The Subtle Art of Go, or Finite Simulations of Infinite Games

Tatta hito-ban to
uchihajimeta wa
sakujitsu nari

Saying `just one game’
they began to play . . .
That was yesterday.

-Senryū (trans. William Pinckard)

A few weeks ago, I found myself in Virginia’s finest book store and made a delightful discovery: a newly published translation of A Short Treatise Inviting the Reader to Discover the Subtle Art of Go, by Pierre Lusson, Georges Perec, and Jacques Roubaud (two mathematicians and two poets, all associates of the French literary workshop Oulipo), originally published in France in 1969.

Go, of course, is a notoriously difficult and abstract board game, invented in China over 2500 years ago and further developed in Korea and Japan. After millennia of being largely unknown outside of East Asia, it has in the last century become popular throughout the world (the publication of this book played a significant role in introducing it to France) and has even been in the news recently, as computer programs using neural networks have defeated some of the best professional go players in highly publicized matches.

The stated goal of Lusson, Perec, and Roubaud’s book is to “[provide], in a clear, complete, and precise manner, the rules of the game of GO” and to “heighten interest in this game.” As a practical manual on the rules and basic tactics and strategy of the game, the modern reader can do much better with other books. As a literary meditation on play, on art, and on infinity, it is dazzling. It is this latter aspect of the book that I want to touch on here today.

A theme running throughout the book is the idea that the practice of go is akin to a journey into infinity, and this theme is expressed both with respect to one’s relationship with other players and with one’s relationship to the game itself.

A joy of learning any game is developing relationships and rivalries with other players, and this is especially true with go, for two main reasons. First, an individual match is not simply won or lost but rather is won or lost by a certain number of points. Second, there is a robust handicapping system whereby a substantially weaker player can legitimately compete with a stronger player, in a match of interest to both players, by placing a specific number of pieces on specific points of the board before the first move. Through these two mechanisms, a rich and rewarding go relationship can thus develop, even between players of unequal skill, over not just one match but a progression of matches, during which handicaps can be finely calibrated and can, indeed, change over time, as the players learn more about each other’s play and about the game in general.

As such, GO, at its limits, constitutes the best finite simulation of an infinite game. The two adversaries battle on the Goban the way cyclists pursue each other in a velodrome.

This is not as much the case with, say, chess, in which the facts that the result of a game is purely binary and that handicap systems are clumsier and more substantially alter the character of the game mean that matches between players of unequal skill will frequently be frustrating for the weaker and boring for the stronger.

It is a cliché to say that one can never truly master a subject, that there is always more to learn. But the richness of go makes it especially true here, and, in a sense, quantifiably so. The number of legal positions in go is more than $2 \times 10^{170}$. This is a truly astounding number, dwarfing the estimated number of atoms in the observable universe ($10^{80}$) or the estimated number of legal positions in chess (a piddling $10^{43}$). The possibilities in go are, for all intents and purposes, infinite. No matter how much one learns, one knows essentially nothing.

Crucially, though, there is a well-developed hierarchy through which one progresses and by which one can measure one’s skill, even if remains practically zero when taking a wider view. Lusson, Perec, and Roubaud write about this better than I could in the following two excerpts, so let us simply listen to them:

The genius of GO stems precisely from what it hides as well as what it reveals, at any moment, at any level, in its different, hierarchized mysteries whose progressive mastery transforms the game every time:

A garden of bifurcating pathways, a labyrinth, the game of Babel, each step forward is decisive and each step forward is useless: we will never have finished learning..

(Note the surely intentional nod to Borges in the last sentence above).

From a beginner to a classified amateur in the bottom ranks of kyu, a player can rise to the top kyus and then, one by one, climb the six amateur echelons, approaching (but only approaching) the inaccessible regions where the true players reign, the professionals…

In this last excerpt, we hear echoes of a mathematical concept from set theory, my personal area of expertise. The authors temper the reader’s (and their own) dreams of go mastery by noting that, no matter how high an amateur go player may ascend in their study of go, they will still never reach the “inaccessible” realm of the true masters of the game. These professionals also inhabit a hierarchy, but it is a separate hierarchy, visible but unreachable from below.

This calls to mind the concept of an inaccessible cardinal, which is an uncountable cardinal number $\kappa$ that cannot be reached from below through the standard procedures of climbing to the next cardinal, applying cardinal exponentiation, or taking unions of a small number of small sets. (More formally, $\kappa$ is (strongly) inaccessible if it is regular, uncountable, and, for all $\lambda < \kappa$, we have $2^\lambda < \kappa$.)

It cannot be proven that inaccessible cardinals exist or are even consistent, and the assumption of the consistency of such cardinals has significant implications for what one can prove (see a previous post for more information about inaccessible and other “large cardinals”). On the simplest descriptive level, an inaccessible cardinal divides the hierarchy of infinite cardinals into two realms that cannot communicate via the standard operations of arithmetic: those above and those below.

(A modern version of the book would surely posit a third separate region of hierarchies in go: that of the neural networks that with stunning swiftness have become stronger than the strongest professionals.)

So why bother? If one can spend one’s whole life studying go and still hardly understand the game, if one can develop to one’s full potential and still be nowhere close to the level of professional players, let alone the newly ascendant artificial intelligences, then why start at all?

The authors consider this question, but ultimately they reject its premises. The study of go is not worthwhile in spite of the fact that it is an “infinite pathway.” It is worthwhile because of it.

And this clearly has implications outside of go. Why devote much of my life to mathematical research if I can never know more than a miniscule portion of what remains undiscovered? Why write if it is physically impossible to write more than about 10,000,000 words in a life, and if everything I may write is already contained in Borges’ Library of Babel anyway? Perhaps because the best life is a finite simulation of an infinite life.

Only one activity exists to which GO may be reasonably compared.

We will have understood it is writing.

PS: We have had occasion to mention chess and its relation to the infinite in a previous post. One of the joys of A Short Treatise… is the exaggerated contempt expressed by its authors for the game of chess. We end by offering you just a taste:

Good news!

One of the best European players, Zoran Mutabzija, abandoned chess, to which he had devoted himself since the age of four, as soon as someone taught him GO!

In related news.

We just received a detailed report concerning a certain Louis A. caught in the act of robbing a gas station attendant on the Nationale 6. According to the report, whose source and information cannot be called into question, Louis A. is a notorious chess player.