On mathematical instrumentalism

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 I Sigma(1) is a conservative extension at the equational theory of Primitive Recursive Arithmetic (PRA). I Sigma(1) has a super-equational 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 I Sigma(1).