Transfer Principles, Klein’s Erlangen Program, and Methodological Structuralism

Source abstract

Abstract The present article investigates Felix Klein’s mathematical structuralism underlying his Erlangen program. The aim here is twofold. The first aim is to survey the geometrical background of his 1872 article, in particular, work on the principle of duality and so-called transfer principles in projective geometry. The second aim is more philosophical in character and concerns Klein’s structuralist account of geometrical knowledge. The chapter will argue that his group-theoretic approach is best characterized as a kind of “methodological structuralism” regarding geometry. Moreover, one can identify at least two aspects of the Erlangen program that connect his approach with present philosophical debates, namely (i) the idea to specify structural properties and structural identity conditions in terms of transformation groups and (ii) an account of the structural equivalence of geometries in terms of transfer principles.