LECTURE 4 (233 a 17-b 32)

THE FINITE AND THE INFINITE ARE FOUND IN BOTH MAGNITUDE AND TIME IN THE SAME WAY. IT IS PROVEN THAT NO CONTINUUM IS INDIVISIBLE

             777. After he has shown that magnitude and time are divided in the same way, he shows here that both the finite and the infinite are found in magnitude and in time in the same way.

             Concerning this he makes three points. First he states his position. Secondly, where he says, 'Hence Zeno's argument . . .' (233 a 21), he answers a difficulty. Thirdly, where he says, 'The passage over . . .' (233 a 32), he proves his position.

             778. He says, therefore, first that if either of these, that is, time and magnitude, is infinite, then so is the other. And they are each infinite in the same way.

             He explains this by distinguishing two types of infinity. If time is infinite in its extremities, then magnitude is also infinite in its extremities. Time and magnitude are said to be infinite in their extremities in the sense that they have no extremities. For example, imagine a line which is not terminated at any points, or a time which is not terminated at any first or last moment.

             Further, if time is infinite by division, then length will be infinite by division. This is the second type of infinity. A thing is said to be infinite by division when it can be divided to infinity. This pertains to the nature [ratio] of continuity, as was said. And if time were infinite in both of these ways, then length would also be infinite in both of these ways.

             These two types of infinity are appropriately contrasted. For the first type of infinity is due to the lack of indivisible extremities. The second type is due to the designation of indivisible points in the middle. For a line is divided by points which are designated within the line.

             779. Next where he says, 'Hence Zeno's argument . . .' (233 a 21), he answers from the foregoing a difficulty of Zeno the Eleatic. Zeno wished to prove that nothing is moved from one place to another, for example from A to B.

             It is clear that between A and B there is an infinity of intermediate points, because a continuum is divisible to infinity. Hence, if something is moved from A to B, it must cross an infinity and touch each one of an infinity of parts. This cannot happen in a finite time. Therefore in no time, however great, as long as it is finite, can anything be moved through any space, however small.

             The Philosopher says that this argument is based on a false judgment. For length and time and any continuum can be said to be infinite in two ways, as was said; that is, infinite by division and infinite in the extremities. Now if there were a mobile body and a space, which are infinite in quantity, that is, infinite in the extremities, they would not touch each other in a finite time. But if they are infinite by division, this could happen. For a time which is finite in quantity is infinite by division. Hence it follows that the infinite is crossed, not indeed in a finite time, but in an infinite time. The infinity of points in the magnitude are crossed in the infinity of 'nows' in time, but not in a finitude of 'nows'.

             It must be realized that this answer is ad hominem, and not to the truth. Aristotle will make this clear below in Book VIII.

             780. Next where he says, 'The passage over . . .' (233 a 32), he proves what he has stated above.

             First he restates his position, and secondly he proves it, where he says, 'This may be shown . . .' (233 a 34).

             He says, therefore, first that no mobile body can cross an infinite space in a finite time, nor can it cross a finite space in an infinite time. Rather, if the time is infinite, the magnitude must be infinite, and vice versa.

             Next where he says, 'This may be shown . . .' (233 a 34), he proves his position.

             First he proves that the time cannot be infinite if the magnitude is finite. Secondly, where he says, 'The same reasoning . . .' (233 b 14), he proves the converse; namely, the time cannot be finite if the length is infinite.

             781. He proves the first point with two arguments. The first is as follows.

             Let there be a finite magnitude A B and an infinite time C. C D is some finite part of this infinite time. Now since the mobile body crosses the finite magnitude A B in the whole time C, then in that part of the time which is C D it crosses some part (B E) of the magnitude. Since the magnitude A B is finite and larger, and B E is finite and smaller, then, if B E is taken many times, it must either measure the whole of A B or fall short of or excel it in measurement. For every finite smaller quantity is related to a finite larger quantity in this way, as is clear in numbers. Three, which is smaller than six, measures six when taken twice. But twice three does not measure five, which is also larger than three, but excels five. For twice three is more than five. Likewise, twice three does not measure seven, but falls short of it. For twice three is less than seven. Nevertheless three times three excels seven. Now B E is related to A B in one of these three ways. For the same mobile body will always cross a magnitude equal to B E in a time equal to C D. But if B E is taken many times, it will either measure the whole of A B or excel it. Therefore, if C D is taken many times, it also will either measure the whole time C or will excel it. And thus the total time C, in which the body crosses the whole finite magnitude, must be finite. For the time must be divided into numerically equal parts, just as the magnitude is.

             782. He gives the second argument where he says, 'Moreover, if it is . . .' (233 b 8). The argument is as follows.

             Although it has been granted that the mobile body crosses the finite magnitude A B in an infinite time, nevertheless it cannot be granted that it crosses every magnitude in an infinite time. For we see that many finite magnitudes are crossed in finite times.

             Hence, let there be a finite magnitude B E which is crossed in a finite time. But since B E is finite, it measures A B, which is also finite. And the same mobile body crosses a magnitude equal to B E in a finite time equal to the time in which it crosses B E. And thus the total of the finite equal times, which are taken to measure or constitute the whole time, will be the same as the total of the equal magnitudes B E, which constitute the whole of A B. Thus it follows that the total time is finite.

             783. This argument is different from the first one. In the first argument B E was given as a part of the magnitude A B. In the second argument B E is given as some other separated magnitude.

             When he adds, 'That infinite time . . .' (233 b 11), he points out the necessity of the second argument given above.

             One might jokingly object to the first argument by saying that, just as the mobile body crosses the whole magnitude A B in an infinite time, it also crosses any part of A B in the same way. And thus it will not cross the part B E in a finite time. But since it cannot be granted that a mobile body crosses every magnitude in an infinite time, it is necessary to bring in the second argument to the effect that B E is some other magnitude which the body crosses in a finite time. And he adds that it is clear that the mobile body does not cross the magnitude B E in an infinite time, 'if the time be taken as limited in one direction' (233 b 13); that is, if we take some magnitude other than the first one. This other magnitude is called B E and is crossed in a finite time. For if the body crosses part of the magnitude in less time than the whole, then this magnitude B E must be finite, 'the limit in one direction being given' (233 b 14); that is, A B. He says, as it were, the following. If the time in which the body crosses B E is finite and is less than the infinite time in which it crosses A B, then B E must be less than A B. And since A B is finite, B E is finite.

             784. Next where he says, 'The same reasoning . . .' (233 b 14), he states that the same proof shows that it is impossible for the length to be infinite and the time finite. For just as a part of an infinite time is finite, likewise a part of an infinite length is finite.

             785. Next where he says, 'It is evident . . .' (233 b 15), he proves that no continuum is indivisible.

             He says, first, that if a continuum is indivisible, an impossibility follows. Secondly, where he says, 'For since the distinction . . .' (233 b 19), he gives a proof which points out this impossibility.

             He says, therefore, first that it is clear from what has been said that neither a line, nor a surface, nor any other continuum is indivisible. From the foregoing it is clearly impossible for a continuum to be composed of indivisible parts, although it can be composed of continuous parts. Another reason is that the indivisible would be divided.

             786. Next where he says, 'For since the distinction . . .' (233 b 19), he gives a proof which brings out this impossibility. He first sets forth certain things which were established above. The first is that in any time there can be faster and slower motion. The second is that a faster body crosses more magnitude in an equal time. The third is that there are diverse proportions between various velocities and between various crossed magnitudes. For example, they can be double, which is a proportion of two to one. Or they can be one and a half [hemioliam]. Another name for this is sexquialtera, a proportion of three to two. Or there can be any other proportion.

             From these suppositions he proceeds as follows. Let one velocity be one and a half times faster than another. This is a proportion of three to two. And let the faster body cross a magnitude A B C D which is composed of three indivisible magnitudes A B, B C, and C D. In the same time the slower body, according to the granted proportion, must cross a magnitude composed of two indivisible magnitudes. This is the magnitude E F G. Now since time and magnitudes are divided in the same way, the time in which the faster body crosses the three indivisible magnitudes must be divided into three indivisible parts. For the body must cross an equal magnitude in an equal time. Therefore, let there be a time J K L M which is divided into three indivisible parts. But since the slower body in the same time is moved through E F G, which are two indivisible magnitudes, the time must be divided in two. And thus it follows that the indivisible is divided. For it will be necessary for the slower body to cross one indivisible magnitude in one and a half indivisible units of time. It cannot be said that it crosses one indivisible unit of magnitude in one indivisible unit of time. For then the first body is not being moved faster than the slower body. Therefore it follows that the slower body crosses an indivisible magnitude in more than one indivisible unit of time and less than two. Thus one indivisible unit of time must be divided.

             And if it be granted that the slower body is moved through three indivisible magnitudes in three indivisible units of time, it follows in the same way that an indivisible magnitude is divided. For in one indivisible unit of time, the faster body will be moved through more than one indivisible magnitude but through less than two.

             Hence it is clear that no continuum can be indivisible.