ZENO'S PARADOXES: ACHILLES AND THE TORTOISE AND THE DICHOTOMY
Some Ancient Greek philosophers have had great impacts on future thinkers whether they are philosophers, mathematicians, scientists, or just an interested public. One such Ancient Greek philosopher was Zeno of Elea. He has fascinated many people with his paradoxes. In fact, he has proposed many paradoxes but the most popular one is "Achilles and the Tortoise." This paradox has two versions, one of which is referred to as "Achilles" and the other version is called "The Dichotomy." Both versions are similar but involve slightly different approaches.
In order to better understand Zeno's two versions of the paradox it would be helpful to have some familiarity with the three philosophers that have been the motivating forces behind the paradox (when I refer to the paradox, it is for both versions). Specifically, the three philosophers that I allude to are Heraclitus (536-470 B.C.), Parmenides (6th-5th century B.C.) and Zeno (490-430 B.C.). All three philosophers were concerned with the concept of motion.
HERACLITUS, PARMENIDES AND ZENO
Heraclitus, Parmenides and Zeno were all concerned with motion because it involved change. In other words, motion implies change. Heraclitus, for example, insisted that everything changes, that is, everything is in a flux. For Heraclitus there was nothing that was permanent except for the "law of change" which itself remains unchanged.
On the other hand, Parmenides and Zeno (a follower of Parmenides) disagreed with Heraclitus. Parmenides believed in permanence and not change. His reasoning was as follows:
Everything else in the world of flux, Parmenides claimed, cannot belong to the real world of permanent being... The permanent cannot change into anything without ceasing to be permanent. What is cannot change into what is not without passing out of existence... What is permanent must remain forever the same. It is what it is, and to become something other than this would involve the contradiction that it has become what it is not. The changing world is what the real, or permanent, world is not... Thus the changing aspect cannot be part of existence, since it does not belong to the real, unchanging aspect and must therefore be nonexistent... Parmenides concluded... "Being is, nonbeing is not," and only the unchanging belongs to the world of Being.
Supporting Parmenides was his disciple Zeno. Zeno claimed that change was basically impossible and that it involved an illusion. Specifically:
Zeno was not disputing that we experience change in the course of our daily lives — in seeing things grow, move around, and change qualities. Rather, he claimed that any attempt to explain change of motion would lead to contradictions and would thus compel acceptance of the Parmenidean philosophy that only the permanent and unchanging are real. The rest could be dismissed as an unfortunate illusion that we can only ignore.
Consequently, in trying to defend Parmenides' philosophy Zeno proposed certain paradoxes claiming that motion cannot occur. His most famous paradox (Version 1) is "Achilles and the Tortoise" ("Achilles"). Zeno's other paradox (Version 2) which is called "The Dichotomy" is similar to Version 1. Both of these versions will be discussed.
Zeno's Version 1 paradox "Achilles and the Tortoise" goes as follows:
If Achilles can run ten times faster than the tortoise, and the tortoise has a ten-yard lead at the outset, then when Achilles has run ten yards to catch up to the tortoise, the latter has moved ahead one yard. When Achilles runs this yard, the slow tortoise has moved on one tenth of a yard. Each time Achilles reaches the position where the tortoise had been, the tortoise has move on some small distance, so that Achilles will never catch up even though he moves so much faster.
Version 2, "The Dichotomy," of Zeno's paradox is similar to Version 1 with a slightly different approach. "The Dichotomy" states:
... that for an object to move from one place to another, it first must move half of the distance involved. But to move half of the distance, it must move half of the half, and so on infinitely. Also, for it to move to each stage will take some time, no matter how slight. Thus, not only will the object go through an infinite number of distances, it will also require an infinite number of time intervals. Therefore, for an object to go from one place to another, no matter how small the distance, will require forever, and according to the argument, in no finite time will it ever be able to traverse the distance.
Different attempts have been proposed for finding a solution to these paradoxes. These attempted solutions include showing the absurdity of Zeno's conclusions, and using logic, mathematics and physics.
PROPOSED SOLUTIONS TO THE PARADOXES
Achilles and the Tortoise — Version 1
To reiterate, in this particular paradox Zeno states that in a race between Achilles and the tortoise Achilles will never be able to catch the tortoise which has a head start. The reason Achilles will not be able to catch up with the tortoise is because when Achilles reaches the tortoise's first position, the tortoise has moved to a new position ahead of Achilles. Then as Achilles reaches the tortoise's second position the tortoise, once again, has moved to another position ahead of Achilles, and so on infinitely.
One problem with this line of reasoning is that Zeno implicitly assumes that both Achilles and the tortoise are running at the same speed (velocity) with neither one running faster than the other. Obviously, in this particular case Achilles will never catch the tortoise. However, Zeno contradicts himself because he admitted at the start of the paradox that the tortoise is slower which is the reason the tortoise was given a head start. What Zeno has done was to change the condition (assumption) of the paradox implicitly (subtlety). Therefore, this paradox is illogical from the very beginning.
One can also see this paradox from a mathematician's or physicist's point of view by the application of the distance formula. The distance formula is d = vt where "d" is the distance, "v" is the velocity, and "t" is the time. Applying this formula to the paradox and solving for the velocity (v) gives the following: v = d/t. According to the paradox, time is not mentioned — it seems to imply that time does not exist. Therefore, in the formula "t" equals zero which becomes meaningless.
Another viewpoint of how Zeno's time element is ignored in the paradox is presented by Saliu. He makes the following presentation:
The argument is NOT logical. Logic assumes that ALL the elements of a system are present. Leaving out just one element makes the system not only ILLOGICAL, but, worse, ABSURD...
Motion assumes two and exactly two elements: Space and Time. Motion is represented relationally by Speed. Speed equals Space over Time (v = s/t). Zeno eliminates one of the sine-qua-non elements: Time. Zeno equation of Speed (or velocity, hence v) leads to the absurdity: v = s/s...
Zeno's paradox takes into account Space only. Time is completely ignored. Achilles will cover one unit of space in less time than the tortoise. Equivalently, Achilles will cover a longer distance than the tortoise in the same time. By the time Achilles reaches the starting point of the tortoise, the tortoise would have moved a shorter distance... Generalizing, the gap gets shorter and shorter as the time progresses. At some point in time — NOT point in space — the gap reverses. The faster competitor surpasses the slower competitor or who had an early start.
On the other hand, if a time element is taken into consideration, then it can be shown mathematically that Achilles does, in fact, catch the tortoise. This involves a geometric series (infinite series) which converges, and, thereby, a finite number can be calculated. Byrd provides just such a calculation. First, he assumes that the distance to be covered in the race is 100 meters. Second, he assumes that the tortoise's speed is 0.8 meters per second, and that Achilles' speed is 10 times faster. The calculation now proceeds as follows:
Now, after 10 sec., Achilles will have run 80 meters, bringing him to the point where the tortoise started. During this time, the tortoise has moved only 8 meters. It will take Achilles 1 sec. more to run that distance, by which time the tortoise will have crawled 0.8 meters farther. Then it'll take Achilles 0.1 sec. to reach this third point while the tortoise moves ahead by 0.08 meters. And so on and so on... Achilles must reach infinitely many points where the tortoise has already been before he catches up, so... he can never overtake the tortoise!
This is obviously wrong, but why?
The total time it would take Achilles to catch up, in seconds, is 10 + 1 + 0.1 + 0.01 + 0.001 +... Technically, this is an infinite series, and yes infinitely many numbers are being added up. But does that mean the total is infinite? Actually:
10 + 1 + 0.1 + 0.01 + 0.001 +... = 11.111...
That's not infinite! In fact, it's obviously between 11 and 12. But what is it exactly? We know 0.333333... = 1/3; dividing both sides by 3 gives 0.111111... = 1/9. So Achilles needs just 11 and 1/9 sec. to catch up.
The flaw in Zeno's argument is his unstated assumption that the sum of an infinite series... cannot be finite.
One can see from this particular paradox that not only illogical and absurd conclusions were revealed, but in reality it was proven mathematically that the paradox was incorrect. The next paradox (Version 2) is similar to the Version 1 paradox. However, Version 2 approaches the concept of motion (movement) from a slightly different perspective.
The Dichotomy — Version 2
In summary this paradox states that basically movement cannot take place. Specifically, to move from one point to another requires moving one-half steps infinitely.
Here Zeno is making the error of not distinguishing between a discrete function and a continuous function. In other words, he assumes incorrectly that the one-half steps required by Achilles to move from one point to another are separate "stop-and-go" movements, that is, movements that can only occur by involving a "jump" from one point to another (a discrete function). The fact is Achilles's movements follow a continuous function. "It is true that if Achilles is to traverse any distance, he must thereby also traverse half that distance, and so on ad infinitum. And it follows from this that there is, in a sense, an infinite number of distances he must cross. But it does not follow that he must cross them one at a time, traversing each of the segments before he can proceed to the next. This is the crucial flaw".
Another problem in Zeno's Dichotomy paradox is that the potential and the actual are not distinguished. Aristotle points out that there is a difference in essence and being. Specifically, he states in Book VIII.8 of the Physics:
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 half-distances, it is different in essence and being.
Aristotle further points out "... that the unlimited cannot be actually, but merely potentially, present in the infinite multiplicability of the numbers and the infinite divisibility of magnitudes". Continuing along this thought process Aristotle introduces the concept of limits. He points out that lines are not made up of points nor are time intervals comprised of instants. Instead, they are limits. Consequently, time and magnitudes are divided into magnitudes but they are not divisible into the limits of points and time intervals. Therefore, this can be applied to Zeno's Dichotomy paradox. As a result, Aristotle's implied response to this paradox would be that the finite rather than the infinite is the real underlying issue.
Once again we can refer to Aristotle for a philosophical solution to Zeno's Dichotomy paradox. Specifically, the principle of contradiction is appropriate in applying to this paradox. The principle of contradiction states that "Of two propositions, one of which affirms something and the other denies the same thing, one must be true and the other false... The same thing cannot be an attribute and not an attribute of the same subject at the same time and in the same way".
The question, therefore, is: "How does the principle of contradiction specifically apply to the Dichotomy paradox?" The way it applies can be seen by the two propositions involved — one is implicit and the other is explicit. The implicit proposition is that a movement takes place by taking one-half steps. The explicit proposition states that movement is impossible because of the infinite number of one-half steps required. The implicit proposition affirms movement, but the explicit proposition denies movement. Therefore, one must be true and the other false. The implicit proposition shows it to be true by observation (a reality). On the other hand, Zeno could respond by stating that the implicit proposition is false because movement is an illusion. However, the fact is Zeno has not proven that movement is an illusion. Consequently, the implicit proposition must be accepted as true and the explicit proposition must be considered as false.
Another approach to providing a solution to the Dichotomy paradox is a mathematical one. There are two mathematical methods that will help to resolve this paradox issue. The first method uses an infinite series computation and the second method uses a geometric series. The objective is to determine whether these computations converge or diverge. If a series of infinite numbers converge, then there is a limit to this infinite series which has a finite sum. On the other hand, if a series of infinite numbers diverges, then there is no limit to this infinite series, that is, it has no finite sum.
Method 1 — Infinite Series Calculation:
The assumption is that the initial distance to be traveled is from zero (the starting point) to one (the first step). The calculation is as follows:
S = 1/2 + 1/4 + 1/8 + 1/16...
S" is the sum of the infinite one-half steps (1/2, 1/4, 1/8, and so on) required to move a distance of one step. The next equation required to solve this series is:
(1/2)S = 1/4 + 1/8 + 1/16...
This equation takes into consideration the first one-half step necessary to start the movement. Subtracting the second equation from the first gives:
S - (1/2)S = 1/2
The reason why 1/2 is the only remaining fraction on the right-hand side of the equation is because all the other fractions after 1/2 cancel each other due to the subtraction. Solving for "S" results in the following :
(1/2)S = 1/2
S = 1
Therefore, the infinite series calculation provides a finite sum of one. What this means is that the sum of the infinite numbers does have a finite sum.
Method 2 — Geometric Series Calculation:
The assumption is the same as for the Method 1: Infinite Series Calculation. The geometric series calculation uses the following formula:
S = a/(1-r)
Once again, "S" is the sum of the infinite numbers. The letter "a" is equal to 1/2 which is the requirement for the first step and is found as the first number in the equation S = 1/2 + 1/4 + 1/8 + 1/16 +... The letter "r" is also equal to 1/2 which represents the ratio of the infinite 1/2 steps required. Solving for the formula gives:
S = 0.5/(1-0.5)
S = 0.5/0.5
S = 1
This calculation also results in a finite sum of one. Therefore, both methods of calculations prove mathematically that Zeno's paradox conclusion is incorrect. In other words, the infinite series converges to a limit.
Zeno's Paradoxes — Versions 1 and 2
Some of the solutions presented in "The Dichotomy" paradox (Version 2) can also apply to the paradox "Achilles and the Tortoise" (Version 1) and vice versa. This section will focus on the possible fallacies committed by both paradoxes (Versions 1 and 2). These logical fallacies consist of the following categories: ignoratio elenchi (also known as missing the point), inductive fallacy of hasty generalization, false cause, formal fallacy by counterexample and non-sequitur.
The fallacy of ignoratio elenchi results from a conclusion different from what the premises propose. Specifically, in the "Achilles and the Tortoise" paradox a race occurs between Achilles and a tortoise with the tortoise having a head start due to being slower. The conclusion is that Achilles never catches up with the tortoise. However, the conclusion is different from the argument that Achilles is the faster runner.
The inductive fallacy of hasty generalization occurs when a conclusion is made without having sufficient knowledge. In the case of both Zeno's paradoxes Zeno did not have an understanding of how an infinite series can have a finite limit. Also, Zeno failed to recognize between potentiality and actuality. In other words, potentially anything can be possible but actuality has limits.
The false cause fallacy states that cause and effect are not properly distinguished. Applying this fallacy to the "Achilles and the Tortoise paradox" Zeno assumes the effect of Achilles never catching up to the tortoise is caused by the tortoise always being slightly ahead of Achilles. It may be true that Achilles never catches up to the tortoise but it does not necessarily follow that the cause was that the tortoise is always slightly ahead of Achilles, i.e. there could be valid alternative causes supporting the conclusion as, for example, Achilles could have twisted his ankles and, thereby, barely move. Similarly, in "The Dichotomy" paradox the effect of no movement taking place is not necessarily caused by the assumed premise of taking infinite one-half steps. Once again, theoretically there could be alternative causes claimed for no movement taking place assuming Zeno's conclusion is accepted. For example, Achilles could have become incapacitated while in the process of taking these steps. In other words, one could argue that other factors could have also impacted the movements of Achilles. The point is that causal evidence is lacking in showing any correlation between Zeno's premises and his conclusions. Zeno is simply making a one-point claim and ignoring other possible explanations. In other words, there could be implicit extenuating circumstances proposed in these paradoxes which could be just as valid.
Another fallacy concerning both of Zeno's paradoxes involves a formal fallacy. Specifically, a counterexample can be given to invalidate Zeno's conclusions. For example, it can be shown experimentally (in actuality) that a person can catch up with and surpass a tortoise. Also, it can be shown that a person, in fact, can move from one place to another. Zeno, on the other hand, could counter that movement is an illusion. However, the problem is that there is no proof that movement is an illusion but simply a claim made by Zeno.
One of the most obvious rules of logic that Zeno's paradoxes violate relates to a non-sequitur fallacy. In other words, Zeno's premises are affirmative (positive) but his conclusions are negative. Specifically, in "Achilles and the Tortoise" Zeno states that both Achilles and the tortoise are moving forward (positive premises) but Achilles never catches up with the tortoise (a negative conclusion with no negative premises). In "The Dichotomy" Zeno again presents positive premises relating to taking one-half steps but his conclusion is, once again, a negative one with no negative premises presented. Therefore, a non-sequitur occurs in both paradoxes because the negative conclusions do not follow from the positive premises.
One interesting observation concerning both of Zeno's paradoxes is that these paradoxes contradict each other. This can be seen by comparing the two paradoxes. In "Achilles and the Tortoise" Zeno has both Achilles and the tortoise constantly moving forward with Achilles trying to catch up with the tortoise. In "The Dichotomy" Zeno claims that movement is impossible because of the infinite one-half steps that are involved. Therefore, this paradox of no movement possible contradicts the "Achilles and the Tortoise" paradox where both Achilles and the tortoise are continuously moving. In addition, the paradox of "Achilles and the Tortoise" should not exist since Zeno claims that movement is an illusion.
As has been presented in this essay Zeno's paradoxes contain mathematical, logical and philosophical errors. For example, Zeno confuses the infinite with the finite. Also, he does not distinguish between the potential and the actual. Furthermore, Zeno's paradoxes contain several logical fallacies. In addition, his paradoxes of "Achilles and the Tortoise" and "The Dichotomy" contradict each other in relation to Zeno's conclusions of movement.
However, on the other hand, both of Zeno's paradoxes are interesting from the viewpoint that they have fascinated people throughout the ages. His paradoxes have stimulated discussion and debate in finding solutions. In fact, Zeno's paradoxes have resulted in applying sophisticated mathematical techniques, which did not exist during Zeno's time, in solving his paradoxes. Furthermore, Zeno's paradoxes have prompted the consideration of applying the space-time continuum concept into his paradoxes.
1. Popkin, p. 101.
2. ibid., pp. 101-102.
3. ibid., p.102.
5. Saliu, p. 2.
6. Byrd, p. 2.
7. Gottlieb, p. 68.
8. Aristotle, p. 440.
9. Zeller, p. 178.
10. Randall, Jr., p.205.
11. Copleston, S.J., p. 283.
12. Based on IB Maths Resources.
13. Nolt, pp. 196, 204, 209, 210, 212.
Aristotle. Physics, in The Complete Works of Aristotle, the Revised Oxford Translation, Volume One (1991). Jonathan Barnes (ed.). Princeton, New Jersey: Princeton University Press.
Byrd, D. "Zeno's "Achilles and the Tortoise" Paradox and The Infinite Geometric Series." Retrieved November 2014 at http://www.informatics.indiana.edu/donbyrd/Teach/Math/Zeno+Footraces+InfiniteSeries.pdf.
Copleston, S.J., F. A History of Philosophy, Volume I, Image Books Edition (1985). New York: Doubleday, a division of Bantam Doubleday Dell Publishing Group, Inc.
Gottlieb, A. The Dream of Reason: A History of Western Philosophy from the Greeks to the Renaissance (2000). New York: W.W. Norton & Company.
IB Maths Resources. "dox — Achilles and the Tortoise." Retrieved November 2014 at ibmathsresources.com/2014/08/27/zenos-paradox-achilles-and-the-tortoise/.
Nolt, J., Rohatyn, D., and Varzi, A. Schaum's Outline of Theory and Problems of Logic (1998, Second Edition). United States of America: McGraw-Hill.
Platonic Realms. "Zeno's Paradox of the Tortoise and Achilles." Retrieved November 2014 at http://platonicrealms.com/encyclopedia/Zenos-Paradox-of-the-Tortoise-and-Achilles.
Popkin, R.H., and Stroll, A. Philosophy Made Simple (1993, Second Edition, Revised). New York: Doubleday, a division of Bantam Doubleday Dell Publishing Group, Inc.
Randall, Jr., J.H. Aristotle (1960). New York: Columbia University Press.
Runes, D.D. (ed.). Dictionary of Philosophy (1962 Edition, Reprinted 1976). Totowa, New Jersey: Littlefield, Adams & Co.
Saliu, I. "Zeno's Paradox, Aporia: Achilles Can't Outrun the Tortoise?! The First-Ever Metaphysical Solution to Zeno's Paradoxes." Retrieved November 2014 at saliu.com/aporia.html.
Thomas, Jr., G.B., and Finney, R.L. Calculus and Analytic Geometry (1992, 8th Edition). United States of America: Addison-Wesley Publishing Company.
Zeller, E. Outlines of the History of Greek Philosophy (1980, Thirteenth Edition). New York: Dover Publications, Inc.