 Other than these broad goals and claims, measurement theory is a highly heterogeneous body of scholarship. It includes works that span from the late nineteenth century to the present day and endorse a wide array of views on the ontology, epistemology and semantics of measurement. Two main differences among mathematical theories of measurement are especially worth mentioning. The first concerns the nature of the relata, or “objects”, whose relations numbers are supposed to mirror. These relata may be understood in at least four different ways: as concrete individual objects, as qualitative observations of concrete individual objects, as abstract representations of individual objects, or as universal properties of objects. Which interpretation is adopted depends in large part on the author’s metaphysical and epistemic commitments. This issue will be especially relevant to the discussion of realist accounts of measurement  (Section 5).  Second, different measurement theorists have taken different stands on the kind of empirical evidence that is required to establish mappings between objects and numbers. As a result, measurement theorists have come to disagree about the necessary conditions for establishing the measurability of attributes, and specifically about whether psychological attributes are measurable. Debates about measurability have been highly fruitful for the development of measurement theory, and the following subsections will introduce some of these debates and the central concepts developed therein.