# Hippasus and the Infinite Descent

Today, dear Reader, we bring you a story. A story of Mathematics and Music, of Reason and Passion, of Drama and Irony. It is the story of Hippasus of Metapontum, of his remarkable life and his equally remarkable death. Before we begin, a note of warning. Not everything presented here is true, but all of it is meaningful.

A Greek philosopher and mathematician from the 5th century BCE, Hippasus was a follower of Pythagoras. The Pythagoreans believed in the transmigration of souls, subscribed to the belief that All is Number (where “Number” is, of course, “Whole Number”), made great strides in the study of musical harmony, and eschewed the eating of beans. Our hero was a particularly illustrious Pythagorean. He performed experiments linking the sizes of metal discs to the tones they emit upon being struck, developed a theory of the musical scale and a theory of proportions, and showed how to inscribe a regular dodecahedron in a sphere. The regular dodecahedron is a twelve-sided solid whose faces are regular pentagons, shapes which were dear to the Pythagoreans and central to our story. The pentagram, the five-sided star formed by extending the sides of a regular pentagon, and whose tips themselves form a regular pentagon, was a religious symbol of the Pythagoreans and a mark of recognition amongst themselves.

A corollary to the Pythagorean doctrine that All is Number was a belief, at the time, that any two lengths are commensurable, i.e., that given any two lengths, they are both whole number multiples of some fixed smaller length. In modern language, this amounts to the assertion that, given any two lengths, their ratio is a rational number, i.e., can be expressed as the ratio of whole numbers. It is only fitting that the first evidence to the contrary would come from the pentagram.

It happened when Hippasus was stargazing. He saw five stars forming a perfect regular pentagon, and inside this regular pentagon he formed a pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram. An infinite web of similar triangles was woven through his mind and, in a flash of insight, he realized something terrible: the lengths of one side of the regular pentagon and one side of the pentagram found inside it are incommensurable.

The next day, he told his fellow Pythagoreans of his discovery, and they were horrified. They could not have this knowledge, which struck at the core of their belief system, getting out into the wider world. So they took Hippasus far out to sea, and they threw him overboard.

Infinite descent is a proof technique that morally dates back to the ancient Greeks but really came into its own in the work of Pierre de Fermat in the 17th century. The idea behind it is simple and immediately appealing. Suppose we want to prove that there are no positive integers satisfying a particular property. One way to prove this would be to show that, given any positive integer $n$ satisfying this property, we could always find a smaller positive integer $n'$ satisfying the same property. Repeating the argument with $n'$ in place of $n$, we could find a still smaller positive integer $n''$ satisfying the same property. Continuing, we would construct a decreasing sequence of positive integers, $n > n' > n'' > n''' > \ldots$, the aforementioned “infinite descent”. But of course there can be no infinite decreasing sequences of positive integers (if a decreasing sequence starts with $n$, it can of course have at most $n$ elements). One thus reaches a contradiction and concludes that there are no positive integers with the given property.

Fermat made great use of the method of infinite descent in his work on number theory. One particularly striking application came in the proof of a special case of his famous Last Theorem: there are no positive integers $a,b,c$ such that $a^4 + b^4 = c^4$. Fermat showed that, if $a_0,b_0,c_0$ are positive integers such that $a_0^4 + b_0^4 = c_0^4$, then we can construct positive integers $a_1, b_1, c_1$ such that $a_1^4 + b_1^4 = c_1^4$ and $c_1 < c_0$. One can then continue, with $a_1, b_1, c_1$ in place of $a_0, b_0, c_0$, and obtain $a_2, b_2, c_2$ with $a_2^4 + b_2^4 = c_2^4$ and $c_2 < c_1$. This of course leads to the infinite descent $c_0 > c_1 > c_2 > \ldots$ and a contradiction.

We will use the method of infinite descent to prove the result of Hippasus mentioned above. This will not be exactly the way the ancient Greeks would have presented the proof, but it is very similar in spirit, and we make no apologies for the anachronism. Let’s get started.

Suppose, for the sake of an eventual contradiction, that we have a regular pentagon whose side length is commensurable with the side length of the inscribed pentagram. Let $s$ denote the side length of the pentagon and $t$ denote the side length of the pentagram (i.e., the diagonal of the pentagon). In modern language, our assumption is that $\frac{t}{s}$ is rational, i.e., that there are positive integers $p$ and $q$ such that $\frac{t}{s} = \frac{p}{q}$. By scaling the pentagon, we can in fact assume that $t = p$ and $s = q$. So let this be our starting assumption: there is a regular pentagon whose side length $s$ and diagonal length $t$ are both positive integers.

Let us label the vertices of the pentagon by the letters A,B,C,D,E. The center of the pentagram forms another regular pentagon, whose vertices we shall call a,b,c,d,e. This is shown in the diagram below. Note that $s$ is the length of the line segment connecting A and B (we will denote this length by |AB|), and $t$ is |AC|. Let $s'$ denote the side length of the inner pentagon, i.e., |ab|, and let $t'$ denote the length of the diagonal of the inner pentagon, i.e., |ac|.

We now make use of the wealth of congruent triangles in the diagram. We first observe that, whenever a pentagram is inscribed into a regular pentagon, the diagonals that form the pentagram exactly trisect the angles of the pentagon. (Exercise: Prove this!) Therefore, the triangle formed by A, E and D is congruent to that formed by A, e, and D, on account of their sharing a side and having the same angles on either end of that side. In particular, we have |Ae| = |AE| = $s$.

Next, consider the triangle formed by A, b, and d. By our observation at the start of the previous paragraph, the angle at b in this triangle has the same measure as the angle at A. But then this triangle is isosceles, so we have |Ad| = |db| = $t'$. Of course, we clearly have |Ad| = |Bd| = |Be| = |Ce| =…, so all of these lengths are equal to $t'$.

Let’s now see what we have. Consider first |AC|. By definition, we have |AC| = $t$. But also |AC| = |Ae| + |Ce|, and we saw previously that |Ae| = $s$ and |Ce| = $t'$. All together, we have $t$ = |AC| = |Ae| + |Ce| = $s + t'$, or $t' = t-s$.

Next, consider |Ae|. We have already seen that |Ae| = $s$. But we also have |Ae| = |Ad| + |de|. We saw previously that |Ad| = $t'$, and by definition we have |de| = $s'$. All together, we have $s$ = |Ae| = |Ad| + |de| = $t' + s'$, or $s' = s - t'$. Since we already know that $t' = t-s$, this yields $s' = 2s - t$.

This might not seem like much, but we’re actually almost done now! The key observation is that, since $s$ and $t$ are integers, and since $s' = 2s - t$ and $t' = t-s$, it follows that $s'$ and $t'$ are also (positive) integers. We started with a regular pentagon whose side length $s$ and diagonal length $t$ were integers and produced a smaller pentagon whose side length $s'$ and diagonal length $t'$ are also integers. But now we can continue this process, producing smaller and smaller regular pentagons and producing infinite descending sequences of positive integers, $s > s' > s'' > \ldots$ and $t > t' > t'' > \ldots$.

This is a contradiction, and we have thus shown that $\frac{t}{s}$ is irrational. But what is this ratio exactly? Well, it turns out to be a fascinating number, easily #2 on the list of Most Famous Ratios of All Time: $\phi$, a.k.a. the golden ratio! But that is a story for another time…

As a short addendum to today’s story, here’s a little-known fact about the end of Hippasus’ life. It turns out that, after being thrown overboard by his fellow Pythagoreans, he has not and will never in fact reach the seabed! For to get there, he would first have to travel halfway down, and then he would have to travel half of the remaining distance, and then half of the still remaining distance, and so on and so forth, completing an endless sequence of tasks, and thus he remains to this date in the midst of an infinite descent of his own.

Notes:

(1) $\sqrt{2}$ is more commonly cited as being the first irrational number discovered by the Pythagoreans, and it is almost always the first number proven to be irrational in classrooms today. However, the proof of the irrationality of $\sqrt{2}$ possessed by the Greeks is not the simple number-theoretic proof used today and is in fact a rather complex elaboration of the proof of the incommensurability of the side and diagonal lengths of a regular pentagon. Given this fact and the centrality of the pentagram in Pythagorean intellectual life, some scholars have suggested that perhaps this was in fact the first proof of the existence of incommensurability and that the proof of the irrationality of $\sqrt{2}$ came later. We have adopted this hypothesis for the purpose of our story today.

(2) This story is derived from legends that significantly post-date the death of Hippasus. It seems unlikely actually to have happened as presented here.

(3) This post was in part inspired by the episode “Drowned at Sea”, from the excellent podcast, Hi-Phi Nation, by philosopher Barry Lam. Check it out!

Cover image: “Rainstorm Over the Sea” by John Constable

# An Infinitude of Proofs, Part 0

Today, we take a break from our recent philosophical musings to return to some good old-fashioned mathematics (indeed, the mathematics today could be seen as all being “old-fashioned,” as the theorem we will be considering dates back to the ancient Greeks).

I’m sure my readers have all been introduced to the prime numbers, but, to refresh any memories, let me remind you that a prime number is a natural number, at least 2, that has no divisors other than 1 and itself. The first few prime numbers are, therefore: 2, 3, 5, 7, 11, 13…

The prime numbers can be thought of as the building blocks of number theory, as the backbone of the natural numbers. They have occupied a central place in mathematics and offered endless fascination for millennia. They are a source of great mystery even today.

In this post, we look at a classic theorem, commonly attributed to Euclid. Euclid’s proof is one of the gems of mathematics, a proof that is taught to every mathematician at the beginning of their true mathematical career. The theorem, as we will state it, is simply this:

Euclid’s Theorem: There are infinitely many prime numbers.

Euclid himself did not state the theorem in exactly this form, perhaps because of the ancient Greeks’ general antipathy towards the existence of infinite sets. He instead put forward the equivalent statement, “For every finite set of prime numbers, there is a prime number not in that set.”

Euclid’s Theorem has collected a vast and varied array of delightful proofs throughout the years, and, in a proposed infinite series of blog posts, I plan to cover all of them. Today, we will look at three proofs, from three very different centuries, all using only elementary number-theoretic techniques. First, of course, is Euclid’s proof itself.

Proof One (Euclid): Suppose that $P = \{p_0, p_1, \ldots, p_n\}$ is a finite set of prime numbers. We will show that there is a prime number that is not in $P$. To do this, let $q = p_0p_2\ldots p_n$, and let $r = q+1$. For all $i \leq n$, $p_i$ divides $q$ with no remainder (as $q = p_i(p_0\ldots p_{i-1}p_{i+1}\ldots p_n)$), so, as $r = q+1$ and $p_i \geq 2$, $p_i$ does not divide $r$. Since $r > 1$ and every integer greater than 1 has prime divisors, there is at least one prime, $p^*$, that divides $r$. But we just saw that no element of $P$ divides $r$, so $p^*$ is a prime that is not in $P$.

The next proof we will consider is due to the eighteenth-century German mathematician Christian Goldbach, largely known today for the statement of the still unproven Goldbach’s Conjecture. This proof (like Goldbach’s Conjecture itself) appears in a letter from Goldbach to Leonhard Euler (whose analytic proof of Euclid’s Theorem we will visit in a future post).

Proof Two (Goldbach): For each natural number $n$, let $F_n = 2^{2^n} + 1$, so $F_0 = 3, F_1 = 5, F_2 = 17$, etc. ($F_n$ is known as the $n^{\mathrm{th}}$ Fermat number.)

Claim 1: For every natural number $n$, $F_{n+1} = F_0F_1\ldots F_n + 2$.

Proof of Claim 1: Suppose that the Claim is false, and let $n$ be the smallest natural number such that $F_{n+1} \neq F_0F_1 \ldots F_n + 2$. Since the Claim can easily be verified by inspection for $n=0$, we may assume $n > 0$. But now we have the following sequence of calculations where, because we are assuming that $n$ is the smallest counterexample to the Claim, we can use the fact that $F_n = F_0F_1\ldots F_{n-1} + 2$.

$\begin{array}{rcl}F_{n+1} & = & 2^{2^{n+1}}+1 \\ & = & 2^{2^n}\cdot 2^{2^n} + 1 \\ & = & (F_n-1)(F_n-1)+1 \\ & = & (F_n)(F_n) - 2F_n + 2 \\ & = & (F_n)(F_0F_1\ldots F_{n-1} + 2) -2F_n + 2 \\ & = & F_0F_1\ldots F_n + 2F_n - 2F_n + 2 \\ & = & F_0F_1\ldots F_n + 2 \end{array}$

But this calculation shows that $n$ is not a counterexample to our Claim, contradicting our assumption and finishing the proof of the Claim.

We now need a very simple number-theoretic Claim.

Claim 2: If $a$ and $b$ are natural numbers, $p$ is a prime, and $p$ divides both $a$ and $a+b$, then $p$ divides $b$.

Proof of Claim 2: Suppose $a = mp$ and  $a+b = np$, where $m$ and $n$ are natural numbers. Then $b = a+b-a = np - mp = (n-m)p$, so $p$ divides $b$.

Claim 3: If $m < n$ are natural numbers, then there is no prime number $p$ such that $p$ divides both $F_m$ and $F_n$.

Proof of Claim 3: Suppose that the Claim is false and that there are natural numbers $m < n$ and a prime number $p$ such that $p$ divides both $F_m$ and $F_n$. Note that $F_m$ is one of the factors in the product $F_0F_1\ldots F_{n-1}$, so $p$ divides $F_0F_1\ldots F_{n-1}$. Since $F_n = F_0F_1\ldots F_{n-1} + 2$, we also conclude that $p$ divides $F_0F_1\ldots F_{n+1} + 2$, so, by Claim 2, we know that $p$ divides $2$. As $2$ is prime, this means that $p = 2$. But $F_m$ and $F_n$ are both odd, so this is impossible.

We are now ready to finish the proof. For each natural number $n$, choose a prime number $p_n$ that divides $F_n$. By Claim 3, for all natural numbers $m < n$, we have $p_m \neq p_n$, so $\{p_n \mid n \in \mathbb{N}\}$ is an infinite set of prime numbers.

Remark: The Fermat numbers were introduced, unsurprisingly, by the seventeenth-century French mathematician Pierre de Fermat (of Fermat’s Last Theorem fame). Fermat conjectured that every Fermat number is in fact prime (this would obviously imply Claim 3 from the previous proof). The first four Fermat numbers (3, 5, 17, 257) are easily verified to be prime, and the next Fermat number, 65537, can be seen to be prime with a bit more work. However, Euler proved in 1732 that $F_5$ is not prime. Indeed, $F_5 = 2^{32}+1 = 4294967297 = 641 \times 6700417$. Many mysteries remain regarding the Fermat numbers. For example, even the following two very basic questions remain unsolved to this day:

• Are there infinitely many prime Fermat numbers?
• Are there infinitely many non-prime Fermat numbers?

The Fermat numbers grow so quickly that, even with computers, it can be hard to analyze even “relatively small” Fermat numbers. For example, it is unknown whether $F_{20}$ or $F_{24}$ are prime. On the other hand, the truly gigantic number $F_{3329780}$ is known to be non-prime: one of its prime factors is $193 \times 2^{3329780} + 1$.

Finally, a proof that combines ideas from Euclid’s and Goldbach’s proofs was given recently by Filip Saidak.

Proof Three: (Saidak) Note first that, for every natural number $n$, we have that $n$ and $n+1$ do not have any shared prime divisors. Now define an infinite sequence of natural numbers, $\langle n_0, n_1, \ldots \rangle$ as follows. Let $n_0$ be any natural number that is at least 2. Let $n_1 = n_0(n_0+1)$. Since $n_0$ and $n_0 + 1$ do not share any prime divisors, and since both $n_0$ and $n_0+1$ have at least one prime divisor, it follows that $n_1$ must have at least 2 distinct prime divisors.

Now let $n_2 = n_1(n_1+1)$. Again, $n_1$ and $(n_1 + 1)$ do not have any shared prime divisors. We have shown that $n_1$ has at least 2 distinct prime divisors, and we know that $n_1 + 1$ has at least one prime divisor, so $n_2$ must have at least 3 distinct prime divisors. Continuing in this way, defining $n_{k+1} = n_k(n_k+1)$ for every natural number $k$, one proves that, for each natural number $k$, the number $n_k$ has at least $k+1$ distinct prime divisors. In particular, there are infinitely many prime numbers.

Stay tuned for a future post, in which we will provide proofs of Euclid’s Theorem from further flung areas of mathematics, including real analysis and topology.

# Zeno, Russell, and Borges on Tortoises

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.