More Dots and Division by Zero

A short post today to point out some happenings elsewhere on the web.


In our last offering, we discussed the art of Yayoi Kusama. The people over at The Atlantic must have noticed, because the next day they posted the following striking photo of Kansas farmland from the dailyoverview Instagram account and noted its affinities with Kusama’s artwork. All part of the neverending cycle of art interacting with the natural (or constructed) world.

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Next, we have a video that has been shared all over the internet in the last few days but deserves a place here as well. Watch as a mechanical calculator enters an infinite loop after being commanded to divide a number by zero.


P.S. The title photo is of one of Damien Hirst’s spot paintings, a widely recognized and widely derided series of paintings that has given rise to some rather amusing journalism, including a Guardian article wondering if the heist of two spot paintings really matters at all and a Business Insider ranking of the all-time worst and best of the series.

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Infinity Dots

When I was a kid and my parents were out of the house, I would sometimes slip into their bathroom. I would do this not to snoop around in the drawers or to try to ascertain the secrets of the lives of adults, but rather to find a piece of infinity.

An explanation of the topography of the bathroom, in particular of the bathroom sink, and especially of the medicine cabinet above the bathroom sink, should elucidate the situation. You see, the medicine cabinet has three partitions, each with a mirrored door, the outer two of which both swing open towards the center. Therefore, if, say, a ten-year-old boy were to open these two doors just the right amount and stick his head between them in just the right place, with his eyes pointing in just the right direction, all of these details of course meticulously fine-tuned during multiple sessions, he would seemingly find himself placed right in the middle of an infinite greenish-silver corridor, alternating images of the front and back of his head lined up as far as he could see. It was intoxicating to find such an expansive space in a room that, at first glance, appears so limited in dimensions.

(This desire to find infinity in unexpectedly small places is perhaps in part responsible for my later infatuation with the short fictions of Jorge Luis Borges, whose techniques, as Lois Parkinson Zamora aptly states in an essay on Borges and trompe l’œil, create “the illusion of infinity in a tightly contained narrative space.”)


In the Mexican War Streets neighborhood of Pittsburgh, located in a former mattress warehouse, there is a remarkable contemporary art museum called, appropriately, Mattress Factory. Among the highlights of the museum are three light installations by James Turrell and two mirrored rooms by Yayoi Kusama, where, as a graduate student, I found more fully realized versions of the infinite corridor in my parents’ bathroom.

The first, Infinity Dots Mirrored Room, is an empty room with a polka-dotted floor and mirrored walls and ceiling, illuminated by black light. The effect is eerie; one feels entirely alone and lost in a formless landscape which grows increasingly indistinct as it recedes to the horizon.

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Yayoi Kusama: Infinity Dots Mirrored Room

The second, Repetitive Vision, is slightly less disorienting and slightly more surreal. The polka dots are all bright red, the light is white rather than black, and the viewer is joined by mannequins, covered in red polka dots themselves.

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Yayoi Kusama: Repetitive Vision

During the 1960s, Kusama, who was born and raised in Japan, was a prominent figure in the New York art scene, working alongside Donald Judd, Claes Oldenburg, and Andy Warhol and romantically involved with Joseph Cornell. She worked obsessively, creating huge installations and staging bizarre happenings in public places, including many in which, anticipating Repetitive Vision, Kusama would paint polka dots on the bodies of nude models.

Her time in New York, though, was also characterized by severe health problems, and she was hospitalized on many occasions. In 1973, at the advice of a doctor, Kusama moved back to Japan, and since 1975 she has resided at the Seiwa Hospital for the Mentally Ill, where she continues to create art. She suffers from depersonalization disorder, in which an individual has recurring feelings of disconnection from their body or thoughts. They may feel as if they are watching their life in a movie. They may have hallucinations. According to Kusama, her artwork is largely an expression of her experiences with mental disease; this seems particularly the case with her mirrored rooms.

Depersonalization disorder is typically thought to be caused by severe traumatic events. In a 1999 interview in Bomb Magazine, which due to Kusama’s residence at a mental hospital was conducted via fax, she describes being abused by her mother as a child:

My mother was a shrewd businesswoman, always horrendously busy at her work. I believe she contributed a great deal to the success of the family business. But she was extremely violent. She hated to see me painting, so she destroyed the canvases I was working on. I have been painting pictures since I was about ten years old when I first started seeing hallucinations.

In her artist’s statement accompanying Repetitive Vision, created and installed in the Mattress Factory in 1996, Kusama describes a hallucination that inspired the work:

One day, I was looking at a tablecloth covered in red flowers, which was spread out on the table. Then I looked up towards the ceiling. There, on the windows and even on the pillars, I would see the same red flowers. They were all over the place in the room, my body, and entire universe. I finally came to a self-obliteration and returned to be restored to the infinity of eternal time and the absoluteness of space. I was not having a vision. It was a true reality. I was astounded. Unless I got out of here, the curse of those flowers will seize my life! I ran frantically up the stairs. As I looked down, the sight of each step falling apart made me stumble. I fell all the way down the stairs and sprained my leg.

There is in the popular imagination a strong link between infinity and madness, in particular a causal link posited from the first to the second, a persistent image of a person staring into the void and never again being quite the same. For example, it is commonly asserted that Georg Cantor went insane from thinking about infinity too much. And while it is true that he spent the end of his life in sanatoria, this likely had little to do with his contemplations of infinity. Rather, experts believe that he suffered from bipolar disorder, which was possibly exacerbated by strong criticism of his work from other mathematicians.

In actuality, I suspect (and I write here with no real authority on the matter) that some amount of immersive experience with infinity is beneficial, that it can help us gain a healthy perspective on our relationship with the world. Much of Kusama’s work, and, I think, much other contemporary art as well, seeks to engender this experience. Standing in one of her mirrored rooms, you temporarily lose yourself in its vastness. You become more aware of the true scale of the world and your size within it. And in my experience this turns out, when taken in small doses, to be a surprisingly comforting sensation.

Kusama herself, in the Bomb interview, puts things a bit more forcefully:

By obliterating one’s individual self, one returns to the infinite universe.

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Yayoi Kusama: Infinity Mirrored Room – The Souls of Millions of Light Years Away

 

 

 

Hypergame and Cantor’s Theorem, Part II

As promised on Monday, we present here William Zwicker’s proof of Cantor’s Theorem. But first: Let’s play Hypergame!

Hypergame

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We will be concerned with two-player games, where the players, who we will name Ada and Bertrand, alternate making plays. Let us say that a game is finite if it is guaranteed to end after a finite number of moves. For example, because of the 50-move rule, under which a game is a draw if 50 moves elapse in which no pawn is advanced and no piece is captured, chess is a finite game. In fact, there is a fixed upper bound (6000 is certainly enough) on the number of moves in a single chess game, but this is not true of all finite games. Consider the game in which the first player names a natural number, N, and the players then play N games of chess. This game is guaranteed to end after finitely many moves (6000 \cdot N is an upper bound), but there is no fixed upper bound before the game starts, as the first player could simply choose a larger number as N.

We are now ready to define Hypergame. The rules are as follows. On her first move, the first player names a finite game. The players then play a round of that game, beginning with the second player. For example, a round of Hypergame between Ada and Bertrand might go something like this:

Ada: Let’s play chess!

Bertrand: f3

Ada: e5

Bertrand: g4

Ada: Qa4#

Now for the crucial question: Is Hypergame a finite game?

Well, yes, obviously. One move of Hypergame consists of naming a finite game. The rest of the moves consist of a round of that finite game, which must, by definition, take only a finite number of moves. One plus a finite number is a finite number, so Hypergame must end after a finite number of moves and is thus a finite game.

But now, armed with the knowledge the Hypergame is a finite game, Ada and Bertrand may play the following round of Hypergame:

Ada: Let’s play Hypergame!

Bertrand: Let’s play Hypergame!

Ada: Let’s play Hypergame!

Bertrand: Let’s play Hypergame!

This is an infinite (and rather boring) round of Hypergame! But didn’t we prove in the previous paragraph that Hypergame is a finite game? What has gone wrong here? I’ll leave you to ponder this instructive question, pointing out that we haven’t really given a formal definition for a “game.” If you try to make this argument more rigorous by providing such a formal definition, does Hypergame actually qualify as a game?

Proof of Cantor’s Theorem

We now turn to Zwicker’s proof of Cantor’s Theorem, where the ideas from the Hypergame paradox will be put to good use. Recall that Cantor’s Theorem states that, for every set X, |X| < |\mathcal{P}(X)|. (Recall also that a review of relevant mathematical definitions is given here.)

Suppose Cantor’s Theorem is false and that there is a set X such that |X| \geq |\mathcal{P}(X)|. This means that there is a function f:X \rightarrow \mathcal{P}(X) that is onto, i.e. a function f such that, for every Y \in \mathcal{P}(X), there is x \in X such that f(x) = Y. We will use this function to derive a contradiction, thus showing that such a function cannot exist and proving Cantor’s Theorem.

We first need a definition. Consider a sequence \langle x_0, x_1, x_2, \ldots \rangle (the sequence could be finite or infinite) consisting of elements of X. We say this sequence is a sequence through f if, for every element x of the sequence, the next element of the sequence, if it exists, comes from f(x), i.e. x_1 \in f(x_0), x_2 \in f(x_1), etc.

As an example, suppose X = \mathbb{N} and we have f(3) = \{2, 17, 9\}, f(9) = \{3, 13, 8\}, and f(8) = \emptyset. Then \langle 3, 9, 8 \rangle is a (finite) sequence through f. If f(4) = \{5\} and f(5) = \{4\}, then \langle 4, 5, 4, 5, 4, \ldots \rangle is an infinite sequence through f.

Back to our general case. We say an element x \in X is infinitary if there is an infinite sequence through f starting with x. x is finitary if there is no such infinite sequence. Let Y be the set of all finitary elements of X. Y \in \mathcal{P}(X), so, since f is onto, there is x \in X such that f(x) = Y.

Is x finitary? Well, yes. This is the same as the argument that Hypergame is a finite game: Suppose (x_0, x_1, x_2, \ldots) is a sequence through f with x_0 = x. Then x_1 \in f(x) and f(x)=Y, so, by our definition of Y, we know that x_1 is finitary. Since (x_1, x_2, \ldots) is a sequence through f starting with x_1 and x_1 is finitary, (x_1, x_2, \ldots) must be a finite sequence. But then (x_0, x_1, x_2, \ldots) must also be finite, as it has just one more element. There are thus no infinite sequences through f starting with x, so x is finitary.

But now we have a problem. In fact, it is the same problem we had with Hypergame. Since x is finitary, we must have x \in Y, by the definition of Y. But this means that (x, x, x, x, \ldots), the infinite sequence, all of whose elements are x, is an infinite sequence through f starting with x. This contradicts the fact that x is finitary and completes our proof.

P.S. Hypergame and Zwicker’s proof of Cantor’s Theorem were brought to my attention by Raymond Smullyan’s excellent book, Satan, Cantor and Infinity, which I would highly recommend to all readers of this blog.

Hypergame and Cantor’s Theorem, Part I

Note: I have added a page, Sets, Functions, and Cardinality, which introduces basic mathematical notions and notations that will be useful for us. Those of you with some mathematical background can safely skip it, possibly referring back to it if something unfamiliar arises. Others may find it helpful to read that page before reading this and future mathematical posts. I will assume in this post that the reader is familiar with the notions of power set and cardinality, which are covered in the linked page. If anything remains unclear, please let me know!


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Georg Cantor

Set theory, and with it the modern mathematical study of infinity, was arguably born in 1874, when Georg Cantor first published his now-celebrated result that the set of all real numbers is strictly larger than the set of all natural numbers. This was the first demonstration that there are multiple sizes of infinity and opened the door to a vast realm of mathematical investigation. In 1891, he published a second proof, introducing what came to be known as the diagonal argument, a beautiful and versatile tool. (First exposure to the diagonal argument is a pivotal moment in the lives of many young mathematicians. I first came across it in high school, in The Man Who Loved Only Numbers, Paul Hoffman’s wonderful book about the Hungarian mathematician Paul Erdős.) In fact, in the 1891 paper, he proved a more general theorem, known today as Cantor’s Theorem.

Cantor’s Theorem: For every set X, the power set of X is strictly larger than X, i.e. |X| < |\mathcal{P}(X)|.

(To see that this theorem does indeed generalize the result that |\mathbb{N}| < |\mathbb{R}|, it suffices to show that |\mathcal{P}(\mathbb{N})| \leq |\mathbb{R}| (in fact, these two sets have the same cardinality). To show this, we will define a one-to-one function f:\mathcal{P}(\mathbb{N}) \rightarrow \mathbb{R}. One way to define such a function is as follows. If X \in \mathcal{P}(\mathbb{N}), then let f(X) = 0.a_0a_1a_2\ldots, where, for every natural number n, a_n = 1 if n \in X and a_n = 0 if n \not\in X. I will leave it to the reader to check that this does in fact define a one-to-one function from \mathcal{P}(\mathbb{N}) into \mathbb{R}.)

By iterating the power set operation, Cantor’s Theorem shows us that the world of infinity is exceedingly rich; there is in fact an infinity of different infinities. For example:

|\mathbb{N}| < |\mathcal{P}(\mathbb{N})| < |\mathcal{P}(\mathcal{P}(\mathbb{N}))| < |\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N})))| \ldots.

(I am in fact understating my case here. Not wanting to become too technical, let me just mysteriously and somewhat vaguely say that the number of infinities is itself larger than any set.)

Cantor’s Theorem was immediately polarizing in the mathematical world. Some influential mathematicians were strong supporters of Cantor’s work who recognized and appreciated the possibilities it opened up. This group included David Hilbert, who famously proclaimed:

No one shall expel us from the paradise that Cantor has created for us.

Many other mathematicians, though, were resistant. Before Cantor, mathematics had dealt with infinity primarily as a potential infinity, an infinity that can be approached but never attained, and had shied away (with some notable exceptions, such as Leibniz and possibly Archimedes) from the notion of an actual infinity, or a completed infinite totality. Many thus objected to what they saw as Cantor’s cavalier application of mathematical and logical methods to infinite sets such as \mathbb{N} and \mathbb{R}. Leopold Kronecker, a great mathematician and arguably Cantor’s arch-nemesis, put his distaste particularly harshly:

I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.

Cantor’s ideas gradually came to to be accepted by the mathematical establishment, but the uneasy feeling that set theory is more theology than mathematics lingered among mathematicians of a certain philosophical disposition. Hermann Weyl, a prominent mathematician, physicist, and philosopher, wrote the following about Cantor’s work and subsequent developments in set theory in his 1946 paper, “Mathematics and Logic.”

But in the resulting system mathematics is no longer founded on logic, but on a sort of logician’s paradise, a universe endowed with an “ultimate furniture” of rather complex structure and governed by quite a number of sweeping axioms of closure. The motives are clear, but belief in this transcendental world taxes the strength of our faith hardly less than the doctrines of the early Fathers of the Church or of the scholastic philosophers of the Middle Ages.

(I think “ultimate furniture” is one of my new favorite expressions.)

Cantor’s diagonal argument is already covered wonderfully in many other places. I can recommend the following TED-Ed video by Dennis Wildfogel:

Here, though, I want to present a slightly different proof, due to William Zwicker, of Cantor’s Theorem. The proof uses ideas from the analysis of Hypergame, a delightful pseudo-paradoxical thought experiment. Come back on Thursday to find out more.

 

Apeiron

The earliest recorded thoughts about infinity come from ancient Greece, in particular from Anaximander and the idea of the apeiron (ἄπειρον). Anaximander was a philosopher who lived in Ionia, in present-day Turkey, in the 6th century BC. As is generally the case with the pre-Socratics, little survives of his work, so much of what we know comes from later testimonies.

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Anaximander (source: Encyclopedia Britannica Online)

Apeiron is typically translated as “infinite,” “eternal,” or “limitless,” and the apeiron was posited by Anaximander as the source of the universe, a formless chaos out of which all else emerged. Xenophanes, later expanding upon Anaximander’s ideas, put forth the following picture of the world:

The upper limit of the Earth borders on Air,

The lower limit of the Earth reaches down to the Unlimited, {i.e. the Apeiron}.

-DK 21B28, translation by Karl Popper

Anaximander had some curious ideas about the shape of the world. He believed the Earth to be a circular cylinder, with diameter exactly three times its height. We live on the top circular surface of this cylinder, which floats unsupported at the center of the universe, surrounded by the apeiron.

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Anaxagoras, depicted in a fresco at the National University of Athens

The apeiron would later play a central role in the cosmology of Anaxagoras, a philosopher also born in Ionia in the late 6th century BC who is typically credited with bringing philosophy to Athens, where he lived during the early life of Socrates. As did Anaximander, Anaxagoras considered the apeiron to be the source of all else and seems to emphasize its infinite and eternal nature.

Anaxagoras … says that the principles are unlimited. He says that almost all of the homogeneous stuffs come to be and pass away in this way (just as fire and water do), viz., only by aggregating and dissociating; they are not generated or destroyed in any other sense, but persist eternally.

-Aristotle’s Metaphysics

Anaxagoras even described a remarkable mechanism for how the apeiron gave rise to the universe, namely that a powerful mind (nous) began rotating the apeiron so that pieces of it broke off to form other entities.

When nous began to move [things], there was separation off from the multitude that was being moved, and whatever nous moved, all this was dissociated; and as things were being moved and dissociated, the revolution made them dissociate much more.

-Fragment 13

Viewed in a certain light, there are striking resonances with our current scientific hypotheses regarding the Big Bang, that moment(?) when a point(?) of infinite(?) density exploded with infinite(?) energy and started in motion the series of events that led to everything else.

P.S. In modern times, Apeiron is perhaps most widely known as the title of two different video games: a currently-under-development fan-made reboot of Star Wars: Knights of the Old Republic and a 1995 remake of Centipede. As research for this post, I downloaded a trial version of the latter. It was aptly titled. The crude and too-small graphics, the aggressively irritating sound effects, and the awkward mechanics indeed made my one session seem endless, and I was relieved when the game abruptly stopped and I was given the message: “If you think we’re still going to let you play for free, I have to ask you, ‘What are you smoking?'” The experience brought to mind another quote from Anaxagoras, recorded in Cicero’s Tusculan Disputations and also used, in a different translation, as the epigraph to William Gass’s The Tunnel.

In truth the roads to the underworld are the same from anywhere.

 

Dreaming of Infinity

Infinity has been with me for most of my life. Of course, there were the games at school, competing with friends to use larger and larger numbers to quantify how much each of us liked a certain shared object of affection. Inevitably, one of us would boldly say “infinity”, after which one of two things would happen. The other might respond with “infinity plus one” and the game would essentially restart, with the additional rule that every play must be prefixed by the phrase “infinity plus.” Or the other might say, “Infinity’s not a number!” The teacher would then be summoned to resolve our dispute, usually with disappointingly unenlightening results.

But there was more than just this. As a child, maybe seven years old, lying awake at night, the darkness of my bedroom impelled my thoughts to the ends of the universe and to the timeless question, “Does the world go on forever?” I was bewildered. Trying to imagine either an ‘edge’ to the universe (what’s on the other side of the edge?) or a universe of infinite expanse, containing an infinite number of people like me thinking these same thoughts at the same time, filled me with a thrilling and irresistible terror. Night after night, I was drawn to this question the way we are drawn to peek ever further over the edge of a precipice.

Later, in a high school calculus class, I became further absorbed by the strangeness of infinity. One day, the teacher presented us with two puzzling scenarios. In the first, he simply drew the line segment

y = 2x     0 \leq x \leq 1

which, by establishing a one-to-one correspondence, shows that there are exactly the same number of real numbers between 0 and 2 as there are between 0 and 1, conflicting with our naive intuition that there should in fact be twice as many.

The second involved Gabriel’s Horn, the surface obtained by rotating the curve

y = \frac{1}{x}     1 \leq x < \infty

about the x-axis. He demonstrated that this surface (together with a circular “cap” at x = 1) encloses a region with finite volume but infinite surface area. Therefore, he claimed, the amount of paint that could fit inside this region would, absurdly, not be enough to coat its interior surface!

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Gabriel’s Horn (A piece of it, at least. The narrow end, continuing to become narrower, extends to infinity.)

Both of these apparent paradoxes are (to a certain extent, at least) easily resolvable, but I was hooked and quickly moved on to other mysteries of the infinite. As a freshman in college, I subjected at least three of my suitemates to impromptu expositions of Cantor’s diagonal argument for the uncountability of the set of real numbers. As a graduate student, I decided to focus on set theory, often described as the mathematical study of infinity. Just as when I was a kid, I now lie in bed dreaming about infinity, though in a more rigorous, technical, and focused manner. Here, though, I want to move back a bit and take in a broader perspective. I want to look at infinity from different angles, through different lenses. Mathematical lenses, to be sure, but also philosophical, artistic, literary, historical, and scientific ones. And I want to share the view with others. I hope you will join me.