Quantities as realistic idealizations

Abstract

The quantitative method is a powerful tool in the natural and social sciences. Depending on the research problem, the use of quantities may give significant methodological gains. Therefore, it is an important task of philosophers to clarify two related questions: in what sense do mathematical objects and structures exist? What justifies the assignment of numbers and numerals to real objects? The former question is discussed in the ontology of mathematics, the latter in the theory of measurement. This paper defends constructive realism in mathematics: numbers and other mathematical entities are human constructions which can be applied to natural and social reality by means of representation theorems. The axioms of such representation theorems are typically idealizations in the sense analyzed by critical scientific realists. If the axioms are truthlike, then quantities can be regarded as realistic idealizations.