Philosophy of Mathematics seminar

Beau Mount, Konstanz

Goliath’s Sword: Repurposing Fictionalism to Tame Choice Sequences  

Traditionally, fictionalism has been a weapon forged by the heterodox for the benefit of the heterodox: its inventors sought to provide nominalists with a way to reap the benefits of classical mathematics while denying that mathematical objects really exist. But history teaches that arms, once invented, can never be kept out of the enemy’s hands for long. After having slain Goliath, David acquired the giant’s sword and then used it himself in battle; I propose that the classical mathematician should follow his example and appropriate the fictionalist’s weapons to interpret deviant mathematics.

At first it might seem that there is little need to explore this option: non-classical logics are usually weaker than classical logic, and nominalistic accounts of mathematical ontology involve fewer commitments than realist ones, leaving no additional anti-classical claims to be explained away. But there is an important exception. Intuitionistic analysis (IA) is not a mere subtheory of classical analysis: it has theorems that directly conflict with classical analysis and even classical first-order logic. The intuitionist accepts the existence of choice sequences: infinite sequences of natural numbers, taken to be temporal objects in fieri rather than atemporal ones in esse, that behave anti-classically. Choice sequences, on the intuitionistic view, conflict with the classical principle of quantified excluded middle (QEM).

The classicist could respond by dismissing choice sequences as nonsense, or by adopting a purely syntactic, ‘if-thenist’ construal of IA’s results. I argue that a better option is to adopt fictionalism: the classicist can say that IA is literally false but nonetheless genuine mathematics—in fact, interesting mathematics—because it represents a coherent fiction about the mathematical universe. It is false that not every mathematical object validates QEM, but it is true that, according to intuitionism, not every mathematical object validates QEM. I sketch a simple system that formalizes this framework, focussing on the Kreisel/Troelstra account of lawless sequences.