Zeno’s Paradoxes refers to a set of ingenious philosophical problems attributed to Zeno of Elea, a pre-Socratic Greek philosopher . These paradoxes were devised in order to defend the teachings of Parmenides of Elea, whom Zeno followed. Parmenides, along with Zeno and Melissus of Samos, are grouped together as members of the Eleatic school of philosophy.

The Eleatics held the view that there is a distinction between appearance and reality, and that what we perceive to be reality is in fact just an appearance. Zeno’s Paradoxes were intended to prove this, and his most famous paradoxes were the four that attacked the concept of motion. These paradoxes aimed to show that although motion seems to be real, it does not in fact exist.

Four of Zeno’s paradoxes against motion, along with several other paradoxes, have been preserved by Aristotle. Nevertheless, many more, which have not survived till this day, were attributed to him. Zeno’s Paradoxes, in particular those against motion, have not only been pondered upon by philosophers, but also by mathematicians, who view them as a mathematical (rather than philosophical) problem.

Plato’s Parmenides on Zeno’s Paradoxes

The primary source of information for Zeno of Elea’s life is Plato’s Parmenides. In this dialogue, Plato states that Parmenides and Zeno visited Athens to attend the festival of the Great Panathenaea. At Athens, Zeno presented his treatise. As it was the first time the treatise was being presented in the city, many were eager to hear it, including Socrates.

The dialogue between Socrates and Zeno follows after the reading of the treatise. Some scholars point out that the conversation reported in Parmenides certainly did not take place and are doubtful that Socrates even met the two Eleatic philosophers.

Parmenides and Zeno visited Athens to introduce Zeno’s Paradoxes. (Hohum / Public Domain )

Nevertheless, Plato also provides the ages of the three men when the supposed conversation took place, and this information is unlikely to be fictional. According to Plato, at the time of the encounter between the Eleatics and Socrates, Parmenides was about 65 years old, while Zeno was around 40.

On the other hand, Socrates is described as being “very young”. Using this information, along with the traditional date of Socrates’ birth, i.e. 470/69 BC, scholars have estimated that Zeno was born around 490 BC.

Apart from the possible date of Zeno’s birth there is little else that we know about this philosopher’s life. From Plato’s Parmenides, for instance, we learn that there was a rumor going around alleging that Zeno had been Parmenides’ young lover. From other ancient authors, we hear of Zeno’s involvement in a plot to overthrow an Elean tyrant by the name of Nearchus, and that he was caught. Subsequently, the tyrant had Zeno tortured, in the hope that he would reveal the identity of his fellow conspirators. Zeno, however, refused to betray his comrades.

Another story states that Zeno was the adopted son of Parmenides.

The Eleatic School

Zeno belonged to the Eleatic school of philosophy, which was based on the teachings of Parmenides. Eleatic philosophy revolved around a concept known as Monism, the term itself being coined much later on. Parmenides’ version of Monism states that change and plurality, as perceived by the five senses, are merely illusions.

Behind this illusion, however, is the ‘absolute truth’, which is one, static, and unchanging. It goes without saying that the philosophy propounded by the Eleatics was controversial, as it called into question the ‘realness’ of one’s sensory perception and, as may be expected, not everyone was convinced by it.

Zeno’s Paradox attempted to defend Plato’s argument of the ‘realness’ of things such as Plato’s ‘Allegory of the Cave’ that states objects that are seen are not real but literally mimic the real forms. (Gothika / CC BY-SA 4.0 )

Zeno neither proposed novel ideas of his own, nor extended those of Parmenides’ or supported them with improved arguments. Instead, he played a more defensive role, and defended the teachings of the Eleatic school through his Paradoxes. Zeno’s Paradoxes are excellent examples of a philosophical technique known as dialectic.

Aristotle even credited Zeno with the invention of this technique and dubbed him the ‘father of dialectic’. According to Aristotle, dialectic was particularly useful at destroying the ideas of an opponent by getting him/her to state a thesis and then ask questions to draw out its implications.

The goal of those employing this technique is to demonstrate that an opponent’s thesis is not tenable because it leads to either false or unacceptable consequences. Zeno’s Paradoxes specifically employed a form of argument known as reductio ad absurdum or reduction ad impossible , which mean ‘reduction to the absurd’ and ‘reduction to the impossible’ respectively.

Zeno is known to have written a book containing paradoxes to defend Parmenides’ teachings. Apparently, the book (which is now lost) contained as many as 40 paradoxes. Unfortunately, most of these are now forgotten and we know only of nine of them. Additionally, the paradoxes were preserved by commentators, who did not quote them word for word, but in paraphrase.

The most famous of Zeno’s Paradoxes is the one known as ‘The Achilles’ or ‘Achilles and the Tortoise’, which is one of the four ‘Paradoxes of Motion’, the other three being ‘The Dichotomy’ (known also as ‘The Stadium’), ‘The Flying Arrow’, and ‘The Stadium’ (known also as ‘The Moving Rows’). Three of Zeno’s ‘Paradoxes of Plurality’ – ‘The Argument from Like and Unlike’, ‘The Argument from Large and Small’, and ‘The Argument from Limited and Unlimited’ have also survived. In addition, two more paradoxes, ‘The Place of Place’ and ‘The Millet Seed’ have been preserved and are grouped as ‘Two Remaining Paradoxes’.

Zeno’s Paradox - The Moving Rows. (Calvinius / CC BY-SA 4.0 )

Zeno’s ‘Paradoxes of Motion’

Zeno’s ‘Paradoxes of Motion’ were aimed at demonstrating that motion, as perceived by the senses, is in fact an illusion. ‘The Achilles’ is the second ‘Paradox of Motion’ found presented in Aristotle’s Physics. According to Aristotle, “ This is to the effect that the slowest as it runs will never be caught by the quickest. For the pursuer must first reach the point from which the pursued departed, so that the slower must always be some distance in front.”

A common way of looking at this argument is to imagine a race between the legendary Greek hero, Achilles and a tortoise. In the race, the tortoise is given a head start (for instance, 3.3 feet (1 meter)) and Achilles needs to first clear this distance before he is able to catch up with the tortoise. By the time Achilles clears the first meter, however, the tortoise would have moved on to a second point and Achilles would need to reach this point in order to catch up with the tortoise.

Zeno’s Paradox doesn’t account for the movement of a tortoise being slower. (Kenogenic / CC BY-SA 4.0 )

By this time, however, the tortoise would have moved forward again and Achilles need to catch up with his opponent once again. This situation is assumed to go on eternally and Achilles is never able to catch up with the tortoise. This is known as an infinite regress argument.

This situation does not seem to make sense, especially if numerical values are thrown in. Assuming that the tortoise has a head start of 0.9 m, Achilles has a constant speed of 1 ms -1, and the tortoise 0.1 ms -1, then Achilles would have travelled a distance of 1 m in a second. By that time, the tortoise would have reached the 1.1 m point, just 0.1 m ahead of Achilles. In the next second, the tortoise would have arrived at the 1.2 m point, whilst Achilles at the 2 m point, thus overtaking his competition.

Zeno, however, saw things a bit differently, as he assumed that the racetrack and the amount of time needed to run, which represent space and time respectively, were infinitely divisible. In other words, the race between Achilles and the tortoise takes place on a racetrack that goes on indefinitely over an infinite period of time.

As the distance of each step taken by the racers decreases, both space and time are divided into smaller and smaller units, which in turn implies that the passage of time is slowing down. As a consequence, Achilles never reaches the moment when he overtakes the tortoise.

Zeno’s Paradox on Achilles and the tortoise. (Calvinius / CC BY-SA 4.0 )

We do know, however, that time does not slowdown in this manner. Furthermore, Zeno’s assumption that space and time are infinitely divisible was not shared by everyone. In any case, the concept of infinity introduced by Zeno in this paradox introduced some interesting mathematics, which only gained attention during the 17th century, when calculus was invented.

Using calculus, mathematicians were able to demonstrate that the distance of the racetrack was finite. Although the number of points taken by Achilles and the tortoise on the racetrack is infinite, the sum of these points is in fact finite. This phenomenon is known as a ‘convergent series’. Interestingly, there is an opposite phenomenon known as a ‘divergent series’, in which the distance between Achilles and the tortoise does not decrease but increases infinitely.

For this phenomenon to occur, the tortoise would need to run twice as fast as Achilles. As opposed to the sum of distance obtained from the calculation of a ‘convergent series’, a ‘divergent series’ would give a sum of infinity, meaning that Achilles would never catch up with the tortoise.

Zeno’s Paradox Against Plurality

As already mentioned, Zeno created several types of paradoxes, and motion was not the only phenomenon he sought to prove as illusory. As a Monist, Zeno believed that all is one and rejected the idea of plurality. One of the paradoxes used by Zeno against plurality is ‘The Argument from Like and Unlike’, which is found in Plato’s Parmenides.

In this paradox, Zeno begins by saying that “If things are many, they must therefore be like and unlike”. Zeno continues the argument with the claim that it is impossible for what is like to be unlike and for what is unlike to be like. Therefore, Zeno concludes, there cannot be many things and plurality was impossible.

This is not a particularly strong argument, however, as it is possible to demonstrate that something can both be like and unlike. If one were to pick two objects, for instance, a and b, one may say that a is not like b, and vice versa. At the same time, a is like a and b like b. Therefore, there is no contradiction in saying that a thing can be like and unlike.

‘The Argument from Like and Unlike’ shows that not all of Zeno’s arguments were necessarily solid. The other two paradoxes, i.e. ‘The Argument form Large and Small’, and ‘The Argument from Limited and Unlimited’ make use of the infinite regress argument that Zeno had employed in ‘The Achilles’.

Two More of Zeno’s Paradoxes

Finally, two more paradoxes by Zeno have survived – ‘The Place of Place’ and ‘The Millet Seed’. The first paradox is roughly as follows: “If place exists, where is it? For everything that exists is in a place. Therefore, if place exists, then place is in a place. This goes on to infinity. Therefore, place does not exist”.

Once again, Zeno uses the infinite regress argument to cast doubt on the reality of what we perceive with our five senses. ‘The Millet Seed’ is preserved in Simplicius’ Commentary on Aristotle’s Physics , and takes on the form of a dialogue between Zeno and Protagoras, a Sophist. The former poses a question about the noise made by the fall of a single millet seed or one ten-thousandth of a seed versus that made of a bushel of millet seeds.

Protagoras answers that while a bushel makes a sound, a single grain does not. Zeno argues that if the fall of a bushel of millet seeds makes a noise, then a single millet seed would make a fraction of that noise, since there is a ratio between the two objects.

Zeno’s Paradox - The Millet Seed. (Fryed-peach / Public Domain )

Zeno points out that since we do not hear the noise of a single millet grain falling, our sense of hearing is in fact reliable. This paradox can be refuted by considering that there is a threshold below which no sound is made and that human sensory perception has its limitations.

To conclude, Zeno’s Paradoxes were indeed tricky problems that puzzled philosophers ever since they were first posed during the 5th century BC, though they were by no means flawless arguments. The legacy of Zeno’s Paradoxes lies in both its form and substance. The former is evident in the extensive use of the dialectic technique by Zeno’s contemporary, Socrates, his student Plato, as well as the Sophists.

In terms of substance, Zeno’s Paradoxes have generated a lot of discussion over the millennia. These paradoxes continue to be relevant in recent times, as they have provoked much discussion and debate about the nature of space, time, and the infinite.

Top image: Zeno of Elea shows Youths the Doors to Truth and False. Source: Singinglemon / Public Domain .

By Wu Mingren