The Utility of Mathematical Symbols
Explanations of why mathematics is useful to empirical research focus on mathematics' role as a representation or model. On platonist accounts, the representational relation is one of structural correspondence between features of the real world and the abstract mathematical structures that represent them. Where real numbers are concerned, however, there is good reason to think the world's correspondence with systems of real number symbols, rather than the real numbers themselves, can be utilized for our representational purposes. One way this can be accomplished is through a paraphrase interpretation of real number symbols where the symbols are taken to refer directly to the things in the world real numbers are supposed to represent. A platonist account of structural correspondence between structures of real numbers and the world can be found in the foundations of measurement where a scale of real numbers is applied to quantities of physical properties like length, mass and velocity. This subject will be employed as a demonstration of how abstract real numbers, traditionally construed as modeling features of the world, are superfluous if their symbols are taken to refer directly to those features.