Finite functions and the necessary use of large cardinals

We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied higher infinities of abstract set theory called large cardinals in an essential way in order to derive results in the context of the natural numbers. The findings raise the specific issue of what consitutes a valid mathematical proof and the general issue of objectivity in mathematics in a down to earth way. Large cardinal axioms, which go beyond the usual axioms for mathematics, have been commonly used in abstract set theory since the 1960’s. We believe that the results reported on here are the early stages of an evolutionary process in which new axioms for mathematics will be commonly used in an essential way in the more concrete parts of mathematics.