, diverse, unlike and unequal, pertain to plurality. For those things are diverse whose substance is not one; those are unlike whose quality is not one; and those are unequal whose quantity is not one.
2001. He now explains the various senses in which these terms are used; and in regard to this he does two things. First, he shows how the modes of those attributes which accompany unity differ from each other. Second (2013) he does the same thing for those attributes which accompany plurality ("It is evident").
In regard to the first part he does two things. First, he explains the various ways in which things are said to be the same; and second (2006), those in which they are said to be like ("Things are like"). He does not make any distinctions as regards equality, however, because there are not many ways in which things are said to be equal, unless perhaps in reference to the various kinds of quantity.
2002. He accordingly gives three ways in which the term same is used. For since same means one in substance, and substance is used of two things, namely, of the supposit itself and of the nature or species of a thing, the term same is used of three things: either (1) of the supposit alone, as this white thing or this musical man, assuming that Socrates is white or musical; or (2) of the nature of the supposit alone, that is, its intelligible expression or species, as Socrates and Plato are the same in terms of humanity; or (3) of both together, as Socrates is the same as Socrates.
2003. Hence, the Philosopher, in giving these three ways in which the term is used, says that the term same is used in many senses. (1) In one sense it means what is numerically the same, which we sometimes express by the term itself, as when we say that Socrates is a man and that he himself is white. For since the pronoun itself is reflexive, and a reflexive term brings back the same supposit, wherever the term itself is used it signifies that the supposit is numerically one and the same.
2004. (2) A thing is said to be the same in another sense if it is one not only by the oneness of the supposit, as this wood and this white thing, but if it is the same both in its intelligible structure and in number, as you are the same as yourself both specifically and materially, inasmuch as matter, which is the principle of individuation is taken for the supposit, and species is taken for the nature of the supposit.
2005. (3) Things are said to be the same in a third sense when "the intelligible structure of the primary substance," i.e., of the supposit, is one, even though there is not one supposit. And these things are the same specifically or generically but not numerically. He gives an example of this in the case of quantity, according to the opinion of those who claimed that quantities are the substances of things; and according to this opinion many straight lines are regarded as many supposits in the genus of substance, and the measure of a line is considered to be its species. This opinion maintains, then, that many straight lines are one, just as distinct supposits are one which have one specific nature in common. And since mathematicians speak of lines in the abstract, for them many equal straight lines are considered as one. And in a similar fashion many "equal quadrangles," i.e., figures which have four angles and are equal in size and "equiangular," i.e., having equal angles, are considered to be the same. And in such things as these equality provides the unity of their specific nature.
2006. Here he reveals the different ways in which things are said to be like, and there are four of these.
(1) The first corresponds to the third way in which things are the same; for since that is the same which is one in substance, and that is like which is one in quality, the basis of likeness must be related to the basis of sameness as quality to substance. And since he has used equality to designate oneness of substance, he uses figure and proportion to designate quality.
2007. It should also be noted that, since quality and quantity are rooted in substance, it follows that wherever there is oneness of substance there is oneness of quantity and quality, although this oneness or unity does not derive its name from quantity and quality but from something more basic, namely, substance. Hence, wherever there is oneness of substance we do not speak of likeness or of equality but only of identity.
2008. Diversity of substance, then, is required for likeness or equality. This is why he says that some things are said to be like even though they are not absolutely the same as to the species of their substance (provided that they are also not without difference in their underlying subject, which is called the supposit) but are specifically the same in some way. Thus a larger quadrangle is said to be like a smaller one when the angles of one are equal to those of the other and the sides containing the angles are proportional. It is evident, then, that this likeness is viewed from the standpoint of oneness of figure and proportion. And in a similar way many unequal straight lines are not the same in an absolute sense even though they are like.
2009. It can also be noted here that, when there is unity in regard to the complete concept of the species, we speak of identity. But when there is no unity in regard to the whole concept of the species, we speak of likeness; so that if someone says that things which are generically one are like, then those which are specifically one are the same, as the examples given above would seem to indicate. For he said that equal straight lines and equal quadrangles are identical with each other, whereas unequal quadrangles and unequal straight lines are said to be like.
2010. (2) Things are said to be like in a second sense when they have in common one form which admits of difference in degree although they participate in that form without difference in degree; for example, whiteness admits of greater and lesser intensity, so that, if some things are equally white without any difference in degree, they are said to be like.
2011. (3) Things are said to be like in a third sense when they have in common one form or affection but to a greater or lesser degree; for example, a thing which is whiter and one which is less white are said to be like because they have "one form," i.e., one quality.
2012. (4) Things are said to be like in a fourth sense when they have in common not merely one quality but many, as those things which are said to be like because they agree in more respects than they differ, either in an absolute sense, or in regard to certain particular attributes; for example, tin is said to be like silver because it resembles it in many respects. And similarly fire is like gold, and saffron like red.
2013. Here he treats the attributes which naturally accompany plurality. First, he considers unlikeness and diversity; and second (2017), he treats difference ("But different").
He accordingly says, first, that, since the terms same and diverse and like and unlike are opposed to each other, and since the terms same and like are used in many senses, it is evident that the terms diverse and unlike are used in many senses; for when, one of two opposites is used in many senses, the other is also used in many senses, as is said in the Topics, Book I.
2014. But omitting the many senses in which the term unlike is used, since it is quite apparent how the senses of this term are taken in contrast to those of the term like, he gives three senses in which the term diverse, or other, is employed. (1) First, the term diverse refers to everything that is other in contrast to the same; for just as everything that is itself is said to be the same, and this is the relation of identity, in a similar fashion everything that is diverse is said to be other, and this is the relation of diversity. Hence everything is either the same as or other than everything else. (2) Second, the term diverse, or other, is used in another sense when the matter and intelligible structure of things are not one; and in this sense you and your neighbor are diverse. (3) The term is used in a third sense in mathematics, as when unequal straight lines are said to be diverse.
2015. [Objection] And since he had said that everything is either the same as or other than everything else, lest someone think that this is true not only of beings but also of non-beings, he rejects this by saying that everything is either the same as or other than everything else in the case of those things of which the terms being and unity are predicated, but not in the case of those things which are non-beings. For same and diverse are not opposed as contradictory terms, of which one or the other must be true of any being or non-being; but they are opposed as contraries, which are only verified of beings. Hence diversity is not predicated of non-beings. But the phrase not the same, which is the opposite of the same in a contradictory sense, is also used of non-beings. However, same or diverse is used of all beings; for everything that is a being and is one in itself, when compared with something else, is either one with it, and then it is the same, or it is capable of being one with it but is not, and then it is diverse. Diverse and same, then, are opposites.
2016. But if someone were to raise the objection that diversity and sameness do not apply to all beings, since sameness is a natural consequence of oneness of substance, and diversity is a natural consequence of plurality of substance, we should have to answer that, since substance is the root of the other genera, whatever belongs to substance is transferred to all the other genera, as the Philosopher pointed out above regarding quiddity in Book VII (1334).
2017. Then he shows how difference and diversity differ. He says that diverse and different mean different things; for any two things which are diverse need not be diverse in some particular respect, since they can be diverse in themselves. This is evident from what has been said above, because every being is either the same as or other than every other being.
2018. But that which differs from something else must differ from it in some particular respect. Hence that by which different things differ must be something that is the same in things which do not differ in this way. Now that which is the same in many things is either a genus or a species. Therefore all things that differ must differ either generically or specifically.
2019. Those things differ generically which have no common matter; for it has been said above, in Book VIII (1697), that although matter is not a genus, still the essential note of a genus is taken from a thing's material constituent; for example, sensory nature is material in relation to the intellectual nature of man. Hence anything that does not possess sensory nature in common with man belongs to a different genus.
2020. And since those things which do not have a common matter are not generated from each other, it follows that those things are generically diverse which are not generated from each other. It was also necessary to add this because of the things which do not have matter, such as accidents, so that those things which belong to different categories are generically diverse, for example, a line and whiteness, neither one of which is produced from the other.
2021. Now those things are said to be specifically diverse which are the same generically and differ in form. And by genus we mean that attribute which is predicated of two things which differ specifically, as man and horse. Moreover, contraries differ, and contrariety is a type of difference.
2022. Then he proves by an induction what he had said above about the formal note whereby things differ, because all things that are different seem to be such that they are not merely diverse but diverse in some particular respect. Some things, for instance, are diverse in genus; some belong to the same category and the same genus but differ in species, and some are the same in species. What things are the same or diverse in genus has been established elsewhere, namely, in Book V of this work (931).
Lesson 5
Contrary
2023 ...In regard to the first he does two things. First, he shows that there is a greatest difference, as follows: there is some maximum in all things which admit of difference in degree, since an infinite regress is impossible. But it is possible for one thing to differ from something else to a greater or lesser degree. Hence it is also possible for two things to differ from each other to the greatest degree; and therefore there is a greatest difference.
2024. Second, he shows by an induction that contrariety is the greatest difference; for all things which differ must differ either generically or specifically.
Now those things which differ generically cannot be compared with each other, being too far apart to admit of any difference of degree between them. This is understood to apply to those things which are changed into each other, because a certain process or way of change of one thing into another is understood from the fact that at first they differ more and afterwards less, and so on until one is changed into the other. But in the case of things which differ generically we do not find any such passage of one thing into another. Hence such things cannot be considered to differ in degree, and so cannot differ in the highest degree. Thus in things which differ generically there is no greatest difference.
2025. However, in the case of things which differ specifically there must be a greatest difference between contraries, because:'reciprocal processes of generation arise from contraries as their extremes. And an intermediate arises from an extreme or vice versa, or an intermediate also arises from an intermediate, as gray is produced from black or from red. Yet generations of this kind do not arise from two things as extremes; for when something passes from black to gray in the process of generation, it can still pass farther to some color which differs to a greater degree. But when it has already become white, it cannot continue farther to any color which differs to a greater degree from black, and there it must stop as in its extreme state. This is why he says that processes of generation arise from contraries as extremes. But it is evident that the distance between extremes is always the greatest. Hence it follows that contraries have the greatest difference among things which differ specifically.
2026. And since we have shown that things which differ generically are not said to have a greatest difference, although there is a greatest difference, it follows that contrariety is nothing else than the greatest difference.
2027. He draws two corollaries from what has been said. The first is that contrariety is the perfect difference....
2030. Here he gives the second corollary. He says that, since the foregoing remarks are true, it is evident that one thing cannot have many contraries....
Lesson 6
Other kinds of opposition
2040. Then he proves his thesis, namely, that the primary contrariety is privation and possession; and he does this in two ways: first, by a syllogism; second (2054), by an induction ("This also").
In regard to the first he does two things. First, he shows that contrariety is not contradiction. He says that among the four kinds of opposition between two things-(1) contradiction, as sitting is opposed to not-sitting; (2) privation, as blindness is opposed to sight; (3) contrariety, as black is opposed to white; and (4) relation, as a son is opposed to his father-the first is contradiction.
2041. The reason is that contradiction is included in all the other kinds of opposition as something prior and simpler; for in any kind of opposition it is impossible that opposites should exist simultaneously. This follows from the fact that one of two opposites contains the negation of the other in its notion; for example, the notion of blind contains the fact of its not seeing, and the notion of black, of its not being white. And similarly the notion of son contains his not being the father of him of whom he is the son.
2042. Moreover, it is evident that there is no intermediate in contradiction; for one must either affirm or deny, as has been shown in Book IV (725). However, it belongs to contraries to have an intermediate; and thus it is clear that contrariety and contradiction are not the same.
2043. Then he shows how privation is related to contradiction by indicating the way in which they are alike and that in which they differ. He says that privation is a kind of contradiction; for the term privation is used in one sense when a thing does not have in any way some attribute which it is capable of having, for example, when an animal does not have sight. And this occurs in two ways: (a) first, if it does not have it in any way at all; and (b) second, if it does not have it in some definite respect, for example, at some definite time or in some definite manner, because privation is used in many senses, as has been stated in Books V (1070) and IX (1784).
2044. It is evident from what has been said, then, that privation is a kind of contradiction; and this is shown from the fact that a thing is said to be deprived of something because it does not have it.
2045. That it is not a simple contradiction but one of a sort is evident from the fact that according to its meaning a contradiction requires neither (~) the aptitude nor the existence of any subject; for it may be truly affirmed of any being or non-being whatsoever. Thus we say that an animal does not see, and that wood does not see, and that a non-being does not see.
A privation, however, necessarily (+) requires some subject, and sometimes it also requires aptitude in a subject; for that which is a non-being in every respect is not said to be deprived of anything.
2046. He says, then, that privation "is found either in a determinate potency," i.e., one with a capacity for possessing something, or at least "is conceived along with something that is susceptible of it," i.e., along with a subject, even though it has no capacity for possessing something. This would be the case, for example, if we were to say that a word is invisible, or that a stone is dead.
2047. (~) Contradiction, then, cannot have an intermediate, whereas in a sense (+) privation has an intermediate; for everything must be either equal or not equal, whether it is a being or a non-being. However, it is not necessary to say that everything is either equal or unequal, but this is necessary only in the case of something that is susceptible of equality.
2048. Hence the opposition of contradiction has no intermediate whatsoever, whereas the opposition of privation has no intermediate in a determinate subject; but it is not without an intermediate in an absolute sense. And from this it is evident that contrariety, which is such as to have an intermediate, is closer to privation than to contradiction. Yet it still does not follow that privation is the same as contrariety.
2049. Third, it remains to be shown that contrariety is privation, and in regard to this he does two things. First, he shows by a syllogism that contrariety is privation. He argues as follows: everything from which a process of generation arises is either a form (i.e., the possession of some form) or the privation of some specifying principle (i.e., some form). He says "everything" because generation is twofold. For things are generated absolutely in the genus of substance, but in a qualified sense in the genus of accidents; for generations arise from contraries in matter. Hence it is evident that every contrariety is a privation; for if in any process of generation one of the two extremes is a privation, and each of the contraries is an extreme in the process of generation (because contraries are generated from each other, as white from black and black from white), then one of the two contraries must be a privation.
2050. Here he proves another assertion made above, that not every privation is a contrariety. He says that the reason for this is that there are many ways of being deprived; for a thing that is capable of having a form and does not have it in any way can be said to be deprived of it, and it makes no difference whether it is proximately or remotely disposed for that form.
Now a contrary is always remotely disposed; for contraries are the sources, in the sense of extremes from which changes arise. Hence it was said above (2038) that they are farthest removed from each other. For whether a thing is yellowish or of some other color, it is said to be deprived of whiteness if it is not white. But it is not on that account called a contrary except when it is farthest removed from whiteness, namely, when it is black. Thus it is clear that not every privation is a contrariety.
2051. And since privation requires nothing else than the absence of form (merely presupposing a disposition in a subject without conferring upon that subject any definite disposition through which the subject is close to a form or distant from it), it is evident that privation does not designate any positive reality in a subject, but presupposes a subject with an aptitude. But a contrary requires a definite disposition in a subject, by which it is farthest removed from a form. Therefore it necessarily designates in a subject some positive reality which belongs to the same class as the absent form, as black belongs to the same class as white.
2052. It should also be noted that privation is of two kinds. (1) There is one which has an immediate relationship to the subject of the form (as darkness has an immediate relationship to the transparent medium), and between a privation of this kind and its opposite form there is (+) reciprocal change; for the atmosphere passes from a state of illumination to one of darkness, and from a state of darkness to one of illumination. (2) And there is another kind of privation which is related to the subject of the form only by means of the form, since it has the nature of a corruption of form; for example, blindness is the corruption of sight, and death the corruption of life. In such cases there is no (~) reciprocal change, as has been pointed out in Book IX (1785).
2053. Therefore, since it has been shown here that contrariety is the privation arising from reciprocal change which involves contraries and privation and form, it is clear that contrariety is not the type of privation which is the corruption of a form, but that which has an immediate relation to the subject of the form. Hence the objection raised in the Categories, that it is impossible to revert from privation to possession, does not apply here. But contraries are changed into each other.
2058. He proves the same point by reducing the other contraries to the primary ones. He says that in order to show that one of two contraries is a privation it is enough if this is found to be true in the case of the primary contraries, which are the genera of the others, for example, one and many.
That these are the primary contraries is evident from the fact that all other contraries are reduced to them; for equal and unequal, like and unlike, same and other, are reduced to one and many. Moreover, difference is a kind of diversity, and contrariety is a kind of difference, as has been said above (2017; 2023). Hence, it is evident that every contrariety is reducible to one and many. But one and many are opposed as the indivisible and the divisible, as has been pointed out above (1983). Therefore it follows that all contraries include privation.
Lesson 7
Equal, large, small
2066. Here he establishes the truth about this question; and in regard to this he does three things. First, he shows that the equal is opposed to the large and to the small in a way different from that of contrariety; and he draws this conclusion from the arguments given above on each side of the question. For the first set of arguments showed that the equal is opposed to the large and to the small, whereas the second showed that it is not contrary to them. It follows, then, that it is opposed to them by some other type of opposition. And after having rejected the type of opposition according to which the equal is referred to the unequal but not to the large and the small, it follows that the equal is opposed to the large and to the small either (1) as their negation or (2) as their privation.
2067. He shows in two ways that in the latter type of opposition the equal is opposed to both of the others (the large and the small) and not merely to one of them. First, he says that there is no reason why the equal should be the negation or the privation of the large rather than of the small, or vice versa. Hence it must be the negation or the privation of both.
2068. He also makes this clear by an example, saying that, since the equal is opposed to both, then when we are making inquiries about the equal we use the term whether of both and not merely of one; for we do not ask whether one thing is more than or equal to another, or whether it is equal to or less than another. But we always give three alternatives, namely, whether it is more than or less than or equal to it.
2069. Second, he indicates the type of opposition by which the equal is opposed to the large and to the small. He says that the particle not, which is contained in the notion of the equal when we say that the equal is what is neither more nor less, does not designate a (~) negation pure and simple but necessarily designates a (+) privation; for a negation pure and simple refers to anything to which its own opposite affirmation does not apply; and this does not occur in the case proposed. For we do not say that everything which is not more or less is equal, but we say this only of those things which are capable of being more or less.
2070. Hence the notion of equality amounts to this, that the equal is what is neither (~) large nor (~) small, but is (+) naturally capable of being either large or small, just as other privations are defined. Thus it is evident that the equal is opposed to both the large and the small as a privative negation.
2071. Third, in concluding his discussion, he shows that the equal is intermediate between the large and the small. In regard to this he does two things. First, he draws his thesis as the conclusion of the foregoing argument. For since it has been said that the equal is what is neither large nor small but is naturally capable of being the one or the other, then anything that is related to contraries in this way is intermediate between them, just as what is neither good nor evil is opposed to both and is intermediate between them. Hence it follows that the equal is intermediate between the large and the small. But there is this difference between the two cases: what is neither large nor small has a name, for it is called the equal, whereas what is neither good nor evil does not have a name.
2072. The reason for this is that sometimes both of the privations of two contraries coincide in some one definite term; and then there is only one intermediate, and it can easily be given a name, as the equal. For by the fact that a thing has one and the same quantity it is neither more nor less. But sometimes the term under which both of the privations of the contraries fall is used in several senses, and there is not merely one subject of both of the privations taken together; and then it does not have one name but either remains completely unnamed, like what is neither good nor evil, and this occurs in a number of ways; or it has various names, like what is neither white nor black; for this is not some one thing. But there are certain undetermined colors of which the aforesaid privative negation is used; for what is neither white nor black must be either gray or yellow or some such color.
Lesson 8
Many & few, one & many
2081. Then he explains the second point: how the many and the few are opposed. He says that the term many is used in two senses. First, it is used in the sense of a plurality of things which is excessive, either (1) in an absolute sense or in comparison with something.
(a) It is used in an absolute sense when we say that some things are many because they are excessive, which is the common practice with things that belong to the same class; for example, we say much rain when the rainfall is above average. It is used in comparison with something when we say that ten men are many compared with three. And in a similar way a few means "a plurality which is deficient," i.e., one which falls short of an excessive plurality.
2082. (b) The term much is used in an absolute sense in a second way when a number is said to be a plurality; and in this way many is opposed only (+) to one, but not (~) to a few. For many in this sense is the plural of the word one; and so we say one and many, the equivalent of saying one and ones, as we say white and whites, and as things measured are referred to what is able to measure. For the many are measured by one, as is said below (2087). And in this sense multiples are derived from many. For it is evident that a thing is said to be multiple in terms of any number; for example, in terms of the number two it is double, and in terms of the number three it is triple, and so on. For any number is many in this way, because It is referred to one, and because anything is measurable by one. This happens insofar as many is opposed to one, but not insofar as it is opposed to few.
2083. Hence two things, which are a number, are many insofar as many is opposed to one; but insofar as many signifies an excessive plurality, two things are not many but few; for nothing is fewer than two, because one is not few, as has been shown above (2078). For few is a plurality which has some deficiency. But the primary plurality which is deficient is two. Hence two is the first few.
2087. Next, he shows how the one and the many are opposed; and in regard to this he does two things. First, he shows that the one is opposed to the many in a relative sense. Second (2096), he shows that an absolute plurality is not opposed to few.
In regard to the first he does three things. First, he shows that the one is opposed to the many relatively. He says that the one is opposed to the many as a measure to what is measurable, and these are opposed relatively, but not in such a way that they are to be counted among the things which are relative of themselves. For it was said above in Book V (1026) that things are said to be relative in two ways: for some things are relative to each other on an equal basis, as master and servant, father and son, great and small; and he says that these are relative as contraries; and they are relative of themselves, because each of these things taken in its quiddity is said to be relative to something else.
2088. But other things are not relative on an equal basis, but one of them is said to be relative, not because it itself is referred to something else, but because something else is referred to it, as happens, for example, in the case of knowledge and the knowable object. For what is knowable is called such relatively, not because it is referred to knowledge, but because knowledge is referred to it. Thus it is evident that things of this kind are not relative of themselves, because the knowable is not said to be relative of itself, but rather something else is said to be relative to it.
2089. Then he shows how the one is opposed to the many as to something measurable. And because it belongs to the notion of a measure to be a minimum in some way, he therefore says, first, that one is fewer than many and also fewer than two, even though it is not a few. For if a thing is fewer, it does not follow that it is few, even though the notion of few involves being less, because every few is a certain plurality.
2090. Now it must be noted that plurality or multitude taken absolutely, which is opposed to the one which is interchangeable with being, is in a sense the genus of number; for a number is nothing else than a plurality or multitude of things measured by one.
Hence one, (1) insofar as it means an indivisible being absolutely, is interchangeable with being; but (2) insofar as it has the character of a measure, in this respect it is limited to some particular category, that of quantity, in which the character of a measure is properly found.
2091. And in a similar way (1) insofar as plurality or multitude signifies beings which are divided, it is not limited to any particular genus. But (2) insofar as it signifies something measured, it is limited to the genus of quantity, of which number is a species.
Hence he says that number is plurality measured by one, and that plurality is in a sense the genus of number.
2092. He does not say that it is a genus in an (~) unqualified sense, because, just as being is not a genus properly speaking, neither is the one which is interchangeable with being nor the plurality which is opposed to it. But it is (+) in some sense a genus, because it contains something belonging to the notion of a genus inasmuch as it is common.
2093. Therefore, when we take the one which is the principle of number and has the character of a measure, and number, which is a species of quantity and is the plurality measured by one, the one and the many are not opposed as contraries, as has already been stated above (1997) of the one which is interchangeable with being and of the plurality which is opposed to it; but they are opposed in the same way as things which are relative, i.e., those of which the term one is used relatively. Hence the one and number are opposed inasmuch as the one is a measure and number is something measurable.
2094. And because the nature of these relative things is such that one of them can exist without the other, but not the other way around, this is therefore found to apply in the case of the one and number. For wherever there is a number the one must also exist; but wherever there is a one there is not necessarily a number. For if something is indivisible, as a point, we find the one there, but not number.
But in the case of other relative things, each of which is said to be relative of role of something measured; for in a itself, one of these does not exist without the other; for there is no master without a servant, and no servant without a master.
2095. Here he explains the similarity between the relation of the knowable object to knowledge and that of the one to the many. He says that, although knowledge is truly referred to the knowable object in the same way that number is referred to the one, or the unit, it is not considered to be similar by some thinkers; for to some, the Protagoreans, as has been said above (1800), it seemed that knowledge is a measure, and that the knowable object is the thing measured. But just the opposite of this is true; for it has been pointed out that, if the one, or unit, which is a measure, exists, it is not necessary that there should be a number which is measured, although the opposite of this is true. And if there is knowledge, obviously there must be a knowable object; but if there is some knowable object it is not necessary that there should be knowledge of it. Hence it appears rather that the knowable object has the role of a measure, and knowledge the sense knowledge is measured by the knowable object, just as a number is measured by one; for true knowledge results from the intellect apprehending a thing as it is.
2096. Then he shows that an absolute plurality or multitude is not opposed to a few. He says that it has been stated before that insofar as a plurality is measured it is opposed to the one as to a measure, but it (~) is not opposed to a few. However, much, in the sense of a plurality which is excessive, (+) is opposed to a few in the sense of a plurality which is exceeded.
Similarly a plurality is not opposed to one in a single way but in two. (1) First, it is opposed to it in the way mentioned above (2081), as the divisible is opposed to the indivisible; and this is the case if the one which is interchangeable with being and the plurality which is opposed to it are understood universally. (2)Second, plurality is opposed to the one as something relative, just as knowledge is opposed to its object. And this is the case, I say, if one understands the plurality which is number, and the one which has the character of a measure and is the basis of number.
Lesson 9
Intermediaries of contraries
2098. Then he carries out his plan; and in regard to this he does three things. First, he shows that intermediates belong to the same genus as contraries. Second (2101), he shows that there are intermediates only between contraries ("But all intermediates"). Third (2098), he establishes his main thesis, that intermediates are composed of contraries ("Now if intermediates").
He accordingly says, first, that all intermediates belong to the same class as the things of which they are the intermediates. He proves this by pointing out that intermediates are defined as that into which a thing undergoing change from one extreme to another first passes.
2099. He makes this clear by two examples. First, he uses the example of sounds; for some sounds are low and some are high and some are intermediate. And strings on musical instruments are distinguished by this distinction of sounds; for those strings which yield low pitched sounds are called "top-strings" because they are the basic ones, and those which yield high pitched sounds are called "bottom-strings." Hence, if a musician wishes to proceed step by step from low sounds to high ones, and so to pass through an intermediate register, he must first come to the intermediate sounds. Second, he makes this clear by using colors. For if a thing is changed from white to black, it must first pass through the intermediate colors before it reaches black. The same thing is true of other intermediates.
Lesson 12
Corruptible & incorruptible are generically different.
2137. The second premise is that the corruptible and the incorruptible are contraries. He proves this from the fact that the incapacity opposed to a definite capacity is a kind of privation, as has been stated in Book IX (1784). Now privation is a principle of contrariety; and therefore it follows that incapacity is contrary to capacity, and that the corruptible and the incorruptible are opposed as capacity and incapacity.
But they are opposed in a different way. For if capacity is taken (1) according to its general meaning, as referring to the ability to act or to be acted upon in some way, then the term corruptible is used like the term capacity, and the term incorruptible like the term incapacity. (2) But if the term capacity is used of something inasmuch as it is incapable of undergoing something for the worse, then contrariwise the term incorruptible is referred to capacity, and the term corruptible is referred to incapacity.
2137a. But although it seems necessary from these remarks to conclude that the corruptible and the incorruptible differ specifically, he concludes that they differ generically. And this is true because, just as form and actuality pertain to the species, so too matter and capacity pertain to the genus. Hence, just as the contrariety which pertains to form and actuality causes difference in species, so too the contrariety which pertains to capacity or potency causes difference in genus.