Commentary Aristotle Physics
Book 6. Division of Motion into Quantitative Parts
Lecture 1. No continuum is composed of indivisible parts
Now, if the terms “continuous,” “in contact,” and “consecutive” are understood as defined above—things being continuous if their extremities are one, in contact if their extremities are together, and consecutive if there is nothing of their own kind intermediate between them—nothing that is continuous can be composed of indivisibles.
For instance, a line cannot be composed of points, the line being continuous and the point indivisible.
For the extremities of two points can be neither one (since there can be no extremity of an indivisible as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct).
Moreover, if that which is continuous is composed of points, these points must be either continuous or in contact with one another, and the same reasoning applies in the case of all indivisibles.
Now, for the reason given above, they cannot be continuous, and one thing can be in contact with another only if whole is in contact with whole or part with part or part with whole.
But, since indivisibles have no parts, they must be in contact with one another as whole with whole.
And, if they are in contact with one another as whole with whole, they will not be continuous, for that which is continuous has distinct parts, and these parts into which it is divisible are different in this way, that is, in distinct places.
Nor, again, can a point be consecutive to a point or a moment to a moment in such a way that length can be composed of points or time of moments.
For things are consecutive if there is nothing of their own kind intermediate between them, but that which is intermediate between points is always a line and that which is intermediate between moments is always a period of time.
Again, if length and time could thus be composed of indivisibles, they could be divided into indivisibles, since each is divisible into the parts of which it is composed. But, as we saw, no continuous thing is divisible into things without parts.
Nor can there be anything of any other kind intermediate between the parts or between the moments, for if there could be any such thing, it is clear that it must be either indivisible or divisible, and if it is divisible, it must be divisible either into indivisibles or into divisibles that are infinitely divisible, in which case it is continuous.
Moreover, it is plain that everything continuous is divisible into divisibles that are infinitely divisible; if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are continuous with one another are one and are in contact.
750. After the Philosopher has finished dividing motion into its species and discussing the unity and contrariety of motions and of states of rest, he proposes in book 6 to discuss the things that pertain to the division of motion precisely as it is divisible into quantitative parts.
The whole book is divided into two parts.
In the first, he shows that motion, as every continuum, is divisible;
in the second, he shows how motion is divided, at “the present also is” (233b33; [787]).
The first part is subdivided into two sections:
in the first, he shows that no continuum is composed of indivisibles;
in the second, that no continuum is indivisible, at “it is evident, then” (233b15; [785]).
The first is further subdivided into two parts:
in the first, he shows that no continuum is composed of indivisibles;
in the second (because the proofs for the first seem to be applicable mainly to magnitudes), he shows that the same proofs apply to magnitudes, to motion, and to time, at “the same reasoning applies” (231b18; [758]).
In regard to the first part, he does two things:
first, he recalls some definitions previously given, with a view to using them in demonstrating his proposition;
second, he proves the proposition, at “for the extremities of” (231a26; [752]).
751. He says first (231a21) that, if the previously given definitions of continuum, of that which is touched, and of that which is consecutive are correct (namely, that continua are things whose extremities are one; touched things are those whose extremities are together; and consecutive things are those between which nothing of the same type intervenes), then it would follow that it is impossible for any continuum to be composed of indivisibles; that is, it is impossible, for example, for a line to be composed of points, provided, of course, that a line is said to be a continuum and that a point is an indivisible.
This proviso is added to prevent other meanings being attached to “point” and “line.”
752. Then, at “for the extremities of” (231a26), he proves the proposition:
first, he gives two proofs of the proposition;
second, he explains things that might be misunderstood in his proofs, at “nor can there be” (231b12; [756]).
In regard to the first proof, he does two things:
first, he shows that no continuum is composed of indivisibles, either after the manner of continuity or after that of contact;
second, nor after the manner of things that are consecutive, at “nor, again, can a point” (231b6; [754]).
In regard to the first, he gives two reasons.
The first is that, whatever things a unit is composed of, either after the manner of continuity or after that of contact, the extremities must either be one, or they must be together. But the extremities of points cannot be one, because an extremity is spoken of in relation to a part, but in an indivisible, it is impossible to distinguish that which is an extremity from something else that is a part.
Similarly, it cannot be said that the extremities are together, because nothing can be the extremity of a thing that cannot be divided into parts, while an extremity must always be distinct from that of which it is the extremity. But in a thing that cannot be divided into parts, there is no way of distinguishing one thing and another. It follows, therefore, that a line cannot be composed of points, either after the manner of continuity or after the manner of contact.
753. The second reason is given at “moreover, if that which” (231a29). If a continuum is composed of points, they must be either continuous with one another or touch (and the same is true of all other indivisibles, that is, that no continuum is composed of them).
To prove that they are not continuous with one another, the first argument suffices.
But, to prove that they cannot touch one another, another argument is adduced, which is the following. Everything that touches something else does so either by the whole touching the other wholly or by a part of one touching a part of the other or the whole of the other. But since an indivisible does not have parts, it cannot be said that part of one touches either a part or the whole of the other. Hence, if two points touch, the whole point touches another whole point.
But, when a whole touches a whole, no continuum can be formed, because every continuum has distinct parts so that one part is here and another there, and is divisible into parts that are different and in distinct places, that is, in position (in things that have position), whereas things that touch one another totally are not distinguished as to place or position. It therefore follows that a line cannot be composed of points that are in contact.
754. Then, at “nor, again, can a point” (231b6), he shows that no continuum is composed of indivisibles after the manner of things that are consecutive. For no point will be consecutive to another so as to form a length, that is, a line. And no now is consecutive to another now such as to form a period of time, because consecutive things are by definition such that nothing of the same kind intervenes between any two. But, between any two points, there is always a line, and so, if a line is composed of points, it would follow that, between any two points there is always another mediate point. The same is true for the nows. If a period of time is nothing but a series of nows, then, between any two nows, there would be another now. Therefore, no line is composed of points, and no time is composed of nows, after the manner of things that are consecutive.
755. The second reason is given at “again, if length” (231b10), and is based on a different definition of continuum—the one given at the beginning of book 3—that a continuum is that which is divisible to infinity. Here is the proof: a line or a time can be divided into whatever it is composed of; so, if each of them is composed of indivisibles, it follows that each is divided into indivisibles; but this is false, since neither of them is divisible into indivisibles, for that would mean they would not be divisible to infinity; therefore, no continuum is composed of indivisibles.
756. Then, at “nor can there be” (231b11), he explains two statements he made in the course of his proofs.
The first of these was that, between two points, there is always a line and that, between two nows, there is always time.
He explains it thus. If two points exist, they must differ in position; otherwise, they would not be two, but one. But they cannot touch one another, as was shown above; hence, they are distant, and something is between them. But no other intermediate is possible, except a line between two points, and time between nows, for if the intermediate between two points were other than a line, that intermediate must be either divisible or indivisible. If indivisible, it must be distinct from the two points—at least in position—and since it touches neither, there must be another intermediate between that indivisible and the original extremities, and so on to infinity until a divisible intermediate is found. However, if the intermediate is divisible, it will be divisible into indivisibles or into what are further divisible. But it cannot be divided into indivisibles, because then the same difficulty returns: how a divisible can be composed of indivisibles. It must be granted, then, that the intermediate is divisible into what are further divisible. But that is what a continuum is. Therefore, that intermediate will be a continuum. But the only continuous intermediate between two points is a line. Therefore, between any two points, there is an intermediate line. Likewise, between two nows there is time, and the same for other types of continua.
757. Then, at “moreover, it is plain” (231b15), he explains the second statement referred to above, that every continuum is divisible into divisibles. For on the supposition that a continuum is divisible into indivisibles, it would follow that two indivisibles would have to be in contact in order to form the continuum. For continua must have an extremity that is one, as appears from its definition; moreover, the parts of a continuum must touch, because, if the extremities are one, they are together, as was stated in book 5. Therefore, since it is impossible for two indivisibles to touch, it is impossible for a continuum to be divided into indivisibles.