12 Jan 2009

Space-Time Continuum and Zeno's Paradox in Aristotle's Physics Book VI

[Aristotle's theory of the continuity of time and space forms the basis for one of his critiques of Zeno's paradoxes. Below is summary of the beginning of his Physics Book VI; after which I place the original text that I summarize and explain.]

Aristotle, Physics, Book VI, section 1:

Things are continuous if their outer parts are united. If their extremities are merely found together, then these different things are just in contact. And if they are separated by something else, then these neighboring things are just in succession.

We consider a point in space as indivisible, because it is not yet one-dimensional. In other words, a point does not have extension. It only has location.

Since points are indivisible, they do not have parts. But if they do not have parts, points cannot be things whose outer-parts are united.

A line is continuous. Its parts must have extremities that are united. But points do not have extremities, so lines cannot be made up of points.

Two similar things are in succession if something dissimilar stands between them. If we have two separate points in space, what stands between them would be a line spanning that space. But if a line connects the points, then there is no longer a gap between them. So aligned points cannot make a succession. Thus an extended length is not composed of a succession of points.

Likewise for points in time. If we have two points in time, what stands between them is a temporal duration. But then, if a temporal duration connects those points, then there is not something of another kind standing between them. Hence time as well is not composed of sequences of moments.

So time is made up of temporal parts, and extension is made up of extended parts. Because its parts are of the same kind, they cannot be linked in succession. Hence time and space are continua of smaller parts.

Indivisibles lack the united extremities to constitute a continuum, so the time and space continua cannot be made up of indivisibles.

But we do not encounter any contradictions when we suppose that time and space are continua divisible into parts that themselves are infinitely divisible.

This holds likewise for motion. Consider the following example.

Person O is walking to Thebes, and must cross three points on the way, points A, B, and G:

There are the moments of motion corresponding to the three points, motion-moments D, E, and Z:

At any moment of the walker's motion, he cannot both
a) be in motion,
and at the same time
b) have completed his motion at the respective distance-point.

If a man is walking to Thebes, he cannot be walking to Thebes and at the same time have completed his walk to Thebes.

Aristotle will now proceed to show that time and space are infinitely divisible. He will demonstrate how it leads to absurdity if we presume that time and space are made up of indivisibles.

So we presume that point A is an indivisible part of extension. When the walker was in his corresponding motion-moment D at place A, we know that he was in motion, because he had not yet arrived at Thebes. So the walker was in an intermediate state of progression. But as intermediate, that means his motion extends between places, and hence is divisible.

If at the hypothetically-supposed indivisible point A the walker was not in an intermediary state of motion, then that means he was at rest. But if that is so, then his walk to Thebes was made up of a continuous rest, or a series of rests, which is absurd.

Likewise for time. If we thought of the walker moving through indivisible time points, then at any one point, he would be at rest, which is absurd.

So, there is a correspondence between the divisibility of time and of space.

And thus, there is an infinitely divisible space-time continuum (or time-space continuum).

Aristotle now has us consider a new set of bodies in motion.

Body A is faster than body B

Now we consider an extent of time, with points G, E, O, and D along it.

As well, we are to consider time moments Z, K, and H.

At the same moment Z, both Body A and Body B start from spatial-point G and move toward spatial-point D. Body A continues moving through time until time-point H, at which time it arrives at spatial-point D. But because Body A is faster than Body B, as soon as A arrives at point D, Body B has only arrived at spatial-point E.

And of course, it would take Body A less time to arrive at previous spatial-point O. Body A arrives at O at time-point K.

Hence we see that the quicker Body A passes over a greater magnitude of space in less time than does the slower Body B.

And given the continuous correlation between time and space, that is to say, given the space-time continuum, faster Body A will cross an equal distance in less time:

Now Aristotle has us imagine that Bodies A and B move at a different ratio of speed, although A is still faster than B. [Aristotle changes the letters here for this new example, but we will keep them the same for consistency. See the original text at the end of this entry for the exact letter arrangements.] In this new example, it takes Body B until time-point H to arrive at spatial-point D. However, because Body A is faster, it only takes until time-point K to arrive at spatial-point D.

But, at time-point K, Body B only made it to spatial point E.

So when we divide Body B's time, we thereby proportionally divide its distance.
Now, because Body A is faster, it arrived at spatial-point E in less time than it took Body B.

Thus we see that when we divide Body A's distance magnitude, we thereby proportionally divide its time extent. This is the proportional correspondence between the continua of time and space. In other words, this is their marriage; it is the space-time continuum. In something's movement, you cannot divide the time without also thereby dividing the distance. And vice versa, you cannot divide the distance without also thereby dividing the time. For, there is a continuous intersection of the dimensions of time and space.

Both time and space are continuous. By "continuous" Aristotle means:

that which is divisible into divisibles that are infinitely divisible

Both time and space go-on infinitely, and they can be divided infinitely. And, each instance of a division of one dimension correlates to a proportional division in the other dimension.

Later Aristotle writes in section 4 that:

For suppose that A is the time occupied by the motion B. Then if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, less again to occupy a further subdivision of the time, and so on to infinity. Again, the time will be divisible similarly to the motion: for if the whole motion occupies all the time half the motion will occupy half the time, and less of the motion again will occupy less of the time.
[Returning now to section 2]

This is why Zeno's paradoxes of motion are erroneous. It is true that a finite amount of space is infinitely divisible. But that does not mean that in a finite amount of time, we cannot traverse an infinite number of divisions. For, what Zeno misses is that time is infinitely divisible as well, and each division of space corresponds to a proportional division of time. So although we can say that a moving object has an infinite number of places to cross, we also must say that it passes from one point to the next in infinitely little time. So it is perfectly conceivable that motion may cross infinitely divided space, so long as we consider also that each division of space corresponds to a proportional division of time. And so long as distance is divisible, so too is too is time divisible. But if time is always divisible, then that means it extends or has duration. Because there is duration, that means an object can be found at different spatial points within that duration. For, the object has different temporal opportunities to be in different places within that span of time. And because we can never attain span-less time periods, we can never take away the opportunity for an object to change its position.

From Aristotle's Physics Book VI:

Book VI
Now if the terms ‘continuous’, ‘in contact’, and ‘in succession’ are understood as defined above things being ‘continuous’ if their extremities are one, ‘in contact’ if their extremities are together, and ‘in succession’ if there is nothing of their own kind intermediate between them-nothing that is continuous can be composed ‘of indivisibles’: e.g. a line cannot be composed of points, the line being continuous and the point indivisible. For the extremities of two points can neither be one (since of an indivisible there can be no extremity 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, i.e. spatially separate.
Nor, again, can a point be in succession 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 in succession if there is nothing of their own kind intermediate between them, whereas 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: for 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.
The same reasoning applies equally to magnitude, to time, and to motion: either all of these are composed of indivisibles and are divisible into indivisibles, or none. This may be made clear as follows. If a magnitude is composed of indivisibles, the motion over that magnitude must be composed of corresponding indivisible motions: e.g. if the magnitude ABG is composed of the indivisibles A, B, G, each corresponding part of the motion DEZ of O over ABG is indivisible. Therefore, since where there is motion there must be something that is in motion, and where there is something in motion there must be motion, therefore the being-moved will also be composed of indivisibles. So O traversed A when its motion was D, B when its motion was E, and G similarly when its motion was Z. Now a thing that is in motion from one place to another cannot at the moment when it was in motion both be in motion and at the same time have completed its motion at the place to which it was in motion: e.g. if a man is walking to Thebes, he cannot be walking to Thebes and at the same time have completed his walk to Thebes: and, as we saw, O traverses a the partless section A in virtue of the presence of the motion D. Consequently, if O actually passed through A after being in process of passing through, the motion must be divisible: for at the time when O was passing through, it neither was at rest nor had completed its passage but was in an intermediate state: while if it is passing through and has completed its passage at the same moment, then that which is walking will at the moment when it is walking have completed its walk and will be in the place to which it is walking; that is to say, it will have completed its motion at the place to which it is in motion. And if a thing is in motion over the whole KBG and its motion is the three D, E, and Z, and if it is not in motion at all over the partless section A but has completed its motion over it, then the motion will consist not of motions but of starts, and will take place by a thing’s having completed a motion without being in motion: for on this assumption it has completed its passage through A without passing through it. So it will be possible for a thing to have completed a walk without ever walking: for on this assumption it has completed a walk over a particular distance without walking over that distance. Since, then, everything must be either at rest or in motion, and O is therefore at rest in each of the sections A, B, and G, it follows that a thing can be continuously at rest and at the same time in motion: for, as we saw, O is in motion over the whole ABG and at rest in any part (and consequently in the whole) of it. Moreover, if the indivisibles composing DEZ are motions, it would be possible for a thing in spite of the presence in it of motion to be not in motion but at rest, while if they are not motions, it would be possible for motion to be composed of something other than motions.
And if length and motion are thus indivisible, it is neither more nor less necessary that time also be similarly indivisible, that is to say be composed of indivisible moments: for if the whole distance is divisible and an equal velocity will cause a thing to pass through less of it in less time, the time must also be divisible, and conversely, if the time in which a thing is carried over the section A is divisible, this section A must also be divisible.
And since every magnitude is divisible into magnitudes-for we have shown that it is impossible for anything continuous to be composed of indivisible parts, and every magnitude is continuous-it necessarily follows that the quicker of two things traverses a greater magnitude in an equal time, an equal magnitude in less time, and a greater magnitude in less time, in conformity with the definition sometimes given of ‘the quicker’. Suppose that A is quicker than B. Now since of two things that which changes sooner is quicker, in the time ZH, in which A has changed from G to D, B will not yet have arrived at D but will be short of it: so that in an equal time the quicker will pass over a greater magnitude. More than this, it will pass over a greater magnitude in less time: for in the time in which A has arrived at D, B being the slower has arrived, let us say, at E. Then since A has occupied the whole time ZH in arriving at D, will have arrived at O in less time than this, say ZK. Now the magnitude GO that A has passed over is greater than the magnitude GE, and the time ZK is less than the whole time ZH: so that the quicker will pass over a greater magnitude in less time. And from this it is also clear that the quicker will pass over an equal magnitude in less time than the slower. For since it passes over the greater magnitude in less time than the slower, and (regarded by itself) passes over LM the greater in more time than LX the lesser, the time PRh in which it passes over LM will be more than the time PS, which it passes over LX: so that, the time PRh being less than the time PCh in which the slower passes over LX, the time PS will also be less than the time PX: for it is less than the time PRh, and that which is less than something else that is less than a thing is also itself less than that thing. Hence it follows that the quicker will traverse an equal magnitude in less time than the slower. Again, since the motion of anything must always occupy either an equal time or less or more time in comparison with that of another thing, and since, whereas a thing is slower if its motion occupies more time and of equal velocity if its motion occupies an equal time, the quicker is neither of equal velocity nor slower, it follows that the motion of the quicker can occupy neither an equal time nor more time. It can only be, then, that it occupies less time, and thus we get the necessary consequence that the quicker will pass over an equal magnitude (as well as a greater) in less time than the slower.
And since every motion is in time and a motion may occupy any time, and the motion of everything that is in motion may be either quicker or slower, both quicker motion and slower motion may occupy any time: and this being so, it necessarily follows that time also is continuous. By continuous I mean that which is divisible into divisibles that are infinitely divisible: and if we take this as the definition of continuous, it follows necessarily that time is continuous. For since it has been shown that the quicker will pass over an equal magnitude in less time than the slower, suppose that A is quicker and B slower, and that the slower has traversed the magnitude GD in the time ZH. Now it is clear that the quicker will traverse the same magnitude in less time than this: let us say in the time ZO. Again, since the quicker has passed over the whole D in the time ZO, the slower will in the same time pass over GK, say, which is less than GD. And since B, the slower, has passed over GK in the time ZO, the quicker will pass over it in less time: so that the time ZO will again be divided. And if this is divided the magnitude GK will also be divided just as GD was: and again, if the magnitude is divided, the time will also be divided. And we can carry on this process for ever, taking the slower after the quicker and the quicker after the slower alternately, and using what has been demonstrated at each stage as a new point of departure: for the quicker will divide the time and the slower will divide the length. If, then, this alternation always holds good, and at every turn involves a division, it is evident that all time must be continuous. And at the same time it is clear that all magnitude is also continuous; for the divisions of which time and magnitude respectively are susceptible are the same and equal.
Moreover, the current popular arguments make it plain that, if time is continuous, magnitude is continuous also, inasmuch as a thing passes over half a given magnitude in half the time taken to cover the whole: in fact without qualification it passes over a less magnitude in less time; for the divisions of time and of magnitude will be the same. And if either is infinite, so is the other, and the one is so in the same way as the other; i.e. if time is infinite in respect of its extremities, length is also infinite in respect of its extremities: if time is infinite in respect of divisibility, length is also infinite in respect of divisibility: and if time is infinite in both respects, magnitude is also infinite in both respects.
Hence Zeno’s argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two senses in which length and time and generally anything continuous are called ‘infinite’: they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.

Aristotle. Physics. Available online at:

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