It's not quite what you have asked for, but very close:
Some facts - and proofs! - in combinatorics can be interpreted as linear algebra over the "field with one element". In this very nicely written article Henry Cohn gives a concrete meaning to this and shows how to make a proof from linear algebra into a proof about a combinatorical statement by rephrasing it into axiomatic projective geometry.
(by the way: Lior's answer is an instance of linear algebra over field with one element)