Saving Proof from Paradox: Godel's Paradox and the Inconsistency of Informal Mathematics

Citation

Tanswell FS (2016) Saving Proof from Paradox: Godel's Paradox and the Inconsistency of Informal Mathematics. In: Andreas H & Verdee P (eds.) Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Trends in Logic Studia Logica Library, 45. Cham, Switzerland: Springer, pp. 159-173. http://www.springer.com/gb/book/9783319402185; https://doi.org/10.1007/978-3-319-40220-8_11

Abstract
In this paper I shall consider two related avenues of argument that have been used to make the case for the inconsistency of mathematics: firstly, Gödel’s paradox which leads to a contradiction within mathematics and, secondly, the incompatibility of completeness and consistency established by Gödel’s incompleteness theorems. By bringing in considerations from the philosophy of mathematical practice on informal proofs, I suggest that we should add to the two axes of completeness and consistency a third axis offormalityandinformality. I use this perspective to respond to the arguments for the inconsistency of mathematics made by Beall and Priest, presenting problems with the assumptions needed concerning formalisation, the unity of informal mathematics and the relation between the formal and informal.  This work was supported by the Caroline Elder Scholarship and a St Andrews/Stirling Philosophy Scholarship. Many thanks to two anonymous referees, Aaron Cotnoir, Benedikt Löwe, Noah Friedman-Biglin, Jc Beall, Graham Priest, Alex Yates, Ryo Ito, Morgan Thomas, Brian King and audiences in St Andrews, Cambridge and Munich.