Number and measure: Hermann von Helmholtz at the crossroads of mathematics, physics, and psychology

In 1887 Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics (Grassmann / Du Bois), on the possibility of quantitative psychology (Fechner / Kries, Wundt / Zeller), and on the meaning of temperature measurement (Maxwell, Mach). Late nineteenth-century scrutinisers of the foundations of mathematics (Dedekind, Cantor, Frege, Russell) made little of Helmholtz’s essay. Yet it inspired two mathematicians with an eye on physics (Poincaré and Hölder), and a few philosopher-physicists (Mach, Duhem, Campbell). The aim of the present paper is to situate Helmholtz’s contribution in this complex array of nineteenth-century philosophies of number, quantity, and measurement.