Playing Games I: Setting Up the Pieces

Life is more fun if you play games.

-Roald Dahl

Combinatory play seems to be the essential feature in productive thought.

-Albert Einstein

Observant readers will have noted the multiple occasions on which games have shown up in our posts here at Point at Infinity. We have examined the paradoxes of Hypergame in pursuit of a proof of Cantor’s Theorem. We have callously decided the fates of prisoners by playing games with hat colors. We have seen mysterious characters engage in a variant of Nim in Last Year at Marienbad. Some may even accuse us of playing a few games ourselves.

There are reasons for this. Games are fun, for one. And, more to the point, games often provide a useful lens through which to view more “serious” topics. So, over the next few weeks, we are going to be taking a deeper look at all kinds of games and the light they can shed on the infinite. We will discover a winning strategy for Marienbad (among other games). We will investigate Conway’s surreal numbers (for real this time) in the context of game values. We will consider the profound and often surprising role infinite games have played in modern set theory, in particular with regard to questions around the strange and counter-intuitive Axiom of Determinacy. We may even venture into muddier philosophical waters to look at Wittgenstein’s language games or James Carse’s ideas about finite and infinite games.

It will be fun, and I hope you will join us. For today, though, just enjoy this video of an instance of Conway’s Game of Life, implemented inside another instance of Conway’s Game of Life:

Cover image: Still from The Seventh Seal.


Infinity at the End of the World

I recently spent five days at the Arctic Set Theory Workshop, in the town of Kilpisjärvi, at the northernmost tip of Finland. There were vast, monochromatic landscapes, endlessly protracted sunrises and sunsets, ephemeral glimpses of the Northern Lights. There was remoteness, and there was good company. It was an ideal place to think about the mathematics of infinity.


In my simplistic, idealized mental image of the European political map, Norway, Sweden, and Finland are three parallel line segments situated above the main bulk of Europe. The place where these three countries meet, then (or, perhaps, where the infinite rays extending the countries meet), should be a point at infinity.


This “point at infinity” actually exists on Earth. (And it’s not the North Pole!)


And we went there:

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Finland (author for scale)
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Reading Assignment I

I mean how could you get them all to stay still?

-Billy Collins, “Cosmology”

Remember Bertrand Russell and his infinite stack of turtles? So does Billy Collins, whose poem “Cosmology” appeared in the September 5 issue of The New Yorker.

Over at Nautilus, Trevor Quirk addresses “The Problem With Science Writing” and the difficulties of accurately and appealingly communicating scientific ideas to the public without being overly reductive (writing this blog has given me some first-hand experience of these difficulties).

One of the works Quirk brings up is David Foster Wallace’s Everything and More: A Compact History of Infinity, a book that I read at the very outset of my graduate study and about which I have very conflicted feelings. On the one hand, I am a big fan of Wallace’s writing, the book’s attempt to explain technical material in a way that is both accessible and integrated into a historical narrative is admirable, and there is some good stuff in the book (particularly the material early on about the Greeks). On the other hand, the book contains a huge number of technical errors and misleading statements, a fact made more egregious by the at times irritatingly arrogant tone of the prose. It sometimes feels as if Wallace is saying, “This is really complicated stuff, and not many people can truly understand it. Let me explain it to you.” before getting something completely wrong. It is a testament to Wallace’s writing and the book’s ambitions that, despite this, I still would not necessarily attempt to dissuade someone from reading it.

Really, though, you should just go read Infinite Jest.

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.


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.

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!

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.