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

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The Greatest Theorem

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

Ordinary words can take on new meanings within math. Trivial, for example – though hardly part of our daily discourse – appears frequently, describing something subtly in-between “true because of definitions” and “true for reasons simpler than befitting the seriousness of this situation”. A priori is perhaps more interesting. Though it’s traditionally a philosophical term – describing facts knowable without appeal to the senses but rather to logic alone – in math, further logical investigation fills empiricism’s vacant role. “A priori, we do not know whether the following property holds,” a mathematician might explain. “But upon appeal to further (i.e., logical!) mathematical arguments, we’ll soon find out that it does.”

It should be no surprise, then, that the apparently innocuous words weaker and stronger will prove sufficiently interesting to occupy our attention for the remainder of this essay. Continue reading

Theorems’ Almanack

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

Theorems are the hallmark of the mathematical trade. In them reside mathematical rigor, mathematics’ unspoken holy grail. They also possess a kind of mystique. “I’m scared of upper-division math classes,” I once overheard a student say. “It’s all proofs.”

Theorems, independent of their mathematical content, are interesting in their own right, because they delineate the mathematical landscape. Open a textbook of mathematics, and you’ll find the difficult universe of mathematics, once uncharted and unknown, arrayed and packaged before your eyes. “Theorem: ___________. Proof: _____________.

I attempt here to understand mathematical theorems, in a purely logical sense: for what they are, for what they do, and for what role they play. Continue reading