LESSON 13

Are Numbers and Continuous Quantities the Substances and Principles of Sensible Things?

Chapter 5: 1001b 26-1002b 11

             275. And connected with these is the question whether numbers and bodies and surfaces and points are substances, or not.

             276. For if they are not, we are in a quandary as to what being is, and what the substances of things are. For affections and motions and relations and dispositions and their complex conceptions do not seem to signify substance; because all are predicated of some subject, and no one of them is a particular thing. And those things which seem to signify substance most of all, as fire, water, earth [and air], of which composite bodies are constituted, their heat and cold and similar affections, are not substances. And it is only the body which undergoes these that remains as a being and is a substance.

             277. Yet a body is a substance to a lesser degree than a surface, and this than a line, and this in turn than a unit and a point; for a body is defined by means of these, and these seem to be capable of existing without a body, but that a body should exist without these is impossible.

             278. For this reason many of the natural philosophers, including the first, thought that substance and being are bodies, and that other things are attributes of this kind of thing; and hence too that the principles of bodies are the principles of beings. But the later philosophers, who were wiser than these, thought that the principles of things are numbers. Therefore, as we have said, if these are not substance, there is no substance or being at all; for the accidents of these things are not worthy to be called beings.

             279. But if it is admitted that lengths and points are substances to a greater degree than bodies, and we do not see to what sort of bodies these belong (because it is impossible for them to exist in sensible bodies), there will then be no substance at all.

             280. Further, all of these seem to be dimensions of bodies, one according to width, another according to depth, and another according to length.

             281. And, similarly, any figure whatever already exists in a solid. Hence if neither Mercury is in the stone, nor one half of a cube in a cube as something segregated, neither will surface exist in a solid; for if this were true of anything whatever, it would also be true of that which divides a thing in half. And the same argument would apply in the case of a line, a point and a unit. If, then, a body is substance in the highest degree, and these things are such to a greater degree than it is, and these do not exist and are not substances, it escapes our understanding as to what being itself is and what the substance of beings is.

             282. For along with what has been said there happen to be certain unreasonable views about generation and corruption. For if substance, not existing before, exists now, or existing before, does not exist afterwards, it seems to suffer these changes through generation and corruption. But it is impossible for points and lines and surfaces either to come to be or to be destroyed, even though they sometimes exist and sometimes do not. For when bodies are joined or divided, at one time, when they are joined, they [i.e., the two surfaces] simultaneously become one, and at another time, when they are divided, two surfaces are produced; because it [i.e., one of the two surfaces in question] is not in the bodies which have been joined but has perished. And when bodies are divided surfaces exist which did not exist before. For the indivisible point is not divided into two, and if things are generated and corrupted, they are generated from something.

             283. And it is similar with regard to the now in time, for this cannot be generated and corrupted. Yet it seems always to exist, although it is not a substance. It is also clear that this is true of points, lines and surfaces, because the argument is the same; for they are all similarly either limits or divisions.

COMMENTARY

             502. Having inquired whether unity and being are the substances of sensible things, the Philosopher now asks whether numbers and continuous quantities are the substances of sensible things; and in regard to this he does three things. First (275:C 502), he presents the question. Second (276:C 503), he argues in support of one side of the question ("For if they are not"). Third (279:C 507), he argues on the other side ("But if it is admitted").

             Accordingly he says, first (275), that "connected with these," i.e., following from the foregoing problem, there is the question whether numbers and continuous quantities, i.e., bodies, surfaces, and their extremities, such as points, are either substances that are separate from sensible things, or are the substances of sensible things themselves, or not. He says that this problem is a result of the foregoing one, because in the foregoing problem it was asked whether unity is the substance of things. Now unity is the principle of number. But number seems to be the substance of continuous quantity inasmuch as a point, which is a principle of continuous quantity, seems to be merely the number one having position, and a line to be the number two having position, and the primary kind of surface to be the number three having position, and a body the number four having position.

             503. For if they are not (276).

             Then he advances an argument to show that these are the substances of sensible things; and in regard to this he does two things. First (276:C 503), he introduces an argument to show that these are the substances of sensible things. Second (278:C 506), he shows how the early philosophers followed out the first arguments ("For this reason").

             In regard to the first he does two things. For, first, he advances an argument to show that body is the substance of things; and second (277:C 504), to show that many other things are substances to an even greater degree ("Yet a body").

             He says, first (276), that if these things are not substances, we are in a quandary as to what being is essentially, and what the substances of beings are. For it is evident that affections and motions and relations and dispositions or arrangements, and their complex conceptions according as they are put into words, do not seem to signify the substance of anything; because all things of this kind seem to be predicated of a subject as something belonging to the genus of quantity, and no one of them seems to signify "this particular thing," i.e., something that is complete and subsists of itself. This is especially evident in regard to the foregoing things, which are not said to be complete things but things whose nature consists in a kind of relation. But of all things those which especially seem to signify substance are fire, earth, and water, of which many bodies are composed. But he omits air, because it is less perceptible; and this is the reason why some thought air to be nothing. But in these bodies there are found certain dispositions, namely, hot and cold and other affections and passible qualities of this kind, which are not substances according to what has been said. It follows, then, that body alone is substance.

             504. Yet a body (277).

             Here he proceeds to examine those things which appear to be substance to an even greater degree than a body. He says that a body seems to be a substance to a lesser degree than a surface, and a surface than a line, and a line than a point or a unit. He proves this in two ways, of which the first is as follows. That by which a thing is defined seems to be its substance, for a definition signifies substance. But a body is defined by a surface, a surface by a line, a line by a point, and a point by a unit, because they say that a point is a unit having position. Therefore surface is the substance of body, and so on for the others.

             505. The second argument runs as follows. Since substance is the primary kind of being, whatever is prior seems to be substance to a greater degree. But a surface is naturally prior to a body, because a surface can exist without a body but not a body without a surface. Therefore a surface is substance to a greater degree than a body. The same reasoning can be applied to all the others in turn.

             506. For this reason (278).

             Then he shows how the earlier philosophers followed out the foregoing arguments. He says that it was because of the foregoing arguments that many of the ancient philosophers, especially the first, thought that body alone was being and substance, and that all other things were accidents of bodies. Hence when they wanted to study the principles of beings, they studied the principles of bodies, as was stated above in Book I (36:C 74) with regard to the positions of the ancient natural philosophers. But the other philosophers who came later, and were reputed to be wiser than the aforesaid philosophers inasmuch as they dealt more profoundly with the principles of things, i.e., the Pythagoreans and Platonists, were of the opinion that numbers are the substances of sensible things inasmuch as numbers are composed of units. And the unit seems to be one substance of things. Hence, according to the foregoing arguments and opinions of the philosophers, it seems that if these things--numbers, lines, surfaces, and bodies--are not the substances of things, there will be no being at all. For if these are not beings, it is unfitting that their accidents should be called beings.

             507. But if it is (279).

             Then he argues in support of the other side of the question; and he gives four arguments, the first of which is as follows. If anyone were to admit that lengths and points are substances to a greater degree than bodies, then supposing that things of this sort are not substances, it also follows that bodies are not substances. Consequently, no substance will exist, because the accidents of bodies are not substances, as has been stated above (C 503). But points, lines and surfaces are not substances. For these must be the limits of some bodies, because a point is the limit of a line, a line the limit of a surface, and a surface the limit of a body. But it is not evident to what sort of bodies these surfaces, lines and points, which are substances, belong. For it is evident that the lines and surfaces of sensible bodies are not substances, because they are altered in the same way as the other accidents in reference to the same subject. Therefore it follows that there will be no substance whatever.

             508. Further, all of these (280).

             Here he gives the second argument, which is as follows. All of the above-mentioned things seem to be certain dimensions of bodies, either according to width, as a surface, or according to depth, as a solid, or according to length, as a line. But the dimensions of a body are not substances. Therefore things of this kind are not substances.

             509. And, similarly (281).

             Here he gives a third argument, which is as follows. Any figure which can be educed from a solid body according to some dimension is present in that body in the same way, i.e., potentially. But in the case of a large piece of stone which has not yet been cut, it is evident that "Mercury," i.e., the figure of Mercury, is not present in it actually but only potentially. Therefore, in like manner, "in a cube," i.e., in a body having six square surfaces, one half of the cube, which is another figure, is not present actually; but it becomes actual in this way when a cube has already been divided into two halves. And since every eduction of a new figure in a solid which has been cut is made according to some surface which limits a figure, it is also evident that such a surface will not be present in a body actually but only potentially. For if each surface besides the external one were actually present in a solid body, then for the same reason the surface which limits one half of the figure would also be actually present in it. But what has been said of a surface must also be understood of a line, a point, and a unit; for these are actually present in the continuum only insofar as they limit the continuum, and it is evident that these are not the substance of a body. But the other surfaces and lines cannot be the substance of a body, because they are not actually present in it; for substance is actually present in the thing whose substance it is. Hence he concludes that of all of these body especially seems to be substance, and that surfaces and lines seem to be substance to a great degree than bodies. Now if these are not actual beings or substances, it seems to escape our comprehension as to what being is and what the substances of things are.

             510. For along with (282).

             Here he gives the fourth argument. First, he states it, and second (283:C 513), he clarifies it by using a similar case ("And it is similar").

             Accordingly, he says, first (282), that along with the other untenable consequences mentioned there also happen to be certain unreasonable views about generation and corruption on the part of those who hold that lines and surfaces are the substances of sensible things. For every substance which at first did not exist and later does exist, or which first was and afterwards is not, seems to suffer this change by way of generation and corruption. This is most evident in the case of all those things which are caused by way of motion. But points and lines and surfaces sometimes are and sometimes are not. Yet they are not generated or corrupted. Neither, then, are they substances.

             511. He then proves each assumption. The first of these, is that they sometimes are and sometimes are not. For it happens that bodies which were at first distinct are afterwards united, and that those which were at first united are afterwards divided. For when bodies which were initially separated are united, one surface is produced for the two of them, because the parts of a continuous body are united in having one common boundary, which is one surface. But when one body is divided into two, two surfaces are produced, because it cannot be said that when two bodies are brought together their surfaces remain intact, but that both "perish," i.e., cease to be. In like manner, when bodies are divided there begin to exist for the first time two surfaces which previously did not exist. For it cannot be said that a surface, which is indivisible according to depth, is divided into two surfaces according to depth; or that a line, which is indivisible according to width, is divided according to width; or that a point, which is indivisible in every respect, is divided in any respect whatsoever. Thus it is clear that two things cannot be produced from one thing by way of division, and that one thing cannot be produced from two of these things by way of combination. Hence it follows that points, lines and surfaces sometimes begin to be and sometimes cease to be.

             512. After having proved this, he proves the second assumption, namely, that these things are neither generated nor corrupted. For everything that is generated is generated from something, and everything that is corrupted is dissolved into something as its matter. But it is impossible to assign any matter whatever from which these things are generated and into which they are dissolved, because they are simple. Therefore they are neither generated nor corrupted.

             513. And it is similar (283).

             Then he makes the foregoing argument clear by using a similar case. For the now in time stands to time as a point to a line. But the now in time does not seem to be generated and corrupted, because if it were its generation and corruption would have to be measured by some particular time or instant. Thus the measure of this now either would be another now and so on to infinity, or would be time itself. But this is impossible. And even though the now is not generated or corrupted, still each now always seems to differ, not substantially but existentially, because the substance of the now corresponds to the mobile subject. But the difference of the now in terms of existence corresponds to the variation in motion, as is shown in Book IV of the Physics. Therefore the same thing seems to be true of a point in relation to a line, and of a line in relation to a surface, and of a surface in relation to a body, namely, that they are neither corrupted nor generated, although some variation is observable in things of this kind. For the same holds true of all of these, because all things of this kind are, in like manner, limits if regarded as at the extremities, or divisions if they are found in between. Hence, just as the now varies existentially as motion flows by, although it remains substantially the same because the mobile subject remains the same, so also does the point vary. And it does not become different because of the division of a line, even though it is not corrupted or generated in an absolute sense. The same holds true of the others.

             514. But the Philosopher will treat this question in Books XIII and XIV. And the truth of the matter is that mathematical entities of this kind are not the substances of things, but are accidents which accrue to substances. But this mistake about continuous quantities is due to the fact that no distinction is made between the sort of body which belongs to the genus of substance and the sort which belongs to the genus of quantity. For body belongs to the genus of substance according as it is composed of matter and form; and dimensions are a natural consequence of these in corporeal matter. But dimensions themselves belong to the genus of quantity, and are not substances but accidents whose subject is a body composed of matter and form. The same thing too was said above (C 501) about those who held that numbers are the substances of things; for their mistake came from not distinguishing between the one which is the principle of number and that which is interchangeable with being.