CERES: An analysis of Fürstenberg’s proof of the infinity of primes

The distinction between analytic and synthetic proofs is a very old and important one: An analytic proof uses only notions occurring in the proved statement while a synthetic proof uses additional ones. This distinction has been made precise by Gentzen’s famous cut-elimination theorem stating that synthetic proofs can be transformed into analytic ones.CERES (cut-elimination by resolution) is a cut-elimination method that has the advantage of considering the original proof in its full generality which allows the extraction of different analytic arguments from it. In this paper we will use an implementation of CERES to analyze Fürstenberg’s topological proof of the infinity of primes. We will show that Euclid’s original proof can be obtained as one of the analytic arguments from Fürstenberg’s proof. This constitutes a proof-of-concept example for a semi-automated analysis of realistic mathematical proofs providing new information about them.