Commentary on Zeno’s Paradoxes
Zeno sets out a group of paradoxes devised to support Parmenides’ theory that ‘all is one’ unchangeable reality. If reality is really one rather than many, then the belief that things are separate and distinct is merely an illusion, likewise, change and motion are also merely part of that illusion. Zeno of Elea’s original work is no longer in existence so to review his paradoxes we have to turn to sources that reference it, namely, Plato’s Parmenides, Aristotle’s Physics and Simplicius’ Commentary on Aristotle’s Physics. The idea is that Zeno is replying to those that argue against Permenides’ theory of the One, such that Zeno presents his paradoxes to demonstrate to believe reality is a plurality is even more absurd and leads to paradox. The paradoxes still have value today because it concerns the infinite divisibility of time and space, and therefore, presents to problem of being able to traverse an infinite. Furthermore we end up with these paradoxes of the infinite because we live with finites and infinites need to be handled with their own mathematics that seems contrary to our experiences. In addition, continuums such as time composed of instants and space composed of points lead to some peculiar results. Ontologically what we believe an instant and a point of space to be changes how we view such paradoxes and their solutions.
Paradoxes of Motion
Achilles and the tortoise
Achilles, who runs fast, is in a race with a tortoise, who moves very slow. Achilles allows the tortoise a head start of a hundred meters. After a finite amount of time has passed, Achilles will have reached where the tortoise began. During which the tortoise has run, say twenty meters, so Achilles will have to run for another shorter period of time to run those twenty meters, by which time the tortoise will have run another portion. By the time Achilles runs that portion, the tortoise will have advanced some more. Whenever Achilles reaches were the tortoise was there is always more to go. For “it is necessary that what is to overtake [something], before overtaking it, first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which is pursing advanced, by virtue of the fact that it is slower.” (Simplicius, p. 34, Zeno to Einstein). Since there is an infinite amount of points Achilles much reach, he will never overtake the tortoise. This paradox is a result of the infinite divisibility of space and so, is similar in kind, to the next paradox.
The dichotomy paradox
Suppose Homer wants to cross the room. Before he can get to the other side of the room, he must cross half the room. Before he can get half way there, he must go a quarter of the way. Before he can go a quarter he must go one eighth, one sixteenth and so forth. This requires him to complete an infinite number of tasks, which is impossible. If there is motion “it is necessary that a moving thing will entirely traverse an infinite number of things in a finite time; but this is impossible, consequently, there is no motion.” (Simplicius, p, 32. Zeno to Einstein). Secondly, there is no first step to take, because that would mean traversing a finite space, which then can be halved and so forth. Homer can neither arrive at the other side of the room nor begin.
Aristotle made the distinction between potential and actual infinites. There are two ways in which length and time are considered infinite, “in respect to their 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 finite but an infinite time, and the contact with infinities is made by means of moments not finite but infinite in number.” (Aristotle, 32-33, Zeno to Einstein). He further states there is not an infinite number of things in an interval actually, but just potentially. For “neither will the moving thing traverse the interval by dividing it into infinite halves, but rather as a single and continuous thing.” (Aristotle, p. 33 Zeno to Einstein). It will traverse a potentially infinite number of parts of the length in a potentially infinite number of instants of time. So Hector traversing an infinitely divisible space in an infinitely divisible amount of time does not quite seem as odd.
What appears to be odd is the nature of the continuum such that it is infinitely divisible. So that even though we have a first term to the series, where Hector begins, and a last term of the series, where he finally ends up, the span as a whole is finite, but infinitely dense. Aristotle says there is a difference traversing the infinite parts of a finite distance. Indeed, for every length covered there is a corresponding time period to do it and so there is enough time. While we would all agree it is impossible to travel an actually infinite distance in a finite time.
Therefore the problem lies in infinitely divisible things for both the dichotomy paradox as well as Achilles and the tortoise. How is it we can have infinite terms add up to a finite length? We have this principle that any sum is greater than its parts. That is using arithmetic suitable to finite series. For example if I began to count 1,2,3... No matter when I stop counting, I end up with a finite series, and never reach infinity. Which means, we would believe, the sum of any infinite collection is itself equal to infinity. Yet clearly that is not true since we have a finite length, even if it is infinitely divisible. In actuality, the sum of an infinite series is the limit, if there is one. If “an infinite sequence of increasing positive numbers eventually gets arbitrarily close to some number, L, without ever exceeding it, then L is the limit of that sequence.” (p. 42, Zeno to Einstein). In these two cases there is a limit. If the room is a 100m then the series would be ½ * 100m+ ¼ * 100m + 1/8 * 100m + … = 1 * 100m. Therefore we have an infinite series of terms that has a finite sum.
The arrow paradox
Zeno states for motion to happen an object must change position from the one it occupies to the next. He uses the example of an arrow in flight, where in any one moment of time, for the arrow to move it must either move to where it is, or it must move to where it is not. It cannot move to where it is not, because it is a single instant, and it cannot move to where it is because it is already there. In any instant there is no motion occurring, because an instant is a slice of the timeline. Thus, if it cannot move in a single instant then it cannot move in any instant, therefore, motion is impossible.
Aristotle stated when Zeno says “that if everything when it occupies an equal space is at rest, and if that which is 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.” (Aristotle, p. 35, Zeno to Einstein).
Instants are like points, with no parts. No motion can occur in an instant then. If the arrow moves during any instant from point A to B through n, then an instant has parts: the part before n is reached and the part after. So we then end up with no motion during instants, all of time is composed of instants, then how can there be any motion? One solution states we can deny “that there is motion during any instant; instead motion between points p and q occurs if at every instant during the journey the arrow is at the appropriate place along the trajectory.” (p. 50, Zeno and Einstein). We might feel that motion occurs “if from one instant to the next the arrow flows from one place to the next, but this would be misleading, because space and time are dense and so there is no ‘next’ point. Instead, objects move simply by being at a continuous series of locations over a continuous interval of instants.” (p. 50, Zeno to Einstein). So we can consistently say at every t (time) the arrow is at some l (location).
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 traveling 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 on 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 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.” (Aristotle Physics, 239b33)
Paradox of Plurality
“if it should be added to something else that exists, it would not make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing. (Simplicius On Aristotle's Physics,139.9)
“But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited.” (Simplicius On Aristotle's Physics, 141.2)
“… whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible will have resulted if it has actually been divided … though perhaps nobody in fact could so divide it. What then will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesi the body was divisible through and through. But if it be admitted that neither a body nor a magnitude will remain … the body will either consist of points (and its constituents will be without magnitude) or it will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus presumably the whole body will be nothing but an appearance. But if it consists of points, it will not possess any magnitude.” (Aristotle On Generation and Corruption, 316a19)
This is taken from two passages in Simplicius taken to show “a line is composed of smallest elements-points-” (p.44, Zeno to Einstein) and part of the problem is understanding how anything without magnitude can exist. Much like the problem of instants. For “either the points have a finite size, in which case an infinity of them has infinite length, and the interval of all of them is infinitely long; or, the points have zero length, so the sum of their lengths is zero, and the interval has zero length.” (p 44, Zeno to Einstein). As Simplicius said “so small as to not have magnitude, so large as to be unlimited.” And so within each finite segment we have a length of zero or infinite. In this case if we were to add one point to another and so forth ad infinitum we would not approach a limit as we did with infinitely divisible magnitudes. Now with the correspondence theory we can compare the size of denumerable infinites, that is infinite series we can put into a one-to-one correspondence with the series of natural numbers. Unfortunately there are more points in a line segment than there are natural numbers, which means we have an uncountable infinite. As such we are not able to even say what length such points would add up to. So if points are dimensionless and have no length are we forced to conclude a segment composed of them would have zero length? It certainly seems if we add zero to zero and so on, we will never get anything else other than zero. However, according to Cantor segments do not get their length from points of which they are composed. There length is a independent metrical property. So we end up with a principle that the length of a segment is equal to the sum of the length of its parts, so long as those parts are also segments. And further all segments contain the same number of points.
The Argument from Denseness
“If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are, are unlimited.” (Simplicius On Aristotle's Physics, 140.29)
Here we have another argument to show the world is a monism and plurality causes problems. If we assume there are many things it would mean they are both limited and unlimited. He states a collection must contain within it a definite number of things. If you have a definite number of things, you must have a finite and limited number of them. He assumes to have an infinite amount of things would be indefinite. Imagine we have a collection of balls lined up. Between any two of the balls is a third, or so Zeno claims, and in-between these three, another two and so on. Therefore the collection is also unlimited, which is a contradiction. Now with points on a line or instants in time, we do have just such a density, between any two points is another point since it is densely ordered. And indeed Zeno might just be thinking along these lines. For although we have a row of distinct balls lined up, space being dense means each ball has point parts and in-between them a finite distance that is infinitely dense. In which case we can word this paradox thus: If we suggest the world to be a plurality that contains many things, then we are faced with paradox, for the collection must be both infinite and finite. Finite because it contains a definite number of things. Infinite because they are dense. Yet we know with transfinite numbers and using the correspondence theory that we can know infinites to be definite, and to even compare the sizes of infinites.
The Paradox of Place
“Zeno's difficulty demands an explanation; for if everything that exists has a place, place too will have a place, and so on ad infinitum.” (Aristotle Physics, 209a23)
Which is actually not so much of a problem when we now accept that an actual infinity can exist, which Aristotle did not. For example we allow that there is an infinite amount of space points actually existing.
The Grain of Millet
“… Zeno's reasoning is false when he argues that there is no part of the millet that does not make a sound; for there is no reason why any part should not in any length of time fail to move the air that the whole bushel moves in falling.” (Aristotle Physics, 250a19)
This is basically in defense of the Parmendean one against the reliability of the senses, which suggest a plurality. If we drop a sack filled with millet we will hear the thud of it hitting the floor. This noise is the noise made by every grain in the sack and the result of the noise made by every part of every grain, thus, every part of every grain makes a noise as it hit’s the ground. Yet when we drop one tiny part of a grain there will be no discernible noise. Therefore our sense of hearing is unreliable and should not be trusted. I would say, yes, the senses cannot always be trusted. And definitely some noises cannot be heard by our ears, but then we have ways to measure noises that we cannot hear and ways to measure noice with more accuracy than we can even within our range of hearing.
Aristotle On Generation and Corruption
Huggett, Nick Space From Zeno To Einstein MIT Press, 2000
Simplicius On Aristotle's Physics