Return to Vignettes of Ancient Mathematics
Aristotle, Physics Z 7.237b23-26
which gets traversed gets traversed in time, and the greater magnitude in more
time, it is impossible that something undergo a finite traversal in the infinite
time, i.e. not undergoing the same traversal repeatedly or some part of the
traversal repeatedly, but the whole traversal in the whole time.
Aristotle, Physics Z 7.237b26-33: if the movement is regular, then a finite distance will be traversed in a finite time and not an inifinite time.
(figure 1) Hence it is clear
that if something should travel uniformly quickly, it is necessary that the
finite get traversed in a finite time (for if a portion is taken which measures
out the whole traversal, what gets traversed will undergo the whole traversal
in as many equal times as there are parts of the change. (figure
2=initial figure) Thus since these parts are limited by each being so much
and being altogether so many, the time would also be limited. For it will be
thus much so-many-times, as the time of the portion of the traversal multiplied
by the number of portions of the traversal).
Aristotle, Physics Z 7.237b34-238a19: if the movement is irregular, then a finite distance will be traversed in a finite time and not an inifinite time.
(figure 1) But it makes no difference if it does not traverse uniformly fast. For let AB be a finite distance which is traversed in the infinite time, and let the infinite time be GD. (figure 2) If it is necessary that one part get traversed before the other (this is clear since one of them gets traversed in an earlier or later time than the other; for the other will always get traversed in additional time, whether or not it changes equally quickly, and no less whether the change increases or diminishes or remains stationary), let some part of the distance AB be taken, AE, (figure 3) which measures out AB. (figure 4) This occurs in some time which is a part of the infinite time. For it cannot occur in an infinite time. For the whole is in an infinite time. (figure 5) And again if I take another part as large as AE, it must be in a finite time. For the whole is in an infinite time. (figure 6) (figure 7 and watch your step) And taking another in this way, since there is no portion of the infinite which measures it (for it is impossible for the infinite to be composed of finite magnitudes, equal or unequal, on account of the fact that the finite things whether in multitude or magnitude are measured out by some one, whether equal or unequal, but no less determinate in magnitude), the finite distance is measured by so many AE's, it would traverse AB in a finite time (it occurs in the same way in the case of coming to rest). Thus something which is the same single thing can neither always be coming to be nor being destroyed.
Aristotle, Physics Z 7.238a20-30: if the movement is regular, then in a finite time only a finite distance will be traversed. Similarly if the movement is irregular, in a finite time only a finite distance will be traversed.
(figure 1)The same argument also arises that it is not possible for something to traverse an infinite or be brought to rest over an infinite in a finite time, whether it is traversed (figure 2 = initial figure) uniformly or (figure 3 and watch your step again) not. For if some part is taken which measures out the whole time, in this time it will pass through some amount which is a part of the magnitude and which is not the whole (for it passes through the whole in the whole time), and again in an equal time it will pass through another, and in each time similarly, whether they are equal or unequal to the initial magnitude. For it makes no difference, if only each is finite. For it is clear that as the time is exhausted, the infinite will not be exhausted, since the subtraction becomes finite both in the amount taken and in the number of times. Thus it will not pass through the infinite in a finite time.
Aristotle, Physics Z 7.238a30-31: an important claim
It makes no difference whether the magnitude is infinite on one side or on both. For the argument is the same.
Aristotle, Physics Z 7.238a32-6: Aristotle now turns to the moving body. A finite body cannot traverse an infinite distance in a finite time.
When these things
are demonstrated it is manifest that a finite magnitude cannot pass through
the infinite in a finite time either, for the same reason. For it passes though
a finite amount in the portion of the time, and in each portion of time in the
same way. Thus it passes through a finite amount in the whole time.
Aristotle, Physics Z 7.238a36-238b16: An infinite body cannot traverse a finite distance.
( figure 1) Since the finite will not pass through the infinite in a finite time, it is clear that the infinite does not pass through the finite. For if the infinite passes though the finite, it is also necessary that the finite pass through the infinite. For it makes no difference which of them is that which traverses. For in both cases the finite passes through the infinite. ( figure 2 = initial figure) For whenever the infinite, A, traverses, there will be some part of it at B which is finite, e.g. GD, ( figure 3) and again a different part and a different part, and always in this way. Thus it will follow at the same time that the infinite traverses the finite and the finite passes through the infinite. For equally it is not possible that the infinite traverse the finite other than by the finite passing through the infinite, whether by moving or by measuring it out. Thus since this is impossible, the infinite could not pass through the finite. Yet the infinite will not pass through the infinite in a finite time either. For if the infinite does, then so does the finite. For the finite inheres in the infinite. Moreover if the time is also taken the demonstration will be the same.
Aristotle, Physics Z 7.238b17-22: Summary
Since neither the finite traverses the infinite nor the infinite the finite nor the infinite the infinite in a finite time, it is manifest that there will be no infinite traversal in a finite time. For what difference does it make whether the traversal or the magnitude yields an infinite? For it is necessary that if either is infinite that the other be infinite. For all movement is in place.