You can’t get there from here. Because first you have to get halfway there. Then you have to go half of the rest of the way. And once you’re there, you have to go half of the rest of the way. And once you’re there, you have to go half of the rest of the way again – over and over and over again without end. So, you can never get there from here. What’s worse, you can’t even get started on the journey from here to there. Because first you have to get halfway there. But before getting halfway there, you have to get halfway to that halfway point. And before getting there, you have to get halfway to that new halfway point – and you have to do this over and over and over again without end. So, you can never even start getting there from here.
These paradoxical arguments originated with the early Greek philosopher Zeno of Elea. Aristotle in Physics VI mentions “the argument about its being impossible to move because what moves must reach the half-way point earlier than the end” (Physics VI 9.239b11-13), and we know that the argument he is describing is Zeno’s from the reference in Topics VIII to “Zeno’s argument that it is not possible to move or to traverse the stadium” (Topics VIII 8.160b7-9). There Aristotle describes it as an example of an argument opposed to ordinary belief that is difficult to resolve – a “paradox” in the etymological sense of being opposed or contrary to (Gr. para) ordinary or common belief (Gr. doxa). It is not entirely clear from Aristotle’s descriptions whether Zeno originally argued, in what has come to be known as the Stadium Paradox, that is impossible to finish traversing the stadium or to begin doing so. The paradox is also called the Dichotomy because it depends on dividing the distance to be traversed successively in halves without limit.
Zeno’s most famous argument is a variant on the Stadium Paradox known as the Achilles. Aristotle presents this argument in Physics VI immediately after the Stadium Paradox: “The slowest runner never will be overtaken by the fastest; for it is necessary for the one chasing to come first to where the one fleeing started from, so that it is necessary for the slower runner always to be ahead some” (Physics VI 9.239b14-18). Zeno apparently aimed to show, again paradoxically, that if a tortoise starts ahead of Achilles in a race, Achilles will never be able to overtake the tortoise. For once Achilles has gotten to where the tortoise began, the tortoise will have moved some distance ahead. And once Achilles has then gotten to where the tortoise is at that moment, the tortoise will have moved some distance ahead. And so again, and again, and again endlessly forever. Each time Achilles gets to where the tortoise was, the tortoise will have moved some distance ahead. And so, Achilles will never be able to overtake the tortoise.
There is some evidence that the paradox of the Achilles appears to have achieved some popular notoriety in Zeno’s own day. The interior of a red-figure drinking cup discovered in the southern Etrurian city of Falerii depicts a figure in a cap and cape nimbly jumping ahead of a large tortoise who looks up at him in apparent surprise. This cup dates to the mid-fifth century BC and now forms part of the collection the National Etruscan Museum housed in the Villa Giulia in Rome. It would appear to be the earliest known response to Zeno’s Achilles paradox, even though it does no better than visually asserting the natural reaction: “Achilles can obviously overtake a tortoise!” But merely asserting belief in the obvious is not an adequate response to the paradox, for it is designed to suggest that there are serious problems with our conception of the physical underpinnings of ordinary motion.
What makes Zeno’s Stadium Paradox and the Achilles so ingenious and even profound is that, although the way they are stated is really quite simple, efforts to resolve them soon require some heavy conceptual and mathematical machinery. It has always been difficult to identify just where these arguments go wrong. Aristotle believed the problem was that while it is in fact impossible to cross the stadium if doing so requires touching an actually infinite number of points in a finite time, it is possible to do so if the infinite is only potential. “Someone moving continuously,” he says, “has passed through infinitely many points incidentally but not unqualifiedly; for being infinitely many halves is incidental to the line, whereas its essence and being are different” (Physics VIII 8.263b6-9).
In considering the Stadium Paradox, one would be right to point out that the infinite series 1/2 + 1/4 + 1/8 + 1/16 etc. converges on 1, but it is not altogether clear how this is supposed to help, either. Zeno is not suggesting that the stadium to be crossed is infinitely long but that one must complete an infinite number of tasks to cross it and that this is impossible. Thus, the problem seems to be that if the physical distance is infinitely divisible in a manner that mirrors this infinite series, then there will in fact be an infinite number of physical points the runner must reach to cross the stadium.
Renewed engagement with Zeno’s paradoxes led thinkers in the nineteenth and twentieth century to suppose that Zeno’s paradoxes call into question whether the structure of space and time mirrors the structure of the mathematical continuum. Some took the lesson of Zeno’s paradoxes to be that physical distances and durations are not composed of points and instants, while others continued to work on the assumption that space and time are structured like the mathematical continuum. It remains an open question to this day whether the mathematical continuum is in fact mirrored in the structure of space and time.
Zeno himself surely could not have imagined the sophisticated responses his ingenious paradoxes would provoke. The little information we have about his life indicates that he was born around 490 B.C. in the southern Italian town of Elea (modern Velia) and that he was active in the Greek-speaking regions of southern Italy known as Magna Graecia during the mid-fifth century B.C. The ancient biographer Diogenes Laertius relates that Zeno died when captured in the midst of plotting to overthrow a local tyrant named Nearchus. According to one version of the story, he said under interrogation that he would only reveal information if he could whisper it in Nearchus’s ear. As Nearchus leaned in, Zeno bit into his ear and did not let go until he was stabbed to death. Diogenes relates other versions of this story as well, though, and it is impossible to tell how much truth there is in any of them.
Similar uncertainty surrounds the information provided by our other major source for Zeno’s life, Plato’s dialogue Parmenides. Here Plato depicts Zeno as a younger associate of the early Greek philosopher Parmenides, also a native of Elea, and portrays both of them in conversation with a young Socrates while visiting Athens on the occasion of a great festival. The whole scenario and conversation are an elaborate fiction, but it may nonetheless contain some kernels of truth. Plato has Zeno say that he wrote a book containing many arguments when he was young and that it was stolen and published without his consent. Zeno indicates that the arguments in his book were designed as a sort of support for Parmenides against those who supposed that his doctrine that all is one leads to absurdity – he meant to show in return that their own commonsense view that there are many things leads to even more ridiculous results (Plato, Parmenides 128c6-d6).
It is noteworthy that Plato has Zeno characterize his arguments as directed against the commonsense presumption of plurality, whereas the arguments by Zeno that Aristotle features most prominently target the possibility of motion. The Parmenides begins with Zeno having just finished reading from his book, whereupon Socrates asks him to read the first hypothesis of the first argument again. Socrates describes the argument as follows: “If the things that are are many, then they must be both alike and not alike, but this is impossible. For things not alike cannot be alike, nor can like things be not alike. … So then if it is impossible for things that are unlike to be like and like things to be not alike, it is impossible for there to be many things. For if there were many things, they would be subject to what is impossible” (Plato, Parmenides 127e).
The argument, if it actually goes back to Zeno, obviously requires some expansion. It nevertheless suggests that he undertook to show that the presumption of plurality leads to contradiction (in this case, that the many things are both like and not alike, which Socrates says is not actually a problem). This is in fact the strategy we find deployed in the major Zenonian arguments against plurality reported by the Alexandrian Neoplatonist Simplicius (6th c. A.D.) comprising antinomies of the general form: if there are many things, they must be both F and not-F, which is impossible; thus, there are not many things.
One of these arguments against plurality, the antinomy of limited and unlimited, Simplicius quotes verbatim as follows: “If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer. But if they are just as many as they are, they will be limited. If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those. And thus, the things that are are unlimited” (Zeno at Simplicius, Commentary on Aristotle’s Physics 140.29-33 Diels). In short, if there are many things, they must be both finitely many and infinitely many, which is impossible; thus, there are not many things. This argument also requires some unpacking, for Zeno appears to by relying on some unstated assumptions, particularly in the latter arm. Although the argument is reminiscent of the limitless division in the Stadium Paradox and the Achilles, here Zeno actually appears to be relying on a postulate to the effect that any two things will be distinct or separate from one another only if there is some other thing between them. Repeated application of this postulate means that there will be limitlessly many things between any two distinct things.
Zeno’s other major argument against plurality, the antinomy of large and small, is a veritable tour de force of early reasoning. In this argument, Zeno purported to show that if there are many things, each of them must both be so small as to have no magnitude and be so large as to have unlimited magnitude. It is again thanks to Simplicius that we know of this argument. Regarding the first arm of the antinomy, Simplicius says only that Zeno argued that none of the many have magnitude since each is the same as itself and one. Although this is not much to go on, Zeno seems to have argued here that if there are many things, they must each be one thing, meaning that each must be the same as itself. Anything with some size or magnitude, though, can be divided into parts so that it is not the same as itself and so not one. Thus, if there are many things, they must each have no size or magnitude.
Simplicius indicates that Zeno made the transition to the second arm of the antinomy via a lemma to the effect that since whatever has no size or magnitude is nothing, each of the many must have some magnitude. Simplicius quotes a significant portion of Zeno’s own argument that what has no magnitude it nothing: “For if it were added to another entity, it would not make it any larger; for since it is of no magnitude, when it is added, there cannot be any increase in magnitude. And so, what was added would just be nothing. But if when it is taken away the other thing will be no smaller, and again when it is added the other thing will not increase, it is clear that what was added and what was taken away was nothing” (Zeno at Simplicius, Commentary on Aristotle’s Physics 139.7-15 Diels).
Simplicius also provides a relatively full report of Zeno’s argument in the antinomy’s second arm for the conclusion that each of the many must have unlimited magnitude. Here we find Zeno pushing to its extreme the idea that any spatially extended magnitude may be divided into parts that are themselves spatially extended magnitudes. Each of the parts into which a thing is divided, he argued, can itself be divided into parts, each of which may be divided into parts, each of which may be divided into parts, which may be so divided again and again endlessly. Simplicius quotes Zeno’s actual argument as follows: “…but if it is, each must have some magnitude and thickness, and one part of it must extend away from another. And the same account applies to the part out ahead. For that part, too, will have magnitude and will have part of it out ahead. Indeed, it is the same to say this once as always to keep saying it; for no such part of it will be last, nor will one part not be related to another” (Zeno at Simplicius, Commentary on Aristotle’s Physics 140.34-141.6). This portion of the argument is very reminiscent of the Stadium Paradox. How Zeno may have continued from here to the arm’s conclusion Simplicius does not indicate, but the thought is presumably this: if the magnitude of a thing is the sum of the infinitely many smaller things that are its parts, each of which has some magnitude, then the magnitude of that thing must itself be infinitely large.
The power of this argument, as with all Zeno’s arguments, consists not in how it compels us to accept its conclusion (for it does not do this), but rather in how it forces us to think more deeply and precisely about the structure of the physical world. Zeno understood that applying a mathematical conception of space and extension to physical magnitudes and bodies naively conceived makes both their composition and their motion highly problematic. Subsequent thinkers felt compelled to respond to his arguments in one way or another. Rejection of the infinite divisibility of matter was a key element in the formulation of the atomism by the early Greek philosopher Leucippus. According to this theory, all bodies are ultimately composed of atoms and void, with atoms being physically indivisible bits of microscopic stuff. (The word “atom” comes from the Greek atomos, which literally means “uncuttable.”) Leucippus thus sought to avoid the unacceptable consequence of Zeno’s argument that each thing must have unlimited magnitude by proposing that the division of bodies into ever smaller parts cannot continue limitlessly but has to stop somewhere.
This was not the only way later philosophers responded to Zeno’s argument. Aristotle himself was a continuum theorist who embraced the infinite divisibility of bodies (as well as space and time). As we have seen, though, he sought to avoid Zenonian paradox by positing that they are not actually but only potentially infinitely divisible – meaning that they can be divided anywhere but not everywhere at once. Before Aristotle, the early Greek philosopher Anaxagoras also developed a physical theory in response to Zeno that accepted the infinite divisibility of matter. Anaxagoras postulated that there was no least magnitude or no lower limit on the size of portions of matter. On the basis of this postulate, he further postulated that every portion of matter contains portions of every kind of stuff and that the characteristics of a thing are a function of the preponderance within it of stuffs with the relevant characters. In making the no least magnitude principle a foundation of his physical theory, Anaxagoras appears to have endorsed all the basic premises of Zeno’s argument that each of the many must be limitlessly large while avoiding the conclusion by recognizing that the sum of infinitely many parts of some magnitude need not in fact be infinite, provided that the parts in question are diminishingly small.
Plato in the Phaedrus describes Zeno as “the Eleatic Palamedes” – Palamedes being a proverbially clever and inventive hero of the Homeric age – because of his ingenious ability to make the same things appear both like and unlike, both one and many, and both moving and at rest (Plato, Phaedrus 261d6-8). Zeno secured his place in the history of philosophy by developing a set of arguments more sophisticated than virtually any previously known. As we have seen, his arguments were designed to show that normal assumptions regarding the phenomena of motion and plurality – the reality of which people naturally take for granted –actually involve numerous and apparent contradictions. Even today grappling with the best of his arguments can still lead one to get clearer about concepts that are fundamental to an adequate understanding of the physical world.