From Jesper Valgreen
In Richard Elwes’s otherwise excellent article on the arithmetical implications of the concept of infinity (14 August, p 34), it is suggested that a finitist theory of arithmetic – one in which there is no infinity – will be truncated, or limited to only finite mathematics. The finitary approach, usually associated with the intuitionist logic of L. E. J. Brouwer and others, stems from a deep unease about treating infinities as real. Yet one is still allowed to consider sets like the natural numbers as potentially infinite.
This leads to an approach to set theory – the branch of mathematics that deals with collections of objects – that considers external relations like “subset of” to be fundamental. Sets are then generalised to objects and the subset relation replaced with a generic operator that maps one set to another. The result is category theory and its significant subclass, topos theory, which is, roughly, a unification of logic and geometry. Rather than being truncated, this is a different and in some ways deeper approach to arithmetic, and the only thing ruled out is transfinite nonsense, as the intuitionists regard it.
Roedovre, Denmark
