What is a proof?

Abstract

To those brought up in a logic-based tradition there seems to be a simple and clear definition of proof. But this is largely a twentieth century invention; many earlier proofs had a different nature. We will look particularly at the faulty proof of Euler's Theorem and Lakatos' rational reconstruction of the history of this proof. We will ask: how is it possible for the errors in a faulty proof to remain undetected for several years—even when counter-examples to it are known? How is it possible to have a proof about concepts that are only partially defined? And can we give a logic-based account of such phenomena? We introduce the concept of schematic proofs and argue that they offer a possible cognitive model for the human construction of proofs in mathematics. In particular, we show how they can account for persistent errors in proofs.