Game of Theorems

This article is part of a series on The Structure of Theorems. See also:
1. Theorems’ Almanack; 2. The Greatest Theorem; 3. Game of Theorems

Few (currently) practicing mathematicians – in my experience – deign to concern themselves with issues surrounding set theory and the foundations of mathematics. In these areas reside the very definitions upon which the rest of our discipline rests; in any case, our discipline proceeds nonetheless, despite its practitioners’ regrettable ignorance. Mathematicians are pragmatic people. In 1949, Bourbaki – perhaps apprehending a subtle need to defend itself – titled a paper Foundations of Mathematics for the Working Mathematician [1].

This state of affairs, unfortunate as it is, explains the surprise and intrigue I often feel when I take time to explore foundational issues. The definition of the so-called Axiom of Determinacy particularly struck me. This set-theoretic axiom is formulated in terms of a certain type of two-player game – an infinite sequential game, in fact, in which two players take turns playing integers, leading ultimately to a sequence of integers of infinite length. The axiom (indeed, it’s something we might choose to suppose) states that every such game – that is, every choice of a victory set, a distinguished collection of possible infinite sequences whose members define the winning outcomes for the first player – is determined, in the sense that one player or the other in the game has a dominant strategy.

I’ll explain these terms below. The important thing, here, is that this mathematical property is defined by the existence of dominant strategies for a certain class of two-player games. This intrusion of an apparently economic, or game-theoretic, notion – that of the two-player game – into mathematics surprised me.

This intrusion, in retrospect, should have been less than surprising. Continue reading