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.
One contribution of 13 to a Discussion Meeting Issue ‘The nature of mathematical proof’.
↵More properly, this would be called ‘Euler's Conjecture’, since he proposed, but did not prove it.
↵Nor is it yet closed, since terms such as surface, system, etc. have still to be defined.
↵It is usually attributed to Richard Dedekind in 1887, although informal uses of induction date back as far as Euclid.
↵For instance, Aaron Sloman, personal communication.
↵Rigorous proofs are sometimes called ‘informal’, but this is a misleading description since, as we have seen, they can be formalized.
- © 2005 The Royal Society