How to get Numbers out of Quantities (and not the Other Way around)
Prima facie, quantitative attributes seem to involve numbers in a way in which other kinds of attributes do not. The plain reason, some have suggested, is that quantities are grounded in simple relations to numbers: my notebook has 1.6 kg mass because it bears the relation mass-in-kilos to the number 1.6 (for different versions of this idea see Mundy 1988, Wheatherson 2006). If this is true, numbers are fundamental and quantities derived. Starting from a critical evaluation of this approach, I suggest to invert the order of fundamentality, by treating numbers as derived and quantities as fundamental. I assume that quantities like mass, charge, temperature etc. are spaces – roughly, rays converging to a limit, which I call point zero. I show that any ray-segment has a real number depending on whether it satisfies certain conditions, formulated in terms of congruence, betweenness and parthood. And I define the fundamental algebraic operations on ray-segments in geometrical terms. The space of numbers can now itself be treated as a ray on which quantity regions are located, that is, as a sort of meta-quantity. This account has a number of interesting features. It employs a Fieldian machinery (Field 1980, 2016) to yield an anti-Fieldian conclusion (numbers exist and we can unproblematically quantify over them). It treats irrational and rational numbers (including integers) on a par. And it completely leaves out of the picture sets, employing geometry instead of set theory as the indispensable theoretical background.
June 23rd / 8:45 / aula magna