QUANTITIES, MAGNITUDES, AND NUMBERS

Quantities are naturally viewed as functions, whose arguments may be c onstrued as situations, events, objects, etc. We explore the question of the range of these functions: should it be construed as the real nu mbers (or some subset thereof)? This is Carnap's view. It has attracti ve features, specifically, what Carnap views as ontological economy. O r should the range of a quantity be a set of magnitudes? This may have been Helmholtz's view, and it, too, has attractive features. It revea ls the close connection between measurement and natural law, it makes dimensional analysis intelligible, and explains the concern of scienti sts and engineers with units in equations. It leaves the philosophical problem of the relation between the structure of magnitudes and the s tructure of the reals. What explains it? And is it always the same? We will argue that on the whole, construing the values of quantities as magnitudes has some advantages, and that (as Helmholtz seems to sugges t in ''Numbering and Measuring from an Epistemological Viewpoint'') th e relation between magnitudes and real numbers can be based on foundat ional similarities of structure.