I’ve been thinking about the chess tree. As I pointed out in an earlier post, Wikipedia informs us that Ernst Zermelo published his proof about the formal structure of chess in 1913 – Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels” (“On an application of set theory to the theory of chess”). That established that chess is a finite game that can be completely represented in the form of a tree where the root is the initial state of the game board and the leaves are completed games. Each path from the root to a leaf describes the course of a single game.
Why the chess tree?
Now, it’s one thing to prove a mathematical theorem. It’s something else to make an informal observation & argument. Purely as an abstract matter, I can imagine that (some) chess players had made some informal argument prior to Zermelo’s proof. But I can just as easily imagine that, no, the observation was new with his proof.
What I’m wondering about is just what is it that would lead one to make such an observation. One can certainly play the game without knowing that the universe of all possible chess games takes the form of tree, much less that the tree is of finite size. One can play tic-tac-toe without knowing that, either, or checkers. I certainly didn’t know that when I played those games years ago (I did play a bit of chess, but never enough to become any good).
One has to have a certain frame of mind to make such and observation and to then develop it into a formal proof. Developing a formal proof is the kind of thing a mathematician would do, and Zermelo was a mathematician. And not just any mathematics either. It’s not simply that Zermelo published his work in 1913, but that he was working in a relatively new mathematical culture, one with concerns quite different from the arithmetic, geometry, algebra, and calculus that had preceded it in earlier eras. While calculus could deal with things unfolding in time, it didn’t deal the kind of iterated actions between agents that’s involved in chess. The important step, it seems to me, is simply realizing that that is the kind of thing around which one can construct some mathematics.
Primitive and sophisticated
Whatever’s going on here, it strikes me as being at one and the same time, sophisticated but also primitive and basic. The frame of mind is sophisticated, but it’s not a sophistication that’s build on a complex body of prior knowledge in the way that understanding calculus requires prior knowledge of algebra, geometry, and trigonometry. Predicate calculus and symbolic logic are like that as well. It doesn’t require any prior mathematical knowledge, but is nonetheless a bit sophisticated. It’s not generally taught at the high school level as algebra, geometry, and trigonometry are. I suspect it could be – and probably is here and there – but I’m not sure with what success.
Now, back to chess. How does knowing the chess tree help you think about chess? What does that explicit knowledge enable you to do that you couldn’t otherwise do? Do the analysis trees that Kotov advocated (Think Like a Grand Master, translated into English in 1971) really help the chess player? I don’t know. Any analysis tree will be local and quite small in relation to the entire tree, which is too large for explicit construction.
What is certain, however, that knowing that chess takes the form of a tree has been central to work on computer chess, which didn’t start until the middle of the 20th century. Without that knowledge, computing chess would have been utterly hopeless. Even with that knowledge it took several decades and the development of extremely large computers before chess had been “solved” in the sense that a computer could beat the very best chess players.
I’m thinking of this in the overall context of cultural evolution. In the late 19th and early 20th century a mathematical culture emerges in which the analysis of chess becomes a matter of intellectual interest. A half century later we have the initial work on computer chess. Roughly 2/3rds of the way in between Alan gives us a formal account of computation in the form of an abstract machine, the Turing machine. That’s the beginning of the era of computational culture. David Hays and I called in Rank 4 in our paper, The Evolution of Cognition.
Tentative comparison: Close-reading
As a point of comparison, I’m thinking about the emergence of close reading in academic literary criticism. It strikes me as being both primitive and sophisticated in a way that’s similar to Zermelo’s proof. It’s primitive in the sense that there is no specific body of prior knowledge that must be mastered before one can undertake close reading. In fact, part of the pedagogical point is that close reading doesn’t require prior knowledge.
Yet, it’s is sophisticated activity as well. It’s not something that comes “naturally,” and it is difficult to do without being “rocket science,” as the phrase goes. What’s going on here? I’ve said a bit about that in my GOAT Literary Critics series: A discipline is founded (sorta’): Brooks & Warren, Northrop Frye, and S. T. Coleridge. Here I just want to note that interpretation of sacred texts is very old. What occasioned the application of hermeneutics to secular texts? There are the pedagogical concerns I sketch out in that essay. But there’s more than pedagogy at stake here.
Even as the activity secures for canonical texts a new position in the cultural landscape, the activity itself aspires to a new kind of knowledge. What’s that about?
More later.
