Abstract

MORMANN, Thomas. Idealization in mathematics. discus.filos [online]. 2012, vol.13, n.20, pp.147-167. ISSN 0124-6127.

The aim of this paper is to elucidate the role of idealizations in the evolution of mathematical knowledge inspired by some ideas of Ernst Cassirer's Neokantian philosophy of science and mathematics. Usually, in contemporary philosophy of science it is taken for granted that the issue of idealization is concerned only with idealizations in the empirical sciences, in particular in physics. In contrast, Cassirer contended that idealization in mathematics as well as in the sciences has the same conceptual and epistemological basis. More precisely, this "sameness thesis" is scrutinized by investigating a variety of examples of idealizations taken from algebra, topology, lattice theory, and physical geometry. Idealizations in mathematical as well as in physical knowledge can be charac­­te­rized by the introduction of ideal elements leading to completions. In both areas these ideal elements play essentially the same role, namely, to replace an incomplete manifold of objects by a complete "idealized" conceptual manifold.

Keywords : Boolean algebras; Cassirer; completions; Dedekind cuts; idealization; Stones spaces.

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