Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

8Reflections on the Purity ofMethod in Hilbert’s Grundlagender GeometrieMICHAEL HALLETT8.1 Introduction: The ‘Purity of Method’ in theGrundlagenThe publication of Hilbert’s monograph Grundlagen der Geometrie in 1899marks the beginning of the modern study of the foundations of mathematics.In the Schlusswort to the book, Hilbert explicitly mentions the ‘purity ofmethod’ question. He states first that his book was guided by ‘the basicprinciple’ of elucidating given problems in such a way as to decide whetherthey can be answered in a ‘prescribed way with certain restricted methods’, or not soanswered, and then designates this as a general principle governing the searchfor mathematical knowledge:This basic principle seems to me to contain a general and natural prescription. Infact, whenever in our mathematical work we encounter a problem or conjecturea theorem, our drive for knowledge [Erkenntnistrieb] is only then satisfied whenwe have succeeded in giving the complete solution of the problem and therigorous proof of the theorem, or when we recognise clearly the grounds for theThis paper is a much expanded version of talks given to the Logic Group of the Department ofPhilosophy at Stanford in April 2004, and to the Annual Meeting of the Association of SymbolicLogicinPittsburghinMay2004. An abbreviated talk based on it was given at the HOPOS BiennialConference in Paris in June 2006. I am grateful to the various participants for discussion andcomments, especially Yvon Gauthier, and to Bill Demopoulos and Dirk Schlimm, who suggestedhelpful modifications to an earlier written version. I am also grateful to my fellow General Editors ofHilbert’s unpublished lectures on foundational subjects, William Ewald, Ulrich Majer, and WilfriedSieg, for discussions on these and other issues over many years. The generous financial support of theSocial Sciences and Humanities Research Council of Canada is gratefully acknowledged.