LESSON 16

Arguments against the View that Ideas Are Numbers

Chapter 9: 991b 9-992a 24

             122. Further, if the Forms are numbers, in what way will they be causes? Will it be because existing things are other numbers, so that this number is man, another Socrates, and still another Callias? In what respect, then, are the former the cause of the latter? For it will make no difference if the former are eternal and the latter are not. But if it is because the things here are ratios of numbers, like a harmony, then clearly there will be one kind of thing of which they are the ratios. And if this is matter, evidently the numbers themselves will be certain ratios of one thing to something else. I mean that, if Callias is a numerical ratio of fire, water, earth and air, [his Idea will also be a ratio of certain things], and man-in-himself, whether it be a number or not, will still be a numerical ratio of certain things and not just a number; nor will it be any number because of these.

             123. Again, one number will come from many numbers, but how or in what way can one Form come from [many] Forms?

             124. But if one number is not produced from them but from the units which they contain, as the units in the number ten thousand, how are the units related? For if they are specifically the same, many absurdities will follow; and if they are not, neither will they be the same as one another nor all the others the same as all.

             125. For in what way will they differ, if they have no attributes? For these statements are neither reasonable nor in accord with our understanding.

             126. Further, [if the Forms are numbers], it is necessary to set up some other class of number: that with which arithmetic deals. And all the things which are said to be intermediate, from what things or what principles in an absolute sense will they come, or why will they be [an intermediate class] between the things at hand and those [in the ideal world]?

             127. Again, each of the units which are contained in the number two will come from a prior two. But this is impossible.

             128. Further, why is a number something composed of these?

             129. And, again, in addition to what has been said, if the units are different, it will be necessary to speak of them in the same way as do those who say that the elements are four or two. For none of them designate as an element what is common, namely, body, but fire and earth, whether body is something in common or not. But now we are speaking of the one as if it were one thing made up of like parts, as fire or water. But if this is the case, numbers will not be substances. Yet it is evident that, if the one itself is something common and a principle, then the one is used in different senses; otherwise this will be impossible.

             130. Now when we wish to reduce substances to their principles, we claim that lengths come from the long and short, i.e., from a kind of great and small; and the plane from the wide and narrow; and body from the deep and shallow.

             131. Yet how will a surface contain a line, or a solid a line or surface? For the wide and narrow is a different class from the deep and shallow. Hence, just as number is not present in these, because the many and few differ from these, it is evident that no one of the other higher classes will be present in the lower. And the broad is not in the class of the deep, for then the solid would be a kind of surface.

             132. Further, from what will points derive being? Plato was opposed to this class of objects as a geometrical fiction, but he called them the principle of a line. And he often holds that there are indivisible lines. Yet these must have some [limit]. Therefore any argument that proves the existence of the line also proves the existence of the point.

COMMENTARY

             239. Here he destroys Plato's opinion about the Forms inasmuch as Plato claimed that they are numbers. In regard to this he does two things. First (122:C 239), he argues dialectically against Plato's opinion about numbers; and second (130:C 254), against his opinion about the other objects of mathematics ("Now when we wish").

             In regard to the first part he gives six arguments. The first (122) is this: in the case of things which are substantially the same, one thing is not the cause of another. But sensible things are substantially numbers according to the Platonists and Pythagoreans. Therefore, if the Forms themselves are numbers, they cannot be the cause of sensible things.

             240. But if it is said that some numbers are Forms and others are sensible things, as Plato literally held (as though we were to say that this number is man and another is Socrates and still another is Callias), even this would not seem to be sufficient; for according to this view the intelligible structure of number will be common both to sensible things and the Forms. But in the case of things which have the same intelligible structure, one does not seem to be the cause of another. Therefore the Forms will not be the causes of sensible things.

             241. Nor again can it be said that they are causes for the reason that, if those numbers are Forms, they are eternal. For this difference, namely, that some things differ from others in virtue of being eternal and non-eternal in their own being considered absolutely, is not sufficient to explain why some things are held to be the causes of others. Indeed, things differ from each other as cause and effect rather because of the relationship which one has to the other. Therefore things that differ numerically do not differ from each other as cause and effect because some are eternal and some are not.

             242. Again, it is said that sensible things are certain "ratios" or proportions of numbers, and that numbers are the causes of these sensible things, as we also observe to be the case "in  harmonies," i.e., in the combinations of musical notes. For numbers are said to be the causes of harmonies insofar as the numerical proportions applied to sounds yield harmonies. Now if the above is true, then just as in harmonies there are found to be sounds in addition to numerical proportions, in a similar way it was obviously necessary to posit in addition to the numbers in sensible things something generically one to which the numerical proportions are applied, so that the proportions of those things which belong to that one genus would constitute sensible things. However, if that to which the numerical proportion in sensible things is applied is matter, evidently those separate numbers, which are Forms, had to be termed proportions of some one thing to something else. For this particular man, called Callias or Socrates, must be said to be similar to the ideal man, called "man-in-himself," or humanity. Hence, if Callias is not merely a number, but is rather a kind of ratio or numerical proportion of the elements, i.e., of fire, earth, water and air, and if the ideal man-in-himself is a kind of ratio or numerical proportion of certain things, the ideal man will not be a number by reason of its own substance. From this it follows that there will be no number "apart from these," i.e., apart from the things numbered. For if the number which constitutes the Forms is separate in the highest degree, and if it is not separate from things but is a kind of proportion of numbered things, no other number will now be separate. This is opposed to Plato's view.

             243. It also follows that the ideal man is a proportion of certain numbered things, whether it is held to be a number or not. For according to those who held that substances are numbers, and according to the philosophers of nature, who denied that numbers are substances, some numerical proportions must be found in the substances of things. This is most evident in the case of the opinion of Empedocles, who held that each one of these sensible things is composed of a certain harmony or proportion [of the elements].

             244. Again, one number (123).

             Here he gives the second argument which runs thus: one number is produced from many numbers. Therefore, if the Forms are numbers, one Form is produced from many Forms. But this is impossible. For if from many things which differ specifically something specifically one is produced, this comes about by mixture, in which the natures of the things mixed are not preserved; just as a stone is produced from the four elements. Again, from things of this kind which differ specifically one thing is not produced by reason of the Forms, because the Forms themselves are combined in such a way as to constitute a single thing only in accordance with the intelligible structure of individual things, which are altered in such a way that they can be mixed together. And when the Forms themselves of the numbers two and three are combined, they give rise to the number five, so that each number remains and is retained in the number five.

             245. But since someone could answer this argument, in support of Plato, by saying that one number does not come from many numbers, but each number is immediately constituted of units, Aristotle is therefore logical in rejecting this answer (124) ("But if one number").

             For if it is said that some greater number, such as ten thousand, is not produced "from them," namely, from twos or many smaller numbers, but from "units," i.e., ones, this question will follow: How are the units of which numbers are composed related to each other? For all units must either conform with each other or not.

             246. But many absurd conclusions follow from the first alternative, especially for those who claim that the Forms are numbers. For it will follow that different Forms do not differ substantially but only insofar as one Form surpasses another. It also seems absurd that units should differ in no way and yet be many, since difference is a result of multiplicity.

             247. But if they do not conform, this can happen in two ways. First, they can lack conformity because the units of one number differ from those of another number, as the units of the number two differ from those of the number three, although the units of one and the same number will conform with each other. Second, they can lack conformity insofar as the units of one and the same number do not conform with each other or with the units of another number. He indicates this distinction when he says, "For neither will they be the same as one another (125)," i.e., the units which comprise the same number, "nor all the others the same as all," i.e., those which belong to different numbers. Indeed, in whatever way there is held to be lack of conformity between units an absurdity is apparent. For every instance of non-conformity involves some form or attribute, just as we see that bodies which lack conformity differ insofar as they are hot and cold, white and black, or in terms of similar attributes. Now units lack qualities of this kind, because they have no qualities, according to Plato. Hence it will be impossible to hold that there is any non-conformity or difference between them of the kind caused by a quality. Thus it is evident that Plato's opinions about the Forms and numbers are neither "reasonable" (for example, those proved by an apodictic argument), nor "in accord with our understanding" (for example, those things which are self-evident and verified by [the habit of] intellect alone, as the first principles of demonstration).

             248. Further, [if the Forms] (126).

             Here he gives the third argument against Plato, which runs thus: all objects of mathematics, which Plato affirmed to be midway between the Forms and sensible substances, are derived unqualifiedly from numbers, either as proper principles, or as first principles. He says this because in one sense numbers seem to be the immediate principles of the other objects of mathematics; for the Platonists said that the number one constitutes the point, the number two the line, the number three surface, and the number four the solid. But in another sense the objects of mathematics seem to be reduced to numbers as first principles and not as proximate ones. For the Platonists said that solids are composed of surfaces, surfaces of lines, lines of points, and points of units, which constitute numbers. But in either way it followed that numbers are the principles of the other objects of mathematics.

             249. Therefore, just as the other objects of mathematics constituted an intermediate class between sensible substances and the Forms, in a similar way it was necessary to devise some class of number which is other than the numbers that constitute the Forms and other than those that constitute the substance of sensible things. And arithmetic, which is one of the mathematical sciences, evidently deals with this kind of number as its proper subject, just as geometry deals with mathematical extensions. However, this position seems to be superfluous; for no reason can be given why number should be midway "between the things at hand," or sensible things, and "those in the ideal world," or the Forms, since both sensible things and the Forms are numbers.

             250. Again, each of the units (127).

             Here he gives the fourth argument, which runs thus: those things which exist in the sensible world and those which exist in the realm of mathematical entities are caused by the Forms. Therefore, if some number two is found both in the sensible world and in the realm of the objects of mathematics, each unit of this subsequent two must be caused by a prior two, which is the Form of twoness. But it is "impossible" that unity should be caused by duality. For it would be most necessary to say this if the units of one number were of a different species than those of another number, because then these units would acquire their species from a Form which is prior to the units of that number. And thus the units of a subsequent two would have to be produced from a prior two.

             251. Further, why is (128).

             Here he gives the fifth argument, which runs thus: many things combine so as to constitute one thing only by reason of some cause, which can be considered to be either extrinsic, as some agent which unites them, or intrinsic, as some unifying bond. Or if some things are united of themselves, one of them must be potential and another actual. However, in the case of units none of these reasons can be said to be the one "why a number," i.e., the cause by which a number, will be a certain "combination," i.e., collection of many units; as if to say, it will be impossible to give any reason for this.

             252. And, again, in addition (129).

             Here he gives the sixth argument, which runs thus: if numbers are the Forms and substances of things, it will be necessary to say, as has been stated before (124:C 245), either that units are different, or that they conform. But if they are different, it follows that unity as unity will not be a principle. This is clarified by a similar case drawn from the position of the natural philosophers. For some of these thinkers held that the four [elemental] bodies are principles. But even though being a body is common to these elements, these philosophers did not maintain that a common body is a principle, but rather fire, earth, water and air, which are different bodies. Therefore, if units are different, even though all have in common the intelligible constitution of unity, it will not be said that unity itself as such is a principle. This is contrary to the Platonists' position; for they now say that the unit is the principle of things, just as the natural philosophers say that fire or water or some body with like parts is the principle of things. But if our conclusion against the Platonists' theory is true--that unity as such is not the principle and substance of things--it will follow that numbers are not the substances of things. For number is held to be the substance of things only insofar as it is constituted of units, which are said to be the substances of things. This is also contrary to the Platonists' position which is now being examined, i.e., that numbers are Forms.

             253. But if you say that all units are undifferentiated, it follows that "the whole," i.e., the entire universe, is a single entity, since the substance of each thing is the one itself, and this is something common and undifferentiated. Further, it follows that the same entity is the principle of all things. But this is impossible by reason of the notion involved, which is inconceivable in itself, namely, that all things should be one according to the aspect of substance. For this view contains a contradiction, since it claims that the one is the substance of all things, yet maintains that the one is a principle. For one and the same thing is not its own principle, unless, perhaps, it is said that "the one" is used in different senses, so that when the senses of the one are differentiated all things are said to be generically one and not numerically or specifically one.

             254. Now when we wish (130).

             Here he argues against Plato's position with reference to his views about mathematical extensions. First (130), he gives Plato's position; and second (131:C 255), he advances an argument against it ("Yet how will").

             He says, first, that the Platonists, wishing to reduce the substances of things to their first principles, when they say that continuous quantities themselves are the substances of sensible things, thought they had discovered the principles of things when they assigned line, surface and solid as the principles of sensible things. But in giving the principles of continuous quantities they said that "lengths," i.e., lines, are composed of the long and short, because they held that contraries are the principles of all things. And since the line is the first of continuous quantities, they first attributed to it the great and small; for inasmuch as these two are the principles of the line, they are also the principles of other continuous quantities. He says "from the great and small" because the great and small are also placed among the Forms, as has been stated (108:C 217). But insofar as they are limited by position, and are thus particularized in the class of continuous quantities, they constitute first the line and then other continuous quantities. And for the same reason they said that surface is composed of the wide and narrow, and body of the deep and shallow.

             255. Yet how will a surface (131).

             Here he argues against the foregoing position, by means of two arguments. The first is as follows. Things whose principles are different are themselves different. But the principles of continuous quantities mentioned above are different, according to the foregoing position, for the wide and narrow, which are posited as the principles of surface, belong to a different class than the deep and shallow, which are held to be the principles of body. The same thing can be said of the long and short, which differ from each of the above. Therefore, line, surface and body all differ from each other. How then will one be able to say that a surface contains a line, and a body a line and a surface? In confirmation of this argument he introduces a similar case involving number. For the many and few, which are held to be principles of things for a similar reason, belong to a different class than the long and short, the wide and narrow, and the deep and shallow. Therefore number is not contained in these continuous quantities but is essentially separate. Hence, for the same reason, the higher of the above mentioned things will not be contained in the lower; for example, a line will not be contained in a surface or a surface in a body.

             256. But because it could be said that certain of the foregoing contraries are the genera of the others, for example, that the long is the genus of the broad, and the broad the genus of the deep, he destroys this [objection] by the following argument: things composed of principles are related to each other in the same way as their principles are. Therefore, if the broad is the genus of the deep, surface will also be the genus of body. Hence a solid will be a kind of plane, i.e., a species of surface. This is clearly false.

             257. Further, from what will (132).

             Here he gives the second argument, which involves points; and in regard to this Plato seems to have made two errors. First, Plato claimed that a point is the limit of a line, just as a line is the limit of a surface and a surface the limit of a body. Therefore, just as he posited certain principles of which the latter are composed, so too he should have posited some principle from which points derive their being. But he seems to have omitted this.

             258. The second error is this: Plato seems to have held different opinions about points. For sometimes he maintained that the whole science of geometry treats this class of things, namely, points, inasmuch as he held that points are the principles and substance of all continuous quantities. And he not only implied this but even explicitly stated that a point is the principle of a line, defining it in this way. But many times he said that indivisible lines are the principles of lines and other continuous quantities, and that this is the class of things with which geometry deals, namely, indivisible lines. Yet by reason of the fact that he held that all continuous quantities are composed of indivisible lines, he did not avoid the consequence that continuous quantities are composed of points, and that points are the principles of continuous quantities. For indivisible lines must have some limits, and these can only be points. Hence, by whatever argument indivisible lines are held to be the principles of continuous quantities, by the same argument too the point is held to be the principle of continuous quantity.