 
Summary: ON FORMALLY MEASURING AND ELIMINATING EXTRANEOUS NOTIONS
IN PROOFS
ANDREW ARANA
ABSTRACT. Many mathematicians and philosophers of mathematics believe some proofs
contain elements extraneous to what is being proved. In this paper I discuss extraneous
ness generally, and then consider a specific proposal for measuring extraneousness syn
tactically. This specific proposal uses Gentzen's cutelimination theorem. I argue that the
proposal fails, and that we should be skeptical about the usefulness of syntactic extrane
ousness measures.
Many mathematicians and philosophers of mathematics think that it's somehow valu
able for a proof to be "pure", that is, for it not to use notions extraneous to what is being
proved. Not worrying about why that is for now, we would like to make better sense of
the suggestion. What does it mean for a notion used in a proof to be extraneous to the
theorem being proved? One way of making this sharper would be to develop a syntactic
way of evaluating extraneousness. I want to consider such a proposal, using Gerhard
Gentzen's cutelimination theorem. I will argue that there are serious obstacles to making
this proposal work.
1. FOUR CLAIMS CONCERNING EXTRANEOUSNESS
Bertrand's postulate states that for every natural number n 1, there is a prime
number between n and 2n. It is so named because it was verified by computation for
