Mathematical Progress: Ariadne’s Thread

Abstract

The aim of this paper is two-fold: 1) to explore a particular variety of mathematical development that exemplifies progress; 2) to examine what bearing, if any, it has on the issue of mathematical realism. Often in the history of mathematics we find mathematicians employing concepts and algebraic techniques that produce a body of successful results, but their procedures only become properly intelligible in the context of definitions, concepts, and structures developed much later. My discussion of this progress to later intelligibility, particularly its bearing on the issue of realism, will take place against the background of analogous discussions in philosophy of science. Steady growth views of scientific progress, according to which scientific development is cumulative — a gradual accretion (without loss) of scientific truths and refinement (without jettisoning) of scientific methods — are taken to support scientific realism. Since Kuhn’s (1962), however, such views of progress, and with them scientific realism, have been constantly challenged. If Kuhn is right, the history of science exhibits such radical conceptual discontinuities that the image of science as revealing more and more truths about the world must be given up.