On the Design of Mathematical Concepts

Abstract

It is one of the deep mathematical insights that foundational systems like firstorder logic or set theory can be used to construct large parts of existing mathematics and formal reasoning. Unfortunately this insight has been used in the field of automated theorem proving as an argument to disregard the need for a diverse variety of representations. While design issues play a major rôle in the formation of mathematical concepts, the theorem proving community has largely neglected them. We argue that this leads not only to problems at the human computer interaction end, but that it causes severe problems at the core of the systems, namely at their representation and reasoning capabilities. In order to improve applicability, theorem proving systems need to take care about the representations used by mathematicians.