Lecture 8 — Opposition between the one and the many

And one might raise similar questions about the one and the many. For if the many are opposed absolutely to the one, certain impossible conclusions will follow.

For one will then be few or a few, for the many are also opposed to the few. Further, two will be many, since the double is multiple, and the double is so designated in reference to two. Hence one will be few; for in relation to what can two be many, except to one, and therefore few? For nothing else is less than this.

Further, if much and little are in plurality what long and short are in length, and if what is much is also many, and what is many is much (unless perhaps there is some difference in the case of an easily bounded continuum), few will be a plurality. Hence one will be a plurality, if it is few; and this will be necessary if two are many.

But perhaps, while many is said in a sense to be much, there is a difference; for example, there is much water but not many waters. But “many” designates those things that are divided.

In one sense, “much” means a plurality which is excessive either absolutely or comparatively. In a similar way, “few” means a plurality which is deficient; and in another sense, it designates number, which is opposed only to one.

For it is in this sense that we say “one” or “many,” just as if we were to say “one” and in the plural “ones” (like white or whites), or to compare what is measured with a measure (that is, a measure and the measurable).

And it is in this sense that multiples are called such, for each number is called many because it is made up of ones and because each number is measurable by one, and number is many as the opposite of one and not of few. So therefore, in this sense, even two is many; but it is not such as a plurality which is excessive either absolutely or comparatively; but two is the first few absolutely, for it is the first plurality which is deficient.

For this reason, Anaxagoras was wrong in speaking as he did when he said that all things were together and unlimited both in plurality and in smallness. He should have said in fewness instead of in smallness, for things could not have been unlimited in fewness, since few is not constituted by one, as some say, but by two.

The one is opposed to the many, then, as a measure is opposed to things measurable, and these are opposed as things that are not relative of themselves. But we have distinguished elsewhere (1056b32) the two senses in which things are said to be relative. Some are relative as contraries, and others as knowledge is relative to the knowable object, because something else is said to be relative to it.

But nothing prevents one thing from being fewer than something else, such as the number two—if it is fewer, it is not few.

And plurality is, in a sense, the genus of number, since number is many measured by one.

And in a sense, one and number are opposed not as contraries, but in the way in which we said that some relative terms are opposed, for they are opposed inasmuch as the one is a measure and the other something measurable.

And for this reason, not everything that is one is a number—for example, anything that is indivisible.

But while knowledge is similarly said to be relative to the knowable object, the relation is not similar. For knowledge might seem to be a measure, and its object to be something measured; but the truth is that, while knowledge is knowable, not all that is knowable is knowledge, because, in a way, knowledge is measured by what is knowable.

And plurality is contrary neither to the few (though the many is contrary to this as an excessive plurality to a plurality which is exceeded) nor to the one in every sense.

But they are contrary in the way we have described, because the one is as something indivisible and the other as something divisible. And in another sense they are relative as knowledge is relative to the knowable object, if plurality is a number and the one is a measure.

2075. Having treated the question which he had raised regarding the opposition of the equal to the large and to the small, here the Philosopher deals with the question concerning the opposition of the one to the many. In regard to this, he does two things.

First (1056b3; [2075]), he debates the question.

Second (1056b14; [2080]), he establishes the truth, at but perhaps.

In regard to the first, he does three things.

First, he gives the reason for the difficulty. He says that, just as there is a difficulty about the opposition of the equal to the large and to the small, so too the difficulty can arise as to whether the one and the many are opposed to each other. The reason for the difficulty is that, if the many without distinction are opposed to the one, certain impossible conclusions will follow unless one distinguishes the various senses in which the term “many” is used, as he does later on (1056b14; [2080]).

2076. For one will (1056b5). He then proves what he had said, for he shows that, if the one is opposed to the many, the one is few or a few. He does this by two arguments:

of which the first is as follows. The many are opposed to the few. Now, if the many are opposed to the one in an unqualified sense and without distinction, then, since one thing has one contrary, it follows that the one is few or a few.

2077. The second argument runs thus. Two things are many. This is proved by the fact that the double is multiple. But the many are opposed to the few. Therefore, two are opposed to few. But two cannot be many in relation to a few, except to [the few as] one, for nothing is less than two except one. It follows, then, that one is a few.

2078. Further, if much (1056b9). Then he shows that this—one is a few—is impossible. For one and a few are related to plurality as the long and the short are to length; for each one of these is a proper passion. But any short thing is a certain length. Hence every few is a certain plurality. Therefore, if one is a few, which it seems necessary to say if two are many, it follows that one is a plurality.

2079. The one, then, will not only be much, but also many, for every much is also many—unless perhaps this differs in the case of fluid things, which are easily divided, as water, oil, air, and the like, which he calls here an easily bounded continuum (for fluid things are easily limited by a foreign boundary). For in such cases, the continuous is also called much, as much water or much air, since they are close to plurality by reason of the ease with which they are divided. But, since any part of these is continuous, that is said to be much (in the singular) which is not said to be many (in the plural). But in other cases, we use the term “many” only when the things are actually divided. For if wood is continuous, we do not say that it is many, but much; when it becomes actually divided, we not only say that it is much, but also many. Therefore, in other cases, there is no difference between saying much and many, but only in the case of an easily bounded continuum. Hence, if one is much, it follows that it is many. This is impossible.

2080. But perhaps (1056b14). Here he solves the difficulty which he had raised; and in regard to this, he does two things.

First, he shows that much is not opposed to one and to a few in the same way.

Second (1056b32; [2087]), he shows how the many and the one are opposed, at the one is opposed to the many, then.

In regard to the first, he does two things.

First, he solves the proposed difficulty;

second (1056b28; [2084]), he rejects an error in the light of what has been said, at for this reason.

And since he had touched on two points above in the objection which he had raised, from which it would seem to follow that it is impossible for much to be many and for many to be opposed to a few, he therefore first of all makes the first point clear. He says that, perhaps in some cases, the term “many” is used with no difference from the term “much.” But in some cases (namely, in that of an easily bounded continuum), “much” and “many” are taken in a different way. For example, we say of one continuous volume of water that there is much water, not many waters. And in the case of things that are actually divided, no matter what they may be, “much” and “many” are both used indifferently.

Many and few; one and many

2081. In one sense (1056b17). 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 that is excessive, either in an absolute sense or in comparison with something. 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 genus—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, that is, one which falls short of an excessive plurality.

2082. The term “much” is used in an absolute sense in a second way when a number is said to be a plurality; in this way, many is opposed only to one, but not to a few. For, in this sense, “many” 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.

2084. For this reason (1056b28). In the light of what has been said, he now rejects an error. For it should be noted that Anaxagoras claimed that the generation of things is a result of separation. Hence he posited that, in the beginning, all things were together in a kind of mixture, but that mind began to separate individual things from that mixture, and that this constitutes the generation of things. And since, according to him, the process of generation is infinite, he therefore claimed that there are an infinite number of things in that mixture. Hence, he said that, before all things were differentiated, they were together, unlimited both in plurality and in smallness.

2085. And the claims which he made about the infinite in respect to its plurality and smallness are true, because the infinite is found in continuous quantities by way of division, and this infinity he signified by the phrase “in smallness.” But the infinite is found in discrete quantities by way of addition, which he signified by the phrase “in plurality.”

2086. Therefore, although Anaxagoras had been right here, he mistakenly abandoned what he had said. For it seemed to him later on that, in place of the phrase “in smallness,” he ought to have said “in fewness,” and this correction was not a true one, because things are not unlimited in fewness. For it is possible to find a first few—namely, two—but not one, as some say. For wherever it is possible to find some first thing, there is no infinite regress. However, if one were a few, there would necessarily be an infinite regress, for it would follow that one would be many, because every few is much or many, as has been stated above (1056b9; [2078]). But, if one were many, something would have to be less than one, and this would be few, and that again would be much; in this way there would be an infinite regress.

2087. The one (1056b32). 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 (1057a12; [2096]), he shows that an absolute plurality is not opposed to few, at and plurality is contrary.

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 that are relative of themselves. For it was said above in book 5 [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. 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. But nothing prevents (1057a1). 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 first says 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, insofar as it means an indivisible being absolutely, is interchangeable with being; but 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, insofar as plurality or multitude signifies beings which are divided, it is not limited to any particular genus. But 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 that are relative, that is, 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 to another, neither is without the other. For there is no master without a servant, and no servant without a master.

2095. But while (1057a7). 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 role something measured; for, in a 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. And plurality (1057a12). 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.

First, it is opposed to it in the way mentioned above [2081], as the divisible is opposed to the indivisible. This is the case if the one which is interchangeable with being and the plurality which is opposed to it are understood universally.

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.