LECTURE 3 (193 b 22-194 a 11)

HOW PHYSICS AND MATHEMATICS DIFFER IN THEIR CONSIDERATION OF THE SAME THING

             157. After the Philosopher has explained what nature is and how many ways the name is used, he here intends to show what it is that natural science considers.

             This section is divided into two parts. First he shows how natural science differs from mathematics. Secondly, where he says, 'Since nature has . . .' (194 a 12), he designates that to which the consideration of natural science extends.

             Concerning the first part he makes three points. First he states the question. Secondly, where he says, 'Obviously physical bodies . . .' (193 b 23), he gives his reasons for [raising] the question. Thirdly, he answers the question where he says, 'Now the mathematician . . .' (193 b 31).

             He says, therefore, first that after the uses of the name 'nature' have been determined, it is necessary to consider how mathematics differs from natural philosophy.

             158. Next where he says, 'Obviously physical bodies . . .' (193 b 23), he gives his reasons for [raising] the question. The first of these is as follows.

             Whenever sciences consider the same subjects, they are either the same science, or one is a part of the other. But the mathematical philosopher considers points and lines and surfaces and bodies, and so does the natural philosopher. (For he proves from the fact that natural bodies have planes, i.e., surfaces, and volumes, i.e., solidity, and lengths and points. Moreover the natural philosopher must consider all things that are in natural bodies.) Therefore it seems that natural science and mathematics are either the same or that one is a part of the other.

             He gives the second reason where he says, 'Further, is astronomy . . .' (193 b 25). In connection with this reason he raises the question whether astronomy is altogether other than natural philosophy or a part of it. For it is clear that astronomy is a part of mathematics. Whence, if it is also a part of natural philosophy, it follows that mathematics and physics agree at least in this part.

             That astronomy is a part of physics he proves in two ways. First by the following argument. To whomever it belongs to know the substances and natures of certain things, also belongs the consideration of their accidents. But it belongs to the natural philosopher to consider the nature and substance of the sun and the moon, since they are certain natural bodies. Therefore it belongs to the natural philosopher to consider their per se accidents.

             He proves this also from the custom of the philosophers. For natural philosophers are found to have treated the shape of the sun and of the moon and of the earth and of the whole world. And these are topics which claim the attention of the astronomers. Therefore astronomy and natural science agree not only in [having] the same subjects but also in the consideration of the same accidents, and in demonstrating the same conclusions. Whence it seems that astronomy is a part of physics, and as a result physics does not differ totally from mathematics.

             159. Next where he says, 'Now the mathematician . . .' (193 b 31), he answers the question raised above. Concerning this he makes two points. First he gives his solution, and secondly he confirms it, where he says, 'This becomes plain . . .' (194 a 1).

             Concerning the first part he makes three points. First he answers the question. Secondly, where he says, 'That is why he separates . . .' (193 b 33), he concludes to a sort of corollary from the above. Thirdly, where he says, 'The holders of . . .' (193 b 35), he excludes an error.

             160. He says, therefore, first that the mathematician and the natural philosopher treat the same things, i.e., points, and lines, and surfaces, and things of this sort, but not in the same way. For the mathematician does not treat these things insofar as each of them is a boundary of a natural body, nor does he consider those things which belong to them insofar as they are the boundaries of a natural body. But this is the way in which natural science treats them. And, it is not inconsistent that the same thing should fall under the consideration of different sciences according to different points of view.

             161. Next where he says, 'That is why he separates . . .' (193 b 33), he concludes to a sort of corollary from what he has just said. Because the mathematician does not consider lines, and points, and surfaces, and things of this sort, and their accidents, insofar as they are the boundaries of a natural body, he is said to abstract from sensible and natural matter. And the reason why he is able to abstract is this: according to the intellect these things are abstracted from motion.

             As evidence for this reason we must note that many things are joined in the thing, but the understanding of one of them is not derived from the understanding of another. Thus white and musical are joined in the same subject, nevertheless the understanding of one of these is not derived from an understanding of the other. And so one can be separately understood without the other. And this one is understood as abstracted from the other. It is clear, however, that the posterior is not derived from the understanding of the prior, but conversely. Hence the prior can be understood without the posterior, but not conversely. Thus it is clear that animal is prior to man, and man is prior to this man (for man is had by addition to animal, and this man by addition to man). And because of this our understanding of man is not derived from our understanding of animal, nor our understanding of Socrates from our understanding of man. Hence animal can be understood without man, and man without Socrates and other individuals. And this is to abstract the universal from the particular.

             In like manner, among all the accidents which come to substance, quantity comes first, and then the sensible qualities, and actions and passions, and the motions consequent upon sensible qualities. Therefore quantity does not embrace in its intelligibility the sensible qualities or the passions or the motions. Yet it does include substance in its intelligibility. Therefore quantity can be understood without matter, which is subject to motion, and without sensible qualities, but not without substance. And thus quantities and those things which belong to them are understood as abstracted from motion and sensible matter, but not from intelligible matter, as is said in Metaphysics, VII.

             Since, therefore, the objects of mathematics are abstracted from motion according to the intellect, and since they do not include in their intelligibility sensible matter, which is a subject of motion, the mathematician can abstract them from sensible matter. And it makes no difference as far as the truth is concerned whether they are considered one way or the other. For although the objects of mathematics are not separated according to existence, the mathematicians, in abstracting them according to their understanding, do not lie, because they do not assert that these things exist apart from sensible matter (for this would be a lie). But they consider them without any consideration of sensible matter, which can be done without lying. Thus one can truly consider the white without the musical, even though they exist together in the same subject. But it would not be a true consideration if one were to assert that the white is not musical.

             162. Next where he says, 'The holders of the theory . . .' (193 b 35), he excludes from what he has said an error of Plato.

             Since Plato was puzzled as to how the intellect could truly separate those things which were not separated in their existence, he held that all things which are separated in the understanding are separated in the thing. Hence he not only held that mathematical entities are separated, because of the fact that the mathematician abstracts from sensible matter, but he even held that natural things themselves are separated, because of the fact that natural science is of universals and not of singulars. Hence he held that man is separated, and horse, and stone, and other such things. And he said these separated things are ideas, although natural things are less abstract than mathematical entities. For mathematical entities are altogether separated from sensible matter in the understanding, because sensible matter is not included in the understanding of the mathematicals, neither in the universal nor in the particular. But sensible matter is included in the understanding of natural things, whereas individual matter is not. For in the understanding of man flesh and bone is included, but not this flesh and this bone.

             163. Next where he says, 'This becomes plain . . .' (194 a 1), he clarifies the solution he has given in two ways, first by means of the difference in the definitions which the mathematician and the natural philosopher assign, and secondly by means of the intermediate sciences, where he says, 'Similar evidence . . .' (194 a 7).

             He says, therefore, first that what has been said of the different modes of consideration of the mathematician and the natural philosopher will become evident if one attempts to give definitions of the mathematicals and of natural things and of their accidents. For the mathematicals, such as equal and unequal, straight and curved, and number, and line, and figure, are defined without motion and matter, but this is not so with flesh and bone and man. Rather the definition of these latter is like the definition of the snub in which definition a sensible subject is placed, i.e., nose. But this is not the case with the definition of the curved in which definition a sensible subject is not placed.

             And thus from the very definitions of natural things and of the mathematicals, what was said above about the difference between the mathematician and the natural philosopher is apparent.

             164. Next where he says, 'Similar evidence . . .' (194 a 7), he proves the same thing by means of those sciences which are intermediates between mathematics and natural philosophy.

             Those sciences are called intermediate sciences which take principles abstracted by the purely mathematical sciences and apply them to sensible matter. For example, perspective applies to the visual line those things which are demonstrated by geometry about the abstracted line; and harmony, that is music, applies to sound those things which arithmetic considers about the proportions of numbers; and astronomy applies the consideration of geometry and arithmetic to the heavens and its parts.

             However, although sciences of this sort are intermediates between natural science and mathematics, they are here said by the Philosopher to be more natural than mathematical, because each thing is named and takes its species from its terminus. Hence, since the consideration of these sciences is terminated in natural matter, then even though they proceed by mathematical principles, they are more natural than mathematical sciences.

             He says, therefore, that sciences of this sort are established in a way contrary to the sciences which are purely mathematical, such as geometry or arithmetic. For geometry considers the line which has existence in sensible matter, which is the natural line. But it does not consider it insofar as it is in sensible matter, insofar as it is natural, but abstractly, as was said. But perspective conversely takes the abstract line which is in the consideration of mathematics, and applies it to sensible matter, and thus treats it not insofar as it is a mathematical, but insofar as it is a physical thing.

             Therefore from this difference between intermediate sciences and the purely mathematical sciences, what was said above is clear. For if intermediate sciences of this sort apply the abstract to sensible matter, it is clear that mathematics conversely separates those things which are in sensible matter.

             165. And from this it is clear what his answer is to the objection raised above concerning astronomy. For astronomy is a natural science more than a mathematical science. Hence it is no wonder that astronomy agrees in its conclusions with natural science.

             However, since it is not a purely natural science, it demonstrates the same conclusion through another method. Thus, the fact that the earth is spherical is demonstrated by natural science by a natural method, e.g., because its parts everywhere and equally come together at the middle. But this is demonstrated by astronomy from the figure of the lunar eclipse, or from the fact that the same stars are not seen from every part of the earth.