On mathematical instrumentalism

Abstract

In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano's Arithmetic known as IΣ₁ is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA), IΣ₁ has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse Mathematics that formalize powerful mathematical principles have only polynomial speed-up over IΣ₁.