Plato's Ghost: The Modernist Transformation of Mathematics

TLDR: In this article, the development of mathematics between 1880 and 1920 as a modernist transformation similar to those in art, literature, and music is discussed, and it is shown that modernism succeeded in mathematics because it connected fruitfully with what mathematicians were doing and with the image they were creating for themselves as an autonomous body of professionals, but also that it steadily raised the stakes by forcing deeper and ultimately unanswerable questions onto the agenda.

Abstract: This book presents the development of mathematics between 1880 and 1920 as a modernist transformation similar to those in art, literature, and music. It is the first to trace the growth of mathematical modernism from its roots in explicit mathematical practice – problem solving and theory building – down to the foundations of mathematics and out to its interactions with physics, philosophy, and theology, the popularisation of mathematics, psychology, and ideas about real and artificial languages. It shows that modernism succeeded in mathematics because it connected fruitfully with what mathematicians were doing and with the image they were creating for themselves as an autonomous body of professionals, but also that it steadily raised the stakes by forcing deeper and ultimately unanswerable questions onto the agenda. Novel objects, definitions, and proofs in mathematics coming from the use of naive set theory and the revived axiomatic method animated debates that spilled over into contemporary arguments in philosophy, and drove an upsurge of popular writing on mathematics and the psychology of learning mathematics. A final chapter looks at mathematics after the First World War: the so-called Foundational crisis, the mechanisation of thought, and mathematical Platonism. Prominent figures in these debates who are seen here for the first time in a broad web of influences include, among the mathematicians, Borel, Dedekind, du Bois-Reymond, Enriques, Hilbert, Holder, Klein, Kronecker, Lebesgue, Minkowski, Peano, and Poincare, as well as Helmholtz, Hertz, Maxwell, and the neglected but important figures of Paul Carus and Wilhelm Wundt.