 Broadly speaking, measurement theory sets out to (i) identify the assumptions underlying the use of various mathematical structures for describing aspects of the empirical world, and (ii) draw lessons about the adequacy and limits of using these mathematical structures for describing aspects of the empirical world. Following Otto Hölder (1901), measurement theorists often tackle these goals through formal proofs, with the assumptions in (i) serving as axioms and the lessons in (ii) following as theorems. A key insight of measurement theory is that the empirically significant aspects of a given mathematical structure are those that mirror relevant relations among the objects being measured. For example, the relation “bigger than” among numbers is empirically significant for measuring length insofar as it mirrors the relation “longer than” among objects. This mirroring, or mapping, of relations between objects and mathematical entities constitutes a measurement scale. As will be clarified below, measurement scales are usually thought of as isomorphisms or homomorphisms between objects and mathematical entities.