HPS 0628  Paradox  
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Zeno's Paradoxes of Motion
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
http://www.pitt.edu/~jdnorton
Zeno of Elea was an earlier contributor to the ancient Greek tradition in philosophy. He worked in the 5th century BCE and his presence and work were recorded in the writings of Plato and Aristotle.
His paradoxes of motion are the oldest and most important of all paradoxes in the totality of writing on paradoxes. They no longer present us with foundational difficulties. However that is so, because their pervasive presence in our tradition has played a role in forcing us to clarify our ideas on infinity and related matters. Untangling the complications Zeno identified continued as late as the 19th century. That is especially so for his paradoxes of measure (which will be discussed in a later chapter). His paradoxes of motion have stimulated variations whose analysis continues in the 20th century and to the present day.
Anyone with a serious interest in paradoxes must make their peace with Zeno!
Zeno was an associate and student of Parmenides and undertook the work of defending his mentor's views. Parmenides argued that things in the world are an unchanging unity, without parts, and that all change is impossible. This may at first seem like an abstruse position that requires great philosophical sophistication to understand. It is not. It an extreme, fantastical form of skepticism.
https://commons.wikimedia.org/wiki/File:Red_Apple.jpg
In ordinary experience, we see change everywhere. Flies buzz about. Apples fall from trees. Leaves slowly turn yellow in the fall and are blown off the trees. And so on in innumerable variation. Parmenides held that all this motion is an illusion. Nothing changes. We just have the mistaken impression that it does. This view is an instance of a perennially reappearing form of skepticism: that reality is other than what our senses overwhelmingly tell us.
The enduring and fatal difficulty of this form of skepticism is that the evidence of our senses is powerful. Perhaps, with tenacity and even some clever sophistry, a skeptic can shake our confidence in the strength of that evidence. However that falls far short of what skeptics need. They must provide positive evidence for their alternative, fantastical conception: that nothing changes; or that we are brains in vats fed spurious perceptions; or whatever other bizarre imaginings they propose.
Zeno's paradoxes of motion are targeted at one particular form of change: locomotionthat a body changes (motion) its position (locus) in space. They seek to show that the idea of ordinary motion is internally contradictory.
It is hard to believe that Parmenides and Zeno really believed that motion is impossible. The evidence of our senses is powerful, unrelenting and, I believe, irrefutable. Someone who genuinely believes that all change in illusion would seem to be massively deluded and in the grip of a mad fantasy.
We can cast a kinder light on Parmenides and Zeno's project if we understand them not to be challenging change, but to be challenging the accounts we give of it. Can we really reason reliably about motion using the concepts we have? We think we can. Zeno says otherwise. Look at them more closely and you will find them to be an internal mess.
If that was Zeno's goal, then his efforts have met
with great success. In one form or another, his paradoxes involve
infinities associated with our normal thinking about motion. His paradoxes
have forced us to think with great rigor
about these infinities and, in immunizing ourselves from his paradoxes, we
have brought greater clarity to these notions.
We do not have Zeno's wording for his paradoxes. Rather, we are in the unfortunate position that our best account of the paradoxes come from a critic of them, Aristotle. Nonetheless, that is what we have and that is what we must work with. Here is the totality of Aristotle's account in his work Physics, Book 6, Part 9:
"Zeno’s
reasoning, however, is fallacious, when he says that if everything when it
occupies an equal space is at rest, and if that which is in locomotion is
always in a now, the ﬂying arrow is therefore motionless. This is false;
for time is not composed of indivisible nows any more than any other
magnitude is composed of indivisibles.
["Dichotomy"]
Zeno’s arguments about motion, which cause so much trouble to those who
try to answer them, are four in number. The ﬁrst asserts
the nonexistence of motion on the ground that that which is in locomotion
must arrive at the halfway stage before it arrives at the goal. This we
have discussed above.
["Achilles"]
The second is the socalled Achilles, and it amounts to
this, that in a race the quickest runner can never overtake the slowest,
since the pursuer must ﬁrst reach the point whence the pursued started, so
that the slower must always hold a lead. This argument is the same in
principle as that which depends on bisection, though it differs from it in
that the spaces with which we have successively to deal are not divided
into halves. The result of the argument is that the slower is not
overtaken; but it proceeds along the same lines as the bisectionargument
(for in both a division of the space in a certain way leads to the result
that the goal is not reached, though the Achilles goes further in that it
afﬁrms that even the runner most famed for his speed must fail in his
pursuit of the slowest), so that the solution too must be the same. And
the claim that that which holds a lead is never overtaken is false: it is
not overtaken while it holds a lead; but it is overtaken nevertheless if
it is granted that it traverses the ﬁnite distance. These then are two of
his arguments.
["Arrow"]
The third is that already given above, to the effect
that the ﬂying arrow is at rest, which result follows from the assumption
that time is composed of moments: if this assumption is not granted, the
conclusion will not follow.
["Stadium"]
The fourth argument is that concerning equal bodies
which move alongside equal bodies in the stadium from opposite
directions—the ones from the end of the stadium, the others from the
middle—at equal speeds, in which he thinks it follows that half the time
is equal to its double. The fallacy consists in requiring that a body
travelling at an equal speed travels for an equal time past a moving body
and a body of the same size at rest. That is false. E.g. let the
stationary equal bodies be AA; let BB be those starting from the middle of
the A’s (equal in number and in magnitude to them); and let CC be those
starting from the end (equal in number and magnitude to them, and equal in
speed to the B’s). Now it follows that the ﬁrst B and the ﬁrst C are at
the end at the same time, as they are moving past one another. And it
follows that the C has passed all the A’s and the B half; so that the time
is half, for each of the two is alongside each for an equal time. And at
the same time it follows that the ﬁrst B has passed all the C’s. For at
the same time the ﬁrst B and the ﬁrst C will be at opposite ends, being an
equal time alongside each of the B’s as alongside each of the A’s, as he
says, because both are an equal time alongside the A’s. That is the
argument, and it rests on the stated falsity.
[Here Aristotle turns to refute Zeno]
Nor in reference to contradictory change shall we ﬁnd anything
impossible e.g. if it is argued that if a thing is changing from
notwhite to white, and is in neither condition, then it will be neither
white nor notwhite; for the fact that it is not wholly in either
condition will not preclude us from calling it white or not white. We
call a thing white or notwhite not because it is wholly either one or
the other, but because most of its parts or the most essential parts of
it are so: not being in a certain condition is different from not being
wholly in that condition. So, too, in the case of being and notbeing
and all other conditions which stand in a contradictory relation: while
the changing thing must of necessity be in one of the two opposites, it
is never wholly in either.
Again, in the case of circles and spheres and everything that moves
within its own dimensions, it is argued that they will be at rest, on
the ground that such things, themselves and their parts, will occupy the
same position for a period of time, and that therefore they will be at
once at rest and in motion. For, ﬁrst, the parts do not occupy the same
place for any period of time; and secondly, the whole also is always
changing to a different position; for the circumference from A is not
the same as that from B or C or any other point except accidentally, as
a musical man is the same as a man. Thus one is always changing into
another, and the thing will never be at rest. And it is the same with
the sphere and everything else which moves within its own dimensions."
"Zeno’s arguments about motion, which cause so much trouble to those who try to answer them, are four in number. The ﬁrst asserts the nonexistence of motion on the ground that that which is in locomotion must arrive at the halfway stage before it arrives at the goal."
Aristotle's account appears innocuous. A runner must first go halfway on the course before arriving at the goal. So far, the account does not generate any discomfort. That changes when we iterate the remark. After the halfway point, the runner must go halfway again to the three quarter point; and so on indefinitely. Each new intermediate point is identified by halving the remaining distance. It is this halving that leads to the name "dichotomy," which derives from the Greek meaning of dividing in two.
Label the start and end points of the course, "0"
and "1." To arrive at the goal "1," the runner must
arrive at an infinity of halfway points and traverse an infinity
of nonzero distances:
• traverse 0 to 1/2, arrive at the 1/2 point;
• traverse 1/2 to 3/4, arrive at the 3/4 point;
• traverse 3/4 to 7/8, arrive at the 7/8 point;
• traverse 7/8 to 15/16, arrive at the 15/16 point;
and so on for infinitely many stages.
There is something discomforting about the infinity involved here and that is supposed to precipitate a paradox. Is it not of the essence of an infinite task that it cannot be completed?
Aristotle presented several responses to Zeno's dichotomy. The most influential response depended on Aristotle's distinction of two types of infinities. A potential infinity is associated with repeated actions, unlimited in number, that are never completed. Were someone to mark the points on the course that the runner must pass, they could mark any finite number of them. But they could never compete the marking of the points. No matter now many they mark, there are always more to mark. Were the impossible to be done, however, all these points would be marked. What would result in their totality is an impossible, actual infinity.
To respond to Zeno's dichotomy, Aristotle presents the course to be run as a single continuous distance. Such a distance can be traversed without problems. The problem arises, Aristotle asserts, because of an artificial construct introduced by Zeno. That is, Zeno conceives the course as consisting of an infinity of distinct, smaller distances to be traversed. This amounts to treating the continuous course as if it were composed of an impossible, actual infinity of distinct distances to be traversed. At best, there is only the potential to mark off an unlimited number of points and distances, that is, a potential infinity. Aristotle denies that there can be the actual infinity that Zeno has proposed.
Here's a way to grasp the point, using an analogy to eating a cookie. This is certainly not Aristotle's example! It is my attempt to capture Aristotle's thinking. If we are offered a cookie and we are very hungry, we might well be able to swallow it in one big hungry chomp.
What if we are presented with a cookie that had been broken in half; and one half has been broken into half again; and one of those halves in half again; and so on indefinitely, until we are offered an infinite collection of broken cookie bits:
The activity of eating cookie is now transformed. We must somehow manage to consume an infinity of cookie bits; and that might well be a problem. Fortunately for cookie eaters, this is not the task routinely presented to them. For the very idea of breaking up a cookie into an actual infinity of bits is a fictional artifice.
Aristotle's suggestion is that Zeno's construction has changed the problem in the same way. What was a simple task has been rendered problematic by Zeno's introduction of an impossible, actual infinity.
Aristotle's solution was influential. However it is not serviceable in the context of modern theories of distances and times. For these magnitudes are represented in modern theories by manifolds of points. They are an infinity pointsmoments in time or positions in spacealong with information about which are near which. These are actual infinities of the type that Aristotle denied in his response to Zeno.
Here is just a portion of Aristotle's rather lengthy, original presentation of his response to Zeno's dichotomy from his Physics, Book 8:
"In the act of dividing the continuous distance into two halves one point is treated as two, since we make it a beginning and an end; and this same result is produced by the act of counting halves as well as by the act of dividing into halves. But if divisions are made in this way, neither the distance nor the motion will be continuous; for motion if it is to be continuous must relate to what is continuous; and though what is continuous contains an inﬁnite number of halves, they are not actual but potential halves. If he makes the halves actual, he will get not a continuous but an intermittent motion. In the case of counting the halves, it is clear that this result follows; for then one point must be reckoned as two: it will be the end of the one half and the beginning of the other, if he counts not the one continuous whole but the two halves. Therefore to the question whether it is possible to pass through an inﬁnite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not. If the units are actual, it is not possible; if they are potential, it is possible. For in the course of a continuous motion the traveller has traversed an inﬁnite number of units in an accidental sense but not in an unqualiﬁed sense; for though it is an accidental characteristic of the distance to be an inﬁnite number of halfdistances, it is different in essence and being."
In the more recent literature, resolutions depend on which modern version of the paradox is addressed and how it produces a contradiction. The first version looks at the distances that the runner must cover. There is an infinity of them and they are all nonzero:
First, 1/2
to the halfway point,
then 1/4 to the
3/4 point,
then 1/8 to the
7/8 point,
and so on infinitely.
The concern is that a sum of infinitely many nonzero distances is itself infinite.
Non  zero distance 
+  Non  zero distance 
+  Non  zero distance 
+  Non  zero distance 
+  Non  zero distance 
+  ... (infinitely) ... 
= 
∞ 
We can represent this as an argument:
1. To complete the course, the runner must traverse in sequence an infinity of nonzero distances of length 1/2, 1/4, 1/8, ... (premise)
2. The sum of an infinity of nonzero distances is itself infinite. (premise)
3. No runner, running at a finitely bounded speed, can traverse an infinite distance (premise)
4. Therefore, the runner cannot complete the
course (in contradiction with our common experience). (From 1, 2, 3)
Why "finitely bounded"? This is to rule out the case of the accelerating spaceship of the Budget of Paradoxes. This spaceship accelerates so that that it covers an infinite distance in a finite time. However at no moment in its motion is its speed infinite. Rather, it is unbounded. That is, eventually it exceeds any finite speed we nominate.
The flaw in the argument is now readily recognized. Premise 2 is not true of all infinities and is false in the particular case of premise 1.
Some infinite sums do add to infinity:
1 + 1 + 1 + 1 + 1 + 1 + ... = ∞
However the infinite sum of the dichotomy case does not:
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
That this is the case scarcely needs any argumentation now. We have all learned in grade school how to sum an infinity of nonzero magnitudes and get a finite result:
1/3 = 0.333333... = 0.3 + 0.03 + 0.003 + 0.0003 + ...
For completeness, we can rehearse a standard argument for the finiteness of the sum. Start with the partial sums:
1/2 = 1  1/2
1/2 + 1/4 = 3/4 = 1  1/4
1/2 + 1/4 + 1/8 = 7/8 = 1  1/8
1/2 + 1/4 + 1/8 + 1/16 = 15/16 = 1  1/16
etc.
The quantities of greatest interest are in boldface. They tell us how close each partial sum comes to 1.
1/2 
1/2 

1/2
+ 1/4 
1/4 

1/2 +
1/4 + 1/8 
1/8 

etc 

1/2
+ 1/4 + 1/8 + 1/16 + ... 









We can see from these partial sums that:
(i) The partial sums grow, with each greater than the one before.
(ii) The partial sums never exceed 1.
(iii) The partial sums come arbitrarily close to 1.
We would like to say that (i) + (ii) + (iii) are sufficient to say that the infinite sum adds up to 1. However there is a subtle complication. In ordinary arithmetic, we have defined how to sum finitely many numbers. This ordinary notion suffices for all the partial sums. It makes sense for 1/2 + 1/4; and 1/2 + 1/4 + 1/8; and 1/2 + 1/4 + 1/8 + ... + 1/1024.
What about the infinite sum? What does it mean to sum an infinity of terms?
It is now recognized that satisfaction of (i) + (ii) + (iii) comprises a definition of the infinite sum (or at least is sufficient for the definition). It is, of course, a very natural definition and one for which we cannot find obvious alternatives. However as a matter of logic, it is an additional supposition of more modern mathematics that this is what we mean when we talk of infinite sums.
This last analysis of Zeno's dichotomy is popular in the literature. However it appears to me to miss the point of Zeno's challenge. Aristotle's cryptic report of Zeno's paradox makes no mention of summing lengths. It merely reports actions the runner must take to complete the course. An alternative and, in my view, more apt formulation of Zeno's dichotomy focuses just on the idea that the runner must complete an infinity of actions; and that is the source of the difficulty.
To see how this creates problems, consider again the notion of infinity. It is grasped well enough just by thinking of the counting numbers. What would it take to count off all of them? We would start counting:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
How could this counting process ever end? There is no "last number" such that when we have arrived at it, the counting process is over. Perhaps we may imagine that the last number is "infinity," so the task is to count off all of:
1,
2, 3,
4, 5,
6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, ..., ∞
The task would be completed if somehow our counting ended with us declaring
..., 10000000000, ...., ∞, finished!
Of course this can never happen. In this counting operation, we can never get to infinity. Where ever you are in the counting process, here is always one more ordinary number to count.
It is this idea of the impossibility of completing an infinity that seems, to me, to be the deeper problem indicated in Zeno's dichotomy. That the points in the formulation come exactly at successive halfway points is inessential to the problem. What matters is that there is an infinity of points to be passed by the runner in completing the course. A better representation of the dichotomy looks like this:
The paradox, then, is that the runner must complete an infinity of actions; and an infinity is in its nature something that cannot be completed. Hence the runner cannot complete the course. In argument form:
1. To complete the course, the runner must complete an infinity of actions: passing point 1, passing point 2, etc. (premise)
2. An infinity of actions cannot be completed. (premise)
3. Therefore, the runner cannot complete the course, (in contradiction with our common experience). (From 1, 2)
To resolve the paradox, we need to examine what it means to complete some sequence of actions. Tacitly, this formulation of the dichotomy gets us to imagine that the runner has been set the task of completing all the actions 1, 2, 3, ..., all the way up to the last one at at infinitythe "infinityieth."
The key is this idea of the "the last one." The paradox depends on employing a notion of completion that is unproblematic when we are dealing with finitely many actions. That notion is:
Completion = Doing all the actions, including the last one.
With finitely many actions, there is always a last one, so this definition is readily satisfied. However the condition "...including the last one" is superfluous. If you do all the actions, they are completed. A sufficient notion is the reduced version
Completion = Doing all the
actions.
It is the notion applicable to infinities of actions since they have no last action. If all are done, what more can be required for completion?
To be valid, the above formulation of the dichotomy is really using the stricter, inapplicable notion of completion, without making its use clear. It is, more fully:
1. To complete the course, the runner must complete an infinity of actions: passing point 1, passing point 2, ..., including the last action. (premise)
2. An infinity of actions has no last member and thus cannot be completed. (premise)
3. Therefore, the runner cannot complete the course, (in contradiction with our common experience). (From 1, 2)
The difficulty is the unnecessary conception of completion in premise 1 that includes the requirement of carrying out the last action. There is no last action. The paradox is escaped by deleting this unnecessary requirement. Then the troublesome conclusion 3 is no longer entailed.
1. To complete the course, the runner must complete an infinity of actions: passing point 1, passing point 2, ..., including the last action. (premise)
2. An infinity of actions has no last member and thus cannot be completed. (premise)
3. Therefore, the runner cannot complete the course, (in contradiction with our common experience). (From 1, 2) (no longer follows from 1,2)
Can the runner complete the infinite sequence of actions required in the run? We can see that the runner can do this, if we dispense with the idea that the runner has to pass the nonexistent last member of the points identified by Zeno. The easy way to see it is to ask when each action is carried out. If we assume that the runner runs at unit speed, then we can assign a time to each actions:
Point 1 at position 1/2 is passed at time 1/2.
Point 2 at position 3/4 is passed at time 3/4.
Point 3 at position 7/8 is passed at time 7/8.
... and so on for all infinity actions.
While it is not apparently part of Zeno's original formulation, there is a variant of the dichotomy. In it, the source of the contradiction can be seen more clearly to lie with supposed problem of completing an infinity. It is not in supposed difficulties of summing an infinity of nonzero distances.
In this reversed version, we consider the runner about to set off on the race:
To complete the course, the
runner must first run to the 1/2 way point.
To run the 1/2 way point, the runner must first run to the 1/4 way point.
To run the 1/4 way point, the runner must first run to the 1/8 way point.
To run the 1/8 way point, the runner must first run to the 1/16 way point.
...
The difficulty, then, is that the runner cannot even start. To advance any distance at all, the runner must first pass an infinity of points closer to the start. For example, to get to the 1/16th way point, the runner must first get to the infinity of points ..., 1/128, 1/64, 1/32. It is the same for any point in the course, no matter how close to the starting point.
The issue here is not related to summing distances. It concerns completing the actions of passing the infinity of points need to advance to any new point on the track.
The difficulty resides again in an inapplicable notion of completion, but now for an infinite sequence of actions, numbered in the reverse order:
Completion of an infinite sequence of actions ..., 5, 4, 3, 2, 1 = Doing all the actions, including the first one.
The catch is that there is no first action in this numbering scheme. The paradox in argument form is:
1. To complete the course, the runner must
complete an infinity of actions:
..., passing point 4 at the 1/8 way point ,
passing point 3 at the 1/4 way point,
passing point 2 at the 1/2 way point,
passing point 1 at the end,
..., including the first action. (premise)
2. This infinity of actions has no first member and thus cannot be completed. (premise)
3. Therefore, the runner cannot even start the course, since the runner cannot leave the starting position (in contradiction with our common experience). (From 1, 2)
The difficulty once again is the unnecessary inclusion of the condition "including the first action." When applied to a finite set of actions, it is not troublesome. It is troublesome when we have an infinite of actions, numbered in the reverse order. It is sufficient to complete all the actions that we complete each one individually:
Completion of an infinite sequence of actions ..., 5, 4, 3, 2, 1 = Doing all the actions, including the first one.
With this amendment, the first premise no longer includes the troublesome condition and the argument to the contradiction fails.
1. To complete the course, the runner must
complete an infinity of actions:
..., passing point 4 at the 1/8 way point ,
passing point 3 at the 1/4 way point,
passing point 2 at the 1/2 way point,
passing point 1 at the end,
..., including the
first action. (premise)
2. This infinity of actions has no first member and thus cannot be completed. (premise)
3. Therefore, the runner cannot even start the race, since the runner cannot leave the starting position (in contradiction with our common experience). (From 1, 2) (no longer follows from 1, 2)
We can ask again whether the runner can complete the infinite sequence of actions required in the run. As before, we can see it can be done if we dispense with the idea that the runner has to pass the nonexistent, first of the points identified by Zeno. We ask when each action is carried out. If we assume that the runner runs at unit speed, then we can assign a time to each actions:
Point 1 at position 1/2 is passed at time 1/2.
Point 2 at position 1/4 is passed at time 1/4.
Point 3 at position 1/8 is passed at time 1/8.
... and so on for all infinity actions.
"["Achilles"]
The second is the socalled Achilles, and it amounts to
this, that in a race the quickest runner can never overtake the slowest,
since the pursuer must ﬁrst reach the point whence the pursued started, so
that the slower must always hold a lead. This argument is the same in
principle as that which depends on bisection, though it differs from it in
that the spaces with which we have successively to deal are not divided
into halves. The result of the argument is that the slower is not
overtaken; but it proceeds along the same lines as the bisectionargument
(for in both a division of the space in a certain way leads to the result
that the goal is not reached, though the Achilles goes further in that it
afﬁrms that even the runner most famed for his speed must fail in his
pursuit of the slowest), so that the solution too must be the same. And
the claim that that which holds a lead is never overtaken is false: it is
not overtaken while it holds a lead; but it is overtaken nevertheless if
it is granted that it traverses the ﬁnite distance. These then are two of
his arguments."
As Aristotle notes, the Achilles is the same in its principle as the dichotomy. All that has changed is that the distance the faster runner must cover is the distance to a slower moving runner. It is customary to call the faster runner Achilles, after the Greek warrior in the Trojan war; and the slower runner, the tortoise.
To catch the tortoise, Achilles must first run to the starting position of the tortoise; he must then run to the new position to which the tortoise has advance; and then to the next new position to which the tortoise has advanced. To catch the tortoise, Achilles must pass through an infinity of these positions. Following the model of the dichotomy, that requires Achilles to complete an infinity of actions, which is impossible.
Aristotle notes that the formulation does not depend on bisection. If we choose distances and speed appropriately, we can convert the Achilles exactly into the dichotomy. To see how, assume that Achilles and the tortoise are on the same course and that:
The course length is 1.
Achilles starts at point zero and runs at unit speed.
The tortoise starts at point 1/2 and runs at 1/2 unit speed.
Time  Achilles' position  Tortoise position  Distance between them 
0  0  1/2  1/2 
1/2  1/2  3/4  1/4 
3/4  3/4  7/8  1/8 
7/8  7/8  15/16  1/16 
...  ...  ...  ... 
These motions can be illustrated in a figure, which shows Achilles arriving at the points successively:
The resolution is the same as whichever is your favorite resolution of the dichotomy. That is, it possible to sum infinitely many nonzero distances and recover a finite distance; or that an infinity of actions can be completed.
The Achilles does not need to use just this numbering scheme. Another popular scheme uses different numbers but comes to the same result:
The course length is 11 1/9.
Achilles starts at point zero and runs at 10 units of speed.
The tortoise starts at point 10 and runs at 1 unit of speed.
Time  Achilles' position  Tortoise position  Distance between them 
0  0  10  10 
1  10  10 + 1 = 11  1 
1+1/10 = 1.1  10+1= 11  11 + 1/10 = 11.1  0.1 
1.1 + 1/100 = 1.11  11 + 1/10 = 11.1  11.1 + 1/100 = 11.11  0.01 
...  ...  ...  ... 
1.11111... = 1 1/9 
11.11111... = 11 1/9 
11.11111... = 11 1/9 
0 
That is, at times 0, 1, 1.1, 1.11, ... Achilles runs to position 0, 10, 11, 11.1, ... and catches the tortoise at time 1.111... The total distance he runs to catch the tortoise is the finite sum of infinitely many non zero distances:
11.11111... = 10 + 1 + 0.1 + 0.01 + 0.001 + 0.0001 + ... = 11 1/9
"Zeno’s
reasoning, however, is fallacious, when he says that if everything when it
occupies an equal space is at rest, and if that which is in locomotion is
always in a now, the ﬂying arrow is therefore motionless. This is false;
for time is not composed of indivisible nows any more than any other
magnitude is composed of indivisibles.
...
["Arrow"]
The third is that already given above, to the effect
that the ﬂying arrow is at rest, which result follows from the assumption
that time is composed of moments: if this assumption is not granted, the
conclusion will not follow.
..."
The essential idea of the paradox is simple to state. Consider a moving arrow. At some moment of time, it simply is where it is. It is fully in the spatial position. But that means that, at that moment, it is not moving. However this is true for all moments of time. Therefore the arrow is never moving.
The temptation is to imagine that, at some moment of time, a moving arrow is not fully present in one position, but is somehow "moving through" that position. That is, we imagine something like
This is an intuitively appealing notion, but it is not an effective response. For Zeno is right to insist that any moment the arrow simply is, fully and completely, just in the one position it occupies.
Here is the Arrow as a formal argument:
1. At any moment of time, the moving arrow is not moving. (premise)
2. This is true for all moments of time. (premise)
3. Therefore, the arrow never moves (in
contradiction with our experience of flying arrows) (From 1, 2)
While this paradox seems less puzzling at first glance, it can be tricky to state clearly where the problem with it lies. The more mathematics one knows, the harder it can be. Those with some calculus will find it almost irresistible to talk of the time derivative of the position of the arrow. That quantity is the arrows speed and it is defined at a moment of time in the arrow's history. This response is in substance correct, but too fancy. It depends on a deeper point that is simpler to state and gives a clearer diagnosis of the problem with Zeno's argument.
The core difficulty is that premise 1. is internally incoherent:
Movement is not a property of a body at one moment of time. It is a property of the set of the positions of a body over some finite time interval.
To attribute motion or rest to a body just at one moment of time is to attribute a property to it that it cannot have or cannot not have. It is rather like considering a single point in space and asking in what direction it is pointing. The answer is that a point does not carry directional properties. To say that it points north; or does not point north makes no sense.
The use of the arrow as the subject of the paradox misdirects us. For we are commonly used to arrows in the air always being in motion. Replace the arrow by a car on a road and the misdirection is avoided.
If all we know is that the car is at midday at the 100 mile marker, we cannot know from this fact alone whether the car is moving or not.
https://commons.wikimedia.org/wiki/File:Car_pictogram.svg
To the question "is it moving?" all we can say is: "We don't know. Can you tell us more about where the car was at other times?"
That is, to be able to discern whether it is moving
or not, we need to know its positions at other times.
• If, at all times in some neighborhood of midday, the car is still at the
100 mile marker, then it is not moving at midday.
• If, at all times in some neighborhood midday, the car was at
different positions on the road, then it is moving at midday.
Once we discard premise 1, the contradiction no longer follows:
1. At any moment of time, the moving arrow is not moving. (premise) (incoherent)
2. This is true for all moments of time. (premise)
3. Therefore,
the arrow never moves (in contradiction with our experience of flying
arrows) (From 1, 2) (No longer follows from 1,2)
Aristotle's response above comes rather close to this resolution. Aristotle makes a bad start with a claim that is no longer tenable:
"...time is not composed of indivisible nows..."
Present theories of time say otherwise. Time is routinely represented as a manifold of points, such as an interval of real numbers. In its place, Aristotle conceived of time intervals as potentially infinitely divisible. That is, each could potentially be divided into ever smaller intervals, each having some nonzero duration. And each of these could be further divided, indefinitely. The division never terminates in indivisible moments. Aristotle would then be correct to say that the arrow can be said to be moving in each of these intervals. Since each consists of some nonzero, if small duration, the moving arrow is changing its position in that small duration. It is moving.
For those who wonder about the calculus, recall that the time derivative of the position of the arrow is defined in terms of the position of the arrow in relation to neighboring positions. If x(t) is its position at time t, we form the ratio (x(t+δ)(x(t))/(t+δt). We then take the limit as δ goes to zero. That limit, if it exists, is the speed of the arrow at time t. This definitions depends essentially on the positions of the arrow at times neighboring t. Without those positions, the ratio cannot be defined and the limit cannot be taken. In short, this derivative is a property of the positions of the arrow over some nonzero time interval.
"["Stadium"]
The fourth argument is that concerning equal bodies
which move alongside equal bodies in the stadium from opposite
directions—the ones from the end of the stadium, the others from the
middle—at equal speeds, in which he thinks it follows that half
the time is equal to its double. The fallacy consists in
requiring that a body travelling at an equal speed travels for an equal
time past a moving body and a body of the same size at rest. That is
false. E.g. let the stationary equal bodies be AA; let BB be those
starting from the middle of the A’s (equal in number and in magnitude to
them); and let CC be those starting from the end (equal in number and
magnitude to them, and equal in speed to the B’s). Now it follows that the
ﬁrst B and the ﬁrst C are at the end at the same time, as they are moving
past one another. And it follows that the C has passed all the A’s and the
B half; so that the time is half, for each of the two is alongside each
for an equal time. And at the same time it follows that the ﬁrst B has
passed all the C’s. For at the same time the ﬁrst B and the ﬁrst C will be
at opposite ends, being an equal time alongside each of the B’s as
alongside each of the A’s, as he says, because both are an equal time
alongside the A’s. That is the argument, and it rests on the stated
falsity."
The paradoxical result recovered from the motions of these bodies passing each other is given in boldface above. We can identify a (nonzero, finite) time interval that is equal in duration to twice itself.
The stadium has proven to be the most recalcitrant of Zeno's paradoxes of motion. Commentator after commentator have struggled to find some reading of paradox that is worthy of the acumen of Zeno. Time and again, however, they arrive at a reading that requires an elementary confusion on Zeno's part. No doubt part of the problem is that our account of the stadium comes from Aristotle. He was unsympathetic to Zeno's views and believed his arguments failed. That does not make him the best source.
Kevin Davey has finally, after over two millennia, found a reading that I believe does justice to Zeno's paradox. We shall see below that he replaces the discrete row of points of the standard reading by an infinity of points; and that turns the paradox into something worth of Zeno.
We will start with the traditional reading of the paradox. The bodies AA, BB and CC are shown as a discrete row of something. They are sometimes letters or (as we will have below) a row of soldiers. They represent the individual points of space, conceived of consisting of just a finite row.
In the figure A and A' designate the leftmost and rightmost points of the first row. A* designates its midpoint. B and B' designate the the leftmost and rightmost points of the second row. C and C' designate the the leftmost and rightmost points of the third row. Initially the rows are aligned so that A*, B' and C line up, as shown. The second row moves to the right and the third row moves to the left, at the same speed. After a short time, all three rows line up: A,B and C line up; and A', B' and C' line up.
The first main result is an obvious one:
Time for B' to
traverse A*A' on the first row.
= Time for B' to traverse CC' on the third row. = 1
Both motions occur at the same time in the process shown in the figure. Let us set this as unit time, as shown.
To get a sharper grip on the timing, we might count off the number of moments of alignment of soldiers in each row. The figure shows the successive alignments by darkening the aligned soldiers:
In traversing A*A' of the first body, B' aligns twice
with soldiers in A*A'.
In traversing CC' of the third body, B' aligns four
times with soldiers in CC'.
If one takes these alignment as counting off moments of time in the traversal, one might be tempted to say that the traversal of A*A' takes half the time of the traversal of CC'. Too many commentators been unable to resist the temptation of ascribing this assertion to Zeno. It commits an obvious error. The timing between the alignments is different. The time elapsed between the alignments as B' traverses A*A' is twice that of the time elapsed between the alignments as B' traverses CC'. Since we take the full motion to require a unit of time, the time elapsed in the first case is 1/2 and in the second 1/4.
That means that we preserve the time equality above by multiplying these times elapsed by the number of alignments. Call this adjustment "calibration":
(calibration)
(1/2) x (2 alignments in B' traversing
A*A'.)
= (1/4) x (4 alignments in B' traversing CC'.) = 1
So far there is nothing paradoxical and commentators have struggled to see what in this Zeno thought leads to trouble.
This analysis considered the simple case of just four discrete elements forming the the lengths. If these are to play a role of atoms of the lengths, then four is far too few. We would need four thousand, or four million or many, many more. However the analysis above would proceed to similar conclusions, but just with many more soldieratoms in their formulations
All this changes if we make the alteration to this account urged by Davey. That is, we do not represent space by a finite set of discrete points (and so also time by a finite set of discrete moments). The discrete line of soldiers in each body is replaced by an infinity points. They are densely distributed along the length, so that there are no empty spaces between them.
Then the motion looks like the figure below.
Once again we can track the alignments of B' with points in A*A' and in CC'. In the discrete case, there were half as many alignments of B' with points in A*A' as in CC'. That has changed. Precisely because there are no gaps between the points, there are exactly as many alignments in both cases. Every time B' aligns with a new point in CC' it also aligns with a new point in A*A'. Since there are infinitely many points in each of A*A' and CC', there are infinitely many alignments. As B' swept past A*A' and CC', it identified a complete one to one matching of the positions in A*A' and CC'. The figure indicates the matching. It can only show a small number of the infinite of matches that exhaust every point on A*A' and CC'.
It is tempting to take the count of the number of alignments in a traversal as a measure of the time elapsed. Because we have a matched infinity of alignments and of points in the two cases, it seems to work. The two infinities are equal. We conclude that the two traversal times are the same, in accord with our ordinary understanding of the motions. We would write:
Time for B' to
traverse A*A'
= (∞ many points in A*A')
= (∞ many points in CC')
= Time for B' to traverse CC'
There is problem, for there is another way of viewing things so that these equalities do not hold. Not all infinities are alike and we may have been too hasty in setting these two infinities equal. The length A*A' is half the length CC'. That means that there are half as many points in the length A*A" as in the length CC'. We can match the points in A*A' oneone to points in half the length CC'.
That is we have for these infinities
(∞ many points in A*A') = 1/2 (∞ many points in CC')
If we now use the count of the number of points as a measure of the length of time of the process, combining these last results we get Zeno's contradiction:
Time for B' to
traverse A*A'
= (∞ many points in A*A')
= 1/2 (∞ many points
in CC')
= 1/2 Time for B'
to traverse CC'
This contradicts the original equality and completes the paradox. We have recovered two argument. One concludes the equality of the two times. Another concludes that one is half the size of the other. We have, in Aristotle's words, that "... half the time is equal to its double."
We might wonder if somehow the consideration of "calibration" above might help us escape the contradiction. It does not help since the time intervals between successive alignments is zero. We might write something like:
(calibration)
(0) x (∞ alignments in B' traversing
A*A'.) = (0) x (∞ alignments in B' traversing CC'.)
The difficulty is that a 0 x ∞ corresponds to adding up infinitely many zeroes. Thatas Zeno has assured us in the paradox of measure elsewherejust gives back zero. That means that an attempt to use calibration just leads to what is obviously the wrong time for the process
0 = 0
This zero time only applies if the bodies are moving infinitely fast.
oOo
Here I make a break with the analysis of this
chapter and do not offer a resolution of this
version of the paradox. The complication is that this version of the
paradox raises issues that require some more advanced ideas for their
resolution. We will turn to them in
a later chapter. There we consider two ways of assessing the size of
a magnitude of infinitely many points: by cardinality and metrically. We
shall see there that the two can come apart in their judgments. That they
do so in this case is, in effect, what the stadium paradox demonstrates.
Many of you likely have already heard of Zeno's paradoxes. How well do the accounts you may have seen fit with what you have read in this chapter?
Which of the three resolutions of the dichotomy do you prefer? (Aristotle's potential infinity, infinite sums can have finite values, an infinity of actions can be completed)
Some sets of infinite tasks cannot be completed. Consider one. Now consider how you would alter it to make it possible to be completed
For those who know calculus, is the calculus based response to the arrow better than the one in the text above?
The stadium does, in my view, need more
sophisticated ideas for its resolution than those presented so far. Or
does it? Can you resolve the paradox?
June 10, September 6, 9, 2021
Copyright, John D. Norton