2.5  Zeno’s Paradoxes

Zeno of Elea was a member of the Eleatic School, which had been founded by his master Parmenides. He was born around 490 B.C.E. in Elea, now Velia in southern Italy. Plato described him as “tall and fair to look upon”, and reported that he has been beloved by Parmenides who took him to Athens, when he was around forty years old, where he also met the younger Socrates (469-399 BC). Zeno brought with him to Athens his book that is a treatise containing about forty arguments to defend the monistic philosophy of his master. Unfortunately, this book has been lost, but some of his arguments managed to reach us through later critics and commentators.

Unlike most Greek philosophers who tried to understand the cosmos from the motion of different objects and celestial spheres, Zeno questioned the mere phenomena of motion and doubted that it has any intrinsic reality. He formulated some “thought experiments” which lead to various kinds of infinity paradoxes, whether we adopt the discrete or the continuum views of space and time. Despite long centuries of research, and despite the evident success of modern mathematics and physics, those paradoxes have never been totally refuted.

Zeno’s paradoxes are simply another practical way to express the monistic thesis of Parmenides. This was instantly noted by Socrates and acknowledged by Zeno himself. Plato, on his part, documented this fact in his dialogue “Parmenides” by quoting a conversation between the three:

“I see Parmenides”, said Socrates, “that Zeno’s intention is to associate himself with you by means of his treatise no less intimately than by his personal attachment. In a way, his book states the same position as your own; only by varying the form he tries to delude us into thinking that his thesis is a different one. You assert in your poem that the ‘all’ is ‘one’, and for this you advance admirable proofs. Zeno, for his part, asserts that it is ‘not a plurality’, and he too has many weighty proofs.”

And thus Zeno confirmed this claim:

“Yes, Socrates”, Zeno replied, “but you have not quite seen the real character of my book. ... The book makes no pretense of disguising from the public the fact that it was written with the purpose you describe, as if such deception were something to be proud of.” Baird and Kaufmann (2000)

Zeno’s original book was never found, but in trying to refute them, Aristotle briefly discussed some of his arguments, later to become known as Zeno’s Paradoxes. A more detailed reflection appeared only a thousand years later, in the works of Proclus (412-485 AD) and Simplicius (490-560 AD) who seem to have had access to the original book. Proclus stated in his commentary on Plato’s “Parmenides” that Zeno produced “not less than forty arguments revealing contradictions”.

Unfortunately, only nine or ten out of those forty arguments managed to reach us indirectly, after various paraphrasing and reconstruction, mainly through the interpretations of Aristotle, Plato, Proclus, and Simplicius. In total, less than two hundred words can be attributed to Zeno, in the form of direct quotations. Otherwise we don’t know how Zeno actually stated his own arguments, and the names and phrasing of the paradoxes were essentially created by various commentators and critics, and not by Zeno himself.

Nevertheless, Zeno’s Paradoxes are considered to be the first examples of a method of proof called reductio ad absurdum, a kind of dialectical syllogism, or proof by contradiction. Although Parmenides himself may have actually been the first to use this style of argument, Zeno became the most famous. In this regard, Aristotle called him the inventor of Dialectic, and Bertrand Russell credited him with having laid the foundations of modern Logic.

Zeno’s arguments are directed against both multiplicity and motion; he maintained that any quantity of space (or time) must either be divisible ad infinitum or composed of ultimate indivisible units. If it is composed of indivisible units, then they must have magnitude, and thus we are faced with the contradiction of a magnitude which cannot be divided. If, however, it is divisible ad infinitum, then we are faced with the different contradictions of supposing that an infinite number of parts can be added up to make a merely finite sum.

Aristotle fervently disagreed with Zeno’s arguments, calling them fallacies, and claiming to have disproved them by inventing the concept of “potential infinity” (discussed briefly in section 5.7) and pointing out that, as the distance decreases, the time needed to cover those distances also decreases. Various other possible solutions have also been offered over the centuries, ranging from Kant, Hume and Hegel, to Newton and Leibniz, who invented mathematical calculus as a method of handling infinite sequences (discussed briefly in section 5.8). Nonetheless, Zeno was definitely the first person in history to show that the concept of infinity is deeply problematic.

Zeno’s paradoxes, however, continued to tease and stimulate thinkers, debating over whether they have been really negated. Bertrand Russell has described them as “immeasurably subtle and profound”:

“In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity’s lack of judgment is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance...” Russell (2013).

The Achilles, the Dichotomy, the Arrow, and the Stadium, out of the ten survived paradoxes, try to attack the phenomena of motion, while the rest are mainly directed against plurality. We shall restrict our discussion here on those four paradoxes of motion and discuss two of the most notable counter arguments: Aristotle’s Potential Infinity and the Standard Analytical Solution.

Each of the four paradoxes of motion challenge all claims that there is real motion at all; the first two lead to logical discrepancies if we suppose space and time to be continuous, while the other two do the same if we suppose them to be discrete. In the Achilles and Dichotomy arguments, Zeno is supposing that space and time are infinitely divisible. They can be easily countered by postulating an atomic theory in which matter (or space, and time) is composed of small indivisible elements. However, the Arrow and Stadium paradoxes cause problems only if we consider that space is made up of indivisible elements that may be cut in indivisible durations of time.