There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.

-Jorge Luis Borges, “Avatars of the Tortoise”

This week, we will consider two classical paradoxes of infinity. Today, we turn to Zeno of Elea, a pre-Socratic Greek philosopher. Zeno, who lived during the 5th century BC, is best known today for his four paradoxes of motion, the most famous of which is that of Achilles and the tortoise.

The paradox goes as follows. Achilles and the tortoise are engaged to run a 100 meter race. Achilles runs ten meters per second, the tortoise runs one meter per second,  and Achilles gives the tortoise a head start of ten meters. An easy algebraic calculation reveals that Achilles will catch up with the tortoise after $1 \frac{1}{9}$ seconds, and Achilles will then quickly pull ahead and win the race. However, consider the sequence of events that must occur before Achilles catches up with the tortoise. First, Achilles must reach the tortoise’s starting point, a position on the racetrack which we will denote $x_0$. By the time Achilles gets to $x_0$, the tortoise has moved ahead one meter, to a position we will denote $x_1$. Achilles must then reach $x_1$, but by the time he reaches $x_1$, the tortoise has moved further ahead, to a point $x_2$. By the time Achilles reaches $x_2$, the tortoise has moved ahead to $x_3$, and so on. Achilles must therefore perform infinitely many actions (namely, repeatedly reaching the position where the tortoise previously was) before catching up with the tortoise. He can thus never catch up with the tortoise, let alone win the race.

The paradox of Achilles and the tortoise has inspired a tremendous amount of thought since its introduction. The great Argentinian writer Jorge Luis Borges devoted not one but two essays to the topic: “Avatars of the Tortoise,” which was quoted above, and “The Perpetual Race of Achilles and the Tortoise.” In the former, he follows the notion of infinite regress through the history of ideas. In the latter, he examines in turn several proposed resolutions of the paradox. After finding the refutations of Aristotle, Hobbes, Mill, and Bergson all lacking in various ways, Borges turns to “the only refutation I know, the only inspiration worthy of the original, a virtue indispensable for the aesthetics of intelligence: the one formulated by Bertrand Russell.”

Let us first consider Mill’s argument, which is a refinement of the arguments of Aristotle and Hobbes. Admitting that Achilles must accomplish infinitely many tasks, we can nonetheless observe that the time it takes to accomplish these tasks becomes successively smaller. Indeed, it will take Achilles $1$ second to reach $x_0$, $\frac{1}{10}$ of a second to travel from $x_0$ to $x_1$, $\frac{1}{100}$ of a second to travel from $x_1$ to $x_2$, and so on. The number of seconds needed for Achilles to accomplish all of these tasks is therefore $1 + \frac{1}{10} + \frac{1}{100} + \ldots$, an infinite sum which, as any student of calculus can determine, converges to precisely $1 \frac{1}{9}$. Mill essentially claims that the convergence of this series explains how it is possible for Achilles to complete infinitely many actions and, in particular, overtake the tortoise.

Borges, however, asserts that this supposed resolution of Zeno’s paradox is little more than a reformulation of the problem. One of the issues here seems to be the distinction between the notions of potential and actual infinity. In Mill’s day, and for most of intellectual history, the only mathematically accepted use of infinity was as a potential infinity, an infinity that can be approached but never attained. Indeed, in the classical treatment of infinite sums, even though they could be said to converge to a particular value, and even though one could get arbitrarily close to this value by adding more and more (but still only ever finitely many) values from the sum, they were never assumed to have been fully completed. From this viewpoint, even though Achilles may get arbitrarily close to catching up with the tortoise, he may never succeed in actually doing so.

By the late nineteenth and early twentieth century, though, by the time Russell was working, the use of actual, or completed, infinities was gaining traction in mathematics. Thinking in terms of completed infinities allows one to shift one’s perspective when considering Zeno’s paradox. The infinity of actions that Achilles must perform can be collected in a single set. One can take the infinite sum all at once rather than simply approaching it from below. Rather than considering the race from its beginning, when Achilles has a daunting, seemingly endless string of obstacles between himself and the tortoise, one can consider it from the moment, exactly $1\frac{1}{9}$ seconds in, when Achilles has successfully overcome all of these obstacles. As Russell puts it in “The Problem of Infinity Considered Historically”:

The apparent force of the argument … lies solely in the mistaken supposition that there cannot be anything beyond the whole of an infinite series, which can be seen to be false by observing that $1$ is beyond the whole of the infinite series $\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \ldots$

This is not to say that we have elucidated all the mysteries of Zeno’s paradox. Russell identifies two kinds of difficulties of infinity. The first kind, which he calls “sham difficulties” and immediately dismisses, consists of “those suggested by confusion of the mathematical infinite with what philosophers impertinently call the ‘true’ infinite. Etymologically, ‘infinite’ should mean ‘having no end.’ But in fact some infinite series have ends, some have not; while some collections are infinite without being serial, and can therefore not properly be regarded as either endless or having ends.” These difficulties are easily overcome. But, he concedes, there are “certain genuine difficulties in understanding the infinite, certain habits of mind derived from the consideration of finite numbers, and easily extended to infinite numbers under the mistaken notion that they represent logical necessities.”

As an example of the type of care one must take when defusing Zeno’s paradox, consider the example of Thomson’s lamp. In 1954, philosopher James F. Thomson posed the following puzzle. Suppose there is a lamp with an on/off button. Suppose that the button is pressed, turning the lamp on. Then, one minute later, the button is pressed again, turning the lamp off. A half minute later, the lamp is turned on. A quarter minute later, the lamp is turned off. An eighth of a minute later, the lamp is turned on, etc. Over the course of two minutes, then, the lamp is turned on and off infinitely many times. At the end of these two minutes, is the lamp on or off? The problem here is that either answer seems absurd, as there is no point before the end of two minutes at which the lamp is turned on and then left on or turned off and then left off. And yet the lamp must be either on or off; there are no other options!

There are a number of similarities between Thomson’s lamp and the race of Achilles and the tortoise, but there are also a number of important differences. Not wanting to make an already too long post even longer, I will leave these for another day or for the reader’s personal contemplation, merely stating that it seems to me that a successful resolution of Zeno’s paradox must also be able to deal with Thomson’s lamp and that naive assaults on Zeno often seem to fail this test.

One can also persuasively argue, as some have done, that any mathematical “resolution” largely misses the point. Zeno’s paradoxes are popularly interpreted as arguing for the impossibility of motion, and it is against this claim that the solutions we have examined set themselves.  Many philosophers, though, argue that the correct interpretation of the paradoxes is as an argument against the existence of a plurality, i.e. an argument in favor of the idea that the world consists solely of a universal, unchanging unity. This idea is a cornerstone of the philosophy of the Eleatic School, founded by Parmenides, Zeno’s teacher. Support for this interpretation can be found in Plato’s Parmenides:

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.

Yes, Socrates, Zeno replied, but you have not quite seen the real character of my book. … The book makes no pretence 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.

In a certain light, then, Zeno’s paradoxes are not problems to be “solved” but rather tools to direct our thinking about the Many and the One. I do not pretend to be sufficiently well-versed in the metaphysical issues involved to make any contributions here, so I will leave the matter, bringing it up only to point the reader toward further investigations and to illustrate the richness contained in these simple thought experiments that have intrigued thinkers for millennia.

Before we leave, let us return once more to the race of Achilles and the tortoise. Let us imagine the starting flag being waved and the protagonists leaping into action, Achilles ten meters behind the tortoise and running ten times as fast. But let us also imagine that, as the race progresses, the runners steadily shrink in size, so that, by the time Achilles has run ten meters and the tortoise one, each is one tenth his original size and running at one tenth his original speed, and by the time Achilles has run eleven meters and the tortoise one and one tenth, they have shrunk to one one hundredth of their original size. From our point of view as spectators, we see Achilles slowly but determinedly catching up with the tortoise, until the runners become so small as to become invisible and we lose interest, returning to other more pressing matters of our lives. But from the point of view of Achilles, he is inexplicably making no progress at all, doomed to be eternally one second behind the tortoise despite running ten times faster.

P.S. The title photo was taken at the amazing Live Turtle and Tortoise Museum in Singapore and brings to mind another story involving Bertrand Russell, shelled reptiles, and infinity:

A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever,” said the old lady. “But it’s turtles all the way down!”

-Stephen Hawking, A Brief History of Time

P.P.S. As indicated by the quote introducing this post, Borges found the notion of infinity both irresistible and deeply troublesome. In “Avatars of the Tortoise,” his unease regarding the infinite leads him to endorse Schopenhauer’s doctrine that “the world is a fabrication of the will.” He ends the essay with this remarkable quote:

“The greatest magician (Novalis has memorably written) would be the one who would cast over himself a spell so complete that he would take his own phantasmagorias as autonomous appearances. Would not this be our case?” I conjecture that this is so. We (the undivided divinity operating within us) have dreamt the world. We have dreamt it as firm, mysterious, visible, ubiquitous in space and durable in time; but in its architecture we have allowed tenuous and eternal crevices of unreason which tell us it is false.