Commentary Aristotle Physics
Lecture 3. How physics and mathematics differ
We have distinguished, then, the different ways in which the term “nature” is used. The next point to consider is how the mathematician differs from the physicist.
Obviously, physical bodies contain surfaces and volumes and lines and points, and these are the subject matter of mathematics.
Further, is astronomy different from physics, or a department of it? It seems absurd that the physicist should be supposed to know the nature of sun or moon but not to know any of their essential attributes, particularly as the writers on physics obviously do discuss their shape also and whether the earth and the world are spherical or not.
Now, the mathematician, though he too treats of these things, nevertheless does not treat of them as the limits of a physical body; nor does he consider the accidents indicated as the accidents of such bodies.
That is why he separates them. For in thought, they are separable from motion, and it makes no difference if they are separated, nor does any falsity result.
The holders of the theory of Forms do the same, though they are not aware of it, for they separate the objects of physics, which are less separable than those of mathematics.
This becomes plain if one tries to state in each of the two cases the definitions of the things and of their accidents. “Odd” and “even,” “straight” and “curved,” and likewise “number,” “line,” and “figure,” do not involve motion. Not so “flesh” and “bone” and “man”—these are defined like “snub nose,” not like “curved.”
Similar evidence is supplied by the more physical of the branches of mathematics, such as optics, harmonics, and astronomy. These are in a way the converse of geometry. While geometry investigates physical lines, but not qua physical, optics investigates mathematical lines, but qua physical, not qua mathematical.
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 the natural philosopher differs from the mathematician;
second, at since “nature” has (194a12; [166]), he designates that to which the consideration of natural science extends.
Concerning the first part, he makes three points.
First, he states the question;
second, at “obviously, physical bodies” (193b23; [158]), he gives his reasons for raising the question;
third, he answers the question at “now, the mathematician” (193b31; [159]).
He therefore first says that, after the uses of the name “nature” have been determined, it is necessary to consider how the mathematician differs from the natural philosopher.
158. Next, at “obviously, physical bodies” (193b24), he gives his reasons for raising the question.
The first of these is as follows. Whenever sciences consider the same subjects, either they are 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 the latter proves from the fact that natural bodies have surfaces and volumes—that is, surfaces and solidity—and lengths and points. Moreover, the natural philosopher must consider all things that are in natural bodies.) Therefore, it seems either that natural science and mathematics are the same or that one is a part of the other.
He gives the second reason at “further, is astronomy” (193b25). 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. Hence, 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 that claim the attention of the astronomers. Therefore, astronomy and natural science agree not only in having the same subjects but also in considering the same accidents and in demonstrating the same conclusions. Hence it seems that astronomy is a part of physics, and as a result, physics does not differ totally from mathematics.
159. Next, at “now, the mathematician” (193b31), he answers the question raised above.
Concerning this, he makes two points.
First, he gives his solution;
second, he confirms it at “this becomes plain” (194a1; [163]).
Concerning the first part, he makes three points.
First, he answers the question;
second, at “that is why he separates” (193b33; [161]), he draws a sort of corollary from the above;
third, at “the holders of” (193b35; [162]), he excludes an error.
160. He first (193b31) says, therefore, that the mathematician and the natural philosopher treat the same things—namely, points, lines, 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 that belong to them insofar as they are the boundaries of a natural body. But this is the way in which natural science considers 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, at “that is why he separates” (193b33), he infers a sort of corollary from what he has just said. Because the mathematician does not consider lines, points, surfaces, and things of this sort, or 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, but 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 in the notion 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, man is not in the notion of animal nor Socrates in the notion 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 that 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 that belong to them are understood as abstracted from motion and sensible matter, but not from intelligible matter, as is said in Metaphysics 7.
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 do not lie in abstracting them in their understanding, 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, at “the holders of the theory” (193b35), he excludes one of Plato’s errors from what he has said. Since Plato was puzzled as to how the intellect could truly separate those things that were not separated in their existence, he held that all things that are separated in the understanding are separated in reality.
Hence, he not only held that mathematical entities are separated because the mathematician abstracts from sensible matter, but he even held that natural things themselves are separated, because 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 included in the understanding of the mathematicals neither in the universal nor in the particular. But some sensible matter, though not individual matter, is included in the understanding of natural things. For in the understanding of man, flesh and bone is included, but not this flesh and this bone.
163. Next, at “this becomes plain” (194a1), he clarifies the solution he has given in two ways:
first, by means of the difference in the definitions that the mathematician and the natural philosopher assign;
second, by means of the intermediate sciences, at “similar evidence” (194a7; [164]).
He therefore first says 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, 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, namely, 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, at “similar evidence” (194a7), he proves the same thing by means of those sciences that are intermediates between mathematics and natural philosophy. Those sciences are called intermediate sciences that take principles abstracted by the purely mathematical sciences and apply them to sensible matter.
For example, perspective applies to the visual line those things that are demonstrated by geometry about the abstracted line, and harmony (or music) applies to sound those things that 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, they are more natural than mathematical sciences, even though they proceed by mathematical principles.
He says, therefore, that sciences of this sort are established in a way contrary to the sciences that are purely mathematical, such as geometry or arithmetic. For geometry considers the line that 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 that is in the consideration of mathematics and applies it to sensible matter, thus treating 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 abstracts those things that 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 middle term.
Thus, the fact that the earth is spherical is demonstrated by natural science by a natural middle term, namely, because its parts equally and in all directions come together at the center. 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.