The Simurgh (Universal Structures I)

To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour

-William Blake, “Auguries of Innocence”

In Persian mythology, the Simurgh is a bird that lives in the mountains of Alborz. Sometimes she has the head or body of a dog, sometimes of a human. She has witnessed the destruction of the world three times. The wind of her beating wings is responsible for scattering seeds from the Tree of Life, creating all plants in the world.

The Simurgh is, in some tellings, the archetype of all birds. Her name resembles the Persian phrase si murg, meaning “thirty birds.”

In The Conference of the Birds, Farid ud-Din Attar’s 12th-century masterpiece, the birds of the world undertake a journey to find the Simurgh. And they succeed.

Their life came from that close, insistent sun
And in its vivid rays they shone as one.
There in the Simorgh’s radiant face they saw
Themselves, the Simorgh of the world – with awe
They gazed, and dared at last to comprehend
They were the Simorgh and the journey’s end.
They see the Simorgh – at themselves they stare,
And see a second Simorgh standing there;
They look at both and see the two are one,
That this is that, that this, the goal is won.

-Farid ud-Din Attar, The Conference of the Birds

The Simurgh is a bird that contains all birds. She is a universal bird.

Unsurprisingly, the Simurgh shows up a number of times in the works of Jorge Luis Borges, in both his short stories and his essays. One reference appears in the masterful story, “The Aleph,” a particularly rich and dense work which you should certainly read for yourself.

“The Aleph” is partly about how we create our own worlds, how we approximate the unknowable universe within our lives and our art. The narrator of the story, also named Borges, grieving the loss of his beloved Beatriz, pays repeated visits to the home of her father and her cousin, the poet Carlos Argentino Daneri. On one of these visits, Carlos Argentino takes Borges to his basement to show him the source of his poetry, the titular Aleph, a single point that contains the universe.

On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realised that this movement was an illusion created by the dizzying world it bounded. The Aleph’s diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror’s face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; I saw a splintered labyrinth (it was London); I saw, close up, unending eyes watching themselves in me as in a mirror; I saw all the mirrors on earth and none of them reflected me; I saw in a backyard of Soler Street the same tiles that thirty years before I’d seen in the entrance of a house in Fray Bentos; I saw bunches of grapes, snow, tobacco, lodes of metal, steam; I saw convex equatorial deserts and each one of their grains of sand; I saw a woman in Inverness whom I shall never forget; I saw her tangled hair, her tall figure, I saw the cancer in her breast; I saw a ring of baked mud in a sidewalk, where before there had been a tree; I saw a summer house in Adrogué and a copy of the first English translation of Pliny — Philemon Holland’s — and all at the same time saw each letter on each page (as a boy, I used to marvel that the letters in a closed book did not get scrambled and lost overnight); I saw a sunset in Querétaro that seemed to reflect the colour of a rose in Bengal; I saw my empty bedroom; I saw in a closet in Alkmaar a terrestrial globe between two mirrors that multiplied it endlessly; I saw horses with flowing manes on a shore of the Caspian Sea at dawn; I saw the delicate bone structure of a hand; I saw the survivors of a battle sending out picture postcards; I saw in a showcase in Mirzapur a pack of Spanish playing cards; I saw the slanting shadows of ferns on a greenhouse floor; I saw tigers, pistons, bison, tides, and armies; I saw all the ants on the planet; I saw a Persian astrolabe; I saw in the drawer of a writing table (and the handwriting made me tremble) unbelievable, obscene, detailed letters, which Beatriz had written to Carlos Argentino; I saw a monument I worshipped in the Chacarita cemetery; I saw the rotted dust and bones that had once deliciously been Beatriz Viterbo; I saw the circulation of my own dark blood; I saw the coupling of love and the modification of death; I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon — the unimaginable universe.

-Jorge Luis Borges, “The Aleph”

Aleph (\aleph) is of course the letter chosen by Georg Cantor to represent transfinite cardinals and the first letter of the Hebrew alphabet. It plays a special role in Kabbalah as the first letter in “Ein Sof,” roughly translated as “infinity,” and in “Elohim,” one of the names of the Hebrew god. We will surely return to these matters.

This is the first installment in a mini-series on what we will call “universal structures,” objects that contain all other objects of their type. We will continue to look at examples from literature and religion, and will delve into the existence of universal structures in mathematics, a topic which continues to drive cutting-edge research to this day. Next week, we will look at a particular universal structure in mathematics, the wonderfully named “random graph.” I hope you will join us.

An Infinitude of Proofs, Part 1

In our previous post, we gave three elementary number-theoretic proofs of the infinitude of the prime numbers. Today, in an unforgivably delayed second installment, we provide two proofs using machinery from more distant fields of mathematics: analysis and topology.

A small word of warning: today’s proofs, though not difficult, require a bit more mathematical sophistication than those of the previous post. The first proof will involve some manipulation of infinite sums, and the second proof uses rudiments of topology, which we introduced here.

Without further ado, let us begin. Our first proof today dates to the 18th century and is due to the prolific Leonhard Euler, of whom Pierre-Simon Laplace, an exceptional mathematician in his own right, once said, “Read Euler, read Euler, he is the master of us all.”

Proof Four (Euler): Suppose there are only finitely many prime numbers, \{p_1, p_2, \ldots, p_n\}. For each prime number p, consider the infinite series \sum_{i = 0}^\infty \frac{1}{p^i} = 1 + \frac{1}{p} + \frac{1}{p^2} + \ldots . As you probably learned in a precalculus class, the value of this infinite sum is precisely \frac{1}{1-\frac{1}{p}}.

Now recall that every natural number greater than 1 can be written in a unique way as a product of prime numbers. Since \{p_1, \ldots, p_n\} are all of the primes, this means that, for each natural number m \geq 1, we can express \frac{1}{m} as \frac{1}{p_1^{s_1}} \cdot \frac{1}{p_2^{s_2}} \cdot \ldots \cdot \frac{1}{p_n^{s_n}} for some natural numbers s_1, s_2, \ldots, s_n. Moreover, it is clear that this gives a one-to-one correspondence between natural numbers m \geq 1 and the corresponding n-tuples of natural numbers, \langle s_1, s_2, \ldots, s_n \rangle. Putting this all together, we obtain the following remarkable formula:

\sum_{m=1}^\infty \frac{1}{m} = \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}}.

The sum on the left hand of this equation is the famous harmonic series, and it is well-known and easily verified that this sum diverges to infinity. On the other hand, the product on the right hand is a finite product of finite numbers and is therefore finite! This gives a contradiction and concludes the proof.

Our next proof was published in 1955 by a young Hillel Furstenburg, who is still active at the Einstein Institute of Mathematics here in Jerusalem.

Proof Five (Furstenberg): For all integers a and b, let U_{a,b} be the set \{a + bk \mid k \in \mathbb{Z} \}, i.e., U_{a,b} contains a and all integers that differ from a by a multiple of b.

Claim: For all pairs of integers (a_0, b_0) and (a_1, b_1), either U_{a_0,b_0} \cap U_{a_1, b_1} = \emptyset or there are a_2, b_2 such that U_{a_0,b_0} \cap U_{a_1, b_1} = U_{a_2, b_2}.

Proof of Claim: If U_{a_0, b_0} \cap U_{a_1, b_1} \neq \emptyset, then let a_2 be any element of the intersection, and let b_2 be the least common multiple of b_0 and b_1. It is easily verified that U_{a_0,b_0} \cap U_{a_1, b_1} = U_{a_2, b_2}, thus finishing the proof of the claim.

It now follows that we can define a topology on \mathbb{Z} by declaring that a set U \subseteq \mathbb{Z} is open if and only if U = \emptyset or U is a union of sets of the form U_{a,b}. In particular, for each prime number p, the set U_{0,p}, which is the set of all multiples of p, is open. However, U_{0,p} is also closed, since it is the complement of the set \bigcup_{0 < a < p} U_{a,p}, which is an open set.

Now suppose that there are only finitely many prime numbers, \{p_1, \ldots, p_n\}. Since each set U_{0,p_k} is closed and a finite union of closed sets is closed, we have that \bigcup_{k = 1}^n U_{0,p_k} is a closed set. Notice that every integer that is not 1 or -1 is a multiple of some prime number, so \bigcup_{k = 1}^n U_{0,p_k} is precisely the set of all integers except 1 and -1. Since it is closed, its complement, which is \{-1,1\}, is open. But every non-empty open set is a union of sets of the form U_{a,b}, each of which is clearly infinite, so there can be no finite non-empty open sets. This is a contradiction and concludes the proof.

Cover Image: Passio Musicae by Eila Hiltunen, a monument to Jean Sibelius in Helsinki, Finland. Photograph by the author.

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.

Infinite Life III: Transfinite Life

In our previous post, we took a detour to consider the shape of space. Today, we return to time, both external and personal, and the question, “What does it mean to be immortal?”

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The distinction between external time and personal time was introduced by Lewis in his 1976 paper, “The Paradoxes of Time Travel.” The term “external time” refers to time itself. “Personal time,” applied to a particular person, is, as Lewis puts it, “roughly, that which is measured by his wristwatch.” It might be more precise to think of personal time as the time measured by one’s internal processes: if, in a period of external time, I go through the physical processes that a typical person would go through in an hour, then, during that period of external time, one hour of personal time has elapsed for me.

In the common view of time, it is shaped like a straight line, stretching infinitely far in front of us. (We will be agnostic here about whether or not infinitely much time has already elapsed.) In this view, the years lying ahead of us form an \omega-sequence, i.e., a sequence ordered like the natural numbers. This year could be called Year 0, next year Year 1, and so on. One natural number for each future year, and one future year for each natural number. Also, in human experience thus far, external time and personal time have pretty much lined up. With this simple view of time and of the relationship between external and personal time, it is also pretty simple to see what it means to be immortal: a person is immortal if they live for all of the years after they are born.

This simple view of immortality probably suffices for practical discussions about its feasibility and desirability. Right now, though, we’re going to indulge ourselves in a bit of conceptual analysis, contemplating improbable thought experiments with the aim of gaining a fuller understanding of immortality or, at the very least, having some fun trying. Today: transfinite life.

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In a delightful 1985 paper, entitled “On Living Forever,” Phillip Bricker argues that he wants not just to live for an infinite number of years, but to live for a transfinite number of years, i.e., more than one \omega-sequence of years. Even if our universe happens to only have an \omega-sequence of years in the future, Bricker argues, it is conceivable that there are universes with more than an \omega-sequence of years, and it would be desirable to live in such a universe and live for a transfinite number of years.

There are a number of points to be considered here. For simplicity, let’s say there is a universe with two \omega-sequences of years, one after the other, and that we are currently considering the beginning of the first of these two \omega-sequences. Let us call this year Year 0A, next year Year 1A, the following year Year 2A, and so on, defining Year NA for every natural number N. After this \omega-sequence of years, there will be the start of the second \omega-sequence. Let us call the first year of this sequence Year 0B, the next year Year 1B, and so on.

The first and most fundamental question is perhaps, “Is this possible? Does this make sense?” More precisely, can the universe of the second \omega-sequence of years really be considered the “same” universe as that of the first \omega-sequence of years? Can a universe extend across multiple \omega-sequences of years? Bricker answers this question affirmatively with a process that might be called “decompression.” He asks the reader to consider the events between 11 PM and midnight on December 31, 1999. There is a possible world, he says, in which the events between 11 and 11:30 take one year to occur, the events between 11:30 and 11:45 take one year to occur, and so on, so that the events of this hour in our world take an entire \omega-sequence of years in this other world. After midnight, the two worlds proceed at the same rate. Thus, if our world has one \omega-sequence of years, this other world has two. Moreover, whatever relations hold between our world pre-midnight and our world post-midnight hold between the first \omega-sequence of the other world and the second \omega-sequence of the other world. So, Bricker argues, the extension of a possible universe across multiple \omega-sequences of years is no more mysterious than the extension of our universe across multiple days.

I admit that this is a somewhat convincing argument, but I find it only convincing of the possibility of a transfinite universe of a very particular sort, namely of a sort that respects some continuity conditions. In the world in which the events of an hour in our world take an entire \omega-sequence of years, things that proceed at a steady pace in our world proceed increasingly slowly. The world undergoes less and less change as this \omega-sequence progresses. In a very precise sense, as these infinitely many years go on, this alternate world converges to a fixed limit. This limit, of course, is the state of our world at midnight, which is the state of the alternate world at the start of the second \omega-sequence. Such a world, though, is a rather uninteresting example of a transfinite universe and would not be particularly appealing for somebody looking for true transfinite immortality. For suppose that such a person lived in this world for both \omega-sequences of years. As the first \omega-sequence went on, everything would slow down, including the person’s movements, internal processes, thoughts. At the end of the first \omega-sequence, even if infinitely many years of external time had occurred, the person’s experiences would only be equivalent to a finite amount of personal time in our world. So this person, living for two \omega-sequences of years, would have the same experience as someone living for one \omega-sequene of years in our world, defeating the purpose of wanting a transfinite existence in the first place.

To illustrate the difficulty, let’s consider a universe, called Universe T, with two \omega-sequences of years, that definitely does not converge, using a classic thought experiment known as Thomson’s Lamp, which we briefly visited in an earlier post in connection with Zeno’s paradoxes. Let’s suppose that you are living in this world, in the first \omega-sequence. Let’s suppose moreover that you live for the entire first \omega-sequence, spending all of the years living in the same house. On your bedside table is a lamp that, miraculously, never breaks and never needs a replacement bulb. Every morning, when you wake, you turn the lamp on, and every night, when you go to bed, you turn the lamp off. Now: at the first moment of the second \omega-sequence of years, is the lamp on or off? (Does the lamp even exist?) It seems impossible to answer this question, or even to make sense of it. If it were the case that, from a certain point onward in the first \omega-sequence of years, the lamp were off, then it would be natural to answer that the lamp is off at the start of second \omega-sequence, and similarly if the lamp were on from a certain point. In this case, though, the lamp does not converge to any single state, so it is hard to see what will happen to it in the second \omega-sequence.

In order to use Bricker’s argument to argue for the plausibility of this world, we would want to apply the opposite of “decompression,” something a person might reasonably call “compression.” To do this, we would want to consider a world, called Universe S, in which the first year’s events in Universe T take half an hour, the second year’s events in Universe T take a quarter of an hour, and so on, so that the events of the first \omega-sequence of years in Universe T take one hour in Universe S. Now, one wants to say, there is no problem with Universe S lasting for more than one hour, so, to figure out what happened to your lamp at the start of the second \omega-sequence in Universe T, just look at its state at the start of the second hour in Universe S. There is an issue here, though. In compressing the events of the first \omega-sequence of Universe T into one hour, certain things, such as the turning on and off of the lamp, happen at faster and faster speeds, indeed at speeds approaching infinity. This would of course be a problem in our universe, in which nothing can travel faster than the speed of light. It might be a problem in every universe. Are there possible universes in which objects can travel arbitrarily fast? If so, what would these universes be like? Are these thought experiments entirely misguided in the first place and driven by inaccurate conceptions of time? These questions certainly lie beyond the scope of this post. For now, let’s just leave them here as possible difficulties in justifying the possibility of interesting transfinite universes.

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Let’s put aside any objections, though, and suppose that there can in fact be a world spanning multiple \omega-sequences of years and that a person could exist across these multiple \omega-sequences in such a way that their personal time would coincide with the external time. Would this in fact be better than simply living for one \omega-sequence? Bricker’s primary reason for answering affirmatively is simply that he wants the pleasures of life (for example, eating Thai food) to be experienced as many times as possible. Eating Thai food provides pleasure; why limit oneself to just an \omega-sequence of Thai meals? Why not two \omega-sequences of Thai meals? Why not uncountably many Thai meals? Inaccessibly many Thai meals? A supercompact number of Thai meals? The more the better!

Another intriguing reason offered by Bricker involves the pursuit of mathematical knowledge. There are unsolved problems in number theory, for example, which could easily be solved in an infinite amount of time simply by checking every number. For example, the Goldbach conjecture, unproven to this day, asserts that every even number greater than 2 can be expressed as the sum of two prime numbers. The conjecture has resisted many attempts at proof by some of the great mathematicians of the world, but, if someone were able to live for more than an \omega-sequence of years, they could prove or refute it very easily: during the first \omega-sequence of years, they could systematically check every even number. If they find one that cannot be expressed as the sum of two prime numbers, they write it down as a counterexample, disproving the conjecture. If, after the first \omega-sequence of years, they have not written down a counterexample, they will know that there is none, and the conjecture is true.

Let’s note a couple of potential issues here. First, humans have finite brains. So, suppose that someone lives for longer than one \omega-sequence of years. When they wake up in Year 0B, the first year of the second \omega-sequence, they will only be able to remember finitely many things from the first \omega-sequence. Thus, as far as their memory is concerned, they will essentially only have lived for a finite number of years. This brings up two questions. First, will they be able to tell that it is Year 0B? If they cannot remember anything past Year 100A, then, as far as they are concerned, it might as well be Year 101A. There are obvious external remedies for this. For example, perhaps they could fashion a clock that moves from 12 to 6 during one year, 6 to 9 during the next, 9 to 10:30 during the next, and so on, so that the clock returning to 12 will mark the end of the first \omega-sequence of years.

Secondly, and perhaps more seriously, suppose that this person engaged themselves in solving Goldbach’s Conjecture by checking all of the numbers. Suppose that, in Year 0B, they find that they have not written down a counterexample. This indicates that the conjecture is true. However, due to their finite memory, they will only remember checking finitely many of the numbers. Can they assure themselves that they did in fact complete all of the calculations and didn’t give up partway through and sink into a decadent life of leisure? Again, there are possible remedies to this. They could set a computer on the task, putting precautions in place to make sure that the computer is not disturbed throughout the first \omega-sequence of years. This would be better but would run into similar objections. How could the person be sure that the computer’s integrity was maintained throughout its calculations, that it didn’t lose power partway through, that a rival mathematician didn’t hack into it, that it didn’t obtain consciousness and sink into a decadent life of leisure? Verifying that an infinite sequence of calculations was carried out correctly could take another infinity of years, in which case we are back to where we started. Such a proof of Goldbach’s Conjecture would be fundamentally different from any proofs done today, and it seems likely that its veracity would always be subject to some amount of doubt.

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We will turn to some other esoteric thought experiments in our next post. Until then, here are some questions, essentially taken from Cody Gilmore’s “The Metaphysics of Mortals: Death, Immortality, and Personal Time,” to ponder in order to test your intuition. We’ll consider possible answers next time; feel free to post thoughts in the comment section.

  • Alfred’s life has an external length of 100 years. However, his personal time passes very differently. The first 50 external years of his life correspond to his first personal year, the next 25 external years correspond to his second personal year, the next 12.5 external years to his third personal year, the next 6.25 to his fourth personal year, and so on. Is Alfred immortal?
  • Betty’s life lasts for an entire \omega-sequence of years of external time (in a world in which there is only one \omega-sequence). The first external year corresponds to one year of personal time, the second external year to half a year of personal time, the third external year to a quarter year of personal time, the fourth external year to an eighth of a year of personal time, and so on. Is Betty immortal?
  • Carmen lives in a world with two \omega-sequences of years. She is born during the first \omega-sequence of years, and her life lasts precisely until the end of the first \omega-sequence of years. Her personal time matches external time. Is Carmen immortal?
  • David lives in the same world as Carmen and is also born during the first \omega-sequence of years, at the same time as Carmen. David lives through the end of the first \omega-sequence of years and then five years into the second \omega-sequence of years before dying. His personal time matches external time. Is David immortal?

Cover image: Salvador Dali, “The Persistence of Memory”

1115

Dante, Einstein, and the Shape of the World

Last week, we began a series of posts dedicated to thinking about immortality. If we want to even pretend to think precisely about immortality, we will have to consider some fundamental questions. What does it mean to be immortal? What does it mean to live forever? Are these the same thing? And since immortality is inextricably tied up in one’s relationship with time, we must think about the nature of time itself. Is there a difference between external time and personal time? What is the shape of time? Is time linear? Circular? Finite? Infinite?

Of course, we exist not just across time but across space as well, so the same questions become relevant when asked about space. What is the shape of space? Is it finite? Infinite? It is not hard to see how this question would have a significant bearing on our thinking about immortality. In a finite universe (or, more precisely, a universe in which only finitely many different configurations of matter are possible), an immortal being would encounter the same situations over and over again, would think the same thoughts over and over again, would have the same conversations over and over again. Would such a life be desirable? (It is not clear that this repetition would be avoidable even in an infinite universe, but more on that later.)

Today, we are going to take a little historical detour to look at the shape of the universe, a trip that will take us from Ptolemy to Dante to Einstein, a trip that will uncover a remarkable confluence of poetry and physics.


One of the dominant cosmological views from ancient Greece and the Middle Ages was that of the Ptolemaic, or Aristotelian, universe. In this image of the world, Earth is the fixed, immobile center of the universe, surrounded by concentric, rotating spheres. The first seven of these spheres contain the seven “planets”: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. Surrounding these spheres is a sphere containing the fixed stars. This is the outermost sphere visible from Earth, but there is still another sphere outside it: the Primum Mobile, or “Prime Mover,” which gives motion to all of the spheres inside it. (In some accounts the Primum Mobile is itself divided into three concentric spheres: the Crystalline Heaven, the First Moveable, and the Empyrean. In some other accounts, the Empyrean (higher heaven, which, in the Christianity of the Middle Ages, became the realm of God and the angels) exists outside of the Primum Mobile.)

boord
An illustration of the Ptolemaic universe from The Fyrst Boke of the Introduction of Knowledge by Andrew Boorde (1542)

This account is naturally vulnerable to an obvious question, a question which, though not exactly in the context of Ptolemaic cosmology, occupied me as a child lying awake at night and was famously asked by Archytas of Tarentum, a Greek philosopher from the fifth century BC: If the universe has an edge (the edge of the outermost sphere, in the Ptolemaic account), then what lies beyond that edge? One could of course assert that the Empyrean exists as an infinite space outside of the Primum Mobile, but this would run into two objections in the intellectual climate of both ancient Greece and Europe of the Middle Ages: it would compromise the aesthetically pleasing geometric image of the universe as a finite sequence of nested spheres, and it would go against a strong antipathy towards the infinite. Archytas’ question went largely unaddressed for almost two millennia, until Dante Alighieri, in the Divine Comedy, proposed a novel and prescient solution.


Before we dig into Dante, a quick mathematical lesson on generalized spheres. For a natural number n, an n-sphere is an n-dimensional manifold (i.e. a space which, at every point, locally looks like n-dimensional real Euclidean space) that is most easily represented, embedded in n+1-dimensional space, as the set of all points at some fixed positive distance (the “radius” of the sphere) from a given “center point.”

Perhaps some examples will clarify this definition. Let us consider, for various values of n, the n-sphere defined as the set of points in (n+1)-dimensional Euclidean space at distance 1 from the origin (i.e. the point (0,0,…,0)).

If n=0, this is the set of real numbers whose distance from 0 is equal to 1, which is simply two points: 1 and -1.

If n=1, this is just the set of points (x,y) in the plane at a distance of 1 from (0,0). This is the circle, centered at the origin, with radius 1.

unit_circle
A 1-sphere

If n=2, this is the set of points (x,y,z) in 3-dimensional space at a distance 1 from the point (0,0,0). This is the surface of a ball of radius 1, and is precisely the space typically conjured by the word “sphere.”

sphere
A 2-sphere

0-, 1-, and 2-spheres are all familiar objects; beyond this, we lose some ability to visualize n-spheres due to the difficulty of considering more than three spatial dimensions, but there are useful ways to think about higher-dimensional spheres by analogy with the more tangible lower-dimensional ones. Let us try to use these ideas to get some understanding of the 3-sphere.

First, note that, for a natural number n, the non-trivial “cross-sections” of an n+1-sphere are themselves n-spheres! For example, if a 1-sphere (i.e. circle) is intersected with a 1-dimensional Euclidean space (a line) in a non-trivial way, the result is a 0-sphere (i.e. a pair of points). If a 2-sphere is intersected with a 2-dimensional Euclidean space (a plane) in a non-trivial way, the result is a 1-sphere (this is illustrated above in our picture of a 2-sphere). The same relationship holds for higher dimensional spheres: if a 3-sphere is intersected with a 3-dimensional Euclidean space in a non-trivial way, the result is a 2-sphere.

Suppose that you are a 2-dimensional person living in a 2-sphere universe. Let’s suppose, in fact, that you are living in the 2-sphere pictured above, with the 1-sphere “latitude lines” helpfully marked out for you. Let’s suppose that you begin at the “north pole” (i.e. the point at the top, in the center of the highest circle) and start moving in a fixed direction. At fixed intervals, you will encounter the 1-sphere latitude lines. For a while, these 1-spheres will be increasing in radius. This will make intuitive sense to you. You are moving “further out” in space; each successive circle “contains” the last and thus should be larger in radius. After you pass the “equator,” though, something curious starts happening. Even though you haven’t changed direction and still seem to be moving “further out,” the radii of the circles you encounter start shrinking. Eventually, you reach the “south pole.” You continue on your trip. The circles wax and wane in a now familiar way, and, finally, you return to where you started.

A similar story could be told about a 3-dimensional being exploring a 3-sphere. In fact, I think we could imagine this somewhat easily. Suppose that we in fact live in a 3-sphere. For illustration, let us place a “pole” of this 3-sphere at the center of the Earth. Now suppose that we, in some sort of tunnel-boring spaceship, begin at the center of the Earth and start moving in a fixed direction. For a while, we will encounter 2-sphere cross-sections of increasing radius. Of course, in the real world these are not explicitly marked (although, for a while, they can be nicely represented by the spherical layers of the Earth’s core and mantle, then the Earth’s surface, then the sphere marking the edge of the Earth’s atmosphere) but suppose that, in our imaginary world, someone has helpfully marked them. For a while, these successive 2-spheres have larger and larger radii, as is natural. Eventually, of course, they will start to shrink, contracting to a point before expanding and contracting as we return to our starting point at the Earth’s core.


Dante’s Divine Comedy, completed in 1320, is one of the great works of literature. In the first volume, Inferno, Dante is guided by Virgil through Hell, which exists inside the Earth, directly below Jerusalem (from where I happen to be writing this post). In the second volume, Purgatorio, Virgil leads Dante up Mount Purgatory, which is situated antipodally to Jerusalem and formed of the earth displaced by the creation of Hell. In the third volume, Paradiso, Dante swaps out Virgil for Beatrice and ascends from the peak of Mount Purgatory towards the heavens.

dantes_earth
Dante’s universe. Image by Michelangelo Caetani.

Dante’s conception of the universe is largely Ptolemaic, and most of Paradiso is spent traveling outward through the larger and larger spheres encircling the Earth. In Canto 28, Dante reaches the Primum Mobile and turns his attention outward to what lies beyond it. We are finally in a position to receive an answer to Archytas’ question, and the answer that Dante comes up with is surprising and elegant.

The structure of the Empyrean, which lies outside the Primum Mobile, is in large part a mirror image of the structure of the Ptolemaic universe, a revelation that is foreshadowed in the opening stanzas of the canto:

When she who makes my mind imparadised
Had told me of the truth that goes against
The present life of miserable mortals —

As someone who can notice in a mirror
A candle’s flame when it is lit behind him
Before he has a sight or thought of it,

And turns around to see if what the mirror
Tells him is true, and sees that it agrees
With it as notes are sung to music’s measure —

Even so I acted, as I well remember,
While gazing into the bright eyes of beauty
With which Love wove the cord to capture me.

When Dante looks into the Empyrean, he sees a sequence of concentric spheres, centered around an impossibly bright and dense point of light, expanding to meet him at the edge of the Primum Mobile:

I saw a Point that radiated light
So sharply that the eyelids which it flares on
Must close because of its intensity.

Whatever star looks smallest from the earth
Would look more like a moon if placed beside it,
As star is set next to another star.

Perhaps as close a halo seems to circle
The starlight radiance that paints it there
Around the thickest mists surrounding it,

As close a ring of fire spun about
The Point so fast that it would have outstripped
The motion orbiting the world most swiftly.

And this sphere was encircled by another,
That by a third, and the third by a fourth,
The fourth by a fifth, the fifth then by a sixth.

The seventh followed, by now spread so wide
That the whole arc of Juno’s messenger
Would be too narrow to encompass it.

So too the eighth and ninth, and each of them
Revolved more slowly in proportion to
The number of turns distant from the center.

This seemingly obscure final detail, that the spheres of the Empyrean spin increasingly slowly as they increase in size, and in distance from the point of light, turns out to be important. Dante is initially confused because, in the part of the Ptolemaic universe from the Earth out to the Primum Mobile, the spheres spin faster the larger they are; the fact that this is different in the Empyrean seems to break the nice symmetry he observes. Beatrice has a ready explanation, though: the overarching rule governing the speed at which the heavenly spheres rotate is not based on their size, but rather on their distance from God.

This is a telling explanation and seems to confirm that the picture Dante is painting of the universe is precisely that of a 3-sphere, with Satan, at the center of the Earth, at one pole and God, in the point of light, at the other. If Dante continues his outward journey from the edge of the Primum Mobile, he will pass through the spheres of the Empyrean in order of decreasing size, arriving finally at God. Note that this matches precisely the description given above of what it would be like to travel in a 3-sphere. Dante even helpfully provides a fourth dimension into which his 3-sphere universe is embedded: not a spatial dimension, but a dimension corresponding to speed of rotation!

(For completeness, let me mention that the spheres of the Empyrean are, in order of decreasing size and hence increasing proximity to God: Angels, Archangels, Principalities, Powers, Virtues, Dominions, Thrones, Cherubim, and Seraphim.)

Dante’s ingenious description of a finite universe helped the Church to argue against the existence of the infinite in the physical world. Throughout the Renaissance, Scientific Revolution, and Enlightenment, this position was gradually eroded in favor an increasingly accepted picture of infinite, flat space. A new surprise awaited, though, in the twentieth century.

‘Paradiso II’; Dante and Beatrice in the sphere of the moon, with Beatrice explaining the nature of the heavens; illustration by Sandro Botticelli, circa 1490
Beatrice explaining the nature of the heavens to Dante. Drawing by Botticelli.

In 1917, Einstein revolutionized cosmology with the introduction of general relativity, which provided an explanation of gravity as arising from geometric properties of space and time. Central to the theory are what are now known as the Einstein Field Equations, a system of equations that describes how gravity interacts with the curvature of space and time caused by the presence of mass and energy. In the 1920s, an exact solution to the field equations, under the assumptions that the universe is homogeneous and isotropic (roughly, has laws that are independent of absolute position and orientation, respectively), was isolated. This solution is known as the Friedmann-Lemaître-Robertson-Walker metric, after the four scientists who (independently) derived and analyzed the solution, and is given by the equation,

ds^2 = -dt^2 + R^2(t)\left(\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2)\right),

where k is a constant corresponding to the “curvature” of the universe. If k = 0, then the FLRW metric describes an infinite, “flat” Euclidean universe. If k < 0, then the metric describes an infinite, hyperbolic universe. If k>0, though, the metric describes a finite universe: a 3-sphere.


PS: Andrew Boorde, from whose book the above illustration of the Ptolemaic universe is taken, is a fascinating character. A young member of the Carthusian order, he was absolved from his vows in 1529, at the age of 39, as he was unable to adhere to the “rugorosite” of religion. He turned to medicine, and, in 1536, was sent by Thomas Cromwell on an expedition to determine foreign sentiment towards King Henry VIII. His travels took him throughout Europe and, eventually, to Jerusalem, and led to the writing of the Fyrst Boke of the Introduction of Knowledge, perhaps the earliest European guidebook. Also attributed to him (likely without merit) is Scoggin’s Jests, Full of Witty Mirth and Pleasant Shifts, Done by him in France and Other Places, Being a Preservative against Melancholy, a book which, along with Boord himself, plays a key role in Nicola Barker’s excellent novel, Darkmans.

Further Reading:

Mark A. Peterson, “Dante and the 3-sphere,” American Journal of Physics, 1979.

Carlo Rovelli, “Some Considerations on Infinity in Physics,” and Anthony Aguirre, “Cosmological Intimations of Infinity,” both in Infinity: New Research Frontiers, edited by Michael Heller and W. Hugh Woodin.

Cover Image: Botticelli’s drawing of the Fixed Stars.

Infinite Life II: The Transhumanists

I’m working on a long post that will hopefully be published later this week, but, in the meantime, here’s another article from The Atlantic about immortality. The piece focuses on practical matters, largely on the transhumanist community, centered in and around Silicon Valley, which is working on scientific and technological methods for radically and, perhaps one day, indefinitely extending human life. There are some nice introductions to arguments for and against this endeavor that we will no doubt return to in more depth in future installments of this series (which, I promise, will not just be a sequence of links to articles in The Atlantic).

Infinite Life I: Apeirophobia

Eternity is a very long time, especially towards the end.

-Woody Allen

During my freshman year of college, a friend asked me if I wanted to live forever and was shocked when, after five seconds of thought, I had not yet given an answer of “No” (an answer that would come after another five seconds). Everlasting life has been an object of intellectual fixation for millennia and has been addressed in countless works of art and mythology – think of Gilgamesh, the Monkey King and the Peaches of Immortality, the Flying Dutchman- most of which reach the inevitable conclusion that immortality is ultimately undesirable. The finiteness of our lives allows them to have meaning, the standard thinking goes. And an infinite life would get awfully boring. The fact that it took me, a reasonably well-educated eighteen-year-old who had undoubtedly read Tuck Everlasting in an elementary school classroom, ten whole seconds to reach this conclusion was remarkable.

In the years since this dorm room conversation, my thinking on this topic has not exactly become clarified. It would certainly take me longer than ten seconds to answer the question today, and I’m not sure I would finally come up with the same answer. And so, in a probably futile attempt to illuminate some of the murky corners of the issue, or at least to enjoy engaging with some fun problems, I will be writing a series of posts dedicated to immortality, to infinite life, drawing inspiration from literature, philosophy, psychology, and, yes, mathematics.

Immortality is unappealing to many, but to some, the prospect of an infinite life (or afterlife) is terrifying. This fear of eternity is known as apeirophobia (note the Greek root ‘apeiron‘) and, according to some anecdotal evidence in this article from The Atlantic, it might not be particularly uncommon. Apeirophobia has not been very well studied, but the article puts forward one explanation by Martin Wiener, a George Mason neuroscientist and psychologist. His idea is that, as children enter adolescence and develop the capacity for long-term planning, they realize both that they themselves will become adults and, one day, die and that this newfound ability for long-term planning would become somewhat useless in the context of eternity. It is impossible to mentally project forward through an infinite stretch of time, and the realization of this fact creates a fear.

The article also highlights an illuminating quote from Pascal:

When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill, and even can see, engulfed in the infinite immensity of spaces of which I am ignorant, and which know me not, I am frightened, and am astonished at being here rather than there; for there is no reason why here rather than there, why now rather than then.

-Blaise Pascal, Pensées

And here’s a nice companion video The Atlantic made for the original article.


A bonus morsel this week: a piece published last week at The Millions about Richard Burgin, who is best known for his book of conversations with Borges. The piece ends with Burgin recounting talking with his son, a budding filmmaker, about infinity:

He asks me why anyone would waste their time with such an idea when there’s so much work to do.

-John Burgin, quoted in “Burgin, Borges, and Infinity”

Cover Image: Joseph Cornell, Soap Bubble Set