The board and the beyond
It is one of the minor trials of academic life, ranking somewhere between the lifeless sandwich at a departmental colloquium and the discovery that your bicycle has been nicked from the porter’s lodge, that we are periodically buttonholed at dinner by a well-meaning person who has just thought of something terribly clever. “I say,” this person will announce, settling into his chair with the air of a man about to solve the accumulated problems of two millennia, “isn’t mathematics really just a game? A closed system of rules, rather like chess, only with more squiggles?” He will then lean back, radiating the quiet satisfaction of one who has put the abstract sciences firmly in their place and will await the gratitude of the professional.
My own response on these occasions has over the years evolved from a tight‑lipped tutorial in the philosophy of language to something closer to a weary, affectionate sigh. The chess analogy is one of those notions that seems, at first sip, to contain a stimulating shot of insight and turns out, upon a second tasting, to be composed largely of damp cardboard steeped in the wishful thinking of those who have never actually spent a wet Wednesday afternoon trying to prove that a certain cohomology group vanishes. It is the intellectual equivalent of mistaking a postage stamp for a landscape painting because both are rectangular and made of paper.
Let us grant, in a spirit of reckless generosity, that the analogy possesses a certain surface charm. Chess is indubitably a closed system. Its rules are finite, explicit and fixed by a governing body that does not, on the whole, issue errata. The board has sixty‑four squares, the pieces move in prescribed ways and the aim, checkmate, is as unambiguous as a slammed door. From these givens, a game tree unfolds, a monstrous but determinate branching of all possible legal continuations. In principle, chess is solvable; every position is either a forced win for White, a forced win for Black or a forced draw and the fact that we have not yet computed the solution is merely a matter of the universe’s insufficient lifespan, not of any deep logical openness.
So far, so tidy. And yet the analogy dissolves the moment we ask what, specifically, is being compared. Mathematics is not a game. A game is an activity undertaken for diversion or competition, governed by a set of constitutive rules that define what counts as a move and what counts as a win. Mathematics has no win condition. There is no moment at which a referee steps forward, raises a little flag and declares that the mathematician has checkmated the Riemann hypothesis and may now retire to the bar for a celebratory gin. A proof is not a coup de grâce delivered to a vanquished opponent; it is a step in a conversation that began long before I was born and will continue, if the lights hold out, long after I am forgotten. To mistake the two mistakes a sonnet for a tennis match because both employ lines and are improved by a good volley.
The proponent of the chess analogy might object that he was speaking only of formal systems, of the axiomatic method in which theorems are derived from axioms by rule‑governed transformations and that in that limited sense mathematics is a closed syntactic game. But even here the comparison stumbles and stumbles badly. A game of chess is played on a board of finite size with a finite set of legal positions. The game tree, though vast, is a completed totality; every branch terminates in a well‑defined outcome. The formal systems that capture even elementary arithmetic are not so obliging. As a young logician demonstrated in the early 1930’s, any consistent system that can express the addition and multiplication of the natural numbers will contain well‑formed statements that are true but unprovable and cannot prove its own consistency. There is no analogue of this in chess. No chess position sits on the board insouciantly declaring, “I cannot be evaluated from within the rules.” The rules of chess do not generate self‑referential paradoxes; they do not require the player to ascend an infinite hierarchy of metalanguages to discuss the truth of a position. Chess is a closed system precisely because it is so impoverished, so carefully clipped of the expressive power that makes mathematics what it is. If we call mathematics a closed game because chess is a closed game, we might as well call a cathedral a greenhouse because both admit light through stained glass. The one is a modest, rule‑bound structure designed for a specific human purpose; the other is a vast, self‑transcending construct whose builders keep discovering that the foundations they laid yesterday were only the crypt.
There is a further impertinence in the analogy in the suggestion that the rules of mathematics are static. The laws of chess are not subject to revision by the players; we cannot, upon finding ourselves in an unpromising middlegame, announce that henceforth knights shall move in an L‑shape plus one square sideways, because that would be cheating. The mathematician is perfectly at liberty to introduce a new axiom, define a new structure or enlarge the ambit of an existing theory, provided the resulting structure is consistent and, with a bit of luck, illuminating. The axiom of choice, the axiom of determinacy, large cardinal postulates, toposes, derivators, these are not cheating; they are the lifeblood of the subject. They represent the permanent openness of mathematics to its own future, a future that is not pre‑contained in any existing rulebook. A chess player cannot, in the midst of a tournament, turn the board into a Möbius strip and declare a new geometry. A mathematician can and does, and sometimes gets a Fields Medal for the trouble.
The chess partisan may make one final stand: surely, he will say, mathematics is like a game in the sense that it is an internally coherent practice, a “language game” whose meaning is determined by its use within a community of rule‑followers. This is, at last, a more sophisticated version of the claim, but it is a Pyrrhic retreat. The idea that mathematics is a language game is says nothing more than that it is a human practice governed by norms, which is true of law, cookery, diplomacy and the selection of a decent claret. It drains the chess analogy of any specific content, leaving behind only the truism that human beings sometimes follow rules. The interesting claim was that mathematics is a closed system in the sense that chess is a closed system, with a finite, complete set of axioms that mechanically determine all truths. That claim, I am afraid, lies in pieces on the board and no amount of fiddling with the pieces will bring it back to life.
There is a kind of person who finds the game metaphor comforting because it promises that mathematics is, at bottom, a tractable puzzle, a closed box that can be placed on a shelf and admired for its ingenuity. The reality is both more exhilarating and more demanding. Mathematics is not a game; it is the activity by which games themselves are transcended. It continually breaks its own rules in the interest of deeper understanding, and its horizons recede at exactly the rate at which we approach them. Treating it as a closed system mistakes a single, frozen snapshot for the whole, living film.
The next time I am cornered at dinner by an enthusiast of the chess analogy, I shall, with a smile of the most exquisite politeness, ask whether he considers his own marriage to be a closed system. If he replies in the negative, I shall suggest that the comparison with mathematics is instructive. If he replies in the affirmative, I shall recommend a good solicitor. Either way, the port will continue to circulate and the conversation will, in the open-ended manner of all good conversations, move on to something more rewarding.