LESSON 7

Are There Certain Other Substances Separate from Sensible Things? Criticism of the Different Opinions Regarding the Objects of Mathematics

Chapters 2 & 3: 997a 34-998a 21

             208. Furthermore, there is the problem whether sensible substances alone must be said to exist, or others besides these. And whether there is one genus or many genera of substances, as is held by those who speak of the Forms and the intermediate entities with which they say the mathematical sciences deal.

             209. Now the way in which we say that the Forms are both causes and substances in themselves has been treated in our first discussions concerning all of these things (69).

             210. But while they involve difficulty in many respects, it is no less absurd to say that there are certain other natures besides those which exist in the heavens, and that these are the same as sensible things, except that the former are eternal whereas the latter are corruptible. For they [i.e., the Platonists] say nothing more or less than that there is a man-in-himself and horse-in-itself and health-in-itself, which differ in no respect [from their sensible counterparts]; in which they act like those who say that there are gods and that they are of human form. For just as the latter made nothing else than eternal men, in a similar way the former make the Forms nothing else than eternal sensible things.

             211. Furthermore, if anyone holds that there are intermediate entities in addition to the Forms and sensible substances, he will face many problems. For evidently there will be, in like manner, lines in addition to ordinary sensible lines, and the same will be true of other classes of things. Therefore, since astronomy is one of these [mathematical sciences], there will be a heaven in addition to the one we perceive, and a sun and moon, and the same will be true of the other celestial bodies. And how are we to accept these things? For it is unreasonable that a heaven should be immobile, but that it should be mobile is altogether impossible. The same thing is true of the things with which the science of perspective is concerned, and of harmonics in mathematics, because for the same reasons it is also impossible that these should exist apart from sensible things. For if there are intermediate sensible objects and senses, evidently there will be intermediate animals between animals-in-themselves and those which are corruptible.

             212. Again, one might also raise the question as to what things these sciences must investigate. For if geometry, which is the art of measuring the earth, differs from geodesy, which is the art of dividing the earth, only in this respect that the latter deals with things which are perceptible by the senses, whereas the former deals with those which are imperceptible, evidently there will be, in addition to the science of medicine, another science midway between the science of medicine itself and this particular science of medicine; and this will be true of the other sciences. But how is this possible? For then there will be certain healthy things besides those which are sensible and besides health-in-itself.

             213. Similarly, neither does it seem that geodesy is concerned with continuous quantities which are sensible and corruptible. For in this case it would be destroyed when they are destroyed.

             214. Nor again will astronomy deal with sensible continuous quantities, or with this heaven. For the lines we perceive by the senses are not such as those of which geometry speaks, since none of the things perceived by the senses are straight or round in this way. For the circle does not touch the rule at a point, but in the way in which Protagoras spoke in arguing against the geometricians. Neither are the motions or revolutions of the heavens similar to the things of which geometry speaks, nor do points have the same nature as the stars.

             215. However, there are also some who say that these intermediate entities, which are below the Forms and above sensible things, do not exist outside of sensible things but in them. But to enumerate all the impossible consequences which follow from this theory would require too long a discussion. It will be sufficient to propose the following consideration.

             216. It is unreasonable that this should be so only in the case of such things, but evidently it is also possible for the Forms to exist in sensible things, because both of these views depend on the same argument.

             217. Furthermore, it would be necessary for two solids to occupy the same place.

             218. And [the objects of mathematics] would not be immobile since they exist in sensible things, which are moved.

             219. Moreover, on the whole, to what end would anyone hold that they exist but exist in sensible things? For the same absurdities as those described will apply to these suppositions. For there will be a heaven in addition to the one which we perceive, although it will not be separate but in the same place; but this is quite impossible.

Chapter 3

             In these matters, then, it is difficult to see how it is possible to have any positive truth.

COMMENTARY

             403. Having debated the questions which pertain to the scope of this science, the Philosopher now treats dialectically the questions which pertain to the substances themselves with which this science is chiefly concerned. In regard to this he does three things. First (208:C 403), he raises the questions. Second (209:C 406), he indicates the source from which arguments can be drawn in support of one side of the question ("Now the way"). Third (210:C 407), he argues on the other side of the question ("But while they involve").

             In regard to the first part of this division he raises two questions. The first question is whether sensible substances alone are found in the universe, as certain of the ancient philosophers of nature claimed, or whether besides sensible substances there are certain others, as the Platonists claimed.

             404. And assuming that besides sensible substances there are certain others, the second question is whether these substances belong to one genus, or whether there are many genera of substances. For he considers both opinions. For some thinkers held that in addition to sensible substances there are only separate Forms, i.e., an immaterial man-in-himself and horse-in-itself and so on for the other classes of things, whereas others held that there are certain other substances midway between the Forms and sensible things, namely, the objects of mathematics, with which they said the mathematical sciences deal.

             405. The reason for this view is that they posited on the part of the intellect a twofold process of abstracting things: one whereby the intellect is said to abstract the universal from the particular, and according to this mode of abstraction they posited separate Forms, which subsist of themselves; and another [whereby the intellect is said to abstract] from sensible matter certain forms in whose definition sensible matter is not given, for example, the abstraction of circle from brass. And according to this mode of abstraction they posited separate objects of mathematics, which they said are midway between the Forms and sensible substances, because they have something in common with both: with the Forms inasmuch as they are separate from sensible matter, and with sensible substances inasmuch as many of them are found in one class, as many circles and many lines.

             406. Now the way in which (209).

             Then he shows how it is possible to argue one side of the question, saying that it has been stated "in our first discussions," i.e., in Book I (69:C 151), how the Forms are held to be both the causes of sensible things and substances which subsist of themselves. Hence, from the things which have been said there in presenting the views of Plato, arguments can be drawn in support of the affirmative side of the question.

             407. But while they involve (210).

             Here he advances reasons for the negative side. He does this, first (210), for the purpose of showing that the Forms are not separate from sensible things; and, second (211:C 410), for the purpose of showing that the objects of mathematics are not separate ("Furthermore, if anyone").

             Now above in Book I (103:C 208) he gave many arguments against those who posited separate Forms; and, therefore, passing over those arguments, he gives the line of reasoning which seems most effective. He says (210) that while the position of those who posit separate Forms contains many difficulties, the position of those which is now given is no less absurd than any of the others, i.e., that someone should say that there are certain natures in addition to the sensible ones which are contained beneath the heavens. For the heavens constitute the limit of sensible bodies, as is proved in Book I of The Heavens and the World. But those who posited the Forms did not place them below the heavens or outside of it, as is stated in Book III of the Physics. Hence, in accordance with this he says that they posited certain other natures in addition to those which exist in the heavens. And they said that these opposite natures are the same as these sensible things both in kind and in their intelligible constitution, and that they exist in these sensible things; or rather they said that those natures are the Forms of these sensible things. For example, they said that a separate man constitutes the humanity of this particular man who is perceived by the senses, and that a man who is perceived by the senses is a man by participating in that separate man. Yet they held that these differ in this respect, that those immaterial natures are eternal, whereas these sensible natures are corruptible.

             408. That they hold those natures to be the same as these sensible things is clear from the fact that, just as man, horse, and health are found among sensible things, in a similar way they posited among these natures "a man-in-himself," i.e., one lacking sensible matter; and they did the same with regard to horse and health. Moreover, they claimed that nothing else existed in the class of separate substances except [the counterpart of] what existed materially in the sensible world. This position seems to be similar to that of those who held that the gods are of human form, which was the position of the Epicureans, as Tully states in The Nature of the Gods. For just as those who held that the gods are of human form did nothing else than make men eternal in nature, in a similar way those who claimed that there are Forms do nothing else than hold that there are eternal sensible things, such as horse, ox, and the like.

             409. But it is altogether absurd that what is naturally corruptible should be specifically the same as what is naturally incorruptible; for it is rather the opposite that is true, namely, that corruptible and incorruptible things differ in kind to the greatest degree, as is said below in Book X (895:C 2137) of this work. Yet it can happen that what is naturally corruptible is kept in being perpetually by Divine power.

             410. Furthermore, if anyone (211).

             Then he argues against those who claimed that the objects of mathematics are midway between the Forms and sensible things. First (211:C 410), he argues against those who held that the objects of mathematics are intermediate entities and are separate from sensible things; and, second (215:C 417), against those who held that the objects of mathematics exist but exist in sensible things ("However, there are").

             In regard to the first he does two things. First, he introduces arguments against the first position. Second (214:C 416), he argues in support of this position ("Nor again").

             He brings up three arguments against the first position. The first argument is this: just as there is a mathematical science about the line, in a similar way there are certain mathematical sciences about other subjects. If, then, there are certain lines in addition to the sensible ones with which geometry deals, by the same token there will be, in all other classes of things with which the other mathematical sciences deal, certain things in addition to those perceived by the senses. But he shows that it is impossible to hold this with regard to two of the mathematical sciences.

             411. He does this, first, in the case of astronomy, which is one of the mathematical sciences and which has as its subject the heavens and the celestial bodies. Hence, according to what has been said, it follows that there is another heaven besides the one perceived by the senses, and similarly another sun and another moon, and so on for the other celestial bodies. But this is incredible, because that other heaven would be either mobile or immobile. If it were immobile, this would seem to be unreasonable, since we see that it is natural for the heavens to be always in motion. Hence the astronomer also makes some study of the motions of the heavens. But to say that a heaven should be both separate and mobile is impossible, because nothing separate from matter can be mobile.

             412. Then he shows that the same view is unacceptable in the case of other mathematical sciences, for example, in that of perspective, which considers visible lines, and "in the case of harmonics," i.e., in that of music, which studies the ratios of audible sounds. Now it is impossible that there should be intermediate entities between the Forms and sensible things; because, if these sensible things--sounds and visible lines--were intermediate entities, it would also follow that there are intermediate senses. And since senses exist only in an animal, it would follow that there are also intermediate animals between the Form animal, and corruptible animals; but this is altogether absurd.

             413. Again, one might (212).

             The second argument [which he uses against the possibility of the objects of mathematics being an intermediate class of entities separate from sensible things] is as follows. If in those classes of things with which the mathematical sciences deal there are three classes of things--sensible substances, Forms and intermediate entities, then since the intelligible structure of all sensible things and of all Forms seems to be the same, it appears to follow that there are intermediate entities between any sensible things at all and their Forms. Hence there remains the problem as to what classes of things are included in the scope of the mathematical sciences. For if a mathematical science such as geometry differs from geodesy, which is the science of sensible measurements, only in this respect that geodesy deals with sensible measurements, whereas geometry deals with intermediate things which are not sensible, there will be in addition to all the sciences which consider sensible things certain [other] mathematical sciences which deal with these intermediate entities. For example, if the science of medicine deals with certain sensible bodies, there will be in addition to the science of medicine, and any like science, some other science which will be intermediate between the science of medicine which deals with sensible bodies and the science of medicine which deals with the Forms. But this is impossible; for since medicine is about "healthy things," i.e., things which are conducive to health, then it will also follow, if there is an intermediate science of medicine, that there will be intermediate health-giving things in addition to the health-giving things perceived by the senses and absolute health, i.e., health-in-itself, which is the Form of health separate from matter. But this is clearly false. Hence it follows that these mathematical sciences do not deal with certain things which are intermediate between sensible things and the separate Forms.

             414. Similarly, neither (213).

             Then he gives the third argument [against the possibility of the objects of mathematics being an intermediate class]; and in this argument one of the points in the foregoing position is destroyed, namely, that there would be a science of continuous quantities which are perceptible; and thus, if there were another science of continuous quantities, it would follow from this that there would be intermediate continuous quantities. Hence he says that it is not true that geodesy is a science of perceptible continuous quantities, because such continuous quantities are corruptible. It would follow, then, that geodesy is concerned with corruptible continuous quantities. But it seems that a science is destroyed when the things with which it deals are destroyed; for when Socrates is not sitting, our present knowledge that he is sitting will not be true. Therefore it would follow that geodesy, or geosophics as other readings say, is destroyed when sensible continuous quantities are destroyed; but this is contrary to the character of science, which is necessary and incorruptible.

             415. Yet this argument can be brought in on the opposite side of the question inasmuch as one may say that he intends to prove by this argument that there are no sciences of sensible things, so that all sciences must be concerned with either the intermediate entities or the Forms.

             416. Nor again will (214).

             Here he argues in support of this position, as follows: it belongs to the very notion of science that it should be concerned with what is true. But this would not be the case unless it were about things as they are. Therefore the things about which there are sciences must be the same in themselves as they are shown to be in the sciences. But perceptible lines are not such as geometry says they are. He proves this on the grounds that geometry demonstrates that a circle touches "the rule," i.e., a straight line, only at a point, as is shown in Book III of Euclid's Elements. But this is found to be true of a circle and a line in the case of sensible things. Protagoras used this argument when he destroyed the certainties of the sciences against the geometricians. Similarly, the movements and revolutions of the heavens are not such as the astronomers describe them; for it seems to be contrary to nature to explain the movements of the celestial bodies by means of eccentrics and epicycles and other different movements which the astronomers describe in the heavens. Similarly, neither are the quantities of the celestial bodies such as the astronomers describe them to be, for they use stars as points even though they are still bodies having extension. It seems, then, that geometry does not deal with perceptible continuous quantities, and that astronomy does not deal with the heaven which we perceive. Hence it remains that these sciences are concerned with certain other things, which are intermediate.

             417. However, there are (215).

             Here he argues against another position. First, he states the point at issue. Second (216:C 418), he brings in arguments germane to his purpose ("It is unreasonable").

             Accordingly, he says, first (215), that some thinkers posit natures midway between the Forms and sensible things, yet they do not say that these natures are separate from sensible things but exist in sensible things themselves. This is clear regarding the opinion of those who held that there are certain self-subsistent dimensions which penetrate all sensible bodies, which some thinkers identify with the place of sensible bodies, as is stated in Book IV of the Physics and is disproved there. Hence he says here that to pursue all the absurd consequences of this position is a major undertaking, but that it is now sufficient to touch on some points briefly.

             418. It is unreasonable (216).

             Then he brings four arguments against this position. The first runs as follows. It seems to be for the same reason that in addition to sensible things the Forms and objects of mathematics are posited, because both are held by reason of abstraction on the part of the intellect. If, then, the objects of mathematics are held to exist in sensible things, it is fitting that not only they but also the Forms themselves should exist there. But this is contrary to the opinion of those who posit [the existence of] the Forms. For they hold that these are separate, and not that they exist anywhere in particular.

             419. Furthermore, it would be (217).

             Here he gives the second argument, which runs thus: if the objects of mathematics differ from sensible things yet exist in them, since a body is an object of mathematics, it follows that a mathematical body exists simultaneously with a sensible body in the same subject. Therefore "two solids," i.e., two bodies, will exist in the same place. This is impossible not only for two sensible bodies but also for a sensible body and a mathematical one, because each has dimensions, by reason of which two bodies are prevented from being in the same place.

             420. And [the objects of mathematics] (218).

             Here he gives the third argument. For when something is moved, anything that exists within it is moved. But sensible things are moved. Therefore, if the objects of mathematics exist in sensible things, it follows that the objects of mathematics are moved. But this is contrary to the intelligible constitution of mathematical objects, which abstract not only from matter but also from motion.

             421. Moreover, on the whole (219).

             Then he gives the fourth argument, which runs thus: no position is thought to be reasonable unless it is based on one of the causes, and especially if a more untenable conclusion follows from such a position. But this position is held without a cause. For the same absurdities face those who hold the objects of mathematics to be intermediate entities and to exist in sensible things, as face those who hold that they do not exist in sensible things, as well as certain other peculiar and greater difficulties, as is clear from what has been said above. Hence, this position is an unreasonable one. In concluding he states that the questions mentioned above involve much difficulty as to what is true in these matters.

             422. Now the Philosopher treats these questions below in Books XII, XIII and XIV of this work, where he shows that there are neither separate mathematical substances nor Forms. The reasoning which moved those who posited the objects of mathematics and the Forms, which are derived from an abstraction of the intellect, is given at the beginning of Book XIII. For nothing prevents a thing which has some particular attribute from being considered by the intellect without its being viewed under this aspect and yet be considered truly, just as a white man can be considered without white being considered. Thus the intellect can consider sensible things not inasmuch as they are mobile and material but inasmuch as they are substances or continuous quantities; and this is to abstract the thing known from matter and motion. However, so far as the thing known is concerned, the intellect does not abstract in such a way that it understands continuous quantities and forms to exist without matter and motion. For then it would follow either that the intellect of the one abstracting is false, or that the things which the intellect abstracts are separate in reality.